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Nonlinear vibration of axially moving beams with internal resonance, speed-dependent tension, and tension-dependent speed

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Abstract

In this paper, a new nonlinear model of an axially accelerating viscoelastic beam is established. The effects of a speed-dependent tension (a time-dependent tension due to perturbation of the speed) and a tension-dependent speed (a time-dependent speed due to perturbation of the tension) are highlighted. However, there was one common characteristic in previous studies about parametric vibration of axially moving beams with time-dependent tensions and time-dependent speeds: Axial tension and axial speed are independent of each other. Another highlight is that the inhomogeneous boundary conditions arising from Kelvin viscoelastic constitutive relation are taken into account. The technique of the modified solvability conditions is employed to resolve the existing solvability conditions failure because of the inhomogeneous boundary conditions. The influences of material’s viscoelastic coefficients, axial speed fluctuation amplitude, axial tension fluctuation amplitude, and dependent and independent models on the steady-state vibration responses are demonstrated by some numerical examples. Furthermore, the approximate analytical results are verified by using the differential quadrature method.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Project No. 11672186) and the Chen Guang Project supported by Shanghai Municipal Education Commission and Shanghai Education Development Foundation (No. 14CG57).

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  1. School of Mechanical Engineering, Shanghai Institute of Technology, Shanghai, 201418, China

    You-Qi Tang & Zhao-Guang Ma

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Appendices

Appendix A1

$$\begin{aligned} \begin{array}{l} f_1 ={\left( {1-\gamma _0^2 } \right) }/{k_{\mathrm{f}}^2 },\;f_2 ={2\mathrm{i}\gamma _0 \lambda _n }/{k_{\mathrm{f}}^2 },\;\\ f_3 ={\lambda _n^2 }/{k_{\mathrm{f}}^2 },\;g_1 =f_1^2 +12f_3 ,\; \\ g_2 =2f_1^3 +27f_2^2 -72f_1 f_3 ,\;\;g_4 =-2f_1 -4g_3^2 , \\ g_3 =\frac{1}{2}\sqrt{-\,\frac{2f_1 }{3}+\frac{\root 3 \of {2}g_1 }{3\root 3 \of {g_2 +\sqrt{g_2^2 -4g_1^3 }}}+\frac{\root 3 \of {g_2 +\sqrt{g_2^2 -4g_1^3 }}}{3\root 3 \of {2}}},\;\; \\ g_5 ={f_2 }/{g_3 },\;\;\;\;\lambda _n =\delta _n +\mathrm{i}\omega _n ,\;\;\;\\ \beta _{1n} =g_1 -0.5\sqrt{g_2 -g_3 },\;\; \\ \beta _{2n} =g_1 +0.5\sqrt{g_2 -g_3 },\;\;\beta _{3n} =-g_1 -\,0.5\sqrt{g_2 +g_3 },\;\;\\ \beta _{4n} =-\,g_1 +0.5\sqrt{g_2 +g_3 }. \\ \end{array}\nonumber \\ \end{aligned}$$
(A1)

Appendix A2

$$\begin{aligned} \begin{array}{l} \zeta _0 =-\,\mathrm{i}\omega _1 \varphi _1 -\gamma _0 \varphi _1 ^{\prime } ,\;\;\zeta _1 =-\mathrm{i}\gamma _0 \varphi _2 \hbox {,}_{xx} \\ \qquad +\,\omega _2 {\varphi }'_2 +0.5x\varOmega _1 {\varphi }''_2 , \\ \zeta _2 =\omega _1 \bar{{\varphi }}_1 ,_x +\mathrm{i}\gamma _0 {\bar{{\varphi }}}''_1 +0.5x\varOmega _1 {\bar{{\varphi }}}''_1 ,\;\;\\ \zeta _3 =-\mathrm{i}\omega _1 \varphi _1 ''''-\gamma _0 \varphi _1 ^{\prime \prime \prime \prime \prime } , \\ \zeta _4 =2{\varphi }''_1 \int _0^1 {{\varphi }'_1 {\bar{{\varphi }}}'_1 } {\mathrm{d}}x+{\bar{{\varphi }}}''_1 \int _0^1 {{\varphi }_1 ^{\prime 2}} {\mathrm{d}}x,\;\;\\ \zeta _5 ={\varphi }''_2 \int _0^1 {{\bar{{\varphi }}}_1 ^{\prime 2}} {\mathrm{d}}x+2{\bar{{\varphi }}}''_1 \int _0^1 {{\varphi }'_2 {\bar{{\varphi }}}'_1 } {\mathrm{d}}x, \\ \zeta _6 ={\varphi }''_1 \int _0^1 {{\varphi }'_2 {\bar{{\varphi }}}'_2 } {\mathrm{d}}x +{\varphi }''_2 \int _0^1 {{\varphi }'_1 {\bar{{\varphi }}}'_2 } {\mathrm{d}}x\\ \qquad +\,{\bar{{\varphi }}}''_2 \int _0^1 {{\varphi }'_1 {\varphi }'_2 } {\mathrm{d}}x, \\ \xi _0 =-\mathrm{i}\omega _2 \varphi _2 -\gamma _0 {\varphi }'_2 ,\;\;\\ \xi _1 =\hbox {i}\gamma _0 {\varphi }''_1 -\omega _1 {\varphi }'_1 +0.5x\varOmega _1 {\varphi }''_1 , \\ \xi _2 ={\left( {\omega _2 {\bar{{\varphi }}}'_2 +\hbox {i}\gamma _0 {\bar{{\varphi }}}''_2 } \right) }/{\varOmega _2 }+0.5x{\bar{{\varphi }}}''_2 ,\;\;\\ \xi _3 =-\mathrm{i}\omega _2 \varphi _2 ''''-\gamma _0 \varphi _2^{ \prime \prime \prime \prime \prime } , \\ \xi _4 =2{\varphi }''_2 \int _0^1 {{\varphi }'_2 {\bar{{\varphi }}}'_2 } {\mathrm{d}}x+{\bar{{\varphi }}}''_2 \int _0^1 {{\varphi }_2 ^{\prime 2}} {\mathrm{d}}x,\;\;\\ \xi _5 ={\varphi }''_1 \int _0^1 {{\varphi }_1 ^{\prime 2}} {\mathrm{d}}x, \\ \xi _6 ={\varphi }''_1 \int _0^1 {{\varphi }'_2 {\bar{{\varphi }}}'_1 } {\mathrm{d}}x+{\varphi }''_2 \int _0^1 {{\varphi }'_1 {\bar{{\varphi }}}'_1 } {\mathrm{d}}x+{\bar{{\varphi }}}''_1 \int _0^1 {{\varphi }'_1 {\varphi }'_2 } {\mathrm{d}}x. \\ \end{array} \end{aligned}$$
(A2)

Appendix A3

$$\begin{aligned} \begin{array}{l} \displaystyle \zeta _1 \leftrightarrow \frac{\int _0^1 {\zeta _1 \bar{{\varphi }}_1 {\mathrm{d}}x} }{\int _0^1 {2\zeta _0 \bar{{\varphi }}_1 {\mathrm{d}}x} },\;\;\zeta _2 \leftrightarrow \frac{\int _0^1 {\zeta _2 \bar{{\varphi }}_1 {\mathrm{d}}x} }{\int _0^1 {2\zeta _0 \bar{{\varphi }}_1 {\mathrm{d}}x} },\;\;\;\\ \zeta _3 \leftrightarrow \frac{\int _0^1 {\zeta _3 \bar{{\varphi }}_1 {\mathrm{d}}x} -\alpha \gamma _0 {\varphi }'''_1 \;{\bar{{\varphi }}}'_1 |_0^1 }{\int _0^1 {2\zeta _0 \bar{{\varphi }}_1 {\mathrm{d}}x} }, \\ \displaystyle \zeta _4 \leftrightarrow \frac{\mathop \int \nolimits _0^1 \zeta _4 \bar{{\varphi }}_1 {\mathrm{d}}x}{\mathop \int \nolimits _0^1 4\zeta _0 \bar{{\varphi }}_1 {\mathrm{d}}x},\;\;\zeta _5 \leftrightarrow \frac{\mathop \int \nolimits _0^1 \zeta _5 \bar{{\varphi }}_1 {\mathrm{d}}x}{\mathop \int \nolimits _0^1 4\zeta _0 \bar{{\varphi }}_1 {\mathrm{d}}x},\;\;\\ \zeta _6 \leftrightarrow \frac{\mathop \int \nolimits _0^1 \zeta _6 \bar{{\varphi }}_1 {\mathrm{d}}x}{\mathop \int \nolimits _0^1 2\zeta _0 \bar{{\varphi }}_1 {\mathrm{d}}x}; \\ \displaystyle \xi _1 \leftrightarrow \frac{\int _0^1 {\xi _1 \bar{{\varphi }}_2 {\mathrm{d}}x} }{\int _0^1 {2\xi _0 \bar{{\varphi }}_2 {\mathrm{d}}x} },\xi _2 \leftrightarrow \frac{\int _0^1 {\xi _2 \bar{{\varphi }}_2 {\mathrm{d}}x} }{\int _0^1 {2\xi _0 \bar{{\varphi }}_2 {\mathrm{d}}x} },\\ \xi _3 \leftrightarrow \frac{\int _0^1 {\xi _3 \bar{{\varphi }}_2 {\mathrm{d}}x} -\alpha \gamma _0 \varphi _2 \hbox {,}_{xxx} \bar{{\varphi }}_2 ,_x |_0^1 }{\int _0^1 {2\xi _0 \bar{{\varphi }}_2 {\mathrm{d}}x} }, \\ \displaystyle \xi _4 \leftrightarrow \frac{\mathop \int \nolimits _0^1 \xi _4 \bar{{\varphi }}_2 {\mathrm{d}}x}{\mathop \int \nolimits _0^1 4\xi _0 \bar{{\varphi }}_2 {\mathrm{d}}x},\xi _5 \leftrightarrow \frac{\mathop \int \nolimits _0^1 \xi _5 \bar{{\varphi }}_2 {\mathrm{d}}x}{\mathop \int \nolimits _0^1 4\xi _0 \bar{{\varphi }}_2 {\mathrm{d}}x},\\ \xi _6 \leftrightarrow \frac{\mathop \int \nolimits _0^1 \xi _6 \bar{{\varphi }}_2 {\mathrm{d}}x}{\mathop \int \nolimits _0^1 2\xi _0 \bar{{\varphi }}_2 {\mathrm{d}}x}. \end{array} \end{aligned}$$
(A3)

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Tang, YQ., Ma, ZG. Nonlinear vibration of axially moving beams with internal resonance, speed-dependent tension, and tension-dependent speed. Nonlinear Dyn 98, 2475–2490 (2019). https://doi.org/10.1007/s11071-019-05105-3

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