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Low-frequency perturbations of rigid body motions of a viscoelastic inhomogeneous bar

Abstract

This paper deals with a low-frequency analysis of a viscoelastic inhomogeneous bar subject to end loads. The spatial variation of the problem parameters is taken into consideration. Explicit asymptotic corrections to the conventional equations of rigid body motion are derived in the form of integro-differential operators acting on longitudinal force or bending moment. The refined equations incorporate the effect of an internal viscoelastic microstructure on the overall dynamic response. Comparison with the exact time-harmonic solutions for extension and bending of a bar demonstrates the advantages of the developed approach. This research is inspired by modeling of railcar dynamics.

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Acknowledgements

J. Kaplunov and A. Shestakova gratefully acknowledge support from the industrial project with AMSTED Rail, USA. J. Kaplunov’s research in the area of mechanics of inhomogeneous solids was supported by National University of Science and Technology “MISiS”, Russia by grant K3-2014-052. The authors also grateful to Dr. D. Prikazchikov for a number of valuable comments.

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Correspondence to A. Shestakova.

Appendix

Appendix

Substitute the formulae (6.1) and (6.2) into the equations of motion (2.1) and (2.2), (2.3) specified for a time-harmonic motion of a homogeneous bar and introduce dimensionless variables. Then, these equations take the form

$$\begin{aligned} u_{\xi\xi}+q_{h}^{2}u=0 \end{aligned}$$
(A.1)

and

$$\begin{aligned} w_{\xi\xi\xi\xi}-q_{v}^{4}w=0, \end{aligned}$$
(A.2)

where \(q_{h}^{2}=\lambda_{h}^{2}(1+i\delta)\) and \(q_{v}^{4}=\lambda _{v}^{2}(1+i\delta)\). Subject them to the boundary conditions corresponding to the problems analyzed in the previous section, i.e.,

$$\begin{aligned} u_{\xi}|_{\xi=-1}=0, \quad\quad u_{\xi}|_{\xi=1}= \frac{F_{2}l(1+i\delta)}{EA} \end{aligned}$$
(A.3)

and

$$\begin{aligned} w_{\xi\xi\xi}|_{\xi=\pm1}=\mp\frac{N_{2}l^{3}(1+i\delta)}{EI},\quad\quad w_{\xi\xi}|_{\xi=\pm1}=0. \end{aligned}$$
(A.4)

The solution of the problem (A.1) and (A.3) is given by

$$\begin{aligned} u(\xi)=-\frac{F_{2}l(1+i\delta)\cosh(q_{h}(1+\xi))}{EAq_{h}\sinh 2q_{h}}. \end{aligned}$$
(A.5)

In this case the horizontal acceleration of the center  (ξ=0)  is given by

$$\begin{aligned} a_{h}=\frac{F_{2}q_{h}}{M\sinh q_{h}}. \end{aligned}$$
(A.6)

Over the low-frequency band λ h ≪1 we get q h ≪1 assuming that δ∼1 (γω) in (5.3). As a result, we arrive at the expansion

$$\begin{aligned} a_{h}=\frac{F_{2}}{M} \biggl(1+\frac{q_{h}^{2}}{6}+\cdots \biggr). \end{aligned}$$
(A.7)

The solution of the problem (A.2)–(A.4) can be written as

$$\begin{aligned} w(\xi)=\frac{N_{2}l^{3}(1+i\delta)}{EI}\frac{\cos q_{v} \cosh\xi q_{v}+\cosh q_{v}\cos\xi q_{v}}{q_{v}^{3}(\cos q_{v} \sinh q_{v}+ \sin q_{v} \cosh q_{v} )}. \end{aligned}$$
(A.8)

The associated acceleration of the center ξ=0, namely

$$\begin{aligned} a_{v}=-\frac{2N_{2}}{M}\frac{q_{v}(\cos q_{v}+\cosh q_{v})}{\cos q_{v}\sinh q_{v}+\sin q_{v}\cosh q_{v}}, \end{aligned}$$
(A.9)

has the following low-frequency expansion

$$\begin{aligned} a_{v}=-\frac{2N_{2}}{M} \biggl(1+\frac{3}{40}q_{v}^{4}+ \cdots \biggr). \end{aligned}$$
(A.10)

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Kaplunov, J., Shestakova, A., Aleynikov, I. et al. Low-frequency perturbations of rigid body motions of a viscoelastic inhomogeneous bar. Mech Time-Depend Mater 19, 135–151 (2015). https://doi.org/10.1007/s11043-015-9256-x

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Keywords

  • Viscoelastic
  • Microstructure
  • Perturbation
  • Rigid body
  • Low-frequency