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[.]
Special Issue Article Dynamic properties of an axially moving sandwich beam with magnetorheological fluid core Advances in Mechanical Engineering 2017, Vol 9(2) 1–9 Ó The Author(s) 2017 DOI: 10.1177/1687814017693182 journals.sagepub.com/home/ade Minghai Wei1,2, Li Sun1,2 and Gang Hu3 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 magnetorheological 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 magnetorheological 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 advantages of these devices, the associated noises and vibrations have impeded their applications In order to control these noises and vibrations, some smart materials such as electrorheological (ER),1–5 magnetorheological fluids (MRFs)/elastomers,6–10 shape memory alloy (SMA),11–14 piezoelectric patches (PEP),15,16 and shear thickening fluid (STF)17–19 have 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 integrated 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 concentrated on the dynamic characteristics, stability, and vibration control of axially moving beams Natural frequencies of axially moving beams with pinned–pinned ends and clampedclamped ends were studied by Oăz and Pakdemirli20 and Oăz,21 respectively Ghayesh and Khadem22 analyzed the free nonlinear transverse vibration of an axially moving beam In their article, the School of Civil Engineering, Dalian University of Technology, Dalian, China School of Civil Engineering, Shenyang Jianzhu University, Shenyang, China 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) 2 natural frequency versus mean velocity and rotary inertia were plotted, and the natural frequency versus the mean velocity and temperature for the first two modes were also given Ding and Chen23 obtained 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.24 For the effect of time-dependent velocity, Rezaee and Lotfan25 investigated 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 materials, MalekzadehFard et al.26 investigated the free vibration 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.2 studied the vibration control 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 rotating three-layered symmetric sandwich beam with a magnetorheological (MR) elastomer core subjected to axial periodic loads was studied by Nayak et al.28 For the PEP sandwich beam, Ozdemir and Kaya29 investigated 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, rotational 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 characteristics 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 natural frequencies of the sandwich beam are analyzed Advances in Mechanical Engineering Figure 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 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 A number of assumptions are made as follows: (a) deformations 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 ðt2 ðdT dV dW Þ = ð1Þ t1 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 Wei et al including the kinetic energy, potential energy, and work done over the sandwich beam terms can be presented as ðt2 the beam, respectively; g(x, t) is the shear displacement of the MRF material and has the following form g ðx, tÞ = ðdTV + dTR dVE, e dVE, b dVE, P dVMR, s dW Þ = t1 ð2Þ where dTV and dTR denote the kinetic energies due to transverse and rotational motions, respectively; dVE,e and dVE,b represent the potential energy due to extensional and bending stresses of the surface plates, respectively; 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 following form28,30,31 TV = ðL ∂wðx, tÞ ∂wðx, tÞ +v rð xÞdx ∂t ∂x ð3aÞ ∂wðx, tÞ uðx, tÞ ∂x ð6Þ Substituting equations (3)–(6) into Hamilton’s equation and integrating by parts, the equations of motion yield ∂2 u ∂2 u ∂w J ðh1 + h2 Þ bh1 E G bh2 u =0 ∂t ∂x ∂x ð7aÞ r ∂2 w ∂4 w ∂2 w 2∂ w + rv + 2EI P ∂t2 ∂x2 ∂x4 ∂x2 ∂2 w ∂2 w ∂u G bh2 + 2rv =0 ∂t∂x ∂x2 ∂x 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 TR = ðL ∂uðx, tÞ ∂t 2 wð0, tÞ = wðL, tÞ = 0, J ð xÞdx ð3bÞ ∂2 wðx, tÞ ∂2 wðx, tÞ jx = = jx = L = ∂x ∂x2 ð8Þ VE, e = Eðh1 + h2 Þ2 bh1 ðL ∂uðx, tÞ ∂t 2 dx ð4aÞ From the classical beam theory, the displacement of beam under a transverse periodic excitation can be written as ðL VE, b = EI ∂w ðx, tÞ ∂x2 2 wðx, tÞ = dx ð4bÞ ðL ðL ∂wðx, tÞ ∂wðx, tÞ VE, P = P dx + Ebh1 dx ð4cÞ ∂x ∂x 0 ðL VMR, s = G bh2 g ðx, tÞ2 dx ð4dÞ ‘ X fn ð xÞ expðivn tÞ ðn = 1, 2, , ‘Þ ð9aÞ n=1 uðx, tÞ = ‘ X un ð xÞ expðivn tÞ where 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 vibration and can be further expressed as follows ðL W= f ðx, tÞwðx, tÞdx ð5Þ where w(x, t) and u(x, t) are the transverse and rotational 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; E and 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 h3 are the thicknesses of the top, core, and bottom layers, respectively; L and b are the length and width of ðn = 1, 2, , ‘Þ ð9bÞ n=1 fn ð xÞ = sinðln xÞ ð10aÞ un ð xÞ = Cn cosðln xÞ ð10bÞ in which Cn denotes the ratio of the rotational and transverse displacement amplitudes, and ln can be written 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, substituting equations (9) and (10) into equation (7a) and simplifying the equation yields Advances in Mechanical Engineering Cn = 2G h2 ln Eh1 ðh1 + h2 Þ2 l2n + 2G h2 ð12Þ According to equations (10a) and (10b), the nth mode shape of the rotational vibration can be written as Cn f n ð xÞ u n ð xÞ = ln ð13Þ Furthermore, the rotational displacement can be written as uðx, tÞ = ‘ X Cn l n=1 n fn ð xÞ qn ðtÞ 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 fi ð xÞ€ qi ðtÞ + 2vfi ð xÞ0 q_ i ðtÞ + n=1 ð16Þ M€q + Cq_ + Kq = where the mass, damping, and stiffness matrices are given, respectively, by mij = fi ð xÞfj ð xÞdx cij = 2v fi ð xÞfj ð xÞ dx 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, elasticity, and plasticity, 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 preyield and post-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 ð15Þ 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 tðx, tÞ = G g ðx, tÞ ð20Þ ð17aÞ 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 ð17bÞ G = G0 ð BÞ + iG00 ð BÞ ðL ðv Þ2 = v2 ð1 + ihÞ v2 P 2EI G bh2 Cn 1 fi ð xÞ00 qi ðtÞ + fi ð xÞ0000 qi ðtÞ fi ð xÞ00 qi ðtÞ = r ln r r Each function fi(x) is used as the weighting function for the residual to equation (15), and the following orthogonal property is used ðL Furthermore, the complex natural frequency is expressed as ð21Þ Cn v P G bh2 kij = ln r ðL ðL 2EI fi ð xÞfj ð xÞ00 dx + fi ð xÞfj ð xÞ0000 dx r where G#(B) is the storage modulus and G$(B) is the loss modulus and are given by32 ð17cÞ ð22aÞ Thus, considering n = 1, an expression for the natural frequency can be obtained 2EIp4 p2 ðv Þ = + rL L r 0 @v P G bh2 @1 2G h2 Eh1 ðh1 + h2 Þ2 p2 L2 + 2G h2 G0 ðBÞ = 3:11 107 B2 + 3:56 104 B + 5:78 101 11 AA ð18Þ G00 ð BÞ = 3:47 109 B2 + 3:85 106 B + 6:31 103 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 Wei et al Figure 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 = Results and discussion The dynamic characteristics and vibration suppression capabilities of the MRF cored sandwich beam with various 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 parameters 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 parameters in Nayak et al.28 The natural frequency of the sandwich beam with h1 = h2 = h3 = 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 natural frequency (v) of the sandwich beam with both the axial velocity (v) and the axial force (P) are shown in Figure 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 compression For instance, when B = 0.5 T, the critical velocities 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 = 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 = are shown in Figure As seen in Figure 3(a) and (b), the suppression capability of the MRF in the natural frequency increases nonlinearly with increasing magnetic 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 frequency 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 = m/s, 27.6% at v = 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 The magnetic field strength (B) is set to 0.5 T, and the skin–core thickness ratio (g) is set to 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 = 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 tension On the contrary, if the axial force is a compression, the natural frequency decreases with the compression The effects of varying axial velocity (v) and magnetic field (B) on the capability of natural frequency suppression of an axial moving MRF cored sandwich beam are shown in Figure 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 = 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, respectively Moreover, when the axial force is compression, the suppression capability of the MRF gradually Advances in Mechanical Engineering Figure 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 = N, 100 N and 500 N Figure 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 = increases with the increasing axial force As v = 20 m/s and B = 2.0 T, the suppression ratios of the natural frequency 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 As the skin–core thickness ratio increases, the natural frequency increases nonlinearly 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 velocity, 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 suppression capability of natural frequency are shown in Figure 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 frequency 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 particular, as B = 2.0 T, comparing case and case and case and case demonstrates that the suppression Wei et al Figure 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 = m/s and 10 m/s and (b) v = 20 m/s Figure Effect of the skin/core thickness ratio g on the natural frequency versus axial force when the magnetic field strength B = 0.5 T ratio 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 corresponding equations of motion are derived based on Hamilton’s principle The variation of natural frequency 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 suppression 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 suppression capability is strong when the beam is under Advances in Mechanical Engineering Figure Effect of skin/core thickness ratio g and magnetic field strength B (T) on the capability of natural frequency suppression: (a) Case and Case and (b) Case and Case action of a tension and moves slowly; the maximum suppression ratio reaches 41.7% The suppression capability 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 suppression capability of the MRF is stronger when the axial force is tension than compression The suppression 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 support 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) References Allahverdizadeh A, Mahjoob MJ, Eshraghi I, et al Effects of electrorheological fluid core and functionally graded layers on the vibration behavior of a rotating composite beam Meccanica 2012; 47: 1945–1960 Wei K, Meng G, 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