The insatiable demand for high performance on variousdynamic systems quantified by high-speed operation, high control accuracy, and lower energy consumption has triggered vigorous resear
Trang 20 1 2 3 4 5 6 7 8 1.5
1 0.5 0 0.5 1 1.5x 103
1.5 1 0.5 0 0.5 1 1.5x 103
5 0 5
5 0 5
5 0 5
Time (s)
Figure 20 Control signals during H∞ control (experimental).
Trang 30 10 20 30 40 50 60 70 80 90 100 110
Figure 22 Application point
0.2 0.2
0.1 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
Flexible structures are difficult to control because theirdynamics are characterized by a large number of vibra-tional modes To reduce computational complexity, con-troller design is typically performed using a reduced order
model The H∞ controller design procedure yields a troller that concentrates the control energy on the modesincluded in the design model The design procedure ac-counts for sensor noise and disturbances resulting fromnonlinearities in the amplifier and piezoelectric actuators.Control input saturation can be avoided by using a highpenalty on the control energy during the controller design
con-The H∞ controller performance is analyzed using a order evaluation model The simulations showed that the
high-H∞controller provides significant increase in damping tothe modes included in the design model, but does affectthe higher-order excluded modes This behavior is ideal,
as it ensures that the structure will not become unstablethrough the excitation of higher-order modes
Trang 4Figure 23 Displacement of node 24 during continuous
distur-bance test with H∞control (experimental).
Experimental results with the truss structure have
con-firmed the validity of the simulations Two tests have been
performed, an impact test and shaker test Comparisons
between the open loop and the closed loop responses show
that the H∞ controller significantly decreases the
vibra-tional mode amplitudes The controller targets its efforts
on the modes retained in the design model
Piezoelectric materials are ideally suited for
construct-ing actuators and sensors for vibration suppression in
flexible structures Polyvinylidene fluoride (PVDF) is
ide-ally suited for sensor construction It is lightweight,
flexi-ble, and provides a high voltage for a given strain
Piezo-ceramic materials are suited to actuator construction
Piezoceramics are stiff, rugged, and provide relatively
Figure 24 Control signals during continuous
dis-turbance test with H∞ control (experimental).
Table 4 Mode Attenuation
BIBLIOGRAPHY
1 J.C Doyle IEEE Trans Autom Contr AC-23(4): 756–757
(1978).
2 M.J Balas IEEE Trans Autom Contr 27(3): 522–535 (1982).
3 J.J Allen and J.P Lauffer J Dyn Syst Meas Contr 119:
(September 1997).
Trang 54 J.C Doyle, K Glover, P.P Khargonekar, and B.A Francis.
IEEE Trans Autom Contr AC-34(8): 831–847 (1989).
5 B.A Francis A Course in H∞Control Theory Lecture Notes
in Control and Information Series, Vol 88 1987.
6 K Zhou, J.C Doyle, and K Glover Robust and Optimal
Con-trol Prentice Hall, Englewood Cliffs, NJ, 1995.
7 S.A Buddie and T.T Goergiu, ¨ U ¨ Ozg ¨uner and M.C Smith Int.
11 E.F Crawley and J de Luis AIAA J 25(10): 1373–1385 (1987).
12 C.K Lee and F.C Moon J Appl Mech 57(6): 434–441 (1990).
13 D.W Miller, S.A Collins, and S.P Peltzman 31st AIAA/
ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conf., Long Beach, CA, 1990.
14 D.W Miller and M.C van Schoor Proc 1st Joint US/Japan
Conf on Adaptive Structures, Maui, Hawaii, 1990,
pp 304–331.
15 A.F Vaz Modelling Piezoelectric Behaviour for Actuator and
Sensor Applications DSS Contract 9F009-0-4140 Canadian
Space Agency, February 1991.
16 A.F Vaz Theoretical Development for an Active Vibration
Damping Experiment DSS Contract 9F009-0-4140 Canadian
Space Agency, March 1991.
17 A.F Vaz Empirical Verification of Interaction Equations for
Flexible Structures with Bonded Piezoelectric Films DSS
Con-tract 9F011-1-0924 Canadian Space Agency, September 1991.
18 A.F Vaz IEEE Trans Instrumentation and Measurement,
De-2094/001-XSD Canadian Space Agency, April 1999.
24 W Weaver and P.R Johnston Structural Dynamics by Finite Elements Prentice Hall, Englewood Cliffs, NJ, 1987.
25 SDRC Corp I-DEAS Master Series Student Guide On World
30 Quanser Consulting Inc MultiQ-3 Programming Manual.
Hamilton, Ontario, Canada, 1999.
31 D.C Hanselman Mastering MATLAB 5: A Comprehensive Tutorial and Reference Prentice Hall, Englewood Cliffs, NJ,
1998.
32 R Bravo, A.F Vaz, S Leatherland, and M Dokanish 1999 CANSMART Workshop, Canadian Space Agency, St Hubert,
Quebec, September 13–14, 1999.
Trang 6The insatiable demand for high performance on various
dynamic systems quantified by high-speed operation,
high control accuracy, and lower energy consumption
has triggered vigorous research on vibrational control of
distributed flexible structures and discrete systems
Numerous control strategies for conventional
electromag-netic actuators have been proposed and implemented to
suppress unwanted vibration However, the successful
empirical realization of electromagnetic actuators may
be sometimes very difficult under certain conditions due
to hardware limitations such as saturation and response
speed This difficulty can be resolved by employing
smart material actuators in vibrational control As is
well-known, smart material technology features actuating
capability, control capability, and computational
capabil-ity (1) Therefore, these inherent capabilities of smart
materials can execute specific functions autonomously in
response to changing environmental stimuli Among many
smart material candidates, electrorheological(ER) fluids,
piezoelectric materials, and shape-memory alloys (SMA)
are effectively exploited for vibrational control in various
engineering applications A viable vibrational control
algo-rithm can be optimally synthesized by integrating control
strategies, and actuating technology, and sensing
technol-ogy, as shown in Fig 1 The design philosophy presented in
Fig 1 contains a very large number of decisions and design
parameters for the characteristics of controllers, actuators,
and sensors Furthermore, the designer seeking a global
optimal solution for the synthesis of a closed-loop smart
structure system must also address other crucial decisions
concerning the time delay of a high-voltage/current
ampli-fier, the speed of the signal converter, and the microchip
hardware of the control software In this article, two
differ-ent flexible smart structures fabricated from ER fluids and
piezoelectric materials are introduced, and vibrational
con-trol techniques for each smart structure are presented In
addition, vibrational control methodology for a passenger
vehicle under various road conditions is given by adopting
an ER damper, followed by vibrational control of a flexible
robotic manipulator that features piezoceramic actuators
VIBRATIONAL CONTROL OF SMART STRUCTURES
ER Fluid-Based Smart Structures
Significant progress has been made in developing smart
structures that incorporate electrorheological(ER) fluids
Typically, this class of smart structures features an
autonomous actuating capability that makes them ideal
for vibrational control applications in variable service ditions and in unstructured environments This may beaccomplished by controlling the stiffness and energy dis-sipation characteristics of the structures This, of course,
con-is possible due to the tunability of rheological properties
of ER fluids by the intensity of the electric field The velopment of ER fluid-based smart structures was initi-ated by Choi et al (2) They completed an experimentalstudy of a variety of shear configurations based on sand-wich beam structures Gandhi et al (3) suggested using an
de-ER fluid as an actuator to suppress deflections of the ible robotic arm structures by avoiding resonance In thiswork, a phenomenological governing equation was derived
flex-by assuming that the structures are viscoelastic materials
A passive control scheme for obtaining a desired transientresponse was developed on the basis of experimentallyobtained phenomenological governing equation, in whichfield-dependent modal properties were used as pseudocon-trol forces (4) Vibrational control logic to minimize thetip deflections of an ER fluid-based cantilever beam struc-tures was illustrated by field-dependent responses in thefrequency domain (5) Coulter and Duclos (6) suggested
a methodology for replacing a conventional viscoelasticmaterial by an ER fluid Following the formulation of
an analytical model for ER fluid-embedded structures viathe conventional sandwich beam theory, they presented
a feasibility that the controllability of the complex shearmodulus of an ER fluid itself can be used to obtain thedesired responses of the structures Rahn and Joshi (7) de-veloped dynamics for an ER fluid-based on the complexshear modulus of the ER fluid and also theoretically sug-gested a feedback controller for transient vibration con-trol Oyadiji (8) developed a theoretical equation to predictthe field-dependent frequency response by treating the ERfluid layer as a constrained damping layer and verified itsvalidity by experiment Choi et al (9) presented a dynamicmodel for an ER fluid-based smart beam, in which the com-plex shear modulus of the ER fluid itself, measured by arotary oscillation test, was taken into consideration To val-idate the methodology, the predicted elastodynamic prop-erties, such as damped natural frequencies and loss fac-tors, were compared with those measured Gong and Lim(10) experimentally investigated the vibrational properties
of sandwich beam structures in which an ER fluid layerwas partially or fully filled as a constraint damping layer.Yalcintas and Coulter (11) proposed a vibrational modelbased on thin-plate theory, and the transverse vibrationresponse of a nonhomogeneous ER smart beam was inves-tigated In addition, the vibrational control capacity of an
ER beam was illustrated by emphasizing mode shape trol associated with an on and off state of the electric field
con-On the other hand, Choi and Park (12) controlled vibration
of ER smart beam structures by using a closed-loop control.The vibrational control technique was empirically realized
by activating a field-dependent fuzzy controller
In this article, the field-dependent fuzzy control scheme
is introduced after briefly explaining the typical block
1085
Trang 7Figure 1 An algorithm for synthesizing a closed-loop
No
Yes Acceptable performance Control performance and characteristics
Control strategies
Actuation technologies
Sensing technologies
Characteristics of desired control performance Start
PID control Optimal control Sliding mode control Adaptive control
Optimal combination and best performance Control accuracy and fastness
Easy implementation and practical feasibility Robustness to unstructured uncertainties Energy consumption and cost effectiveness
Electrorheological fluid Magnetorheological fluid Piezoelectric material Shape memory alloy
Piezoelectric material Fiber optics
Accelerometer Strain gage
diagram for vibrational control of ER fluid-based smart
structures, shown in Fig 2 The control system consists of
a set of sensors, signal converters, microprocessor,
high-voltage amplifier, and control algorithm Most of the
sen-sors currently available such as accelerometers can be
adapted to measure the dynamic response of ERfluid-based
smart structures The microprocessor which includes A/D
(analog–digital) and D/A(digital–analog) signal converters
plays a very important role in closed-loop control time The
Figure 2 Schematic diagram for controlling
the vibration of an ER fluid-based smart
struc-ture.
High voltageamplifier
Face structure
Insulator ER fluid
Sensor signals
Microprocessor(control algorithm)A/D
D/AInput
field
microprocessor should have at least 12 bits to realize trol software and also should take into account a high sam-pling frequency up to 10 kHz The high-voltage amplifiershould have enough power to generate the required EReffect in smart structures Furthermore, the response time
con-of the high-voltage amplifier to the source input controlvoltage should be fast enough not to delay the control action
of the feedback control system Typically, a smart structureconsists of two host(face) structures, insulators, and an ER
Trang 8fluid layer, as shown in Fig 2 For the composite laminate,
the lay-up angle of the laminates can be selected as a
de-sign parameter to investigate the effect of the ER fluid for
different stiffnesses The insulator is a seal to maintain the
integrity of the structure and is also used to adjust the
vol-ume fraction of the ER fluid relative to the total volvol-ume of
the structure For certain elastodynamic purposes, a smart
structure can be constructed that consists of multilayers of
ER fluids whose rheological properties are different
The elastodynamic properties of an ER fluid-based
smart structure vary with the level of the electric field,
as shown in Fig 3 This implies that the natural frequency
of each vibrational mode can be adjusted by tailoring the
electric field and that, consequently, vibration in real time
can be effectively suppressed in the presence of resonant
disturbances(excitations) In other words, the desired
re-sponse for minimizing the vibrational magnitude can be
obtained by selecting the lowest envelope in the frequency
range considered The desired electric field corresponding
to the desired response can be expressed as a fuzzy
con-trol algorithm (12): if ω i ≤ ω < ω i+1, then E d = E j The
variableω denotes the disturbance frequency, and E d is
the desired electric field Note that the variation
poten-tial of elastodynamic properties with respect to the applied
field may be different upon operating conditions such as
the magnitude of excitation when are altered Therefore,
the frequency bandwidth and the corresponding desired
field for the control algorithm should be modified From
fuzzy logic, the control field for the case shown in Fig 3
is determined as follows: if 0≤ ω < ω1, then E d = E2; if
ω1 ≤ ω < ω2, then E d = 0; if ω2≤ ω < ω3, then E d = E1; if
ω3 ≤ ω < ω4, then E d = E2; ifω4 ≤ ω < ω5, then E d= 0; and
ifω5 ≤ ω < ω6, then E d = E1 Figure 4 presents the tip
de-flection of a cantilevered ER beam in the frequency
do-main, which has been experimentally obtained by
imple-menting the fuzzy control logic (12) It is evident from this
figure that there are effective vibrational suppressions in
the neighborhood of the resonant frequencies However,
a small nonzero vibrational magnitude exists across a
broad frequency range This indicates that ER fluids do
not provide an actuating force but change the stiffness
and the damping properties to avoid resonance To improve
vibrational control performance of the fuzzy control logic,
01234567
Frequency (Hz)
UncontrolledControlled
Figure 4 Forced vibrational responses of a cantilevered ER
beam.
appropriate membership functions for the excitatory nitudes and frequencies can be used to determine the elec-tric fields desired
mag-On the other hand, it is known that an ER fluidcontained in a distributed parameter structural system un-der continuous and periodic small deformations remains
in the preyield state, which shows viscoelastic propertiesrepresented by a complex shear modulus (7,9) The com-
plex shear modulus Gf∗ of an ER fluid is expressed by
f + iG l , where i=√−1 Here, G s
f is defined as thestorage shear modulus (in-phase), a measure of the en-ergy stored, relating to the stiffness of the structure that
contains the ER fluid G lis the loss shear modulus of-phase), a measure of the energy dissipated The shearloss factor is the ratio of the energy lost to the energystored in a cycle of deformation and denotes the damp-ing characteristic of the ER fluid-embedded smart struc-ture The complex shear modulus of the ER fluid is nor-mally measured by employing the oscillation mode of anelectrorheometer (9) The measured complex shear mod-ulus is integrated with a sixth-order partial differentialequation, which is obtained by adopting conventional sand-wich beam theory (13) Then, the field-dependent elas-todynamic properties of the structure such as the nat-ural frequency are determined through a finite-element
(out-model which is governed by [M ] {¨u(t)} + [C(E )]{˙u(t)} + [K(E )]{u(t)} = { f (t)} The global mass, damping, and stiff- ness matrices are denoted by [M ], [C(E )], and [K(E )], re- spectively Clearly, both the stiffness matrix [K(E )] and damping matrix [C(E )] are functions of the electric field (E) applied to the ER fluid domain Thus, these matrices
can be tuned as functions of the electric field The able{u(t)} is a displacement vector, (·) is the time deriva-
vari-tive, and { f (t)} represents the external(or disturbance)
force vector By introducing modal coordinates and alsoadopting mode shape characteristics of the smart struc-ture, the finite-element model can be rewritten in a typi-
cal form of state space representation as follows (4): ˙x(t)=
modal coordinates and the matrix B indicates the fluence matrix of the disturbance A represents the
in-system matrix in the absence of an electric field, and
A(E ) denotes the additional system matrix due to the
elec-tric field This implies that the desired response of the
Trang 90.0 0.2 0.4 0.6 0.8 1.0
−2.2
−1.1 0.0 1.1 2.2
Time (sec)
Uncontrolled Controlled
Figure 5 Transient vibrational responses of a cantilevered ER
beam.
structure can be achieved by tuning the field-dependent
A(E ) In transient vibrational control without an
exter-nal disturbance, the desired eigenvalues of the system,
which directly indicate the desired natural frequencies and
damping ratios of the system, can be obtained by adjusting
the intensity of the electric field in the matrix A(E) One
of the effective control algorithms for achieving this goal
is a so-called pseudostate feedback controller proposed by
Choi et al (4) In this method, the state equation is
modi-fied to fit a PD (proportional-derivative) controller in which
the proportional gain is related to the field-dependent
nat-ural frequency, and the derivative gain is related to the
field-dependent damping ratio In addition, we can easily
shift the desired eigenvalues of the system to avoid
reso-nant phenomena by employing this control algorithm
Fig-ure 5 presents the transient vibrational control response
of a cantilevered ER beam (4) The first mode eigenvalues
of the structure are calculated from−1.7313 ± i91.167 in
the absence of an electric field However, the desired
eigen-values of−11.44 ± i122.114 are achieved by employing
ap-propriate control parameter, which indicate the intensity
of the electric field
An ER beam structure for vibrational control can
eas-ily be extended to an ER plate structure In the vibrational
control of flexible plate structures, the significance of mode
shape control is no less important than vibrational
magni-tude control When we consider large flexible structures
such as aircraft wings and helicopter blades, the mode
shape is directly related to lift distribution and
stabil-ity due to internal and external disturbances and other
aeroelastic problems in a stringent environment
There-fore, much research on the mode shape control of plate
structures have been undertaken by using smart
mate-rial actuators (14) Choi et al (15) proposed an ER plate
and investigated its field-dependent mode shapes Figure 6
presents the measured mode shape of an ER plate which
has clamped-clamped boundary conditions (15) It is clearly
observed that the magnitude of each mode shape is
effi-ciently suppressed by applying a control electric field Note
that we can also control the mode shape in part of the plate
structure (15) by partitioning the ER plate and applying an
electric field to the specific portion By doing this, we may
alter the twist/camber of an airfoil in the aircraft wing,
which in turn controls the lift distribution, to produce
de-sirable performance by real-time control
Uncontrolled
0.4
0
0.20
0.20.4
0
0.2Y(m)
Mode (1,1)
0.4
0
0.20
0.20.4
0.2Y(m)
Y(m)
X(m)
X(m)
0.42.4 ×10−2
Mode (2,2)
Figure 6 Measured mode shapes of an ER plate.
Smart Structures That Feature Piezoelectric Actuators and Sensors
So far, many natural and synthetic materials that exhibitpiezoelectric properties have been proposed and devel-oped Natural materials include quartz, ammonium phos-phate, paraffin, and bone; synthetic materials include leadzironate titanate (PZT), barium titanate, lead niobate,lithium sulfate, and polyvinylidene fluoride (PVDF).Among these materials, PZT and PVDF are the most pop-ular and commercially available Both classes of materialsare available in a broad range of properties suited to vibra-tional control applications as actuators or sensors One ofthe salient properties of a piezoelectric material is that itresponds very fast to voltage, and hence has a wide control
Trang 10bandwidth In addition, we can fabricate simple, compact,
low-power devices that feature a set of piezoelectric
actu-ators or/and sensors Applications that use piezoelectric
materials include vibrational control of flexible structures
such as beams, plates, and shells; noise control of cabins;
positional control of structural systems such as flexible
ma-nipulators; vibrational control of discrete systems such as
engine mounts; ultrasonic motors; and various type of
sen-sors, including accelerometers, strain gauges, and sound
pressure gauges The successful development of a
tech-nology that incorporates piezoelectric materials involves
several issues When we fabricate smart structures that
use piezoelectric actuators and sensors, we must consider,
the fabrication method (surface bonding or embedding), the
curing temperature when embedding, insulation between
piezoelectric layers, and the harness of electric wires The
important issues to considered in modeling
piezoelectric-based smart structures include structural dynamics,
ac-tuator dynamics, sensor dynamics, the bonding effect, the
hysteresis phenomenon, the optimal location of actuators
and sensors, and the number of actuators and sensors
The control technique for vibrational control of
piezoelectric-based smart structures is very similar to that
of a conventional vibrational control system, except that
it uses a voltage amplifier, as shown in Fig 7 The
re-sponse time of the voltage amplifier, which normally has
an amplification factor of 200, should be fast enough so
that it does not deteriorate the dynamic bandwidth of the
piezoactuators The microprocessor that has A/D(analog to
digital) and D/A(digital to analog) signal converters needs
to have at least a 12-bit memory, and also needs to
ac-count for high sampling frequency up to 10 kHz Most of
the currently available control algorithms for
piezoactua-tors are realized in an active manner Therefore, a wide
range of control techniques has been proposed for using
piezoelectric material to control the vibration of flexible
structures actively Bailey and Hubbard (16) applied a
piezofilm as an active vibrational damper for distributed
Microprocessor(control scheme)
Voltage amplifier
A/D
D/A
Host structurePiezosensor
Piezoactuator
Sensorsignal
Inputvoltage
x
y
z
Figure 7 Schematic diagram for vibrational
con-trol of a smart structure that features a ator and a sensor.
piezoactu-structural systems Simulations and experimental tigations of transient vibrational control of a cantileverbeam were conducted They derived two types of controllersbased on Lyapunov stability: a constant-amplitude con-troller (CAC) and a constant-gain controller (CGC) Favor-able vibrational suppression was achieved by implement-ing these two controllers It has been also shown that theCAC is more effective than the CGC for the same maxi-mum voltage However, when the CAC is employed, unde-sirable residual vibration is generated in the settled phasedue to the excessive supply of control voltages from the in-evitable time delay of the hardware system Baz and Poh(17) proposed a modified independent modal space controlmethod to suppress actively the unwanted vibration of aflexible beam structure that features piezoelectric actu-ators The effects of the bonding layer material and theactuator location on the vibrational control performancewere evaluated by numerical simulation Tzou and Gadre(18) derived a physical model for vibrational control, inwhich a piezofilm slab was sandwiched between two otherplates The effectiveness of active vibrational control hasbeen demonstrated by implementing CGC Tzou (19) alsoapplied a piezofilm for vibrational control of arbitrarilyshaped shells Control performance of the distributed sys-tems was successfully evaluated through computer simu-lations by using the CAC and the CGC Baz et al (20) in-tegrated the independent modal space control method andthe positive position feedback method Vibrational controlperformance was enhanced by argumenting the so-calledtime sharing strategy, and its effectiveness was validated
inves-by showing multimode controllability inves-by a single tric actuator On the other hand, Choi et al (21) proposed
piezoelec-a multistep constpiezoelec-ant-piezoelec-amplitude controller (MCAC) to duce undesirable chattering in the settled phase They ex-perimentally demonstrated the effectiveness of the MCAC
re-by comparing the vibrational control response of the CAC.Choi and Kim (22) also proposed a new type of discrete-time, fuzzy, sliding mode controller to reduce unwanted
Trang 11vibrational magnitude favorably in the settled phase Yang
and Lee (23) developed three neural networks for smart
structures that feature a PZT actuator and sensor, one
for system identification, the second for on-line state
es-timation, and the third for vibrational suppression The
effectiveness of the proposed neural networks was
demon-strated by experimentally undertaking transient
vibra-tional control of a cantilevered beam structure Meyer
et al (24) proposed two control methodologies for
vibra-tional control of large flexible structures: positive position
feedback (PPF) and linear quadratic Gaussian (LQG) It
has been shown that PPF is effective in providing high
damping for a particular mode, and LQG is very effective in
meeting specific requirements such as minimization of tip
motion On the other hand, it is generally known that
flexi-ble structures are easily subjected to parameter variations
in practice Therefore, a robust vibrational control
tech-nique that can guarantee favorable structural performance
under system uncertainties needs to be developed Tang
et al (25) proposed an active–passive hybrid piezoelectric
method that used a sliding mode controller to suppress
unwanted vibration of flexible structures A robust sliding
mode controller that compensates for parameter variations
such as material frequency and the hysteretic
nonlinear-ity of the piezoactuator was designed and successfully
im-plemented; it substantially reduced the vibrational
mag-nitude Choi et al (26) formulated a robust quantitative
feedback theory (QFT) controller to suppress the
tion of a flexible structure subjected to parameter
vibra-tions and hysteretic nonlinearity It has been demonstrated
through experiment that the QFT controller is very
ef-fective for robust vibrational control of piezoelectric-based
smart structures
In this article, the CAC, CGC, and MCAC schemes,
which are relatively easy to implement and very
effec-tive for vibrational control of piezoelectric-based smart
structures, are introduced by considering the simple
cantilevered beam structure shown in Fig 7 From the
figure, it is seen that the control objective is to reduce the
vibration in the y direction by activating the
piezoactua-tor To stabilize the structural system, a positive-definite
Lyapunov function F, which is basically a measure of the
energy (potential and kinetic) in the system, is adopted
as follows: 2F=L
Here, L is the length of the beam, and y(x , t) is the
deflection of the beam Minimizing the time derivative of
the function, vibrational control is achieved by bringing
the system to equilibrium Taking the time derivative
of the function and substituting the governing equation
of the beam yields the following (21): ∂ F/∂t =L
0{(1 −
elastic modulus, inertia, density, and cross-sectional area
of the beam, respectively V(t) is the control voltage, and
c is a constant that implies the bending moment per volt.
This constant is normally determined by the geometric
and material properties of the structure It is clear that
the control voltage V(t) should be chosen so that the
second term of the time derivative equation is always as
negative as possible Therefore, two types of control laws
are easily synthesized: (1) C AC: V(t) = −K · sign( ˙V), and
(2) CGC: V(t) = −K2( ˙Vf ) Here K1 and K2 are feedback
gains The variable Vf(t) represents the output
volt-age produced from the piezoelectric sensor The sensor
voltage Vf(t) is proportional to the sign of the angular
displacement at the tip of the beam (16) It has beenexperimentally verified that the CAC is more effectivethan the CGC at the same maximum voltage (16) This
is due to the fact that a square wave has more areathan a sine wave of equal magnitude However, fromthe practical point of view, the CAC causes undesirableresidual oscillations in the settled phase that are at-tributed to the excessive control voltage from the timedelay of the hardware system This problem becomes moreserious when small vibrational levels are considered atrelatively high control voltages On the other hand, theCGC also has some shortcomings under forced vibrationalcontrol Due to insufficient control forces, the suppressionefficiency is degraded This problem becomes more seriouswhen large vibrational levels are considered at relativelylow control voltages The multistep constant-amplitudecontroller (MCAC) has been also proposed to circumventthe drawbacks of the conventional CAC and CGC (21) The
MCAC is given as follows: MCAC: V(t) = −K1· sign( ˙Vf), for
(Vf)m≤ [(Vf)max/a1]; V(t) = −K3· sign( ˙Vf)[(Vf)m/(V f)max], for
(Vf)m≤ [(Vf)max/a2] The variable K i is feedback gain, a iis
a switching constant to determine an appropriate voltage
magnitude, (Vf)max is the initial angular displacement in
the absence of the control voltage, and (Vf)m is the trolled angular deflection at a certain time The feedback
con-gain K iis chosen so that the maximum voltage amplitude
does not exceed the voltage limit Vmax This limit dependsupon the breakdown voltage of the piezoactuator The
determination of the switching constant a i is the keyissue that makes the MCAC algorithm effective Thechattering magnitudes in the settled phase are normallyexperimentally evaluated with respect to imposed initialvibrational magnitudes and also to applied magnitudes
of the control voltage in the CAC And then, from thisinformation, the switching constants are appropriatelychosen so that undesirable chattering can be minimizedfor a certain initial magnitude and corresponding controlvoltage The MCAC may be able to self-tune the voltage
magnitude via the ratio (Vf)m/(V f)max Implementing thistype of controller provides a relatively large control force
to suppress large oscillations at the beginning of thecontrol action and subsequently a small control force
to remove undesirable chattering in the settled phase.Figure 8 schematically presents the types of control inputvoltage for the CAC, the CGC and the MCAC, respectively.Figure 9 presents the measured, transient vibra-tional control responses of a cantilevered beam that fea-tures a piezofilm actuator and sensor (21) The transientvibrational response characteristics were obtained by ex-citing the beam using the first-mode natural frequencyand subsequently removing this excitation and feedbackvoltage applied It is clearly observed that the CAC ismore effective than the CGC but shows unwanted resid-ual vibration (chattering) in the settled phase The chat-tering phenomenon arises from the combined effect of theexcessive supply of control voltage on the relatively small
Trang 12CGCTime
Time
(d)
MCAC
Figure 8 Vibrational control algorithms for a smart structure
that features a piezoactuator and sensor.
oscillation and inevitable time delay of the hardware
sys-tem However, chattering was fairly well eliminated by
employing the MCAC algorithm This implies that the
MCAC produces a relatively small adverse control force for
the time delay in the settled phase Note that the feedback
signal from the piezofilm sensor represents the angular
0
Time (sec)Open-loop (0V)
Figure 9 Transient vibrational responses of a cantilevered beam
that features a piezoactuator and sensor.
displacement at the tip of the cantilevered beam Thus, thedistributed-parameter sensor catches the correspondingresponse caused by all of the vibrational modes Further-more, the CGC, the CAG, and the MCAC are derived with-out modal truncation of the plant model These inherent
Trang 13Figure 10 Simultaneous controllability of various vibrational
modes in a piezofilm-based smart beam.
characteristics of the distributed sensor and control laws
allow one the possibility of controlling all transverse
vibrational modes at once, hence avoiding problems of
spillover of uncontrolled vibrational modes (16) Figure 10
presents the measured transfer function, which is
ob-tained from the ratio of the excitatory input force
mea-sured by the accelerometer to the tip deflection meamea-sured
by the piezofilm sensor (21) It is clearly observed that both
the first and second modes are effectively controlled by
applying the MCAC algorithm without causing spillover
problems
The vibrational control of the piezoelectric-based smart
beam structures can be extended without difficulties to
vi-brational control of the plate or shell structures In the
vibrational control of large structures, determining the
op-timal location for piezoactuators or/and sensors is very
im-portant to suppress effectively unwanted vibration caused
by random disturbances which lead to exciting several
mode shapes simultaneously Furthermore, in practice,
large flexible structures can easily be subjected to
pa-rameter variations such as natural frequency Therefore,
a robust control algorithm should be formulated for the
piezoactuators to protect the robust vibrational control
per-formance from these system uncertainties
VIBRATIONAL CONTROL OF SMART SYSTEMS
Vehicle Suspension Using ER Damper
Recently, a great deal of attention was focussed on a
damper design that significantly suppressed the vibration
of a vehicle system (27) The vehicle vibration was to be
attenuated for various road conditions This is normally
accomplished by employing a suspension system So far,
three types of suspensions were proposed and successfully
implemented: passive, active, and semiactive The passive
suspension system that features an oil damper (or shock
absorber) is simple to design and cost-effective However,
performance limitations are inevitable On the other hand,
the active suspension system provides high control
perfor-mance across a wide frequency range However, the active
suspension requires large power, sources, many sensors,
servovalves, and sophisticated control logic One way to
re-solve these requirements of the active suspension system
is to use a semiactive suspension system The semiactivesuspension system offers desirable performance that isgenerally enhanced in the active mode without large powersources and expensive hardware Recently, a very at-tractive and effective semiactive suspension system fea-turing ER fluids was proposed by many investigators(28–34)
One of the salient properties of an ER fluid is its sponds fast to an electric field, and hence it has a widecontrol bandwidth This inherent feature has triggeredtremendous research activities in the development of var-ious engineering applications including dampers for con-trolling the vibration of vehicles Sturk et al (27) proposed
re-a high-voltre-age supply unit thre-at hre-as re-an ER shock re-absorberand proved its effectiveness via a quarter-car suspensionsystem Nakano (28) constructed a quarter-car suspen-sion model using an ER damper and proposed a propor-tional control algorithm to isolate vibration Petek et al.(29) constructed a semiactive full suspension system thatuses four ER dampers and evaluated the suspension per-formance by implementing a skyhook control algorithmthat considers the heave, pitch, and roll motions of the carbody Gordaninejad et al (30) proposed a cylindrical ERdamper that has multielectrodes and proved its favorablecapability for vibration control by implementing a bang-bang and a linear proportional controller Sims et al (31)proposed an ER valve-controlled vibrational damper, andobtained the linear behavior of the damping force withrespect to the velocity by using a proportional feedbackcontrol gain Peel et al (32) proposed a long-stroke ERdamper for effective vibrational control Choi et al (33)proposed a cylindrical ER damper for a passenger car, andproved its controllability of damping force by implement-ing a skyhook controller Recently, Choi et al (34) devel-oped a sliding mode controller for a full car suspension inusing four ER dampers They constructed a full-car model,and evaluated its vibrational control performance via thehardware-in-the-loop simulation The field test for the ERsuspension system has also been undertaken (35)
In this article, a cylindrical ER damper shown inFig 11 is introduced to evaluate the vibrational control
V
ER Duct
Inner electrode
ER Fluid
Gas chamber Diaphragm
Orifice
Outer electrode Insulator
Outer cylinder
Voltage source
Figure 11 Schematic configuration of an ER damper.
Trang 14Piston velocity (m/s)
Figure 12 Field-dependent damping force of an ER damper.
performance in a passenger vehicle The ER damper is
di-vided into upper and lower chambers by a piston, that is
filled with ER fluid The ER fluid flows by the piston’s
mo-tion through the duct between the inner and outer
cylin-ders from one chamber to the other A positive voltage is
produced by a high-voltage supply unit connected to the
inner cylinder, and the negative voltage is connected to the
outer cylinder The gas chamber located outside the lower
chamber acts as an accumulator of the ER fluid induced
by the piston’s motion If no electric field is applied, the
ER damper produces a damping force caused only by fluid
resistance However, if a certain level of the electric field
is supplied to the ER damper, the ER damper produces an
additional damping force owing to the yield stress of the ER
fluid This damping force of the ER damper can be
continu-ously tuned by controlling the intensity of the electric field
The damping force F of the ER damper shown in Fig 11
can be obtained as follows (34): F = keXP + ceXP˙ + FER The
variable k e is the effective stiffness due to gas pressure,
ce is the effective damping due to the fluid viscosity, XPis
the excitation displacement, and FERis the field-dependent
High voltageamplifier
Microprocessorcontrolalgorithms
AccelerometersgyroscopeLDT sensor
Semi-activeconditions
A/D
ER Shockabsorber
ER shockabsorber
Vertical motionpitch motionroll motion
Discretesignals
Inputvoltage
Modifiedinput voltage
Amplifiedinput field
D/A
Figure 13 Vehicle configuration for an ER
sus-pension test.
damping force which is tunable as a function of applied
electric field E By adopting the Bingham model for the
ER fluid, the controllable damping force FER can be
ex-pressed as FER= (2L/h)(AP− Ar)αE β sign( ˙ XP ) AP and Ar
represent piston and piston rod area, respectively, L is the electrode length, h is the electrode gap, and α and β
are intrinsic values of the ER fluid to be experimentallyevaluated
Figure 12 presents the measured damping force of acylindrical ER damper for a passenger vehicle (35) As seen
in the figure, the damping force increases as the electricfield increases For instance, the damping force is increased
up to 1000 N at a piston velocity of 0.25 m/s by applying anelectric field of 3 kV/mm Note that the level of the damp-ing force of a conventional passive oil damper is almost thesame as this one at 0 kV/mm Thus, we can expect improvedsuspension performance of the vehicle system by control-ling the damping force To evaluate the vibrational controlperformance of the vehicle system using the ER damper,
we can construct a closed-loop control vehicle system, asshown in Fig 13 A portable computer (microprocessor)equipped with a DSP (digital signal processor) board is nor-mally positioned beside the driver’s seat Four pairs (onefor the car body and the others for the wheels) of accelero-meters are installed on four independent suspensions tomeasure the vertical motions of the vehicle The signalsfrom the accelerometers, gyroscope, and LDT (linear dif-ferential transformer) are fed back to the microprocessor,and depending upon the control algorithm employed, therequired control input voltages are determined and ap-plied to the four ER dampers through four high-voltageamplifiers positioned at four corners in the trunk Amongmany controllers are candidates for the vehicle suspension,the skyhook control algorithm, which can be easily imple-
mented, is frequently adopted and given as follows: u i=
C i |˙z si |, for ˙z si (˙z si − ˙z usi)> 0; u i = 0, for ˙z si (˙z si − ˙z usi)> 0.
The variable u i is the control damping force FER, ˙zsidenotes
the vertical velocity of the car body, and ˙zusirepresents the
Trang 15vertical velocity of the wheel The control gain C i needs
to be determined depending upon the road excitation In
the final stage for practical use, the high-voltage
ampli-fier should have short response time and should be
inte-grated with an electronic control unit (ECU) Note that
once the control input u i is determined, the control
elec-tric field to be applied to the ER damper is obtained from
the relationship between the electric field and the damping
force
The control characteristics for suppressing the vibration
of the full-car suspension system are evaluated under two
types of road excitation The first excitation normally used
to reveal the transient response characteristic is a bump
In bump excitation, the vehicle travels over the bump at
a constant velocity of 3.08 km/h (= 0.856 m/s) The second
type of road excitation normally used to evaluate the
fre-quency response is a stationary random process In
ran-dom excitation, the values of road irregularity are chosen
assuming that the vehicle travels on a paved road at a
con-stant velocity of 72 km/h (= 20 m/s) Figure 14a presents
the temporal responses of the ER suspension system to
the bump excitation (34) It is generally known that
ver-tical acceleration of the sprung mass and tire deflection
are used to evaluate the ride comfort and the road
hold-ing of the vehicle, respectively It is seen that both vertical
acceleration of the sprung mass and tire deflection are
sub-stantially reduced by employing the control electric field
This implies that the ER suspension system can
simulta-neously provide both good ride comfort and driving safety
to a driver by applying a control electric field to the ER
dampers Figure 14b presents frequency responses to
ran-dom excitation (34) The frequency responses are obtained
from the power spectral density (PSD) for the suspension
travel and tire deflection As expected, the power spectral
densities for the suspension travel and tire deflection are
substantially reduced in the neighborhood of body
reso-nance (1–12 Hz) It is also observed that tire deflection
is substantially reduced at wheel resonance (10–15 Hz)
This indicates significant enhancement of the steering
sta-bility of the vehicle
Note that most currently employed control algorithms
for vibrational attenuation that use an ER fluid-based
ac-tuator are dubbed semiactive The semiactive control
sys-tem offers desirable performance generally enhanced in
the active mode without requiring large power sources
One of the most popular control logics for the semiactive
control system is the skyhook control algorithm because
it is easy to formulate and implement in practice
Possi-ble candidates for active controllers for the semiactive
con-trol system are the sliding mode concon-trol, neural network
control, Lyapunov-based state feedback control, and
op-timal control However, because the semiactive actuator
cannot increase the mechanical energy of the control
sys-tem, special attention (semiactive conditions in Fig 13)
should be given when these active control strategies are
adopted On the other hand, we can construct an active
control system using an ER fluid by employing a hydraulic,
closed-loop, ER valve–cylinder system In this case, control
logics adapted to conventional hydraulic servomechanism
can be applied without any modification The only
differ-ence is replacing the electromagnetic servovalve by the ER
valve
UncontrolledControlled
−3.0
−1.50.01.53.0
Time (sec)
Uncontrolled Controlled
Bump response(a)
0.0010.0020.0030.004
Uncontrolled Controlled
Random response(b)
Figure 14 (a) Bump and (b) random responses of a passenger
vehicle using ER dampers.
Flexible Manipulator That Features Piezoactuators
Though flexible robotic manipulators have some inherentadvantages over conventional rigid robots, they have posedmore stringent requirements on the control system design,such as accurate end-point sensing and fast suppression
of transient vibration during rapid arm movements thermore, model parameter variations such as natural fre-quencies and damping ratios may easily arise in practicedue to a wide spectrum of various conditions in the design
Trang 16Fur-and manufacturing process, dynamic modeling, Fur-and
oper-ating conditions Numerous control strategies for flexible
manipulators have been proposed in an attempt to find a
successful and practical feedback control Many of the
pre-viously proposed control strategies are based on optimal
control theory (36,37) A few investigators strove to achieve
effective control logics that accounted for the sensitivity
of the control to parameter variations and extraneous
dis-turbances A robust control that guarantees stable system
performance for all possible variations of the parameters
was designed by employing the properties of the uniformly
and ultimate uniformly boundedness of solving the
sys-tem state equation (38) There were also several studies
on sliding mode controllers (39,40) and an H∞ controller
(41) for the feedback control of flexible manipulators
sub-jected to system uncertainties The input torque of the
motor in most of these control techniques for flexible
mani-pulators is determined by simultaneously considering both
the rigid body mode and finite elastic modes The
success-ful experimental realization of this type of torque may be
very difficult under certain conditions due to hardware
lim-itations such as saturation of the motor, computer speed,
and signal noises from the motor and sensors
Further-more, so-called spillover problems will occur because only
some finite elastic modes are considered for controlling a
distributed parameter system of infinite order Other
prob-lems that plague existing conventional control methods
in-clude accurate estimation or measurement of state
vari-ables and the complexity of the control algorithm which
makes on-line implementation infeasible
Recently, a hybrid actuator control scheme that consists
of two types of actuators, motors mounted at the hubs and
piezoactuators bonded to the surface of the flexible links,
has been proposed to resolve some of the existing problems
and hence, to achieve accurate end-point position by
sup-pressing unwanted vibration (42–44) In this article, this
control technique is introduced by considering the
elasto-dynamic flexural response in the horizontal plane (no
grav-ity effect) of a two-link flexible manipulator that features
surface-bonded piezoceramics (PZT) and piezofilms, as
shown in Fig 15 Piezoceramics on the right faces play
the role of actuators, and piezofilms have the role of
dis-tributed sensors to measure elastic deflections caused by
vibrational modes The arm consists of two links connected
by a revolute joint Two links are normally modeled as a
Sliding mode controllers for the motors
Feedforward compensator
Equivalent rigid-body manipulator system
Amplitude controllers for the piezoactuators
Ti+
−
+ +
+ +
Figure 16 Control block diagram for a
two-link flexible manipulator that features piezoactuators.
Figure 15 A two-link flexible manipulator that features
piezo-electric actuators and sensors.
continuous and uniform beam It is also generally assumedthat the beams are flexible only in the direction transverse
to their length in the plane of motion, so that there is noout-of-plane deflection or axial elongation of the links asthe arms move The first flexible link is clamped on thehub of the shoulder motor, and the second flexible link isclamped on the hub of the elbow motor at one end andhas a concentrated tip mass at the other end The motortorque that produces a desired angular position is obtained
by employing the sliding mode controller on the rigid-linkdynamics that have the same mass as that of the flexiblelink Then, the torque is applied to the flexible manipu-lator to activate the commanded motion However, during
this control action, undesirable oscillations w i (x , t) occur
due to the applied torques based on rigid-link dynamics.Subsequently, these vibrations are to be suppressed by ap-plying the feedback voltage to the piezoceramic actuators
As a result, the desired tip motion is achieved favorably.Figure 16 presents a control block diagram that featureshybrid actuators, motors and piezoceramics In the figure,
Tfiis the feed-forward term that compensates for the torque
disturbance [d i (t)] The control torque T i is determined sothat the actual angular motionθ i (t) tracks the desired mo-
tionθid (t) well Because the design procedure for the ing mode controller for the control torque T iis the same as
Trang 17slid-Figure 17 Schematic diagram of the
experimen-tal apparatus for end-point control.
High voltage amp.
Counter
Servo driver
A/D Tip mass
Hub
Motor 2
Hub Motor 1
Piezofilm sensor 2 Piezoactuator 2
Piezofilms
Piezoceramics
Encoders
Motors Piezofilm sensor 1
Piezoactuator 1
D/A
D/A
Microprocessor (controller)
that for the rigid-link robot, we can explain only the
for-mulation of the controller for the piezoactuator As
men-tioned earlier, one of the potential controller candidates
for achieving favorable vibrational control of
piezoelectric-based flexible structures is a constant-amplitude controller
as follows: V i (t) = −K i · sign[ ˙ Vfi (t)] , i = 1, 2 Here, K i is a
feedback gain, and Vfi(t) is the time derivative of the output
signal voltage Vfi(t) from the piezofilm sensor bonded to the
other surface of the flexible link The output voltage
pro-duced by the piezofilm sensor is obtained by integrating the
electric charge developed at a point on the piezofilm along
the entire length of the film surface The feedback gain K i
of the controller for the piezoactuator is chosen by
consid-ering the material property of the piezoceramic actuator
as well as the geometry of the flexible link Furthermore,
the feedback gain should be determined so that the flexible
manipulator system is stable To investigate the stability of
the system, we normally adopt a positive definite Lyapunov
function that is basically a measure of the potential and
kinetic energy of the system The function is given as
fol-lows (43): 2F= ˙zTM ˙z + zTKz Here, M is the system mass
matrix, K is the system stiffness matrix associated with the
link elasticity, and z is the generalized coordinate vector
that consists of the angular displacementθ iand the
elas-tic deflection w i (x , t) We can guarantee the stability of a
flexible manipulator system by choosing a proper feedback
gain K i that makes the time derivative of the Lyapunov
function negative-definite However, the stability of a
flex-ible manipulator system can be violated by fast motions or
by the acceleration phase of the hubs which in turn result
in large oscillations of the flexible links It is known that
when the hub motions are in the deceleration phase, the
stability of the system is satisfied by employing any
posi-tive feedback gain K i On the other hand, if the hub motions
completely stop, the flexible links can be treated just as
cantilever beams Thus, in this case, Lyapunov stability is
also satisfied by employing any positive feedback gain K i
The proper determination of K i normally depends on the
magnitudes of the elastic vibration and angular velocity
Noted that the controller for the piezoactuator is
for-mulated on the basis of the distributed parameter model
without truncating the vibrational mode This allows one
the possibility of simultaneously controlling all of the brational modes Therefore, the control spillover problem,which may occur in the truncated model from uncontrolledvibrational modes, can be avoided We also note that thediscontinuous property may cause undesirable chatteringassociated with time delay and hardware limitations in theactual implementation of constant-amplitude controllers
vi-To remove the chattering effectively, we may use a so-calledmultistep amplitude controller that proportionally tunesthe magnitude of the control voltage according to the out-put signal (21) In practice, we can measure the angulardisplacements by built-in optical encoders in the motorsand the elastic deflections by the distributed piezofilm sen-sors Therefore, we see that no state estimator, which may
be inevitably necessary in most of the conventional controlmethods, is needed for implementing the hybrid actuatorcontrol scheme
Figure 17 presents a typical experimental apparatusfor implementing the hybrid actuator control scheme Thedisplacements of the motors are obtained from the opticalencoders and sent to the microprocessor through the en-coder board The vibrational signals of the flexible linksare measured by the piezofilm sensors and fed back tothe microprocessor through the low-pass analog filter andA/D converter Input torques determined from the slidingmode controller are applied to the motor through a D/Aconverter and a servodriver, and the input voltages deter-mined from the constant-amplitude controller are supplied
to the piezoceramic actuator through the D/A converterand high-voltage amplifier Figure 18 presents the elasticdeflections of the two-link flexible manipulator during theregulating control action (43) It is clear that the unwanteddeflections are significantly reduced by applying feedbackvoltages to the piezoceramic actuators This result directlyindicates that the undesirable tip deflection of each flex-ible link can be effectively suppressed by employing thehybrid actuator control strategy that features servomotorsand piezoactuators Note that the deflection of each linkcould be reduced more by increasing the feedback gain
K i of the constant-amplitude controller However, in thiscase, the breakdown voltage of the piezoactuator should beconsidered
Trang 18−0.010.000.010.02
−0.02
−0.010.000.01
0.02
DesiredActual
Time (sec)
DesiredActual
Time (sec)
Desired Actual
1 M.V Gandhi and B.S Thompson, Smart Materials and
Struc-tures Chapman & Hall, London, 1992.
2 S.B Choi, B.S Thompson, and M.V Gandhi, Proc Damping
’89 Conf., West Palm Beach, FL, Feb 1988, 1, pp CAC.1–
CAC.14.
3 M.V Gandhi, B.S Thompson, S.B Choi, and S Shakir,
ASME J Mech Transmissions Autom Design 111(3): 328–
Elec-and Noise ASME Publication 75, New York, NY, pp 159–167.
8 S.O Oyadiji, J Intelligent Mater Syst Struct 7: 541–549
12 S.B Choi and Y.K Park, J Sound Vib 172(3): 428–432 (1994).
13 D.J Mead and S Markus, J Sound Vib 10(2): 163–175 (1969).
14 H.S Tzou and G.L Anderson, Intelligent Structural Systems.
Kluwer Academic, London, 1992.
15 S.B Choi, Y.K Park, and S.B Jung, J Aircraft 36(2): 458–464
(1999).
16 T Bailey and J.E Hubbard, Jr., J Guidance, Control Dynamics
8(5): 605–611 (1985).
17 A Baz and S Poh, J Sound Vib 126(2): 327–343 (1988).
18 H.S Tzou and M Gadre, J Sound Vib 136(3): 477–490 (1990).
19 H.S Tzou, ASME J Dynamic Syst Meas Control 113: 494–
25 J Tang, K.W Wang, and M Philen, Proc SPIE Conf Smart
Struct Integrated Syst Newport Beach, CA, Mar 1999, 3668,
Trang 1935 H.K Lee, M.S Thesis, Department of Mechanical
Engine-ering, Inha University, Korea, 1999.
36 E Schmitz, Ph D Thesis, Department of Aeronautics and
Astronautics, Stanford University, 1985.
37 Y Sakawa, F Matsno, and S Fukushima, J Robotic Syst 2(4):
42 S.B Choi and H.C Shin, J Robotic Syst 13(6): 359–370 (1996).
43 H.C Shin, Ph D Dissertation, Department of Mechanical
Engineering, Inha University, Korea, 1998.
44 D Sun and J.K Mills, Proc IEEE Int Conf Robotics Autom.
The dominant sources of noise radiation in water are
from ship engines and machinery—the propeller
cavita-tion noise, the noise radiacavita-tion from propeller blades, and
the hydrodynamic pressure fluctuations induced by
tur-bulent water flow along the ship’s hull At speeds below
propeller cavitation inception, a ship’s acoustic signature
is generally dominated by structurally transmitted noise
from onboard machinery Reduction or control of ship noise
has traditionally been implemented by passive means,such as by the use of vibration isolation mounts, flexiblepipe-work, and interior acoustic absorbing materials How-ever, these passive noise control techniques are effec-tive mostly for attenuating high-frequency noise; they aregenerally ineffective for controlling low-frequency noise.There are, on the other hand, active noise control methodsthat have been proven to be effective in controlling low-frequency and tonal noise These active control methodscould be used instead of, or in combination with, passivetechniques, for controlling or reducing ship noise
Active noise control (ANC) involves the reduction orelimination of noise by modification of the dynamic prop-erties of a system or by noise cancellation through linearsuperposition of a secondary noise field of equal but oppo-site strength An active noise control system will typicallyconsist of all or some of the following ingredients: sensors,actuators, and controllers
FUNDAMENTAL CONCEPTS OF SHIP NOISE CONTROL Noise Sources and Transmission Paths in Ship Structures
There are many sources of noise within a ship’s structure.Among these are the propulsion systems, exhaust stacks,and various onboard equipment The principal noise source
is the engine system A typical ship engine along withits mounting system is schematically depicted in Fig 1.The figure shows the various vibroacoustic paths throughwhich the engine vibration is transmitted to the ship struc-ture and eventually radiated into the surrounding medi-ums The various vibroacoustic paths transmit noise indifferent ways For example:
rThe noise from the exhaust stack and the fuel intakeand the cooling systems can be viewed as duct andpiping noise In this mechanism the pressure wave inthe duct is excited and transmitted as noise
rThe mounting systems, consisting of the enginecradle, isolation mount, raft, and foundation, are me-chanical connections between the ship hull and themachine Vibration is transmitted from the enginemotion to the ship hull through these connections.The induced hull vibration is transmitted to the sur-rounding medium and is radiated as acoustic noise.This noise transmission mechanism is referred to asstructural acoustic radiation
rThe engine vibration leads to airborne radiationwithin the enclosure, which may induce an acousticload on the ship hull This resulting excitation is ra-diated to the surrounding medium as acoustic noise.The objective of ship noise control is the minimization ofthe acoustic radiation from the ducting and piping systemsand from the ship’s hull, and appendages to the surround-ing water
Passive and Active Ship Noise Control
In general, then, passive and active control methods aretwo distinct methods that can be used to reduce acoustic
Trang 20As Tank(fuel, )
AS
Propeller
SSSS
SSSA
2
2 2
2 2
2
2
2 2
2 2
1 1
1
1
1
1
Figure 1 Typical marine diesel engine mounted on a ship hull AA: acoustic to acoustic coupling,
SS: structural to structural coupling, AS: acoustic to structural coupling, SA: structural to acoustic coupling The relative importance of energy coupling for radiation into seawater is illustrated by a number (1) for more important and (2) for less important.
noise and radiation Passive noise control essentially
reduces unwanted noise by utilizing the absorption
prop-erty of materials In this approach, sound absorbent
materials are mounted on or around the primary source of
noise or along the acoustic paths between the source and
the receivers of noise At low frequencies, however, passive
control techniques are not effective because the long
acous-tic wavelength of the noise requires large volumes of the
passive absorbers (1)
Active noise control involves the injection of secondary
sound by actuators, which by linear superposition is
addi-tive, to the primary sound field It operates on the
princi-ple of superposing waveforms, by generating a canceling
waveform whose amplitude and envelope match those of
the unwanted noise, but whose phase is shifted by 180◦
(2) The main features of an active control system are
il-lustrated in Fig 2 The basic components are the physical
system (this encompasses the plant, the sensors and the
actuators) and the electronic control system (3) The main
features are:
1 The primary source of noise/disturbance and the
sys-tem to be controlled This is usually referred to as theplant
2 The input and error sensors The input sensors
are the electroacoustic (microphones) or chanical (accelerometers, tachometers) devices thatmeasure the disturbance from the primary sourceand communicate it to the controller They areoften referred to as reference sensors The error
Actuator
Error sensor
Figure 2 Main features of an active
noise control system.
sensor monitors the performance of the activecontroller
3 The actuators These are the electroacoustic or tromechanical devices that generate the secondarynoise or anti-noise in order to reduce or cancel theprimary noise In some cases, the actuators mod-ify the dynamic properties of the system in order
elec-to reduce their noise radiation efficiencies Examples
of actuators include speakers, piezoelectric material,and vibration shakers The actuators, plant, and thesensors are collectively referred to as the physicalsystem
4 The active controller This is the signal processor(usually a digital electronic system) that gives com-mand to the actuators The controller bases its output
on sensor signals (primary noise sensor/error sensor)and usually on some knowledge of how the plant re-sponds to the actuator
The performance of an active controller depends on thephysical arrangements of the control sources (actuators)and the sensors, causality, controllability, observability,and the stability of the control system Active noise con-trol methodologies (ANC) can be classified into two maincategories, namely feedforward control (FFC) and feedbackcontrol (FBC) A summary of the description of the twomethodologies is given by (4) The controllers that havebeen used in active noise control methodologies (FFC andFBC) have evolved over the years from analog to digitaldesigns
Trang 21SENSORS AND ACTUATORS FOR ACTIVE NOISE
AND VIBRATION CONTROL (ANVC)
Piezoelectric Materials
Piezoelectric materials are the oldest and most reliable
ma-terials used in high-speed sensor and actuator
technolo-gies Piezoelectricity was discovered by the Curie
broth-ers in 1880 When a piezoelectric material is subjected to
a mechanical stimulus, an electrical charge or voltage is
induced in the material This is called the “direct
piezo-electric effect,” which enables the material to be used as
a sensor On the other hand, when the piezoelectric
mate-rial is subjected to an electrical charge or voltage, a
me-chanical force or strain is induced in the material This is
called the “converse piezoelectric effect”, which enables the
material to be used as an actuator The induced strain is
directly proportional to the applied electric field and the
linear piezoelectric constitutive equations are given by
where T , E, S, and D are the vectors of stress, electric
field, strain, and electric displacement (charge per unit
area), c , e, and ε are the matrices of the elastic stiffness
coefficients, piezoelectric stress constants, and dielectric
coefficients, respectively Equations (1a) and (1b) describe
the direct and converse effects, respectively Piezoelectric
materials are also called soft ceramics because they are
characterized by high dissipation factors (dielectric losses)
As a result, they have high hysteresis in the displacement
versus voltage curves (5)
The three most important piezoelectric materials are
lithium niobate (Li NbO3), polyvinylidene flouride (PVDF
or PVF2), and lead zirconate titanate (PZT) (6) LiNbO3is
a crystal with a high electromechanical coupling and very
low acoustical attenuation Piezoelectricity is obtained
from a strip of PVDF by stretching it under a high voltage
PVDF, originally discovered in 1969, is known for its
flex-ibility, lightweight, durability, and relatively low acoustic
impedance PZT is by far the most commonly used
piezo-electric material This is a ferropiezo-electric ceramic material
with direct and converse piezoelectric properties A wide
variety of PZT formulations have been developed, with
PZT-5 being one of the most widely used formulations for
actuator applications (7–10) PZT can be used as sensors
or actuators For actuators, the device usually consists of
a stack of many layers of the PZT, alternatively connected
to the positive and negative terminals of a high voltage
source (7)
The mechanical, dielectric and electromechanical
coupling properties of some piezoelectric materials are
shown in Table 1 Many studies, both theoretical and
ex-perimental, have been focused on the application of
piezo-electric materials for vibration control of flexible structures
(6,11–14) Of direct relevance to the noise control problem
that is of interest in this study is the work by Sumali and
Cudeny (15) who developed an actuator from a stack of
layers of piezoelectric material in an actively controlled
engine mount that was designed to reduce structural
vibrations Most of the theoretical and experimental ies on the use of piezoelectric materials for active noisecontrol have been directed at aircraft cabin noise control.For instance, Grewal et al (16) have investigated the use
stud-of piezoceramic elements to reduce cabin noise in the deHavillan Dash-8 series 100/200 aircraft Their study showsthat by judicious actuator and sensor design considerationssystems using bonded piezoelectric actuators and vibra-tion sensors alone are capable of simultaneously providingsignificant noise reduction as well as vibration suppres-sion Other studies include the works of Sutliff et al (17)
on active noise control of low-speed fan rotor-stator modes;and Simonich (5) on the application of rainbow piezoce-ramic actuators (18) for active noise control of gas turbineengines
Electrostrictive Materials
Electrostrictive materials are similar to piezoelectric terials When a mechanical force or strain is applied tothe material, an electric charge or voltage is induced;conversely, when an electric field is applied across anelectrostrictive material, a mechanical strain is induced.Hence, electrostrictive material can also be used as sensors
ma-or actuatma-ors However, there are several differences tween electrostrictive and piezoelectric materials In elec-trostrictive materials, the induced mechanical strain isproportional to the square of the electric field, whereas it
be-is proportional to the electric field in piezoelectric rials Thus electrostrictive materials always produce pos-itive displacements regardless of the polarity As a re-sult, they are always in compression when doing work andavoid typical weakness of ceramics in tension (19) Elec-trostrictive materials exhibit microsecond recovery timeupon withdrawal of the electric field, compared to millisec-onds for piezoelectric materials Electrostrictive materialshave lower dissipation factors (and low displacements andhysteresis) compared to piezoelectric materials, and areregarded as hard ceramics (5) The most commonly usedelectrostrictive material is lead magnesium niobate (PMN)ceramic material
mate-Magnetostrictive Materials
Magnetostrictive materials are those materials that dergo an induced mechanical strain when subjected to amagnetic field On the other hand, when a mechanicalstress (or strain) is applied to the material, it undergoes do-main changes that yield a magnetic field These materialscan thus be used as sensors and actuators due to the directand converse effects The most common magnetostrictivematerial is TERFENOL (consisting of TERbium, FE (iron)and dysprosium, which was developed by NOL, the NavalOrdinance Laboratory) The most commonly used formu-lation is TERFENOL-D Magnetostrictive materials canexhibit strains of up to 0.2% at reasonably low magneticfield strength (20)
un-A detailed description of a magnetostrictive actuator
is presented by Giugiutu et al (7), who show the struction of a terfenol actuator The actuator consists of
con-a TERFENOL-D rod inside con-an electric coil thcon-at is enclosed
Trang 22Table 1 Advantages and Disadvantages of Various Sensor and Actuator Technologies
rUsed as sensors and actuators rRelatively low strain and low displacement
rVery large frequency range capability (typically, less than 0.1% strain,
rQuick response time and 1–100 microns displacement for
rVery high resolution and dynamic range stack actuators)
rPossibility of integration in the structure rActuators require relatively costly
for thin PZT actuators and PVDF voltage amplifiers sensors rLow recoverable strain (0.1%)
rPossibility of shaping PVDF sensors rPiezoelectric ceramics are brittle
(spatial filtering) rCannot measure direct current
rSusceptible to high hysteresis
and creep when strained in direction
of poling (e.g., stack actuators)
Electrostrictive materials
Example:
Lead-magnesium
niobate (PMN)
rUsed as sensor and actuators rMore sensitive to temperature
rLower hysteresis and creep variations than piezoelectrics
compared to piezoelectric
rPotentially larger recoverable strain
than piezoelectric
Magnetostrictive materials
Example:
Terfenol-D
rHigher force and strain capability than rLow recoverable strain (0.15%)
piezoceramics (typically, 1000 rOnly for compression components
microstrain deformation) rNonlinear behavior
rSuited for high-precision applications
rSuited for compressive load carrying
components
rVery durable
Shape-memory alloys (SMA)
Example:
NITINOL
rLarge recoverable strain (8%) rSuited for low-frequency (0–10 Hz)
used largely for actuation due to large and low-precision application force generation rSlow response time
rLow voltage requirements rComplex constitutive behavior
with large hysteresis
Optical fibers
Examples:
Bragg grating,
Fabry-Perot
rSuited for remote sensing of structures rUsed for sensing alone
rCorrosion resistant rBehavior is complicated by thermal strains
rImmune to electric interference
rSmall, light, and compatible with
advanced composite
Electrorheological fluids (ER)
Example:
Alumino-silicate
in paraffin oil
rSimple and quiet devices rLow-frequency applications
rSuitable for vibration control rNonlinear behavior
rOffers significant capability and rCannot tolerate impurities
flexibility for altering structural response rFluid and solid phases tend to separate
rLow density rNot suitable for low temperature applications
rHigh-voltage requirements (2–10 kV)
rHigherη p /τ2
y ratio than MR*
Magnetorheological rSimple and quiet devices rNonlinear behavior
fluids (MR) rQuick response time rHigher density than ER
rSuitable for vibration control
rOffers significant capability and
flexibility for altering structural response
rLow voltage requirements
rBehavior not affected by impurities
rSuitable for wide range of temperatures
rLowerη p /τ2
yratio than ER*
rLarge dynamic range rNeed to achieve directionality in
rExcellent linearity some active control systems (e.g., ducts)
rNeed protection to dust,
moisture, high temperature
(cont.)
Trang 23rGood low-frequency sensitivity (0–10 Hz) rLow-frequency range (typically, below 100 Hz)
rNoncontacting measurement (proximity probe) rLow dynamic range (typically, 100 : 1)
rWell suited to measurement of relative rLow resolution
displacement in active mounts
Velocity sensors rNoncontacting measurement rLow dynamic range (typically 100 : 1)
(magnetic) rWell suited to measurement of relative rLow resolution
velocity in active mounts rHeavy
Accelerometers rLarge dynamic range rLow sensitivity in low frequency (0–10 Hz)
rExcellent linearity rRequire relatively expensive charge
amplifiers (piezoelectric accelerometers)
Loudspeakers rLow cost rNonlinear behavior if driven close to maximum power
rSpace requirement (backing enclosure)
rNeed protection to dust, moisture, high
temperature, corrosive environment
Electrodynamic and rRelatively large force/large rMay need a large reaction mass to
electromagnetic displacement capability transmit large forces
rExtended frequency range
Hydraulic and rLarge force/large rLow-frequency range (0–10 Hz for
pneumatic actuators displacement capability pneumatic; 0–150 Hz for hydraulic)
rNeed for hydraulic or compressed air power supply
rNonlinear behavior
rSpace requirement
in an annular armature When the coil is activated, the
TERFENOL rod expands and produces a displacement
The TERFENOL-D bar, coil, and armature are
assem-bled between two steel washers and put inside a
protec-tive wrapping to form the basic magnetoacprotec-tive induced
strain actuator unit (7) The main advantage of terfenol is
its high-force capability at relatively low cost (21) It also
has the advantage of small size and light weight, which
makes it suitable for situations where no reactive mass is
required such as in stiffened structures of aircraft and
sub-marine hulls The disadvantages of TERFENOL include
its brittleness and low tensile strength (100 MPa)
com-pared to compressive strength (780 MPa) Its low
displace-ment capability is also a major disadvantage especially in
the low-frequency range (less than 100 Hz) In addition, it
also exhibits large hysteresis resulting in a highly
nonlin-ear behavior that is difficult to model in practical
applica-tions (20,21) Tani et al (20) have reviewed of studies on
modeling the nonlinear behavior of TERFENOL-D as well
as its application in smart structures Ackermann et al
(22) developed a transduction model for magnetostrictive
actuators through an impedance analysis of the
electro-magneto-mechanical coupling of the actuator device This
model provided a tool for in-depth investigation of the
frequency-dependent behavior of the magnetostrictive
ac-tuator, such as energy conversion, output stroke, and force
The feasibility of using embedded magnetostrictive mini
actuators (MMA) for vibration suppression has been
in-vestigated by (20)
Shape-Memory Alloys (SMAs)
Shape-memory alloys (SMAs) are materials that undergoshape changes due to phase transformations associatedwith the application of a thermal field When a SMAmaterial is plastically deformed in its martensitic (low-temperature) condition, and the stress is removed, it re-gains (memory) its original shape by phase transforma-tion to its austenite (high-temperature) condition, whenheated SMAs are considered as functional materials be-cause of their ability to sense temperature and stressloading to produce large recovery deformations with forcegeneration TiNi (nitinol), which is an alloy comprisingapproximately 50% nickel and 50% titanium, is the mostcommonly used SMA material Other SMA material in-cluding FeMnSi, CuZnAl, and CuAlNi alloys have also beeninvestigated (20,23)
Typically, plastic strains of 6% to 8% can be completelyrecovered by heating nitinol beyond its transition temper-ature (of 45–55◦C) According to Liang and Rogers (24) re-straining the material from regaining its memory shapecan yield stresses of up to 500 MPa for 8% plastic strainand a temperature of 180◦C By transformation from themartensite to austenite phase, the elastic modulus of niti-nol increases threefold from 25 to 75 GPa, and its yieldstress increases eightfold from 80 to 600 MPa (25).SMAs can be used for sensing or actuation, althoughthey are largely used for actuation due to their largeforce generation capabilities They have very low voltage
Trang 24requirements for operation and are very suited for
low-frequency applications However, their use is limited by
their slow response time, which makes them suitable for
low-precision applications only Also, they exhibit complex
constitutive behavior with large hysterises, which makes it
difficult to understand their behavior in active structural
systems To provide a better understanding of the behavior
of SMAs, several researchers have focused on the
develop-ment of constitutive models for SMAs Some of the most
prominent and commonly used ones are those by Tanaka
(26), Liang and Rogers (24), and Boyd and Lagoudas (27)
These models are derived from phenomenological
consid-erations of the thermomechanical behavior of the SMAs
Because of the numerous advantages they offer, several
investigations on the application of SMAs have been
car-ried out within the present decade Reviews of these
ap-plications, focusing on fabrication of SMA hybrid
com-posites, analytical and computational modeling, active
shape control, and vibration control, are presented in
(20,23)
Optical Fibers
For many applications, ideal sensors would have such
at-tributes as low weight, small size, low power,
environmen-tal ruggedness, immunity to electromagnetic interference,
good performance specifications, and low cost The
emer-gence of fiber-optic technology, which was largely driven by
the telecommunication industry in the 1970s and 1980s, in
combination with low-cost optoelectronic components, has
enabled fiber-optic sensor technology to realize its potential
for many applications (28–30) A wide variety of fiber-optic
sensors are now being developed to measure strain,
tem-perature, electric/magnetic fields, pressure, and other
mea-surable quantities Many physical principles are involved
in these measurements, ranging from the Pockel, Kerr, and
Raman effects to the photoelastic effect (31) These sensors
use intensity, phase, frequency, or polarization modulation
(32) In addition, multiplexing is largely used for
many-sensor systems Fiber-optic many-sensors can also be divided in
discrete sensors and distributed sensors to perform
spa-tial integration or differentiation (33) Three types of
fiber-optic strain sensors are reviewed in the following: extrinsic
interferometric sensors, Bragg gratings, and sensors based
on the photoelastic effect
The most widely used phase modulating fiber-optic
sen-sors are the extrinsic interferometric sensen-sors Two fibers
and directional couplers are generally used for these
sen-sors One of the fibers acts as a reference arm, not affected
by the strain, while the other fiber acts as the sensing arm
measuring the strain field By combining the signals from
both arms, an interference pattern is obtained from the
optical path length difference This interference pattern
is used to evaluate the strain affecting the sensing arm
(e.g., by fringe counting) These sensors have a high
sen-sitivity and can simultaneously measure strain and
tem-perature One interferometer now being used in industrial
applications is the Fabry-Perot interferometer, where a
sensing cavity is used to measure the strain (34) This
sen-sor uses a white-light source and a single multiple mode
Multimodefiber
Cavitylength
Weldedspot
DielectricmirrorsMicrocapillary
Bragg grating reflectors can be written on an optical
fiber using a holographic system or a phase mask to ate a periodic intensity profile (35) These sensors can beused as point or quasi-distributed sensors The reflectedsignal from these sensors consist of frequency componentsdirectly related to the number of lines per millimeter ofeach grating reflector and, thus, to the strain experienced
gener-by the sensor Fiber-optic sensors based on Bragg gratingsare used to measure strain and temperature, either si-multaneously or individually (36) The Bragg gratings aretraditionally interrogated using a tunable Fabry-Perot or
a Mach-Zender interferometer Recently, long-period ings have been used to interrogate Bragg sensing gratings(37) Bragg gratings have been used to measure vibrationseither directly or through the development of novel ac-celerometers A typical fiber Bragg grating (FBG) system
grat-is illustrated in Fig 4
The principle of operation of the sensors based on the
photoelastic effect is a phase variation of the light passing
through a material (fiber) that is undergoing a strainvariation This phase variation can be produced by twoeffects on the fiber: (1) the variation of the length produced
by the strain; (2) the photo-elastic effect and the modaldispersion caused by the variation of the diameter of thefiber These sensors are classified in modal interferometricsensors and polarimetric sensors As it integrates thestrain effect over its length, the modal interferometricsensor can act as a spatial filter if the propagation constant
is given a spatial weighting (38)
Reflectedwave
Bragggrating
Transmittedwave
Incidentwave
Figure 4 Bragg grating on an optical fiber.
Trang 25Electrorheological Fluids (ER)
Electrorheological fluids (ER) are a class of controllable
fluids that respond to an applied electric field with a
dra-matic change in rheological behavior The essential
cha-racteristic of ER fluids is their ability to reversibly change
from free-flowing linear viscous liquids to semisolids
hav-ing controllable yield strength in milliseconds when
ex-posed to an electric field (23) The ER fluids provide very
simple, quiet and rapid response interfaces between
elec-tronic controls and mechanical systems They are very
suit-able for vibration control because of the ease with which
their damping and stiffness properties can be varied with
the application of an electric field
ER materials consist of a base fluid (usually a low
vis-cosity liquid) mixed with nonconductive particles, typically
in the range of 1 to 10 m diameter These particles become
polarized on the application of an electric field, leading to
solidification of the material mixture Typical yield stresses
in shear for ER materials are about 5 to10 kPa The most
common type of ER material is the class of dielectric oils
doped with semiconductor particle suspensions, such as
aluminosilicate in paraffin oil The material exhibits
non-linear behavior, which is still not completely understood by
the research community This lack of understanding has
hindered efforts in developing optimal applications of ER
materials However, electrorheological fluids may be
suit-able for many devices, such as shock absobers and engine
mounts (23,25)
Magnetorheological Fluids (MR)
Magnetorheological fluids (MR) are similar to ER
materi-als in that they are materi-also controllable fluids These materimateri-als
respond to an applied magnetic field with a change in the
rheological behavior MR fluids, which are less known than
ER materials, are typically noncolloidal suspensions of
micron-sized paramagnetic particles The key differences
between MR and ER fluids are highlighted in Table 1 In
general, MR fluids have maximum yield stresses that are
20 to 50 times higher than those of ER fluids, and they
may be operated directly from low-voltage power supplies
compared to ER fluids which require high-voltage (2–5 kV)
power supplies Furthermore, MR fluids are less sensitive
to contaminants and temperature variations than are ER
fluids MR fluids also have lower ratios ofη p /τ2
y than ERmaterials, whereη pis the plastic viscosity andτ ythe max-
imum yield stress This ratio is an important parameter in
the design of controllable fluid device design, in which
min-imization of the ratio is always a desired objective These
factors make MR fluids the controllable choice for recent
practical applications Several MR fluid devices developed
by Lord Corporation in North Carolina under the
Rheo-netic trade name (23)
Microphones
Microphones are usually the preferred acoustic sensors in
active noise control applications Relatively inexpensive
microphones (electret or piezoelectric microphones) can be
used in most active noise control systems because the
fre-quency response flatness of the microphones is not critical
Microphonesupport section
Detection pipesection
Absorbing material
Microphone
between two insulating washer
Figure 5 Sound pressure and particle velocity sensing.
in digital active control systems, as it is compensated in theidentification of the control path The most common types
of microphones are omni-directional, directional, and probemicrophones
Whenever turbulent flow is present in the acousticmedium (e.g., a turbulent flow in a duct conveying a gas or
a fluid), turbulent random pressure fluctuations are ated in the flow, adding to the disturbance pressure field.The most common way of reducing the influence of turbu-lent noise is to use a probe tube microphone consisting of
gener-a long, ngener-arrow tube with gener-a stgener-andgener-ard microphone mounted
at the end The walls of the tube are porous or containholes or an axial slit The probe tube microphone must beoriented with the microphone facing the flow Probe tubemicrophones are convenient as reference sensors in activecontrol systems in ducts because they act as both direc-tional sensors and turbulence filtering sensors Details onthe principle of operation can be found in (39) Low-costmicrophone probes for hot corrosive industrial environ-ments are also available from Soft dB Inc Figure 5 shows amicrophone adapted for such environments
Displacement and Velocity Transducers
Although their dynamic range is usually much less thanthat for accelerometers, displacement and velocity trans-ducers are often more practical for very low frequencies(0–10 Hz) where vibration amplitudes can be of the order
of a millimeter or more for heavy structures whose sponding accelerations are small Also, in low-frequencyactive control systems, displacement or velocity ratherthan acceleration can be the preferred quantities to min-imize The displacement and velocity transducers are de-scribed below
corre-Proximity probes are the most common type of
displace-ment transducers There are two main types of proximityprobes, the capacitance probe and the Eddy current probe.Proximity probes allow noncontact measurement of vibra-tion displacements They are well suited to vibration dis-placement measurements on rotating structures The dy-namic range of proximity probe is very small—typically
100 : 1 for low-frequency applications (<200 Hz) The
res-olution varies from 0.02 to 0.4 mm
The linear variable differential transformer (LVDT) is a
displacement transducer that consists of a single primaryand two secondary coils wound around a cylindrical bobbin
Trang 26A movable nickel iron core is positioned inside the
wind-ings, and it is the movement of this core that is measured
The dynamic range of an LVDT is typically 100 : 1, with
a resolution ranging from 0.01 to 1 mm The frequency
range is typically dc to 100 Hz The total length of the
sen-sor varies from 30 to 50 mm for short stroke transducers
to about 300 mm for long stroke transducers
The linear variable inductance transformer (LVIT) is a
displacement transducer based on the measurement of
in-ductance changes in a cylindrical coil The coil is excited
at about 100 kHz, and the inductance change is caused by
the introduction of a highly conductive, nonferrous coaxial
rod sliding along the coil axis It is the movement of this
coaxial rod that is measured This type of transducer is
particularly suited for measuring relative displacements
in suspension systems Transducer sizes vary from
dia-meters of a few millidia-meters to tens of millidia-meters
Often used among the velocity transducers is the
non-contacting magnetic type consisting of a cylindrical
perma-nent magnet on which is wound with an insulated coil A
voltage is produced by the varying reluctance between the
transducer and the vibrating surface This type of
trans-ducer is generally unsuitable for absolute measurements,
but it is very useful for relative velocity measurement such
as needed for active suspension systems The frequency
range of operation is 10 Hz to 1 kHz; the low-resonance
frequency of the transducer makes it relatively heavy
Velocity transducers cover a dynamic range between 1 and
100 mms−1 Low-impedance, inexpensive voltage
ampli-fiers are suitable
Accelerometers
Accelerometers are the most employed technology for
vi-bration measurements They provide a direct
measure-ment of the acceleration, usually in the transverse
direc-tion of a vibrating object The acceleradirec-tion is a quantity
well correlated to the sound field radiated by the
vibrat-ing object Therefore, accelerometers can be a convenient
alternative to microphones as error sensors for active
structural acoustic control Accelerometers usually have a
much larger dynamic range than displacement or velocity
sensors A potential drawback of accelerometers, in
low-frequency active noise control systems, is their low
sensi-tivity at low frequency (typically 0–10 Hz)
Small accelerometers can measure higher frequencies,
and they are less likely to affect the dynamics of the
struc-ture by mass loading it However, small accelerometers
have a lower sensitivity than bigger ones
Accelerome-ters range in weight from miniature 0.65 g for high-level
vibration amplitudes up to 18 kHz on lightweight
struc-tures, to 500 g for low-level vibration amplitudes on
heavy structures up to 1 kHz Because of the
three-dimensional sensitivity of piezoelectric crystals,
piezoelec-tric accelerometers are sensitive to vibrations at right
an-gle to their main axis The transverse sensitivity should be
less than 5% of the axial sensitivity There are two main
types of accelerometers: piezoelectric and piezoresistive
A piezoelectric accelerometer consists of a small
seis-mic mass attached to a piezoelectric crystal When the
ac-celerometer is attached to a vibrating body, the inertia force
due to the acceleration of the mass produces a mechanicalstress in the piezoelectric crystal that is converted into anelectric charge on the electrodes of the crystal Providedthat the piezoelectric crystal works in its linear regime,the electric charge is proportional to the acceleration of theseismic mass The mass may be mounted to produce eithercompressive or tensile stress, or alternatively, shear stress
in the crystal A piezoelectric accelerometer should be usedbelow the resonance of the seismic mass–piezoelectric crys-tal system Since piezoelectric accelerometers essentiallybehave as electric charge generators, they must generally
be used with high-impedance charge amplifiers The cost ofsuch amplifiers can represent a significant amount of thetotal cost of an active control system when a large number
of accelerometers are used
Piezoresistive accelerometers rely on the measurement
of resistance change in a piezoresistive element usuallymounted on a small beam and subjected to stress Piezore-sistive accelerometers are less sensitive than piezoelectric.They require a stable, external dc power supply to excitethe piezoresistive elements However, piezoresistive ac-celerometers have a better sensitivity at low frequency, andthey require less expensive, low-impedance voltage ampli-fiers The piezoresistive element is sometimes replaced by apiezoelectric polymer film (PVDF), and the electric chargeacross the electrodes of the PVDF is collected as the sen-sor output Such a PVDF accelerometer has a sensitivityand frequency response similar to the piezoresistive ac-celerometer, and it is less expensive than the piezoelectricaccelerometer
Loudspeakers
The electrodynamic loudspeaker is the most commonly ployed actuator technology for active noise control applica-tions When selecting a loudspeaker for an active noise con-trol system, the important parameter is the cone volumevelocity required to cancel the primary sound field (21).For small systems, (small-duct, low-noise, domesticventilation system), active acoustic noise control can beachieved with small commercial medium-quality speakers(radio-type speaker) However, for bigger systems, precau-tions have to be taken
em-Electrodynamic loudspeakers exhibit a nonlinear havior when they are driven close to maximum power ormaximum membrane deflection It can significantly de-grade the performance of active control systems based onlinear filtering techniques It is thus important that loud-speakers should be driven at a fraction of the maximumpower or maximum deflection specifications, especially insituations where single-frequency or harmonic noise has
be-to be attenuated For random noise, the peak cone velocityrequirements for active control are likely to be four or fivetimes the estimated rms velocity requirements (39)
In active control of single-frequency noise, it is desirable
to design the loudspeaker so that its mechanical resonancelies close to the frequency of interest This resonance fre-quency can be adjusted to suit a particular application ei-ther by adding mass to the cone (to reduce the frequency) or
by adding a backing enclosure to the speaker (to increasethe frequency)
Trang 27Standard speaker
Perforated metal sheets
Figure 6 Protective system for loudspeaker membrane.
Operation in industrial environments requires
consid-erable precautions In high-humidity, high-temperature
and corrosive environments, the loudspeaker cone must be
protected with a heat shield Soft dB used a Teflon
brane and a perforated metal sheet to protect the
mem-brane of the speaker from corrosive gas (see Fig 6)
Electromagnetic Actuators
For vibration control purposes, electromagnetic actuators
can be classified into electrodynamic shakers and
elec-trical motors The latter can be used for low-frequency
vibration control Electrodynamic shakers are generally
defined as devices having a central inertial core (usually
a permanent magnet) surrounded by a winding This type
of inertial actuator applies a point force to a structure by
reacting against the inertial mass As in a loudspeaker, a
time-varying voltage is applied to the coil in order to move
the inertial mass and to force the movement of the
struc-ture onto which the shaker is attached
Electricmotors
SynchronousASynchronous
Figure 7 Classification of electric motors (42).
Other inertial type actuators are available which use,for example, the piezoelectric effect, instead of a coil, tomove the inertial mass Proof-mass actuators (also calledinertial actuators) are very similar in their operation toelectrodynamic shakers They usually consist of a massthat is moved by an alternating electromagnetic field.These devices can generate relatively large forces anddisplacements and can be good alternatives to costlyelectrodynamic shakers The devices can excite very stiffstructures such as electrical power transformers Anotheradvantage of proof-mass actuators is that their resonantfrequency can be easily tuned for optimal efficiency at agiven frequency
Electrical Motors
The advent of new control strategies and digital controllershas revolutionized the way electrical motors can be usedand now allows for the use of motor technologies that werepreviously difficult to implement in practical applications.Simple motor drives were traditionally designed withrelatively inexpensive analog components that suffer fromsusceptibility to temperature variations and componentaging New digital control strategies now allow for the use
of electrical motors in active vibration control applications.These efficient controls make it possible to reduce torqueripples and harmonics and to improve dynamic behavior
in all speed ranges The motor design is optimized due tolower vibrations and lower power losses such as harmoniclosses in the rotor Smooth waveforms allow an optimiza-tion of power elements and input filters Overall, these im-provements result in a reduction of system cost and betterreliability
Electrical motors can be divided into motors with a manent magnet rotor (ac and dc motors) and motors with acoiled rotor Figure 7 illustrates a detailed classification ofthe electrical motors With the advent of new controllers,the tendency is to classify electrical motors under ac or dcaccording to the control strategy
per-Due to its high reliability and high efficiency in a
re-duced volume, the brushless motor is actually the most
Trang 28interesting motor for application to active vibration control
(40) Although the brushless characteristic can be applied
to several kinds of motors, the brushless dc motor is
con-ventionally defined as a permanent magnet synchronous
motor with a trapezoidal back EMF waveform shape, while
the brushless ac motor is conventionally defined as a
per-manent magnet synchronous motor with a sinusoidal back
EMF waveform shape New brushless and coreless motors
are now available which are very linear over a wide speed
range (41) The brushless motor control consists of
generat-ing variable currents in the motor phases The regulation
of the current to a fixed 60◦reference can be realized in two
modes: pulse width modulation (PWM) or hysteresis mode.
Shaft position sensors (incremental, Hall effect, resolvers)
and current sensors are used for the control Linear
per-manent magnet motors are also available that, in addition
to the linear action, allow better magnetic dissipation in
the core as it is distributed in space
If volume is not a major concern, a second type of motor
to be used in active vibration control is the induction or
ac motor (41) As for the brushless motor, the performance
of an ac motor is strongly dependent on its control DSP
controllers enable enhanced real time algorithms There
are several ways to control an induction motor in torque,
speed, or position; they can be categorized in two groups:
the scalar and the vector control Scalar control means that
variables are controlled only in magnitude, and the
feed-back and command signals are proportional to dc
quanti-ties The vector control is referring to both the magnitude
and phase of these variables Pulse width modulation
tech-niques are also used for the control of induction motors, and
indirect current measurement (using a shunt or Hall effect
sensor) is used as a feedback information for the controller
The third electrical motor used for active vibration
con-trol is the switched reluctance motor (40) This motor is
widely used mainly because of its simple mechanical
con-struction and associated low cost and secondarily because
of its efficiency, its torque/speed characteristic and its very
low requirement for maintenance This type of motor,
how-ever, requires a more complicated control strategy The
switched reluctance motor is a motor with salient poles on
both the stator and the rotor Only the stator carries
wind-ings One stator phase consists of two series-connected
windings on diametrically opposite poles Torque is
pro-duced by the tendency of its movable part to move to a
position where the inductance of the excited winding is
maximized There are two ways to control the switched
re-luctance motor in torque, speed and position Torque can
be controlled by the current control method or the torque
control method The pulse width modulation (PWM)
strat-egy is used in both current and torque control approaches
to drive each phase of the switched reluctance motor
ac-cording to the controller signal
Hydraulic and Pneumatic Actuators
Hydraulic and pneumatic actuators are good candidate
technologies when low frequency, large force, and
displace-ments are required Hydraulic actuators consist of a
hy-draulic cylinder in which a piston is moved by the action
of a high-pressure fluid The main advantage of hydraulic
actuators is their large force and large displacement bility for a relatively small size The disadvantages includethe need for a hydraulic power supply (which can requirespace and generate noise), the high cost of servo-valves,the nonlinear relation between the servo-valve input volt-age and the output force or displacement produced by theactuator, and the limited bandwidth of the actuator (0–
capa-150 Hz) Hydraulic actuators have been used in the design
of active dynamic absorbers for ship structures (42,43).The principle of operation of pneumatic actuators is verysimilar to hydraulic actuators, except that the hydraulicfluid is replaced by compressed air Due to the higher com-pressibility of air, the bandwidth of pneumatic actuators
is reduced (typically 0–10 Hz), which restricts the tion to nonacoustic problems Pneumatic actuators may be
applica-an attractive option when applica-an existing air supply is alreadyavailable
APPLICATIONS OF NOISE CONTROL
IN SHIP STRUCTURES
A typical marine diesel engine mounted on a ship hull
is schematically depicted in Fig 1 The figure shows thevarious vibroacoustic paths through which the engine vi-bration is transmitted to the ship structure, and eventu-ally radiated into seawater In the figure the coupling be-tween structural and acoustic energy is classified using thefollowing symbols: AA: acoustic to acoustic coupling, SS:structural to structural coupling, AS: acoustic to structuralcoupling, SA: structural to acoustic coupling The relativeimportance of energy coupling for radiation into seawater
is illustrated by a number As shown, there are five ble energy transmission paths, including (1) the mountingsystem, consisting of the engine cradle, isolation mounts,raft, and foundation; (2) the exhaust stack; (3) the fuel in-take and cooling system; (4) the drive shaft; and (5) theairborne radiation of the engine In this study, these fivepaths are grouped into four categories, corresponding togeneric active control problems:
possi-rPath 1: Active vibration isolation (mounting system).
rPath 2: Active control of noise in ducts and pipes haust stack; fuel intake and cooling system)
(ex-rPath 3: Active control of vibration propagation inbeam-type structures (drive shaft)
rPath 4: Active control of enclosed sound fields borne radiation of the engine)
(air-Path 1: Active Vibration Isolation
Active vibration isolation involves the use of an active tem to reduce the transmission of vibration from one body
sys-or structure to another (e.g., transmission of periodic bration from a ship’s engine to the ship’s hull) Such anactive isolation system will be used in practice to comple-ment passive, elastomeric isolation mounts between theengine and supporting structure An active isolation sys-tem is usually much more complex and expensive thanits passive counterpart, but has the advantage of offer-ing better low-frequency isolation performances, and can
Trang 29vi-be designed for a vi-better static stability of the supported
equipment
The first class of system involves the control of
sys-tem damping, and is often referred as a semiactive
isola-tion system, Fig 8(a) The damping modificaisola-tion is usually
achieved by a hydraulic damper with varying orifice sizes
This system is often used for active suspensions in cars
Such a system involves control time constants significantly
longer than the disturbance time constants, with the
ad-vantage of a simpler and less expensive implementation
However, low-frequency performance is much less than for
fully active systems described in the following
A second class of system involves an active control
ac-tuator in parallel with a passive system, with the acac-tuator
Vibrating body
Spring
(a)
Variabledamper
Figure 8 Active vibration isolation systems: (a) semiactive
sys-tem with variable damper; (b) active syssys-tem with control force
ap-plied to both vibrating body and base structure; (c) active system
with control force in series with passive mount.
exerting a force on either the base structure or the rigidmass, Fig 8(b) In this parallel configuration, the actua-tor is not required to withstand the weight of the machine;
as compared to the configuration of Fig 8(c), the requiredcontrol force is smaller above the natural frequency of thesystem (44) The main disadvantage of this configuration
is that at higher frequency (outside the frequency rangewhere the actuator is effective), the actuator itself can be-come a transmission path At low frequency, the large dis-placement/large force requirements for heavy structurespreclude the use of piezoelectric, magnetostrictive actua-tors Instead, hydraulic, pneumatic, or electromagnetic ac-tuators (with their associated weight, space, and possiblyfluid supplies problems) must be used As far as practicalapplication of active control is concerned, the use of an ac-tuator in parallel with a passive isolation stage could havedistinct advantages In a given application, if an actuatorcan be found that provides a control force of the order of theprimary force exciting the machine, then it may be possible
to use of much higher mounted natural frequency ated with the passive isolation stage than would be other-wise possible This in turn has advantages for the stability
associ-of the mounted machine
A third configuration with the active system in serieswith the passive mount is shown on Fig 8(c) Such a sys-tem has several advantages over the parallel configuration.The active system is now isolated from the dynamics of thereceiving structure, which simplifies the control in the case
of a flexible base structure, and the use of an intermediatemass creates a two-stage isolation system that offers betterisolation performance in higher frequency
Path 2: Active Control of Noise in Ducts and Pipes
The reduction of duct noise is the first-known application
of active noise control Active control systems for duct noiseare now a mature technology, with several commercial sys-tems available for ventilation systems, chimney stacks, orexhausts All existing commercial systems are based onfeedforward adaptive control systems In the case of ductscontaining air or a gas, loudspeakers are generally used ascontrol sources, and microphones as error sensors.Two important classes of systems must be distin-guished, depending on the frequency and the cross-sectional dimension of the duct:
1 Systems for which only plane wave propagation ists in the duct Such systems will necessitate asingle-channel control system (one control source andone error sensor)
ex-2 Systems for which higher-order acoustic modes agate in the duct Such systems will require a multi-channel control system
prop-The occurrence of higher-order modes in a duct depends
on the value of the cut-on frequency For a rectangular
duct, the cut-on frequency is given by f c = c0/2d, where d
is the largest cross-sectional dimension and c0is the speed
of sound in free space For a circular duct, f c = 0.586 c0/d,
where d is the duct diameter Higher-order modes will agate at frequencies larger than f
Trang 30prop-For active noise control (ANC) in the large duct,
mul-tichannel acoustical ANC systems are necessary, and M
error sensors have to be used to control M modes for
high-order propagation cases The error sensors should not be
located at the nodal lines (observability condition) (45) For
a rectangular duct, the location of the error sensors is
rel-atively simple because the nodal lines are fixed along the
duct axis However, in circular ducts, the location of the
nodal lines changes along the duct axis, since the modes
usually spin as a function of the frequency, temperature,
and speed (46,47) Those variations of the nodal lines may
explain why ANC of high-order modes in circular or
ir-regular ducts appears to be difficult (48) Instead of using
the modal approach (i.e., the shape of the modes to be
con-trolled) to determine the error sensors location, an
alterna-tive strategy has recently been proposed by A L’Esp´erance
(49)—the error sensor plane concept This concept calls
for a quiet cross section to be created in the duct so that,
based on the Huygen’s principle, the noise from the
pri-mary source cannot propagate over this cross section A
multichannel ANC in a circular duct accords with this
strategy (50)
The principles of active control of noise propagating in
liquid-filled ducts are much the same as in air ducts (51)
The higher speeds of sound in liquids means that plane
wave propagation occurs in a larger frequency range than
in air ducts However, considerable care must be exercised
to the possible transmission of energy via the flexible duct
walls in this case, as a result of the strong coupling between
the duct walls and the interior fluid
Path 3: Active Control of Vibration Propagation
in Beam-Type Structures
The active control of vibration in one-dimensional systems
such as beams, rods, struts, and shafts can be approached
from two different perspectives, depending on the
descrip-tion of the structural response The response can be
de-scribed in terms of vibration modes or in terms of waves
propagating in the structure The modal perspective is
more appropriate to finite, or short, beams and to global
reduction of the vibration The description of the response
in terms of structural waves is more appropriate to
infi-nite, or long, beams and to reducing energy flow from one
part of the beam to another (control of vibration
trans-mission) The wave description is then more appropriate
to the case of the transmission of vibration from a ship’s
engine via the drive shaft, since in this case the source
of vibration is known and the objective is to block the
vi-bration transmission along the shaft The active control
of vibration in beams is widely covered in the literature
(21,44) The following presentation is mostly limited to
feedforward control systems, since it is assumed that for
the problem of vibration transmission along a marine drive
shaft, an advanced signal correlated to the disturbance,
or a measurement of the incoming disturbance, wave is
possible
Simultaneous Control of All Wave Types (Flexural,
Longitudinal, Torsional) In a general adaptive feedforward
controller used for the active control of multiple wave types
in a beam, sensor arrays (e.g., accelerometer arrays) areused to measure the different types of waves propagatingupstream (detection array) or downstream (error array) ofthe control actuators, and an array of actuators is used toinject and control the various wave types in the beam (44).Wave analyzers are necessary to extract the indepen-dent wave types (assumed uncoupled) from the sensor ar-rays, and wave synthesisers are necessary to generate theappropriate commands to the individual actuators Thisapproach has the advantage that independent control fil-ters can be used to control the flexural, longitudinal, andtorsional waves However, it necessitates excellent phasematching of the sensors and a detailed knowledge of thestructure in which the waves propagate An experimen-tal laboratory implementation of this approach has beenconducted by (52), on a thin beam, for the control of twoflexural wave components and one longitudinal wave usingPZT actuators Another, easier option avoids implementingwave analyzers and synthesisers by simply minimizing thesum of squared output of the error sensors to control thedifferent wave types This approach, however, requires afully coupled multichannel control system This approachhas been tested for the control of two flexural waves andone longitudinal wave in a strut using three magnetostric-tive actuators (53,54)
Control of Flexural Waves The dispersive nature of
flex-ural waves implies that a control force applied transversely
to the beam generates propagative waves as well as cent waves localized close to the point of application of theforce If one transverse control force is applied at somelocation on the beam, it generates downstream and up-stream propagating waves plus downstream and upstreamevanescent waves This actuator can minimize the total,transmitted downstream wave, but it generates a reflectedwave toward the source and two evanescent componentsthat may be undesirable A total of four actuators will benecessary to control downstream and upstream, propagat-ing, and evanescent components Therefore, the control of
evanes-flexural waves in beams will in general require actuator
arrays (55) Combinations of force and moment actuators
can also be used in the actuator array The simplest ward control system uses only one control force and one er-ror accelerometer, together with one reference accelerom-eter to measure the incoming wave This system has beenstudied theoretically (56), and tested experimentally (57).Physical limits of this system have been identified Thefirst limit is associated with the detection of the controlactuator evanescent wave by the error sensor that puts alimit on the actuator-error sensor separation: in practice,the sensor should be at least 0.7 from the control actua-tor (λ being the flexural wavelength) The second limit is
feedfor-related to the delay between detection and actuation thatshould be sufficient to allow the active control system toreact at the control actuator location before the primarywave has propagated from the detection sensor to the con-trol actuator This puts a limit on the reference sensor–actuator separation, which depends on the characteristics
of the control system
Similarly to control actuator arrays, error sensor arraysneed to be implemented for the control of flexural waves
Trang 31x
Accelerometer probeerror sensor
Laservibrometer
Piezoelectricpatch controlactuator
Figure 9 Typical experimental setup for the control of active
structural intensity.
to distinguish between the various propagating waves and
evanescent waves at the error sensor locations This is
par-ticularly needed if the error sensors must be located at
a short distance from the control actuators In this case,
an array of four accelerometers can discriminate between
the two propagating waves and the two evanescent waves
at the location of the error sensor array, and extract the
components that need to be reduced (e.g., the downstream
propagating wave)
Other sensing strategies have also been suggested, such
as measuring and minimizing the structural intensity due
to flexural waves (58,59) Structural intensity can be
mea-sured in practice using an array of four or more closely
spaced accelerometers, as presented in Fig 9
Practical Implementations There are a limited number
of practical implementations of these principles to large,
machinery structures Semiactive or active devices have
been used to attenuate the transmission of longitudinal
vibration on a large tie-rod structure (60) The tie-rod is
similar to that found in marine machinery to maintain the
alignment of a machinery raft A tunable pneumatic
vi-bration absorber was used as the semiactive device, and
an electrodynamic shaker or a magnetostrictive actuator
was used as the active device A load cell was used as the
error sensor, such that the force applied by the tie-rod to a
receiving bulkhead was minimized
The suppression of vibration that is generated on
ro-tating machinery with an overhung rotor has been
pre-sented (61) In this case, the vibration of the rotor-shaft
system is controlled by active bearings The active
bear-ings consist of a bearing housing supported elastically by
rubber springs and controlled actively by electromagnetic
actuators These actuators are controlled by displacement
sensors at the pedestal and/or the roller and can apply an
electromagnetic force that suppresses any vibration of the
roller The active vibration control (AVC) of rotating
ma-chinery utilizing piezoelectric actuators was also
investi-gated (62) The AVC is shown to significantly suppress
vi-bration through two critical speeds of the shaft line
Path 4: Active Control of Enclosed Sound Fields
There exists a vast body of literature on the subject of active
control of enclosed sound fields Only the previous work
re-levant to the problem of canceling the sound field radiated
by a ship engine in its enclosed space will be reviewed here.More comprehensive presentations of the generic problemcan be found in (3,21) Active control of enclosed soundfields has found applications essentially for automobile in-terior noise (63,64) and for aircraft interior noise (65–67),leading in some cases to commercial products
There are two main categories of active control systemsrelated to enclosed sound field minimization:
rActive control of sound transmission through elasticstructures into an enclosure
rActive control of sound field into rigid enclosures.Only the second category will be reviewed here The activecontrol of sound transmission has been investigated usingessentially modal approaches (68,69) The same type of an-alytical approach based on modes of the acoustic enclosurecan be used to investigate the active control of sound fieldinto rigid enclosures It should be mentioned, however, thatfinite element approaches have also been used to studythe active control of sound field into enclosures of com-plex geometries (70,71) Additionally, the objective of the
active control in an enclosure can be to minimize the sound
field globally, or locally Only the approaches directed
to-ward global attenuation of the sound field are reviewedhere In this respect, some important physical aspects ofthis problem are discussed in the following These physi-cal aspects depend primarily on the modal density of theenclosure
Enclosures with a Low Modal Density For enclosures with
a low modal density (i.e., a small enclosure, or at low quency), the active control will usually consist of placing
fre-a series of control loudspefre-akers in the enclosure; the speakers are driven to minimize the sound pressure mea-sured by discrete error microphones In the case of anenclosed acoustic space, the performance metrics for thecontrol should be the acoustic potential energy integratedover the volume of the enclosure,
where p(r) is the local sound pressure, ρ0 is the density
of the acoustic medium, and c0is the speed of sound Theactive control scheme should aim at reducing the acousticpotential energy as much as possible
It has been shown that active control of sound fields
in lightly damped enclosures is most effective at the onance of the acoustic modes (72) In these instances, theproblem is essentially the control of a single mode Sig-nificant attenuation of the acoustic potential energy isobtained using a single control source and a single er-ror microphone (provided that neither the control sourcenor the error microphone is located on a nodal surface ofthe acoustic mode) For a multiple-mode (off-resonance)response of the cavity, the number of control sourcesand error microphones should be increased However, thepotential for attenuation is never as large as at a resonancefrequency
Trang 32res-The number and placement of control sources and error
sensors are critical for multiple-mode control The
corre-sponding optimization problem is nonlinear and usually
involves many local minima Optimization processes, such
as multiple regression (21) or genetic algorithms (73), are
used As a general rule is that the number and locations
of the control sources should be such that the secondary
sound field matches as closely as possible the primary
sound field in the enclosure
Enclosures with a High Modal Density As the frequency
increases or the enclosure becomes larger, global
attenua-tion of the sound field becomes more difficult to achieve
us-ing an active control system To quantify these limitations,
there are some approximate formulas, which are
summa-rized here These formulas are approximate, but they give
useful expected performance of an active control system in
a high modal density enclosure
First, assuming a single primary point source and a
sin-gle secondary point source in the enclosure, it is possible
to derive the ratio of the minimized potential energy (after
control) to the original potential energy (before control),
(74):
E p ,0 = 1 −1+ π
2M(ω)−2,
where M( ω) is the modal overlap of the cavity, which
quantifies the likely number of resonance frequencies of
other modes lying within the 3 dB bandwidth of a given
modal resonance For a rigid rectangular enclosure and for
oblique acoustic modes, namely three-dimensional modes,
such as the (1,1,1) mode,
whereζ is the damping ratio in the enclosure (assumed
identical for all acoustic modes),ω is the angular frequency
of the sound field, and V is volume of the enclosure.
If the modal density is low (at low frequency),
E p ,0 ≈ π M(ω),
which means that the achievable attenuation is dictated
by the modal overlap (and hence the modal density and
damping of the enclosure)
If the modal density is large (at high frequency),
which means that no attenuation can be obtained
after control Another expression can be derived from
the asymptotic expression of modal overlap in high
frequency (75),
E p ,0 = 1 − sin c2kd,
where k is the acoustic wave number and d is the
separa-tion between the primary and control sources Thus, as thecontrol source becomes remote from the primary source,
such that kd ≥ π, any global attenuation of the sound field
becomes impossible This provides an explicit analyticaldemonstration that the global control of enclosed soundfields of high modal density is only possible with closelyspaced compact noise sources In other words, assuming anextended primary source such as a ship engine, the only vi-able solution in this case is to distribute control loudspeak-ers around the engine and in the close vicinity of it (within
a fraction of the acoustic wavelength)
Advanced Sensing Strategies Recently, alternatives to
sensing and minimizing squared sound pressure have beensuggested in active control of enclosed spaces Sensingstrategies based on total acoustic energy density minimiza-tion instead of sound pressure minimization have been sug-gested (76,77) The advantage of sensing the total energydensity is that the control is less sensitive to the sensor lo-cations, and in general, a superior attenuation is obtained.The energy density can be measured using combinations ofmicrophones (2 to 6); in this case, finite differences betweenindividual microphones are applied to obtain approximatemeasurements of the pressure gradient in several direc-tions Precise measurements of the pressure gradient re-quires an excellent phase matching of the individual micro-phones, which can result in more expensive microphones.Associated adaptation algorithms for the minimization ofenergy-based quantities have been derived (78)
RECOMMENDATIONS ON SENSORS AND ACTUATORS FOR ANVC OF MARINE STRUCTURES
Steps in Design of Active Control Systems
of the vibroacoustic behavior of the system on which activecontrol is to be applied This involves carefully identifyingand ranking the various paths along which vibroacousticenergy flows This may imply addressing questions such
as the transmission of moments or in-plane forces throughthe engine mounting, or the relative contribution of fluid-borne and structure-borne energy along pipes This earlyphase is crucial in determining the active control strategy
to be implemented A number of experimental techniquesand numerical simulation tools can be used to estimatethe relative contribution of the various paths at a givenreceiving point (e.g., in water) Based on some contractors’previous experiences, a major transmission path appears
to be the engine-mounting system
Trang 33Phase 1–Understanding the vibroacoustics of thesystem
Identifcation of the transmission pathsRanking of the transmission pathsActive control strategy
•
•
Exact quadratic optimizationError sensor configuration forglobal control
•
•
•Control paths transfer functionmeasurements
•
Figure 10 Suggested design steps of an active control system.
The second phase will determine the type, number, and
locations of the control actuators When global control is
desirable (e.g., when attenuation of the sound field is
de-sired at all positions in water), these parameters are
deter-mined by the requirement that the sound field generated
by the control actuators should spatially match the
pri-mary sound field The type of control actuators to be used
will be based primarily on the frequency of the disturbance
and the magnitude of the disturbance at the actuator
loca-tion (for simplicity, the control actuators need to generate a
secondary field with a magnitude equal to the disturbance
at the actuator location) Once the type, number, and
loca-tions of the control actuators are known, extensive transfer
function measurements need to be taken between
individ-ual actuators and field points (vibratory or acoustic), with
the primary source turned off Since this may involve a
con-siderable experimental task, numerical simulations can be
of a great help here
The third phase will address the error sensors Again, if
global control is desirable, the type, number, and locations
of the error sensors are dictated by the requirement that
if the control actuators are driven to minimize the signal
at the error sensors, then the resulting sound field is
glob-ally reduced The measured transfer functions between
in-dividual actuators at field points and the magnitude of
the primary disturbance at these field points are used,
in conjunction with classical exact quadratic optimization
techniques, to calculate the optimal control variables (i.e.,the required inputs of the control actuators) that minimizethe error signals for a given error sensor arrangement Thefinal phase will be to test the active control with a realcontroller
Recommended Sensor and Actuator Technologies for Various Ship Noise Paths
Path 1: Active Vibration Isolation In selecting sensors
and actuators for active vibration isolation of engine noise,due consideration has to be given to the size and weight
of the structure (engine) being isolated Since the engine
is a heavy structure weighing over 6000 kg, it is sary that that the actuators are capable of delivering veryhigh control forces In addition, the nature of the noisethrough this path is nonacoustic, and hence nonacousticsensors and actuators have to be used Based on these con-siderations, the recommended sensors and actuators are(1) accelerometers and force transducers for sensing and(2) hydraulic and electrodynamic actuators for actuation.The recommendations are summarized in Table 3 For in-creased efficiency, the control systems must be designed
neces-to provide control forces in translational and rotational rections, since engine vibrations could take place in all di-rections Furthermore, the active control systems should
di-be used in conjunction with passive control systems, to duce cost as well to provide fail-safe designs
re-Path 2: Active Control of Noise in Ducts and Pipes The
feedforward algorithm has been recommended for the trol of noise associated with a marine diesel engine where areference signal is accessible (4) For ducts, generally asso-ciated with large cross-sectional dimensions, higher-ordermodes are more likely to exist, requiring a large number
con-of sensors and actuators with an appropriate positioningstrategy For pipes, generally associated with small cross-sectional dimensions, it is expected that only plane wavepropagation will exist, thereby limiting the number of ele-ments needed to one sensor and one actuator The followingsensing configurations are possible: microphones, piezo-electric sensors, or accelerometers For actuation, loud-speakers and inertial actuators are recommended
Path 3: Active Control of Vibration Propagation in Beam-Type Structures Feedforward control was recom-
mended for the vibration control of a propeller shaft (4).The configuration of sensors and actuators to be used willdepend on the excitation source and on the modal behavior
of the shaft For modal control, the sensors and actuatorscan be located either on the shaft itself or connected to
it by a stationary mechanical link, such as by a bearingmounted on the shaft Potential mounted actuators includecurved piezoelectric actuators (PZT) and magnetostric-tive actuators For wave transmission control, sensors andactuators arrays are required to measure the downstreampropagating and evanescent waves and to inject the controlwaves in the structure
Mounted sensors to be used include piezoelectric(PVDF) sensors to measure the strain and accelerometers,
Trang 34Table 2 Properties of Selected Piezoelectric Materials
Note: γ0= 8.85 × 10−12farad/m, electric permittivity of air.
if the rotation speed permits, for acceleration
measure-ment The mounted actuators include piezoelectric (PZT)
actuators to induce strain in the structure For
robust-ness, it is recommended that the actuators be combined
with passive control elements such as a viscoelastic layer
bonded to the shaft
Table 3 Recommended Sensors and Actuators for Ship Noise Control
Recommended Sensors and Actuators
Path 1: Active vibration isolation rForce transducers rHydraulic actuators
rAccelerometers rElectrodynamic actuators
Path 2: Active control of noise rMicrophones rLoudspeakers
in ducts and pipes rPiezoelectric sensors rElectric motors
rAccelerometers
Path 3: Active control of vibration rPiezoelectric sensors rPiezoelectric actuators
propagation in beam-type structures rAccelerometers rMagnetostrictive actuators
rElectrodynamic shakers rElectrodynamic shakers
Radiated noise into sea rPiezoelectric sensors rPiezoelectric actuators
rAccelerometers rMagnetostrictive actuators
Path 4: Active Control of Radiated Sound Fields There
are two types of radiated noise to be controlled for shipstructures These are the airborne engine noise into an en-closure, and the noise radiated by the noise into the sea
As stated in (4) both cases require the use of global trol techniques that involve multiple input and multiple
Trang 35con-output transducers Control of radiated noise can be
achieved either by active noise cancellation (ANC) or by
active structural acoustic control (ASAC) techniques For
active cancellation, the following sensors and actuators
are recommended: (1) combination microphones and
ac-celerators as sensors and (2) loudspeakers as actuators
For active structural acoustic control the following
sors and actuators are recommended: (1) piezoelectric
sen-sors (shaped or not) and accelerometers as sensen-sors, and (2)
piezoelectric and magnetostrictive materials as actuators
SUMMARY AND CONCLUSIONS
Among the wide range of sensor and actuator materials
that could be used for active noise and vibration control in
ship structures are piezoelectric and electrostrictive
ma-terials magnetostrictive mama-terials, shape-memory alloys,
optical fibers, electrorheological and magnetoeheological
fluids, microphones, loudspeakers, electrodynamic
actua-tors, and hydraulic and pneumatic actuators In making
the selection, due consideration must be given to factors
such as cost, frequency of the disturbance, operating
(ma-rine) environment, experience in other applications, ease
of implementation, and the expected performance In
gen-eral, the following recommendations are made:
1 Nonacoustic sensors and actuators (e.g.,
accelero-meters, force transducers, hydraulic actuators, electric materials, and electrodynamic actuators) arebest for nonacoustic paths, namely for the engine-mounting system, the drive shafts, and mechanicalcouplings
piezo-2 Acoustic sensors and actuators (e.g., microphones
and loudspeakers) are best for acoustic paths, namelyfor the exhaust stacks and piping systems, and theair-borne noise
It was also recommended that the active control strategies
be combined with passive treatments whenever possible,
to increase the robustness of the control system and to
pro-vide a fail-safe design
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Vibration is present almost everywhere we travel in
mod-ern society Vibrationally induced failures are very
com-mon in products such as television sets and computers
that are shipped by trains and trucks Vibrational failure
in a television set may be just an inconvenience However,
vibrational failure in a large passenger airplane can lead to
many deaths Methods of vibrational analysis are available
that are accurate and can reveal weak structural areas
Steps can then be taken either to repair or replace critical
items Vibrational analysis is a combination of science and
art The science uses sophisticated computers extensively
to solve large complex problems This method requires
ex-tensive training and often takes a long time to reach a
satisfactory solution The art uses approximations, short
cuts, and test data to reduce the time needed to reach a
satisfactory solution The approximations and short cutscan sharply reduce the time required for a solution, but italso reduces the accuracy of the analysis Vibrational anal-ysis can be used to make some materials work smarter
by making small changes in their physical properties.These changes can often increase the fatigue life of criti-cal structural members without a significant increase inthe size, weight, cost, or impact on production and deliveryschedules
VIBRATIONAL REPRESENTATION
In a broad sense, vibration means an oscillating motion,where something moves back and forth If the motion re-peats itself, it is called periodic If continuous motion neverrepeats itself, it is called random motion Simple harmonicmotion is the simplest form of periodic motion, and it istypically represented by a sine wave, as shown in Fig 1.The reciprocal of the period is known as the frequency, and
it is measured in cycles per second, or hertz (Hz) The mum displacement is called the amplitude of the vibration
maxi-DEGREES OF FREEDOM
A coordinate system is usually used to locate the positions
of various elements in a system When only one element isinvolved, it is restricted to moving along only one axis, andonly one dimension is required to locate the position of theelement at any instant, then it is called a single-degree-of-freedom system The same is true for a torsional system.When one element is restricted to rotating about one axis sothat only one dimension is required to locate the position ofthe element at any instant, it is a single-degree-of-freedomsystem Two degrees of freedom requires two coordinates
to locate the positions of the elements, and so on
A single rigid body is usually considered to have sixdegrees of freedom, translation along each of the three
orthogonal x, y, and z axes and rotation about each of the
same three axes Real structures are usually considered tohave an infinite number of degrees of freedom
VIBRATIONS OF SIMPLE STRUCTURES
The natural frequency (often called the resonant quency) of a simple single-degree-of-freedom system can
Trang 37C K
Chassis orPCBM
Figure 2 Single-degree-of-freedom spring-mass system.
often be obtained from the strain energy and the kinetic
energy of the system Consider the single spring and mass
system shown in Fig 2 When there is no damping in the
system, then no energy is lost, and the strain energy must
be equal to the kinetic energy This results in the natural
g= 9.80 m/s2(386 in/s2), the acceleration of gravity and
Ystin meters (inch) is the static displacement
Sample Problem: Natural Frequency of a Simple Structure
When the static displacement of a structure Yst= 1.27 ×
10−5m (0.00050 in), its natural frequency is 140 Hz
The natural frequency is important because it is often
considered the heart of a vibrating system It influences
the number of fatigue cycles and the displacement, which
affect the fatigue life of a system It also influences the
damping, which affects the dynamic acceleration Q level,
and the stress level, which also affects the fatigue life
NATURAL FREQUENCIES OF UNIFORM
BEAM STRUCTURES
Natural frequencies of uniform beam structures can be
de-termined by equating the strain energy to the kinetic
en-ergy without damping This method of analysis leads to
simple solutions and very little error because beam types
of structures normally have very little damping The
re-sulting equations for natural frequency apply to uniform
beams that are forced to bend only in the vertical axis
with-out bending in the horizontal axis and withwith-out torsion or
twisting The beam equation is (1)
a = 3.52 for a cantilevered beam,
a = π2= 9.87 for a beam that is supported (hinged) at
each end,
a = 22.4 for a beam that is clamped (fixed) at both ends,
for beam material,
cross section,
g = 9.80 m/s2(386 in/s2), the acceleration of gravity,
W in Ns (N) (lb) is the total weight of the beam, and
L in m (in.) is the length of the beam between supports.
Section AA
(2.0 in.)0.0508 meters
0.0254 meters (1.0 in.)
Figure 3 Uniform beam simply supported at each end.
Sample Problem: Natural Frequency of a Simply Supported Uniform Beam
For example, consider the simply supported (hinged)
alu-minum beam shown in Fig 3, where E = 6.894 × 1010
N/m2(10× 106lb/in2), L = 0.254 m (10.0 in), I = 6.937 ×
10−8m4(0.1667 in4), and W = 8.896 N (2.0 lb) The
result-ing natural frequency is 890 Hz
NATURAL FREQUENCIES OF UNIFORM PLATES AND CIRCUIT BOARDS
The natural frequencies of different types of flat, uniformplates that have different types of supports can often beobtained by using trigonometric or polynomial series (1).Again, when damping is ignored, the strain energy can
be equated to the kinetic energy of the bending plate
to obtain the natural frequency A printed circuit board(PCB) that supports and electrically interconnects variouselectronic components can be analyzed as a flat rectan-gular plate, often simply supported (hinged) on all foursides, that has a uniformly distributed load across its sur-face The natural frequency for this type of installation
h in m (in) is the plate thickness,
µ is Poisson’s ratio, dimensionless,
2/m3(lb s2/in3), the mass per unit area, (5)
g = 9.80 m/s2(386 in/s2), the acceleration of gravity,
W in newtons (lb), is the total weight of the PCB,
a in m (in) is the length of the plate,
b in m (in) is the width of the plate, and
second harmonic m = 2, n = 1;
third harmonic m = 1, n = 2; fourth harmonic m = 2,
Trang 38X
Xb
aSupported
Z
Figure 4 Uniform flat plate simply supported on four sides.
Sample Problem: Natural Frequency of a Rectangular PCB
(see Fig 4).
Consider a flat rectangular epoxy fiberglass PCB,
sup-ported (hinged) on four sides, where E = 1.379 × 1010N/s
m2, (2.0×106lb/in2), h = 0.00157 m (0.062 in), µ = 0.12
di-mensionless, D = 4.53 N (40.1 lb in), W = 4.448 N (1.0 lb),
a = 0.203 m (8.0 in), b = 0.178 m (7.0 in), ρ = 12.56 Ns2/m3
(0.463×10−4lb s2/in3) The resulting natural frequency for
the first harmonic (m = 1, n = 1) is 52.6 Hz.
METHODS OF VIBRATIONAL ANALYSIS
Hand calculations are still being used extensively for
sim-ple sinusoidal and random vibrational analyses in small
companies due to the high costs of the computers, the
spe-cialized computer software, and the skilled personnel to
operate the computers Many reference books are available
that show how to perform simplified vibrational analyses
on different types of simple structures However, when
large complex structures are involved, hand calculations
are not adequate to ensure reasonable accuracy Small
com-panies often subcontract the work to outside consulting
organizations that specialize in these areas Sometimes it
can be cheaper, faster, and more accurate to build a model
of the structure, so it can be examined in a vibrational test
laboratory
Most large companies rely extensively on various types
of computers and specially formulated finite element
mod-eling (FEM) software programs for vibrational analyses
Their computers are usually networked together, so each
has access to the wide variety of software analytical
pro-grams available on the network The new desktop personal
computers (PC) are very popular for vibrational analyses
using FEM They are more powerful and faster than the
large main frame computers of a few years ago
PROBLEMS OF VIBRATIONAL ANALYSIS
Almost all computers and computer software FEM
pro-grams for vibrational analysis agree within about 2% when
they are used to determine eigenvalues (resonant cies) and eigenvectors (mode shapes) for many types ofcomplex structures However, sample problems solved byusing different FEM software programs have shown sig-nificant variations in their stress values The stress val-ues from four different FEM programs had a total varia-tion of about 60% This was 30% above the average stressvalue of the four programs and 30% below the averagevalue for similar models of the same structure, subjected tothe same type of vibrational excitation Different computerFEM programs typically use different algorithms to definethe building blocks for their various beam, plate, and brickelements These algorithmic variations probably cause thevariations in the stress values Because the fatigue life of
frequen-a structure is closely relfrequen-ated to its stress vfrequen-alue, significfrequen-antvariations in the calculated stress levels can result in dra-matic changes in the calculated fatigue life of a structure.For example, the results of this investigation showed thatthe fatigue life at the critical point in the structure can beexpected to vary across a wide range because of the vari-ations in the calculated stress values The fatigue life inthe lead wires of PCB electronic component parts can be
as much as five times greater than the average calculatedfatigue life, or it can be as little as one-fifth of the averagecalculated fatigue life [See Ref 5, Chap 12, Figs 12.1–12.19 for more detailed information on finite elementmodeling.]
The results shown before may vary substantially Onlyfour different FEM software programs were involved inthis investigation At least several dozen new softwareprograms are available now When the different modelingtechniques of different computer analysts are considered,these factors are expected to have a significant impact onthe computer calculated stress values and the resultingcalculated fatigue life
PROBLEMS OF MATERIAL PROPERTIES
Material properties are often difficult to evaluate for brational environments The life of any structure excited
vi-by vibration depends on the fatigue properties of the mostcritical materials used in fabricating and assembling thestructure When structural elements are forced to bend andtwist back and forth, perhaps millions of times in severevibrational environments, three very important factorshave to be defined:
1 the very basic fatigue properties of the materials used
mate-plotted on log–log curves of stress (S) against the ber of cycles (N ) to failure Only one average straight-line
Trang 39usually represents the fatigue life properties of a material
(1,3–5) When all of the failure test data points for all of
the test samples are plotted, a wide variation in the
fa-tigue life is revealed Because these are log–log plots, the
spread in the possible variation in fatigue life of virtually
identical parts can be very great, sometimes reaching
val-ues of 10 to 1 Engineers involved in vibrational and
fa-tigue life analysis do not like to reveal this type of data to
upper management personnel Personal experience with
nontechnical upper management people is that they often
expect mechanical designers and analysts to predict the
fa-tigue life of their structures to within plus and minus 20%
This is an almost impossible task, when all of the possible
variations are considered
To compensate for these large variations in fatigue life
of virtually identical structural elements, safety factors
(sometimes called factors of ignorance) must be used when
these structures are being designed and analyzed
Build-ing models for vibrational life testBuild-ing in a laboratory can
be a great help in estimating the fatigue life of a structure
However, if no tests are run or if the number of samples
tested is low, there is always the danger of erratic bursts
of high failure rates in the production units because of the
large scatter associated with fatigue
Next, consider the effects of manufacturing tolerances
on the physical dimensions of the structural elements in an
assembly Mass-produced products always show some
vari-ations in the physical dimensions of what appear to be
iden-tical parts Even die cast parts that are made from the same
mold have slightly different physical dimensions Some
manufactured devices, like the automatic transmission in
an automobile, can have many precision gears, ground to
very close tolerances Holding very tight manufacturing
tolerances can be very expensive Therefore, looser
toler-ances are used in production parts that do not require tight
tolerances for precision assembly work because they this
reduce costs When manufactured parts that have loose
tol-erances are exposed to severe vibration, the failure rates
often go up and down erratically Changes in the physical
dimensions of load-carrying structural members can alter
the load path through the structure, which can change the
dynamic loads and stresses in it It is too expensive to keep
track of manufactured parts that have extremes in theirdimensional tolerances These parts can be anywhere inlarge production programs This means that failures whichare difficult to predict and to control, may occur randomly
in harsh environments
To reduce costs, for example, the electronics industrytends to use very loose tolerances in the dimensions thatcontrol the external physical sizes of the length, width, andthickness of their printed circuit boards (PCBs) and elec-tronic component parts These large variations in tolerance
of these parts further increase the difficulty in trying topredict the fatigue life accurately of electronic assembliesthat are exposed to different vibrational environments
RELATION OF DISPLACEMENT TO ACCELERATION AND FREQUENCY
Vibrational displacements are often very small, so theyare difficult to observe during vibrational tests Becausethese displacements are small, it does not mean that theresulting stresses are also small Vibrational environmentsusually impose alternating displacements and alternat-ing stresses on various structural load-carrying elementswithin a system If the vibrating system experiences manythousands of stress reversals, fatigue failures can occur
in critical structural members, even at relatively low placements and stress levels This is the nature of fatiguefailures that occur at relatively low stress levels near smallholes, small notches, and sharp bends These geometricshapes are known as stress concentration factors, whichcan increase peak stress levels in these areas by a factor of
dis-3 or 4 or more (4)
When vibrational tests are run in a laboratory, the mal procedure is to use small accelerometers to monitorthe resulting acceleration values in different parts of thestructure When an electrodynamic shaker is used to gen-erate a sinusoidal wave for the vibrational test, the elec-tronic control system will show the frequency of the im-posed wave in cycles per second, or hertz (Hz) With thistype of setup, the test engineer will know the accelerationlevel and the frequency at any instant This information
nor-is often incomplete without the resulting dnor-isplacement atany instant The resulting displacement at any instant can
be obtained by considering a rotating vector that generates
a sinusoidal wave based on the full relationship (1),
where
Y is the displacement at any time, Y0is the maximum single amplitude displacement fromzero to peak, and
= 2π( f ) rad/s, the frequency.
The acceleration a can be obtained from the second
derivative of the displacement with respect to time fromthe preceding equation The maximum acceleration occurswhen the sine function is one It is convenient to represent
Trang 40the acceleration in terms of gravity units G:
g (gravity units, dimensionless), (7)
where
g = 9.80 m/s2(386 in/s2), the acceleration of gravity
The final results show the displacement Y0in terms of
the frequency f in Hz and the number of dimensionless
G is the acceleration, in gravity units, dimensionless
(same in English units), and
f is the frequency in cycles/s (Hz) (same in English
units)
Sample Problem: Finding the Displacement
from the Frequency and the G Level
For example, when the acceleration G level is 3.0
dimen-sionless gravity units and the frequency is 120 Hz, the
sin-gle amplitude displacement is 0.0000517 m (0.00204 in)
This equation is probably the most important
relation-ship in the entire field of dynamics It shows that when
any two of the parameters of Y0, G or f , are known, then
the third parameter is automatically known This equation
can be used for sine vibration, random vibration, shock,
and acoustics (1)
EFFECTS OF VIBRATION ON STRUCTURES
Vibrational environments can dramatically magnify the
dynamic forces and stresses in different types of structures,
when the structural natural frequencies are excited Forces
and stresses can be magnified and amplified by factors of
10, 30, and even 100 in many different types of structures
for different types of vibrational excitation The magnitude
of the magnification, called the transmissibility Q, often
depends on the amount of damping in the vibrating system
Figure 6 shows damping for a single-degree-of-freedom
sys-tem There are very few single-degree-of-freedom systems
in the real world For example, consider a
two-degree-of-freedom system for an electronic assembly where the
chas-sis is mass1 The plug-in PCBs are attached to the chaschas-sis
so they are mass 2 The response of mass 1 will be the input
to mass 2 Testing experience, including different damping
methods, has shown that the transmissibility Q of PCBs as
mass 2 will depend far more on the dynamic coupling phase
relation and frequency ratio between mass 1 and mass 2
than the damping in either mass 1 or mass 2 because the
transmissibility Q’s between masses 1 and 2 do not add,
they multiply
00.10.20.40.60.81.02
46810
00.100.20
Figure 6 Effects of damping on the transmissibility Q plots.
The Q of a system is defined as the ratio of the
out-put (or response of the system) divided by the inout-put Theoutput and the input are usually defined in terms of thedisplacements, or the acceleration values If the damp-ing in a simple system is zero, the vibration theory states
that the value of the transmissibility Q will be infinite.
If the transmissibility Q is infinite, the resulting dynamic
forces and stresses will also be infinite However, because
all real systems have some damping, Q can never be nite However, in lightly damped systems, Q can be very high A high Q will result in high forces, displacements, and
infi-stresses, which can sharply reduce the fatigue life of thestructure
ESTIMATING THE TRANSMISSIBILITY
Q IN DIFFERENT STRUCTURES
The transmissibility Q is strongly influenced by the
damp-ing in a vibratdamp-ing structure One form of dampdamp-ing is theconversion of kinetic energy into heat This can be shown
by rapidly bending a metal paper clip back and forth about
20 times through a large angle Immediately place your ger on the paper clip in the bending area This area will bequite warm It may even be hot The strain energy of bend-ing has been converted into heat energy, which cannot beconverted back into strain energy It is lost energy Whenheat energy is lost, it means there is also a loss of kinetic en-ergy Therefore,when damping is increased in a vibratingsystem, there is less energy available to convert into kineticenergy Less kinetic energy means that there is less energyavailable to excite the structure at its natural frequency, so
fin-that the transmissibility Q is decreased Conversely, when
there is a decrease in the damping, this makes more kineticenergy available to excite the structure, so the transmissi-
bility Q is increased.