Tài liệu Handbook of Machine Design P45 doc

ficient of viscous damping c is the force provided by the damper opposing the motion per unit velocity.. The system at rest is given an impact which provides initial velocity tothe syste

CHAPTER 38VIBRATION AND CONTROL

R B Bhat, Ph.D.

Associate Professor Department of Mechanical Engineering

Concordia University Montreal, Quebec, Canada

38.2 SINGLE-DEGREE-OF-FREEDOMSYSTEMS

38.2.1 Free Vibration

A single-degree-of-freedom system is shown in Fig 38.1 It consists of a mass m strained by a spring of stiffness k, and a damper with viscous damping coefficient c The stiffness coefficient k is defined as the spring force per unit deflection The coef-

con-FIGURE 38.1 Representation of a

single-degree-of-freedom system.

ficient of viscous damping c is the force provided by the damper opposing the

motion per unit velocity

If the mass is given an initial displacement, it will start vibrating about its librium position The equation of motion is given by

equi-mx + cjc + kx = O (38.1) where x is measured from the equilibrium position and dots above variables repre- sent differentiation with respect to time By substituting a solution of the form x = e 81

into Eq (38.1), the characteristic equation is obtained:

Depending on the value of £, four cases arise

Undamped System (£= O) In this case, the two roots of the characteristic equation

are

and the corresponding solution is

x = A cos GV + B sin GV (38.5) where A and B are arbitrary constants depending on the initial conditions of the

motion If the initial displacement is Jt0 and the initial velocity is V0, by substituting

these values in Eq (38.5) it is possible to solve for constants A and B Accordingly,

the solution is

x = Jt0 cos GV + — sin GV (38.6)

03«

Here, G)n is the natural frequency of the system in radians per second (rad/s), which

is the frequency at which the system executes free vibrations The natural frequency

UnderdampedSystem (O <£< 1) When the system damping is less than the

criti-cal damping, the solution is

x = [exp(-^GV)] (A cos GV + B sin GV) (38.11)

where

co, = con(l-C2)"2 (38.12)

is the damped natural frequency and A and B are arbitrary constants to be

deter-mined from the initial conditions For an initial amplitude of Jt0 and initial velocity V0,

x = [exp (-CcOnOl I *o cos co/ + —— sin GV (38.13)

V co, /

which can be written in the form

x = [exp (-^OV)] X cos (co/ - 6)

x L-^ft m '* o+v °Yr (3814)

^T0 +I co, J Jand

e.ta^itetZo

CO,

An underdamped system will execute exponentially decaying oscillations, as showngraphically in Fig 38.2

FIGURE 38.2 Free vibration of an underdamped single-degree-of-freedom system.

The successive maxima in Fig 38.2 occur in a periodic fashion and are marked

XQ, Xi 9 X 2 , The ratio of the maxima separated by n cycles of oscillation may be

obtained from Eq (38.13) as

^ = exp(-»6) (38.15)

^Owhere

, _ 27CC

(1-O" 2

is called the logarithmic decrement and corresponds to the ratio of two successive

maxima in Fig 38.2 For small values of damping, that is, £ « 1, the logarithmicdecrement can be approximated by

6 = 27iC (38.16)Using this in Eq (38.14), we find

^- = exp (-2roiQ - 1 - 2iwC (38.17)

AO

FIGURE 38.3 Variation of the ratio of displacement maxima with damping.

The equivalent viscous damping in a system is measured experimentally by usingthis principle The system at rest is given an impact which provides initial velocity tothe system and sets it into free vibration The successive maxima of the ensuingvibration are measured, and by using Eq (38.17) the damping ratio can be evalu-ated The variation of the decaying amplitudes of free vibration with the damping

ratio is plotted in Fig 38.3 for different values of n.

Critically Damped System (£ = 1) When the system is critically damped, the roots

of the characteristic equation given by Eq (38.3) are equal and negative real ties Hence, the system does not execute oscillatory motion The solution is of the form

and after substitution of initial conditions,

x=[x 0 + (v 0 + X^ n )I] exp (-GV) (38.19)

This motion is shown graphically in Fig 38.4, which gives the shortest time to rest

Overdamped System (£> 1) When the damping ratio £ is greater than unity, there

are two distinct negative real roots for the characteristic equation given by Eq.(38.3) The motion in this case is described by

jc - exp KGV) [A exp coA/C - 1 + B exp (-GvV£ - I)] (38.20)

FIGURE 38.4 Free vibration of a single-degree-of-freedom system under different values of

All four types of motion are shown in Fig 38.4

If the mass is suspended by a spring and damper as shown in Fig 38.5, the springwill be stretched by an amount 5sf, the static deflection in the equilibrium position Insuch a case, the equation of motion is

mx + ex+ k(x + 8rf) = mg (38.21)

FIGURE 38.5 Model of a

single-degree-of-freedom system showing the static deflection

due to weight.

Since the force in the spring due to the static equilibrium is equal to the weight, or

k$ st = mg = W, the equation of motion reduces to

38.2.2 Torsional Systems

Rotating shafts transmitting torque will experience torsional vibrations if the torque

is nonuniform, as in the case of an automobile crankshaft

In rotating shafts involving gears, the transmitted torque will fluctuate because ofgear-mounting errors or tooth profile errors, which will result in torsional vibration

of the geared shafts

A single-degree-of-freedom torsional system is shown in Fig 38.6 It has a

mass-less shaft of torsional stiffness k, a damper with damping coefficient c, and a disk

with polar mass moment of inertia / The torsional stiffness is defined as the ing torque of the shaft per unit of angular twist, and the damping coefficient is theresisting torque of the damper per unit of angular velocity Either the damping can

resist-be externally applied, or it can resist-be inherent structural damping The equation ofmotion of the system in torsion is given

/e + c9 + £0 = 0 (38.24)

FIGURE 38.6 A representation of a

one-freedom torsional system.

Equation (38.24) is in the same form as Eq (38.1), except that the former deals withmoments whereas the latter deals with forces The solution of Eq (38.24) will be of

the same form as that of Eq (38.1), except that / replaces m and k and c refer to

tor-sional stiffness and tortor-sional damping coefficient

38.2.3 Forced Vibration

System Excited at the Mass A vibrating system with a sinusoidal force acting on

the mass is shown in Fig 38.7 The equation of motion is

Substituting in Eq (38.26), we find that the steady-state solution can be obtained:

(F,/*) sin (a*-9)

Xs [(l-(o2/(B2)2 + №(on)2]''2 ( ™-Z/>

Using the complementary part of the solution from Eq (38.19), we see that the plete solution is

com-x = com-xs + ecom-xp (-Co)nO [A exp (oy V^T) + B exp (-oy V^2 - I)] (38.28)

If the system is undamped, the response is obtained by substituting c = O in Eq.

(38.25) or £ = O in Eq (38.28) When the system is undamped, if the exciting

fre-quency coincides with the system natural frefre-quency, say co/co« = 1.0, the systemresponse will be infinite If the system is damped, the complementary part of thesolution decays exponentially and will be nonexistent after a few cycles of oscilla-tion; subsequently the system response is the steady-state response At steady state,the nondimensional response amplitude is obtained from Eq (38.27) as

Eq (38.30) is plotted in Fig 38.9 For smaller forcing frequencies, the response isnearly in phase with the force; and in the neighborhood of the system natural fre-quency, the response lags behind the force by approximately 90° At large values offorcing frequencies, the phase is around 180°

FREQUENCY RATIO w/u>n

FIGURE 38.8 Displacement-amplitude frequency response due to

oscil-lating force.

Steady-State Velocity and Acceleration Response The steady-state velocity

response is obtained by differentiating the displacement response, given by Eq.(38.27), with respect to time:

And the steady-state acceleration response is obtained by further differentiation

and is

These are shown in Figs 38.10 and 38.11 and also can be obtained directly from Fig.38.8 by multiplying the amplitude by co/con and (co/co«)2, respectively

Force Transmissibility The force F T transmitted to the foundation by a system

subjected to an external harmonic excitation is

FREQUENCY RATIO uj/a>

FIGURE 38.9 Phase-angle frequency response for forced motion.

Substituting the system response from Eq (38.27) into Eq (38.36) gives

^-rsin(co^-0) (38.37)

FQ where the nondimensional magnitude of the transmitted force T is given by

The transmissibility T is shown in Fig 38.12 versus forcing frequency At very low

forcing frequencies, the transmissibility is close to unity, showing that the appliedforce is directly transmitted to the foundation The transmissibility is very large inthe vicinity of the system natural frequency, and for high forcing frequencies thetransmitted force decreases considerably The phase variation between the transmit-ted force and the applied force is shown in Fig 38.13

Rotating Imbalance When machines with rotating imbalances are mounted on

elastic supports, they constitute a vibrating system subjected to excitation from the

FREQUENCY RATIO u/uj

FIGURE 38.10 Velocity frequency response.

rotating imbalance If the natural frequency of the system coincides with the quency of rotation of the machine imbalance, it will result in severe vibrations of themachine and the support structure

fre-Consider a machine of mass M supported as shown in Fig 38.14 Let the ance be a mass m with an eccentricity e and rotating with a frequency GD Consider the motion x of the mass M-m, with x m as the motion of the unbalanced mass m rel- ative to the machine mass M The equation of motion is

imbal-(M - m)'x + m(x + x m ) + ex+ kx = O (38.40)

The motion of the unbalanced mass relative to the machine is

Substitution in Eq (38.40) leads to

Mx + cx+kx = me® 2 sin cor (38.42)

FREQUENCY RATIO u/u> n

FIGURE 38.11 Acceleration frequency response.

This equation is similar to Eq (38.25), where the force amplitude F0 is replaced byweco2 Hence, the steady-state solution of Eq (38.42) is similar in form to Eq (38.27)and is given nondimensionally as

FREQUENCY RATIO w/w n

FIGURE 38.12 Transmissibility plot.

x = exp -CGV {A exp [(? - 1)1/2 oy] + 5 exp [-(£2 - 1)1/2 ov]}

mg(cQ/con)2sin(cor-e)M[(l - G)2/co2)2 + (2Cco/co«)2]1/2 ^ ;

System Excited at the Foundation When the system is excited at the foundation,

as shown in Fig 38.15, with a certain displacement u(f) = U 0 sin otf, the equation ofmotion can be written as

mx + c(x - u) + k(x - u) = O (38.46)

This equation can be written in the form

mx + ex+ kx = Cw0CO cos otf + ku 0 sin co? (38.47)

= F sin (cor + 0)

FREQUENCY RATIO u/u> n

FIGURE 38.13 Phase angle between transmitted and applied forces.

FIGURE 38.14 Dynamic system subj ect to

FIGURE 38.15 A base excited system.

where

F Q = U 0 (k 2 + c2 co 2 ) 1/2 (38.48) and

4> = tan- 1 — (38.49)

CCO

Equation (38.47) is identical to Eq (38.25) except for the phase § Hence the

solu-tion is similar to that of Eq (38.25) If the ratio of the system response to the basedisplacement is defined as the motion transmissibility, it will have the same form asthe force transmissibility given in Eq (38.38)

Resonance, System Bandwidth, and Q Factor A vibrating system is said to be in

resonance when the response is maximum The displacement and accelerationresponses are maximum when

whereas velocity response is maximum when

In the case of an undamped system, the response is maximum when co = con, where

con is the frequency of free vibration of the system For a damped system, the quency of free oscillations or the damped natural frequency is given by

The Q factor is defined as

FIGURE 38.16 Resonance, bandwidth, and Q factor.

which is equal to the maximum response in physical systems with low damping Thebandwidth is defined as the width of the response curve measured at the "half-power" points, where the response is /?maxA/2 For physical systems with £ < 0.1, thebandwidth can be approximated by

ACO = 2CcOn = -^ (38.54)

Forced Vibration of Torsional Systems In the torsional system of Fig 38.3, if the

disk is subjected to a sinusoidal external torque, the equation of motion can be

not be expressed as a simple analytical function, then the numerical method is theonly recourse to obtain the system response.

The differential equation of motion of a system can be expressed in the form

Let Wy denote an approximation to Jt1- (t}) for each ; = 0,1,2, and / = 1,2 For the

initial conditions, set H>I,O = *o and w2,o = X 0 Obtain the approximation Wy + \ y given all

the values of the previous steps w^ as [38.1]

Wy +1 = Wy + - (ku + 2k24 + 2k3j + fcy) i = l,2 (38.57) where ku = HF1(I1 + Wij9 W2J)

Note that /cu and k12 must be computed before we can obtain /c2,i

Example Obtain the response of a generator rotor to a short-circuit disturbance

Since coi = 182 rad/s, the period i = 2rc/182 = 0.00345 s and the time interval h must

be chosen to be around 0.005 s Hence, tabulated values of f(t) must be available for

t intervals of 0.005 s, or it has to be interpolated from Fig 38.17.

FIGURE 38.17 Short-circuit excitation form.

38.3 SYSTEMSWITHSEVERAL

DEGREES OF FREEDOM

Quite often, a single-degree-of-freedom system model does not sufficiently describethe system vibrational behavior When it is necessary to obtain information regard-ing the higher natural frequencies of the system, the system must be modeled as amultidegree-of-freedom system Before discussing a system with several degrees offreedom, we present a system with two degrees of freedom, to give sufficient insightinto the interaction between the degrees of freedom of the system Such interactioncan also be used to advantage in controlling the vibration

38.3.1 System with Two Degrees of Freedom

Free Vibration A system with two degrees of freedom is shown in Fig 38.18 It

consists of masses Jn 1 and W2, stiffness coefficients ki and A;2, and damping cients C1 and C2 The equations of motion are

coeffi-W1Jt1 + (GI + C 2 )Xi + (ki + /T2)Jt1 - C2Jt2 - k 2 x 2 = O

(38.60)

FIGURE 38.18 Two-degree-of-freedom system.

Assuming a solution of the type

solution will consist of four constants which can be determined from the four initialconditions Je1, Jc2, Jc1, and X2 If damping is less than critical, oscillatory motion occurs,

and all four roots of Eq (38.63) are complex with negative real parts, in the form

So the complete solution is

Je1 = exp (-nit) (Ai cos pit+A 2 sin/J1J)

+ exp (-n 2 t) (Bi cos p 2 t + B 2 sin p 2 t)

(38.65)

Je2 = exp (-nit) (A{ cos pit + A2 sin pit)

+ exp (-«20 (Bi cosp2t + B2 sinp2t) Since the amplitude ratio AIB is determined by Eq (38.62), there are only four inde-

pendent constants in Eq (38.65) which are determined by the initial conditions ofthe system

Forced Vibration Quite often an auxiliary spring-mass-damper system is added to

the main system to reduce the vibration of the main system The secondary system is

called a dynamic absorber Since in such cases the force acts on the main system only, consider a force P sin cor acting on the primary mass m Referring to Fig 38.18, we

see that the equations of motion are

(38.66)

Jn2X2 + C2X2 + k2x2 - C2Xi - k2Xi = O

Assuming a solution of the type

= A 1 cos cor+A 2 sin cor

(38.68)03!(2Z)1CoC2CO2-Z)2CQ2Q

Responses may also be written in the form

Jc1 = B1 sin (cor - G1) X2 = B2 sin (cor - 02) (38.71)