ADE693182 1 9 Special Issue Article Advances in Mechanical Engineering 2017, Vol 9(2) 1–9 � The Author(s) 2017 DOI 10 1177/1687814017693182 journals sagepub com/home/ade Dynamic properties of an axial[.]
Trang 1Advances in Mechanical Engineering
2017, Vol 9(2) 1–9
Ó The Author(s) 2017 DOI: 10.1177/1687814017693182 journals.sagepub.com/home/ade
Dynamic properties of an axially
moving sandwich beam with
magnetorheological fluid core
Abstract
Dynamic properties and vibration suppression capabilities of an axially moving sandwich beam with a magnetorheological fluid core were investigated in this study The stress–strain relationship for the magnetorheological fluid was described
by a complex shear modulus using linear viscoelasticity theory First, the dynamic model of an axially moving magnetor-heological fluid beam was derived based on Hamilton’s principle Then, the natural frequency of the sandwich beam for the first mode was determined Later, the effects of the speed of the axial movement, axial force, applied magnetic field, skin–core thickness ratio, and their combination on the dynamic properties of the sandwich beam with a magnetorheo-logical fluid core were investigated It was found that these parameters have significant effects on the dynamic properties
of the sandwich beam Moreover, the results indicate that the active control ability of magnetic field has been influenced
by the axial force, moving speed, and increasing skin–core thickness ratio
Keywords
Sandwich beam, magnetorheological fluid, axially moving beam, natural frequency, vibration suppression capabilities
Date received: 12 August 2016; accepted: 21 December 2016
Academic Editor: Chi-man Vong
Introduction
Axially moving beams can represent many engineering
devices, such as mechanical arms, automotive belts,
band saw blades, and so on Despite the many
advan-tages of these devices, the associated noises and
vibra-tions have impeded their applicavibra-tions In order to
control these noises and vibrations, some smart
materi-als such as electrorheological (ER),1–5
magnetorheolo-gical fluids (MRFs)/elastomers,6–10 shape memory
alloy (SMA),11–14piezoelectric patches (PEP),15,16 and
shear thickening fluid (STF)17–19have been applied in
these structures However, there have been very few
dynamic analyses of axially moving structures that
have incorporated smart materials Therefore, in this
article, how movements affect the dynamic properties
of an axially moving sandwich beam that has
inte-grated an MRF is investigated, and the capability of an
MRF core to suppress vibrations is evaluated as well
In the past decades, several studies have concen-trated on the dynamic characteristics, stability, and vibration control of axially moving beams Natural fre-quencies of axially moving beams with pinned–pinned ends and clamped–clamped ends were studied by O¨z and Pakdemirli20and O¨z,21 respectively Ghayesh and Khadem22analyzed the free nonlinear transverse vibra-tion of an axially moving beam In their article, the
1
School of Civil Engineering, Dalian University of Technology, Dalian, China
2
School of Civil Engineering, Shenyang Jianzhu University, Shenyang, China
3
Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Kowloon, Hong Kong Corresponding author:
Li Sun, School of Civil Engineering, Shenyang Jianzhu University, Shenyang
110168, China.
Email: sunli2009@163.com
Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 3.0 License
(http://www.creativecommons.org/licenses/by/3.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/ open-access-at-sage).
Trang 2natural frequency versus mean velocity and rotary
iner-tia were plotted, and the natural frequency versus the
mean velocity and temperature for the first two modes
were also given Ding and Chen23obtained the natural
frequencies for nonlinear coupled planar dynamics of
an axially moving beam in the supercritical speed
regime via discrete Fourier transform The effects of
both non-ideal boundary conditions and axial velocity
on the natural frequencies of an axially moving beam
were investigated by Bag˘datlı and Uslu.24For the effect
of time-dependent velocity, Rezaee and Lotfan25
inves-tigated the natural frequencies, complex mode shapes,
and responses of an axially moving nanoscale beam by
directly using the multiple-scale method and a power
series method
On the sandwich structures with some smart
materi-als, MalekzadehFard et al.26investigated the free
vibra-tion and buckling of a sandwich panel with MRF layer
under simply supported boundary conditions The
effects of magnetic field and geometrical parameters on
the dynamic properties of the first four mode shapes
are also discussed Wei et al.2studied the vibration
con-trol of a flexible rotating sandwich beam with an ER
core Meanwhile, they assessed the influences of both
various electric field strengths and rotating speeds on
the natural frequencies of the rotating ER beam Wei
et al.27 investigated the vibration characteristics of a
sandwich beam with an MRF under different magnetic
field intensities and different rotating speeds using the
finite element theory The dynamic stability of a
rotat-ing three-layered symmetric sandwich beam with a
magnetorheological (MR) elastomer core subjected to
axial periodic loads was studied by Nayak et al.28For
the PEP sandwich beam, Ozdemir and Kaya29
investi-gated the extension and flapwise bending vibrations of
a rotating piezolaminated composite Timoshenko
beam Numerical results were obtained to investigate
the effects of the applied voltage, ply orientation,
rota-tional speed, and hub radius on the natural frequencies
and tip deflection It can be seen that some efforts have
been made on rotating sandwich beams with various
smart material cores However, the dynamic properties
of an axially moving sandwich beam with an MRF
core are yet to be explored
In this study, the sandwich structure with an MRF
is employed to control vibrations of axially moving
beams The MRF, which is sandwiched between two
elastic layers, acts as a viscoelastic-damping layer with
controllable shear modulus First, a dynamic model of
the axially moving sandwich beam is developed based
on Hamilton’s principle, with the material
characteris-tics of the MRF and the dynamic stiffening caused by
axial motion taken into consideration Then, the effects
of axial velocity, axial force, skin–core thickness ratio,
magnetic field, and combinations of them on the
natu-ral frequencies of the sandwich beam are analyzed
Model formulation of a sandwich beam with MRF core
Axially moving model of a sandwich beam
An axially moving sandwich beam with an MRF core and conductive aluminum skins subjected to periodic excitation is shown in Figure 1 A number of assump-tions are made as follows: (a) deformaassump-tions of top and bottom layers obey the Euler–Bernoulli beam theory, (b) the external rigidity of the moving beam is large enough to render the longitudinal displacement from the preload tension negligible, (c) the MRF core deforms only due to shear, (d) the three layers have the same transverse displacement z, (e) there is no slippage and delamination between the adjacent layers during deformation, and (f) the axial load is less than the buckling load of the beam
Based on the above assumptions, the governing equations of motion for the sandwich beam are obtained using the extended Hamilton’s principle, which states that
ðt 2
t 1
dT dV dW
where dT, dV, and dW are kinetic energy, potential energy, and the work done by external force, respectively
If the strain level of the MRF is considered to be
\1%, then its rheological property is in the pre-yield regime and can be described by the linear viscoelastic theory Thus, the final form of Hamilton’s principle
Figure 1 Schematic diagram of MRF-embedded cored sandwich moving beam: (a) axially moving sandwich beam with MRF core subjected to axial load and (b) configuration of a sandwich beam with MRF.
Trang 3including the kinetic energy, potential energy, and
work done over the sandwich beam terms can be
presented as
ð t 2
t 1
dT V + dT R dV E, e dV E, b dV E, P dV MR, s dW
ð2Þ where dTV and dTR denote the kinetic energies due to
transverse and rotational motions, respectively; dVE
and dVE,brepresent the potential energy due to
exten-sional and bending stresses of the surface plates,
respec-tively; dVE,P denotes the potential energy due to axial
force; and dVMR,s is the potential energy due to shear
stresses of the MR core
Furthermore, these terms can be expressed in the
fol-lowing form28,30,31
TV=1
2
ðL
0
∂w x, tð Þ
∂t + v
∂w x, tð Þ
∂x
r xð Þdx ð3aÞ
TR=1
2
ðL
0
∂uðx, tÞ
∂t
J xð Þdx ð3bÞ
VE, e=1
4E hð 1+ h2Þ2bh1
ðL
0
∂uðx, tÞ
∂t
VE, b= EI
ðL
0
∂w2ðx, tÞ
∂x2
VE, P=
ðL
0
P ∂w x, tð Þ
∂x
dx + Ebh1
ðL
0
∂w x, tð Þ
∂x
dx ð4cÞ
VMR, s= 1
2G
bh2
ðL
0
g x, tð Þ2dx ð4dÞ
W =
ðL
0
f x, tð Þw x, tð Þdx ð5Þ
where w(x, t) and u(x, t) are the transverse and
rota-tional displacements of the sandwich beam at location
x and time t, respectively; r(x) is the density of the
beam; J(x) is the mass moment of inertia; v is the axial
velocity; P is the axial force; f(x, t) is the external force;
Eand I are Young’s modulus and moment of area of
the top and bottom layer materials, respectively; G* is
the complex shear modulus of the MRF core; h1, h2,
and h3are the thicknesses of the top, core, and bottom
layers, respectively; L and b are the length and width of
the beam, respectively; g(x, t) is the shear displacement
of the MRF material and has the following form
g x, tð Þ = ∂w x, tð Þ
∂x u x, tð Þ ð6Þ Substituting equations (3)–(6) into Hamilton’s equa-tion and integrating by parts, the equaequa-tions of moequa-tion yield
J∂
2u
∂t2 1
2ðh1+ h2Þ2bh1E∂
2u
∂x2 Gbh2
∂w
∂x u
= 0 ð7aÞ
r∂
2w
∂t2 + rv2∂
2w
∂x2 + 2EI∂
4w
∂x4 ∂
2w
∂x2P + 2rv∂
2w
∂t∂x Gbh2
∂2w
∂x2 ∂u
∂x
= 0
ð7bÞ
The hinge supports are assumed to be used at the two ends of sandwich beam Therefore, the associated mechanical boundary conditions can be written as
w 0, tð Þ = w L, tð Þ = 0,∂
2w x, tð Þ
∂x2 jx = 0=∂
2w x, tð Þ
∂x2 jx = L= 0
ð8Þ From the classical beam theory, the displacement of beam under a transverse periodic excitation can be writ-ten as
w x, tð Þ = X‘
n = 1
fnð Þ exp ivx ð ntÞ ðn = 1, 2, , ‘Þ ð9aÞ
u x, tð Þ = X‘
n = 1
unð Þ exp ivx ð ntÞ ðn = 1, 2, , ‘Þ ð9bÞ
where vn is the nth natural frequency of the beam,
fn(x) is the nth mode shape of the transverse vibration, and un(x) is the nth mode shape of the rotational vibra-tion and can be further expressed as follows
fnð Þ = sin lx ð nxÞ ð10aÞ
unð Þ = Cx n cos lð nxÞ ð10bÞ
in which Cn denotes the ratio of the rotational and transverse displacement amplitudes, and lncan be writ-ten as
ln=np
L ðn = 1, 2, , ‘Þ ð11Þ Since the transverse vibration frequency is much higher than the rotational vibration frequency for the sandwich beam with MRF core, the influence of the rotational inertia force can be ignored.32 Thus, substi-tuting equations (9) and (10) into equation (7a) and simplifying the equation yields
Trang 4Cn= 2G
h2ln
Eh1ðh1+ h2Þ2l2n+ 2Gh2
ð12Þ
According to equations (10a) and (10b), the nth mode
shape of the rotational vibration can be written as
unð Þ =x Cn
lnfnð Þx 0 ð13Þ Furthermore, the rotational displacement can be
written as
u x, tð Þ = X‘
n = 1
Cn
lnfnð Þx0qnð Þt ð14Þ where qn(t) is the generalized displacement function
Substituting equations (9a) and (14) into equation
(7b) and using Galerkin’s approach, the dynamic model
of an axially moving sandwich beam can be obtained
X‘
n = 1
fið Þ€x qið Þ + 2vft ið Þx 0_qið Þ +t v
2 P
r fið Þx00qið Þ +t 2EI
r fið Þx 0000qið Þ t G
bh2
r 1Cn
ln
fið Þx00qið Þt
= 0 ð15Þ
Each function fi(x) is used as the weighting function
for the residual to equation (15), and the following
orthogonal property is used
M€q + C _q + Kq = 0 ð16Þ where the mass, damping, and stiffness matrices are
given, respectively, by
mij=
ðL
0
fið Þfx jð Þdxx ð17aÞ
cij= 2v
ðL
0
fið Þfx jð Þx 0dx ð17bÞ
kij=1
r v
2 P Gbh2 1Cn
ln
ðL
0
fið Þfx jð Þx00dx +2EI
r
ðL
0
fið Þfx jð Þx 0000dx
ð17cÞ
Thus, considering n = 1, an expression for the
natu-ral frequency can be obtained
v
ð Þ2=2EIp
4
rL4 + p
2
L2r
v2 P Gbh2 1 2G
h2
Eh 1 ð h 1 + h 2 Þ 2 p 2
L 2 + 2Gh2
0
@
1 A
0
@
1 A ð18Þ
Furthermore, the complex natural frequency is expressed as
v
ð Þ2= v2ð1 + ihÞ ð19Þ where v is the natural frequency and h is the system loss factor
MRF material
MRF is one of the materials of controllable rheological properties These properties, such as viscosity, elasti-city, and plastielasti-city, can undergo instantaneous and reversible changes when subjected to a magnetic field Because the rheological response has a yield point, the rheology of MRFs is approximately modeled in pre-yield and post-pre-yield regimes Moreover, since the strain level of an MRF is \1% in the pre-yield regime, the model in the pre-yield regime can be used as a linear
viscoelastic model Thus, the model considered in this study is based on the pre-yield rheological properties of
MR materials
The shear stress of the MRF layer can be expressed as
t x, tð Þ = Gg x, tð Þ ð20Þ where g(x, t) is the shear strain and G* is the complex shear modulus
The complex shear modulus G* is a function of the magnetic field strength applied on the MRFs and can
be written in the form
G= G0ð Þ + iGB 00ð ÞB ð21Þ where G#(B) is the storage modulus and G$(B) is the loss modulus and are given by32
G0ð Þ = 3:11 3 10B 7B2+ 3:56 3 104B + 5:78 3 101
ð22aÞ
G00ð Þ = 3:47 3 10B 9B2+ 3:85 3 106B + 6:31 3 103
ð22bÞ
in which B is the magnetic field strength, and its unit is Tesla
Substituting equations (19)–(22) into equation (18),
it can be seen that the natural frequency of the axially moving sandwich beam has been influenced by axial velocity, axial force, thickness ratio of skin–core, and the controllable magnetic field strength
Trang 5Results and discussion
The dynamic characteristics and vibration suppression
capabilities of the MRF cored sandwich beam with
var-ious system parameters are investigated in this section
Since the moving velocity of the beam does not exceed
the critical velocity, the stable/unstable region of the
moving beam is not studied The other physical
para-meters and material properties of the beam are set as
follows: E1= E3= 72 GPa, L = 416 mm, b = 30 mm,
and r = 2700 kg/m3, which are consistent with the
para-meters in Nayak et al.28 The natural frequency of the
sandwich beam with h1= h2= h3= 1 mm under the
combined effects of the axial force and magnetic field
are first investigated Then, the combined effects of the
axial velocity and magnetic field are evaluated Finally,
the effects of both the skin–core thickness ratio and
magnetic field on the nature frequency are studied
For the case of the magnetic field B = 0.5 T and the
skin–core thickness ratio g = 1, variations in the
natu-ral frequency (v) of the sandwich beam with both the
axial velocity (v) and the axial force (P) are shown in
Figure 2 It can be seen that for a constant axial force
(either compression or tension), the natural frequency
increases nonlinearly with the axial velocity If the
beam is subjected to compression, dynamic properties
of the sandwich beam depend on a critical velocity
Meanwhile, the critical velocity increases with
compres-sion For instance, when B = 0.5 T, the critical
veloci-ties for P = 100 and 500 N are 12.3 and 23.4 m/s,
respectively On the contrary, if the beam is subjected
to tension, dynamic properties of the sandwich beam is
independent of the critical velocity For example, with
B= 0.5 T, when v = 0 m/s, the natural frequencies for
P= 210N and 2500 N are 1.06 and 3.05 rad/s, respectively
The effects of axial force (P) and varying magnetic field strength (B) on the capability of natural frequency suppression when the skin–core thickness ratio g = 1 are shown in Figure 3 As seen in Figure 3(a) and (b), the suppression capability of the MRF in the natural frequency increases nonlinearly with increasing mag-netic strength for a constant axial force Consider
P= 100 N and v = 15 m/s as example (Figure 3(b)), the natural frequency of the sandwich beam is suppressed
by 3.4% at B = 0.5 T, 7.8% at B = 1.0 T, and 27.2% at
B= 2.0 T However, the suppression capability decreases gradually with increasing axial velocity As
P= 100 N and B = 2.0 T (Figure 3(b)), the suppression ratio of the natural frequency is 27.2% at v = 15 m/s, 8.3% at v = 20 m/s, and 4.4% at v = 25 m/s Moreover, comparing Figure 3(a) and (b) shows that the MRF core has significant control ability on the natural fre-quency of the sandwich beam when the axial force is tensional and the axial velocity is much lower Referring to Figure 3(a), as P = 2100 N and B = 2.0 T, the suppression ratio of the natural frequency is 41.7%
at v = 0 m/s, 27.6% at v = 5 m/s, and 14.4% at
v= 10 m/s, respectively
The natural frequency of the sandwich beam versus the axial force (P) with varying axial velocity (v) is plotted in Figure 4 The magnetic field strength (B) is set to 0.5 T, and the skin–core thickness ratio (g) is set
to 1 Referring to Figure 4, the natural frequency curve moves toward the right with increasing axial velocity, which indicates that the natural frequency increases with the velocity under a constant axial force As
v= 0 m/s, the natural frequency of the sandwich beam only exists when the axial force is tensional, and the natural frequency increases with the tension Furthermore, as the velocity is 0, for example, v = 10 and 20 m/s, the natural frequency varies with the type
of the axial force More specifically, if the axial force is
a tension, the natural frequency increases with the ten-sion On the contrary, if the axial force is a compres-sion, the natural frequency decreases with the compression
The effects of varying axial velocity (v) and magnetic field (B) on the capability of natural frequency suppres-sion of an axial moving MRF cored sandwich beam are shown in Figure 5 For a specified velocity curve, it can be found that the suppression capability of the MRF on the natural frequency nonlinearly increases with increasing magnetic strength As v = 10 m/s and
P= 0 N (see Figure 5(a)), the natural frequency of the sandwich beam is suppressed by 4.3% at B = 0.5 T, 13.4% at B = 1.0 T, and 42.5% at B = 2.0 T, respec-tively Moreover, when the axial force is compression, the suppression capability of the MRF gradually
Figure 2 Effect of initial axial force P (N) on the natural
frequency versus axial velocity when the magnetic field strength
B = 0.5 T and the skin/core thickness ratio g = 1.
Trang 6increases with the increasing axial force As v = 20 m/s
and B = 2.0 T, the suppression ratios of the natural
fre-quency are 8.3% for P = 100 N, 14.3% for P = 200 N,
and 42.7% for P = 300 N However, when the axial
force is a tension, the suppression capability only has a
slight decrease as the tension increases For the case
with v = 20 m/s and B = 2.0 T, the suppression ratios
of the natural frequency are 4.7%, 3.9%, and 3.2% for
P= 2100, 2200, and 2300 N, respectively It is
worthwhile to note that the suppression capability of the MRF on the natural frequency is greater when the axial force is expressed as compression than tension with the same magnitude, and the difference becomes more significant as the axial force increases When
v= 20 m/s and B = 2.0 T, the suppression ratios are 8.3% for P = 100 N, 4.7% for P = 2100 N, 42.7% for
P= 300 N, and 3.2% for P = 2300 N
The effects of skin–core thickness ratio on the nature frequency of the axial moving MRF cored sandwich beam are shown in Figure 6 As the skin–core thickness ratio increases, the natural frequency increases nonli-nearly However, the increases become nearly linear as the thickness ratio exceeds 1.0, which implies that the effect of the thickness ratio on the natural frequency is negligible In addition, the nonlinearity behavior of increase is independent of both the axial force and velo-city, while it has a significant effect on the natural frequency
The influences of the skin–core thickness ratio (g) and varying magnetic field strength (B) on the suppres-sion capability of natural frequency are shown in Figure 7 It can be seen that the suppression capability
of the MRF exhibits a nonlinear decrease trend with thickness ratio When v = 20 m/s, P = 2100 N, and
B= 2.0 T, the suppression ratio of the natural fre-quency is 10.3%, 4.8%, and 2.3% for the skin–core thickness ratio g = 0.5, 1.0, and 2.0, respectively Furthermore, the axial velocity and the axial force have non-negligible effects on the suppression ratio In par-ticular, as B = 2.0 T, comparing case 2 and case 4 and case 3 and case 4 demonstrates that the suppression
Figure 3 Effect of axial force P (N) and magnetic field strength B (T) on the capability of natural frequency suppression when the skin/core thickness ratio g = 1: (a) P = –500 N and –100 N and (b) P = 0 N, 100 N and 500 N.
Figure 4 Effect of axial velocity v (m/s) on the natural
frequency versus axial force when the magnetic field strength
B = 0.5 T and the skin/core thickness ratio g = 1.
Trang 7ratio of the natural frequency is reduced by 26.2% and
27.5% at g = 0.5, 9.3% and 9.8% at g = 1.0, and
4.0% and 5.8% at g = 2.0, respectively
Conclusion
In this article, the dynamic model of an axial moving
sandwich beam filled with MRF core has been studied
Considering the stress–strain relationship of the MRF described using linear viscoelasticity theory, the corre-sponding equations of motion are derived based on Hamilton’s principle The variation of natural fre-quency for different system parameters such as axial velocity, axial force, the skin–core thickness ratio, and magnetic field strength has been investigated Moreover, the vibration suppression capabilities of an MRF in the sandwich beam have been evaluated for different system parameters
The natural frequency increases nonlinearly with the axial velocity When the axial force is compression, the dynamic properties of the sandwich beam depend on a critical velocity Furthermore, the critical velocity increases as the compression increases The effect of the axial velocity on the dynamic properties is complicated When the axial velocity is small, the sandwich beam exhibits dynamic properties only when the axial force is tension When the axial velocity is large, the natural frequency of the sandwich beam varies depending on whether the axial force is a tension or compression
In addition, the skin–core thickness ratio has negligible effect on the natural frequency However, when the thickness ratio is small, the natural frequency decreases nonlinearly with increasing ratio
For a constant axial force or axial velocity, the sup-pression capability of an MRF on the natural frequency nonlinearly increases with increasing magnetic field strength However, increasing the axial velocity reduces the suppression capability More importantly, the sup-pression capability is strong when the beam is under
Figure 5 Effect of axial velocity v (m/s) and magnetic field strength B (T) on the capability of natural frequency suppression when the skin/core thickness ratio g = 1: (a) v = 0 m/s and 10 m/s and (b) v = 20 m/s.
Figure 6 Effect of the skin/core thickness ratio g on the
natural frequency versus axial force when the magnetic field
strength B = 0.5 T.
Trang 8action of a tension and moves slowly; the maximum
suppression ratio reaches 41.7% The suppression
capa-bility is increased as the compression increases when the
axial force is compression However, when the axial
force is tension, the suppression capability only slightly
decreases with increasing tension In addition, the
sup-pression capability of the MRF is stronger when the
axial force is tension than compression The
suppres-sion capability decreases nonlinearly with increasing
skin–core thickness ratio
Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with
respect to the research, authorship, and/or publication of this
article.
Funding
The author(s) disclosed receipt of the following financial
sup-port for the research, authorship, and/or publication of this
article: This work was supported by the National Natural
Science Foundation of China (grant nos 51578347 and
51608335), Natural Science Foundation of Liaoning Province
(grant no 2015020578), China Postdoctoral Science
Foundation (grant no 2016M591432), and the Thousand and
Ten Thousand Talent Project of the Liaoning Province (grant
no 2014921045).
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