Nonlinear dynamics of a viscoelastic beam traveling with pulsating speed, variable axial tension under two-frequency parametric excitations and internal resonance

Bamadev Sahoo

doi:10.1007/s11071-019-05264-3

Nonlinear Dynamics

pp 1–35 | Cite as

Nonlinear dynamics of a viscoelastic beam traveling with pulsating speed, variable axial tension under two-frequency parametric excitations and internal resonance

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Bamadev Sahoo

Original paper

First Online: 16 October 2019

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Abstract

This paper investigates nonlinear combined parametric transverse vibrations of a traveling viscoelastic beam. The combined parametric excitations originate from the time dependency of axial velocity as well as axial tension. Two parametric excitations are enforced into the system amid the internal resonance. Two-frequency parametric resonance is assumed to be comprised of combination parametric resonance of first two modes due to the time dependency of axial velocity, and the principal parametric resonance of first mode due to the variable tension in the axial direction in the presence of internal resonance for viscoelastic beam is considered for the first time. The higher-order integro-partial differential equation of motion is solved through direct method of multiple scales. Continuation algorithm is employed to explore the stability and various bifurcations of the nonlinear dynamic system. Focus has been made to study the effect of variations of fluctuating tension component, fluctuating velocity component independently and when combined, internal and parametric frequency detuning parameters and damping on the system response. Frequency response equilibrium curves are complex and unique in shapes which are embodied with various bifurcations. Such steady-state behavior is not seen in the existent literature. With variation in fluctuating velocity component, the number of steady-state nontrivial equilibrium curves increases to three and with variation in fluctuating axial tension, they become four. In this process, significant changes in stability, number and position of various bifurcations like supercritical and subcritical pitchfork, Hopf and saddle node are observed. Unlike the previous study, the shape, stability and bifurcations of equilibrium curves under the combined effect of axial velocity and tension closely match with the case of fluctuating axial tension component. The effect of variation in internal and parametric frequency detuning parameter is more realized for second mode compared to first mode. A comparison of the present work with a previous one where axial tension is variable reveals many qualitative and quantitative similarities and dissimilarities. But when compared with earlier work where axial velocity is constant, significant dissimilarities are surfaced. The system displays a wide ranging dynamic behavior including stable periodic, quasiperiodic and unstable chaotic behavior. The numerical computation depicts various nonlinear characteristics and oscillatory behaviors which are not found so far in the existent literature.

Keywords

Fluctuating axial tension component Stability Bifurcation Chaos Multi-frequency parametric excitations Internal resonance

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Notes

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

Compliance with ethical standards

Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this paper.

Appendix

$$\begin{aligned} \varGamma _1= & {} -2 i\omega _1 {A}'_1 \phi _1 -2 v_0 {A}'_1 {\phi }^{{\prime }}_1 -2i\mu \omega _1 A_1 \phi _1 \\&-2i\alpha \omega _1 A_1 {\phi }^{{\prime }{\prime }{\prime }{\prime }}_1 +\frac{1}{2}v_\mathrm{l} ^{2} \left\{ 2A_1^2 \bar{{A}}_1 {\phi }^{{\prime }{\prime }}_1 \int \limits _0^1 {\phi }^{{\prime }}_1 {\bar{{\phi }}}'_1 \hbox {d}x\right. \\&+A_1^2 \bar{{A}}_1 {\bar{{\phi }}}^{{\prime }{\prime }}_1 \int \limits _0^1 {{\phi }_1^{{\prime }2}\hbox {d}x} +2A_1 A_2 \bar{{A}}_2 {\bar{{\phi }}}^{{\prime }{\prime }}_2 \int \limits _0^1 {{\phi }^{{\prime }}_1 {\phi }^{{\prime }}_2 \hbox {d}x} \\&+\left. {2A_1 A_2 \bar{{A}}_2 {\phi }^{{\prime }{\prime }}_1 \int \limits _0^1 {{\phi }^{{\prime }}_2 {\bar{{\phi }}}'_2 \hbox {d}x} +2A_1 A_2 \bar{{A}}_2 {\phi }^{{\prime }{\prime }}_2 \int \limits _0^1 {{\phi }^{{\prime }}_1 {\bar{{\phi }}}'_2 \hbox {d}x} } \right\} \\ \varGamma _2= & {} \frac{1}{2}v_\mathrm{l} ^{2}\left\{ {2\bar{{A}}_1 ^{2}A_2 {\bar{{\phi }}}^{{\prime }{\prime }}_1 \int \limits _0^1 {{\phi }^{{\prime }}_2 {\bar{{\phi }}}'_1 \hbox {d}x} +\bar{{A}}_1 ^{2}A_2 {\phi }^{{\prime }{\prime }}_2 \int \limits _0^1 \bar{\phi }^{\prime {2}}_1 \hbox {d}x} \right\} \ \\ \varGamma _3= & {} \bar{{A}}_1 \left\{ { v_1 \omega _1 \bar{{\phi }}_1 ^{\prime }-\frac{ v_1 \varOmega }{2} {\bar{{\phi }}}'_1 +i v_0 v_1 {\bar{{\phi }}}^{{\prime }{\prime }}_1 } \right\} \\ \varGamma _4= & {} \bar{{A}}_1 \left\{ {-i\frac{p_1 }{2}\bar{{\phi }}_1 ^{\prime \prime }} \right\} \ \\ \end{aligned}$$

$$\begin{aligned} \varGamma _5= & {} -2 i\omega _2 {A}'_2 \phi _2 -2 v_0 {A}'_2 {\phi }^{{\prime }}_2 -2\mu i\omega _2 A_2 \phi _2 \\&\quad -2\alpha i\omega _2 A_2 {\phi }^{{\prime }{\prime }{\prime }{\prime }}_2 +\frac{1}{2}v_\mathrm{l}^{2}\left\{ A_2^2 \bar{{A}}_2 {{\bar{{\phi }}}^{{\prime }{\prime }}}_2 \int \limits _0^1 {\phi }^{{\prime {2}}}_2 \hbox {d}x\right. \\&\quad +2A_1 \bar{{A}}_1 A_2 {\phi }^{{\prime }{\prime }}_2 \int \limits _{0}^1 {{\phi }^{{\prime }}_1 }{\bar{{\phi }}}'_1 \hbox {d}x+2A_1 \bar{{A}}_1 A_2 {\bar{{\phi }}}^{{\prime }{\prime }}_1 \int \limits _0^1 {{\phi }^{{\prime }}_1 {\phi }^{{\prime }}_2 \hbox {d}x}\\&\quad \left. {+2A_2^2 \bar{{A}}_2 {\phi }^{{\prime }{\prime }}_2 \int \limits _0^1 {{\phi }^{{\prime }}_2 {\bar{{\phi }}}'_2 \hbox {d}x} +2A_1 \bar{{A}}_1 A_2 {\phi }^{{\prime }{\prime }}_1 \int \limits _0^1 {{\phi }^{{\prime }}_2 {\bar{{\phi }}}'_1 \hbox {d}x} } \right\} \\ \varGamma _6= & {} \frac{1}{2}v_\mathrm{l} ^{2}\left\{ {A_1^3 {\phi }^{{\prime }{\prime }}_1 \int \limits _0^1 {{\phi }^{{\prime {2}}}_1\hbox {d}x} } \right\} \\ \varGamma _7= & {} A_2 \left\{ {-i\frac{p_1 }{2}\phi _2^{\prime \prime }} \right\} \\ \varGamma _8= & {} \bar{{A}}_2 \left\{ { v_1 \omega _2 {\bar{{\phi }}}'_2 -\frac{ v_1 \varOmega }{2}{\bar{{\phi }}}'_2 +i v_0 v_1 {\bar{{\phi }}}^{{\prime }{\prime }}_2 } \right\} \\ \varGamma _9= & {} A_1 \left\{ {-i\frac{p_1 }{2}\phi _1 ^{\prime \prime }} \right\} \\ S_1= & {} \frac{\frac{1}{16}v_\mathrm{l} ^{2}\left\{ {2\int \limits _0^1 {{\phi }^{{\prime }{\prime }}_1 \bar{{\phi }}_1 \hbox {d}x} \int \limits _0^1 {{\phi }^{{\prime }}_1 {\bar{{\phi }}}'_1 \hbox {d}x} +\int \limits _0^1 {{\bar{{\phi }}}^{{\prime }{\prime }}_1 \bar{{\phi }}_1 \hbox {d}x} \int \limits _0^1 {{\phi }^{{\prime {2}}}_1 \hbox {d}x} } \right\} }{-\left\{ {i\omega _1 \int \limits _0^1 {\phi _1 \bar{{\phi }}_1 \hbox {d}x} +v_0 \int \limits _0^1{{\phi }^{{\prime }}_1 \bar{{\phi }}_1 \hbox {d}x} } \right\} }\\ \end{aligned}$$

$$\begin{aligned} S_2= & {} \frac{\frac{1}{8}v_\mathrm{l}^{2}\left\{ {\int \limits _0^1 {{\bar{{\phi }}}^{{\prime }{\prime }}_2 \bar{{\phi }}_1 \hbox {d}x} \int \limits _0^1 {{\phi }^{{\prime }}_1 {\phi }^{{\prime }}_2 \hbox {d}x} +\int \limits _0^1 {{\phi }^{{\prime }{\prime }}_1 \bar{{\phi }}_1 \hbox {d}x} \int \limits _0^1 {{\phi }^{{\prime }}_2 {\bar{{\phi }}}'_2 \hbox {d}x} +\int \limits _0^1 {{\phi }^{{\prime }{\prime }}_2 \bar{{\phi }}_1 \hbox {d}x} \int \limits _0^1 {{\phi }^{{\prime }}_1 {\bar{{\phi }}}'_2 \hbox {d}x} } \right\} }{-\left\{ {i\omega _1 \int \limits _0^1 {\phi _1 \bar{{\phi }}_1 \hbox {d}x} + v_0 \int \limits _0^1 {{\phi }^{{\prime }}_1 \bar{{\phi }}_1 \hbox {d}x} } \right\} } \\ S_3= & {} \frac{\frac{1}{8}v_\mathrm{l} ^{2}\left\{ {\int \limits _0^1 {{\phi }^{{\prime }{\prime }}_2 \bar{{\phi }}_2 \hbox {d}x} \int \limits _0^1 {{\phi }^{{\prime }}_1 {\bar{{\phi }}}'_1 \hbox {d}x} +\int \limits _0^1 {{\bar{{\phi }}}^{{\prime }{\prime }}_1 \bar{{\phi }}_2 \hbox {d}x} \int \limits _0^1 {{\phi }^{{\prime }}_1 {\phi }^{{\prime }}_2 \hbox {d}x} +\int \limits _0^1 {{\phi }^{{\prime }{\prime }}_1 \bar{{\phi }}_2 \hbox {d}x} \int \limits _0^1 {{\phi }^{{\prime }}_2 {\bar{{\phi }}}'_2 \hbox {d}x} } \right\} }{-\left\{ {i\omega _2 \int \limits _0^1 {\phi _2 \bar{{\phi }}_2 \hbox {d}x} + v_0 \int \limits _0^1 {{\phi }^{{\prime }}_2 \bar{{\phi }}_2 \hbox {d}x} } \right\} } \\ S_4= & {} \frac{\frac{1}{16}v_\mathrm{l} ^{2}\left\{ {2\int \limits _0^1 {{\phi }^{{\prime }{\prime }}_2 \bar{{\phi }}_2 \hbox {d}x} \int \limits _0^1 {{\phi }^{{\prime }}_2 {\bar{{\phi }}}'_2 \hbox {d}x} +\int \limits _0^1 {{\bar{{\phi }}}^{{\prime }{\prime }}_2 \bar{{\phi }}_2 \hbox {d}x} \int \limits _0^1 {{\phi }^{{\prime {2}}}_2 \hbox {d}x} } \right\} }{-\left\{ {i\omega _2 \int \limits _0^1 {\phi _2 \bar{{\phi }}_2 \hbox {d}x} + v_0 \int \limits _0^1 {{\phi }^{{\prime }}_2 \bar{{\phi }}_2 \hbox {d}x} } \right\} } \ \\ C_1= & {} \frac{-i\omega _1 \int \limits _0^1 {\phi _1 \bar{{\phi }}_1 \hbox {d}x} }{-\left\{ {i\omega _1 \int \limits _0^1 {\phi _1 \bar{{\phi }}_1 \hbox {d}x} + v_0 \int \limits _0^1 {{\phi }^{{\prime }}_1 \bar{{\phi }}_1 \hbox {d}x} } \right\} } \\ C_2= & {} \frac{-i\omega _2 \int \limits _0^1 {\phi _2 \bar{{\phi }}_2 \hbox {d}x} }{-\left\{ {i\omega _2 \int \limits _0^1 {\phi _2 \bar{{\phi }}_2 \hbox {d}x} +v_0 \int \limits _0^1 {{\phi }^{{\prime }}_2 \bar{{\phi }}_2 \hbox {d}x} } \right\} } \\ e_1= & {} \frac{-i\omega _1 \int \limits _0^1 {\phi }^{{\prime }{\prime }{\prime }{\prime }}_1 \bar{{\phi }}_1 \hbox {d}x}{-\left\{ {i\omega _1 \int \limits _0^1 {\phi _1 \bar{{\phi }}_1 \hbox {d}x} +v_0 \int \limits _0^1 {{\phi }^{{\prime }}_1 \bar{{\phi }}_1 \hbox {d}x} } \right\} } \ \\ e_2= & {} \frac{-i\omega _2 \int \limits _0^1 {\phi }^{{\prime }{\prime }{\prime }{\prime }}_2 \bar{{\phi }}_2 \hbox {d}x}{-\left\{ {i\omega _2 \int \limits _0^1 {\phi _2 \bar{{\phi }}_2 \hbox {d}x} +v_0 \int \limits _0^1 {{\phi }^{{\prime }}_2 \bar{{\phi }}_2 \hbox {d}x} } \right\} } \\ \end{aligned}$$

$$\begin{aligned} g_1= & {} \frac{\frac{1}{16}v_\mathrm{l} ^{2}\left\{ {2\int \limits _0^1 {{\bar{{\phi }}}^{{\prime }{\prime }}_1 \bar{{\phi }}_1 \hbox {d}x} \int \limits _0^1 {{\phi }^{{\prime }}_2 {\bar{{\phi }}}'_1 \hbox {d}x} +\int \limits _0^1 {{\phi }^{{\prime }{\prime }}_2 \bar{{\phi }}_1 \hbox {d}x} \int \limits _0^1 {{\bar{{\phi }}}^{{\prime }2}_1 \hbox {d}x} } \right\} }{-\left\{ {i\omega _1 \int \limits _0^1 {\phi _1 \bar{{\phi }}_1 \hbox {d}x} +v_0 \int \limits _0^1 {{\phi }^{{\prime }}_1 \bar{{\phi }}_1 \hbox {d}x} }\right\} } \\ g_2= & {} \frac{\frac{1}{16}v_\mathrm{l} ^{2}\left\{ {\int \limits _0^1 {{\phi }^{{\prime }{\prime }}_1 \bar{{\phi }}_2 \hbox {d}x} \int \limits _0^1 {{\phi }^{{\prime {2}}}_1 \hbox {d}x} } \right\} }{-\left\{ {i\omega _2 \int \limits _0^1 {\phi _2 \bar{{\phi }}_2 \hbox {d}x} +v_0 \int \limits _0^1 {{\phi }^{{\prime }}_2 \bar{{\phi }}_2 \hbox {d}x} } \right\} } \ \\ K_4= & {} \frac{\frac{1}{2}\left\{ {v_1 \omega _2 \int \limits _0^1 {{\bar{{\phi }}}'_2 \bar{{\phi }}_1 \hbox {d}x} -\frac{v_1 \varOmega }{2}\int \limits _0^1 {\bar{{\phi }}_2 ^{\prime }} \bar{{\phi }}_1 \hbox {d}x+i v_0 v_1 \int \limits _0^1 {{\bar{{\phi }}}^{{\prime }{\prime }}_2 \bar{{\phi }}_1 \hbox {d}x} } \right\} }{-\left\{ {i\omega _1 \int \limits _0^1 {\phi _1 \bar{{\phi }}_1 \hbox {d}x} +v_0 \int \limits _0^1 {{\phi }^{{\prime }}_1 \bar{{\phi }}_1 \hbox {d}x} } \right\} } \\ K_5= & {} \frac{\frac{1}{2}\left\{ {v_1 \omega _1 \int \limits _0^1 {{\bar{{\phi }}}'_1 \bar{{\phi }}_2 \hbox {d}x} -\frac{v_1 \varOmega }{2}\int \limits _0^1 {{\bar{{\phi }}}'_1 \ } \bar{{\phi }}_2 \hbox {d}x+i v_0 v_1 \int \limits _0^1 {\bar{{\phi }}_1 ^{\prime \prime }\bar{{\phi }}_2 \hbox {d}x} } \right\} }{-\left\{ {i\omega _2 \int \limits _0^1 {\phi _2 \bar{{\phi }}_2 \hbox {d}x} +v_0 \int \limits _0^1 {{\phi }^{{\prime }}_2 \bar{{\phi }}_2 \hbox {d}x} } \right\} } \ \\ K_6= & {} \frac{ \frac{1}{2}\left\{ {-i\frac{p_1 }{2}\int \limits _0^1 {\bar{{\phi }}_1 ^{\prime \prime }\bar{{\phi }}_1 \hbox {d}x} } \right\} }{-\left\{ {i\omega _1 \int \limits _0^1 {\phi _1 \bar{{\phi }}_1 \hbox {d}x} +v_0 \int \limits _0^1 {{\phi }^{{\prime }}_1 \bar{{\phi }}_1 \hbox {d}x} } \right\} }\\ K_7= & {} \frac{ \frac{1}{2}\left\{ {-i\frac{p_1 }{2}\int \limits _0^1 {\phi _2 ^{\prime \prime }\bar{{\phi }}_1 \hbox {d}x} } \right\} }{-\left\{ {i\omega _1 \int \limits _0^1 {\phi _1 \bar{{\phi }}_1 \hbox {d}x} +v_0 \int \limits _0^1 {{\phi }^{{\prime }}_1 \bar{{\phi }}_1 \hbox {d}x} } \right\} }\\ K_8= & {} \frac{\frac{1}{2} \left\{ {-i\frac{p_1 }{2}\int \limits _0^1 {\phi _1 ^{\prime \prime }\bar{{\phi }}_2 \hbox {d}x} } \right\} }{-\left\{ {i\omega _2 \int \limits _0^1 {\phi _2 \bar{{\phi }}_2 \hbox {d}x} +v_0 \int \limits _0^1 {{\phi }^{{\prime }}_2 \bar{{\phi }}_2 \hbox {d}x} } \right\} } \\ \end{aligned}$$

References

1.
Oz, H.R., Pakdemirli, M.: Vibrations of an axially moving beam with time-dependent velocity. J. Sound Vib. 227(2), 239–257 (1999)CrossRefGoogle Scholar
2.
Parker, R.G., Lin, Y.: Parametric instability of axially moving media subjected to multi-frequency tension and speed fluctuations. ASME J. Appl. Mech. 68(1), 49–57 (2001)zbMATHCrossRefGoogle Scholar
3.
Yang, X.D., Tang, Y.Q., Chen, L.Q., Lim, C.W.: Dynamic stability of axially accelerating Timoshenko beams: averaging method. Eur. J. Mech. A/Solids 29, 81–90 (2010)MathSciNetCrossRefGoogle Scholar
4.
Ba ğdatli, S.M., Uslu, B.: Free vibration analysis of axially moving beam under non-ideal conditions. Struct. Eng. Mech. 54(3), 597 (2015)CrossRefGoogle Scholar
5.
Ding, H., Chen, L,Q.: Stability of axially accelerating viscoelastic beams: multi-scale analysis with numerical confirmations. Eur. J. Mech. A/Solids 27, 1108–1120 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
6.
Saksa, T., Jeronen, J.: Dynamic analysis for axially moving viscoelastic Poynting–Thomson beams. Math. Model Optim. Complex Struct. 40, 131–151 (2016)MathSciNetGoogle Scholar
7.
Oz, H.R., Pakdemirli, M., Boyaci, H.: Non-linear vibrations and stability of anaxially moving beam with time-dependent velocity. J. Non-Linear Mech. 36, 107–115 (2001)zbMATHCrossRefGoogle Scholar
8.
Ding, H., Chen, L.Q.: Nonlinear dynamics of axially accelerating viscoelastic beams based on differential quadrature. Acta Mech. Solida Sin. 22, 267–275 (2009)CrossRefGoogle Scholar
9.
Chen, L.H., Zhang, W., Yang, F.H.: Nonlinear dynamics of higher dimensional system for an axially accelerating viscoelastic beam. J. Sound Vib. 329, 5321–5345 (2010)CrossRefGoogle Scholar
10.
Ghayesh, M.H., Balar, S.: Nonlinear parametric vibration and stability analysis for two dynamic modes of axially moving Timoshenko beams. Appl. Math. Model. 34, 2850–2859 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
11.
Chen, L.Q., Ding, H., Lim, C.W.: Principal parametric resonance of axially accelerating viscoelastic beams: multi-scale analysis and differential quadrature verification. Shock Vib. 19(4), 527–543 (2012)CrossRefGoogle Scholar
12.
Wang, Y.B., Ding, H., Chen, L.Q.: Nonlinear vibration axially accelerating hyperelastic beams. Int. J. Non-Linear Mech. 99, 302–310 (2018)CrossRefGoogle Scholar
13.
Tang, Y.Q., Luo, E.B., Yang, X.D.: Complex modes and traveling waves in axially moving Timoshenko beams. Appl. Math. Mech. Engl. Ed. 39(4), 597–608 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
14.
Wang, B.: Effect of rotary inertia on stability of axially accelerating viscoelastic Rayleigh beams. Appl. Math. Mech. Engl. Ed. 39(5), 717–732 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
15.
Mao, X.Y., Ding, H., Chen, L.Q.: Parametric resonance of a translating beam with pulsating axial speed in the supercritical regime. Mech. Res. Commun. 76, 72–77 (2016)CrossRefGoogle Scholar
16.
Mao, X.Y., Ding, H., Chen, L.Q.: Dynamics of a super critically axially moving beam with parametric and forced resonance. Nonlinear Dyn. 89(2), 1475–1487 (2017)CrossRefGoogle Scholar
17.
Zhu, W.D., Chen, Y.: Theoretical and experimental investigation of elevator cable dynamics and control. ASME J. Vib. Acoust. 128(1), 66–78 (2005)CrossRefGoogle Scholar
18.
Zhu, H., Hu, Y.M., Zhu, W.D.: Dynamic response of a front end accessory drive system and parameter optimization for vibration reduction via a genetic algorithm. J. Vib. Control 24(11), 2201–2220 (2018)MathSciNetCrossRefGoogle Scholar
19.
Mokhtari, A., Mirdamadi, H.R.: Study on vibration and stability of an axially translating viscoelastic Timoshenko beam: non-transforming spectral element analysis. Appl. Math. Model. 56, 342–358 (2018)MathSciNetCrossRefGoogle Scholar
20.
Ding, H., Wang, S., Zhang, Y.W.: Free and forced nonlinear vibration of a transporting belt with pulley support ends. Nonlinear Dyn. 92, 2037–2048 (2018)CrossRefGoogle Scholar
21.
Ding, H., Lim, C.W., Chen, L.Q.: Nonlinear vibration of a traveling belt with non-homogeneous boundaries. J. Sound Vib. 424, 78–93 (2018)CrossRefGoogle Scholar
22.
Hu, Y., Rong, Y.: Primary parametric resonance of an axially accelerating beam subjected to static loads. J. Theor. Appl. Mech. 56(3), 815–828 (2018)CrossRefGoogle Scholar
23.
Dehadrai, A.R., Sharma, I., Gupta, S.S.: Stability of vertically traveling, pre-tensioned, heavy cables. J. Comput. Nonlinear Dyn. 13(081003), 1–9 (2018)Google Scholar
24.
Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, New York (1979)zbMATHGoogle Scholar
25.
Riedel, C.H., Tan, C.A.: Coupled, forced response of an axially moving strip with internal resonance. Int. J. Nonlinear Mech. 37, 101–116 (2002)zbMATHCrossRefGoogle Scholar
26.
Chen, S.H., Huang, J.L., Sze, K.Y.: Multidimensional Lindstedt–Poincare method for nonlinear vibration of axially moving beams. J. Sound Vib. 306, 1–11 (2007)CrossRefGoogle Scholar
27.
Ozkaya, E., Bagdatli, S.M., Oz, H.R.: Nonlinear transverse vibrations and 3:1 internal resonances of a beam with multiple supports. J. Vib. Acoust. 130(2), 1–11 (2008)CrossRefGoogle Scholar
28.
Bagdatli, S.M., Oz, H.R., Ozkaya, E.: Nonlinear transverse vibrations and 3:1 internal resonances of a tensioned beam on multiple supports. Math. Comput. Appl. 16(1), 203–215 (2011)MathSciNetGoogle Scholar
29.
Ozhan, B.B., Pakdemirli, M.: A general solution procedure for the forced vibrations of a continuous system with cubic nonlinearities: primary resonance case. J. Sound Vib. 325, 894–906 (2009)CrossRefGoogle Scholar
30.
Ozhan, B.B., Pakdemirli, M.: A general solution procedure for the forced vibrations of a system with cubic nonlinearities: three-to-one internal resonances with external excitation. J. Sound Vib. 329, 2603–2615 (2010)CrossRefGoogle Scholar
31.
Huang, J.L., Su, R.K.L., Li, W.H., Chen, S.H.: Stability and bifurcation of an axially moving beam tuned to three-to-one internal resonances. J. Sound Vib. 330, 471–485 (2011)CrossRefGoogle Scholar
32.
Ghayesh, M.H., Kafiabad, H.A., Reid, T.: Sub- and super-critical nonlinear dynamics of a harmonically excited axially moving beam. Int. J. Solids Struct. 49, 227–243 (2012)CrossRefGoogle Scholar
33.
Zhang, D.B., Tang, Y.Q., Ding, H., Chen, L.Q.: Parametric and internal resonance of a transporting plate with a varying tension. Nonlinear Dyn. (2019). https://doi.org/10.1007/s11071-019-04981-z CrossRefGoogle Scholar
34.
Zhang, D.B., Tang, Y.Q., Chen, L.Q.: Irregular instability boundaries of axially accelerating viscoelastic beams with 1:3 internal resonance. Int. J. Mech. Sci. 133, 535–543 (2017)CrossRefGoogle Scholar
35.
Zhang, D.B., Tang, Y.Q., Chen, L.Q.: Internal resonance in parametric vibrations of axially accelerating viscoelastic plates. Eur. J. Mech. A. Solids 75, 142–155 (2019)MathSciNetzbMATHCrossRefGoogle Scholar
36.
Mao, X.Y., Ding, H., Chen, L.Q.: Forced vibration of axially moving beam with internal resonance in the supercritical regime. Int. J. Mech. Sci. 131–132, 81–94 (2017)CrossRefGoogle Scholar
37.
Mao, X.Y., Ding, H., Chen, L.Q.: Internal resonance of a supercritically axially moving beam subjected to the pulsating speed. Nonlinear Dyn. 95(1), 631–651 (2019)CrossRefGoogle Scholar
38.
Mao, X.Y., Ding, H., Chen, L.Q.: Steady-state response of a fluid-conveying pipe with 3:1 internal resonance in supercritical regime. Nonlinear Dyn. 86, 795–809 (2016)CrossRefGoogle Scholar
39.
Huang, J.L., Zhu, W.D.: A new incremental harmonic balance method with two time scales for quasi-periodic motions of an axially moving beam with internal resonance under single-tone external excitation. ASME J. Vib. Acoust. 139, 021010–1 (2017)CrossRefGoogle Scholar
40.
Ding, H., Huang, L., Mao, X., Chen, L.Q.: Primary resonance of traveling viscoelastic beam under internal resonance. Appl. Math. Mech. 38(1), 1–14 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
41.
Zhu, B., Dong, Y., Li, Y.: Nonlinear dynamics of a viscoelastic sandwich beam with parametric excitations and internal resonance. Nonlinear Dyn. 94, 2575–2612 (2018)CrossRefGoogle Scholar
42.
Tang, Y.Q., Zhang, Y.X., Yang, X.D.: On parametric instability boundaries of axially moving beams with internal resonance. Acta Mech. Solida Sin. 31, 470 (2018)CrossRefGoogle Scholar
43.
Sahoo, B., Panda, L.N., Pohit, G.: Stability, bifurcation and chaos of a traveling viscoelastic beam tuned to 3:1 internal resonance and subjected to parametric excitation. Int. J. Bifurc. Chaos. 27(2), 1750017 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
44.
Sahoo, B., Panda, L.N., Pohit, G.: Combination, principle parametric and internal resonances of an accelerating beam under two frequency parametric excitation. Int. J. Non-Linear Mech. 78, 35–44 (2016)CrossRefGoogle Scholar
45.
Sahoo, B., Panda, L.N., Pohit, G.: Two-frequency parametric excitation and internal resonance of a moving viscoelastic beam. Nonlinear Dyn. 82, 1721–1742 (2015)MathSciNetCrossRefGoogle Scholar
46.
Chen, L.Q., Tang, Y.Q.: Combination and principal parametric resonances of axially accelerating viscoelastic beams: recognition of longitudinally varying tensions. J Sound Vib. 330, 5598–614 (2011)CrossRefGoogle Scholar
47.
Chen, L.Q., Tang, Y.Q.: Parametric stability of axially accelerating viscoelastic beams with the recognition of longitudinally varying tensions. ASME J. Vib. Acoust. 134(1), 245–6 (2012)CrossRefGoogle Scholar
48.
Tang, Y.Q., Chen, L.Q., Zhang, H.J., Yang, S.P.: Stability of axially accelerating viscoelastic Timoshenko beams: recognition of longitudinally varying tensions. Mech. Mach. Theory 62, 31–50 (2013)CrossRefGoogle Scholar
49.
Tang, Y.Q., Zhang, D.B., Gao, J.M.: Parametric and internal resonance of axially accelerating viscoelastic beams with the recognition of longitudinally varying tensions. Nonlinear Dyn. 83(1–2), 401–18 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
50.
Abdelhafez, H.M.: Resonance of a nonlinear forced system with two-frequency parametric and self-excitations. Math. Comput. Simul. 66, 69–83 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
51.
Ozhan, B.B.: Vibration and stability analysis of axially moving beams with variable speed and axial force. Int. Struct. Stab. Dyn. 14(6), 1450015 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
52.
Sahoo, B.: Nonlinear dynamics of a viscoelastic traveling beam with time dependent axial velocity and variable axial tension. Nonlinear Dyn. 97(1), 269–296 (2019)CrossRefGoogle Scholar
53.
Wickert, J.A.: Nonlinear vibration of a traveling tensioned beam. Int. J. Non-Linear Mech. 27, 503–517 (1992)zbMATHCrossRefGoogle Scholar
54.
Chakraborty, G., Mallick, A.K., Hatwal, H.: Non-linear vibration of traveling beam. Int. J. Non-Linear Mech. 34, 655–670 (1999)zbMATHCrossRefGoogle Scholar
55.
Chen, L.Q., Yang, X.D.: Steady-state response of axially moving viscoelastic beams with pulsating speed: comparison of two nonlinear models. Int. J. Solids Struct. 42, 37–50 (2005)zbMATHCrossRefGoogle Scholar
56.
Chakraborty, G., Mallick, A.K.: Non-linear vibration of a traveling beam having an intermediate guide. Nonlinear Dyn. 20, 247–265 (1999)CrossRefGoogle Scholar
57.
Ghayesh, M.H., Kafiabad, H.A., Amabili, M.: Coupled longitudinal-transverse dynamics of an axially moving beam with an internal resonance. Mech. Mach. Theory 52, 18–34 (2012)CrossRefGoogle Scholar
58.
Yang, T., Fang, B., Chen, Y., Zhen, Y.: Approximate solutions of axially moving viscoelastic beams subject to multi-frequency excitations. Int. J. Non-Linear Mech. 44, 230–238 (2009)CrossRefGoogle Scholar
59.
Ghayesh, M.H., Amabili, M., Farokhi, H.: Coupled global dynamics of an axially moving viscoelastic beam. Int. J. Non-Linear Mech. 51, 54–74 (2013)zbMATHCrossRefGoogle Scholar
60.
Chin, C.M., Nayfeh, A.H.: Three-to-one internal resonance in parametrically excited hinged-clamped beams. Nonlinear Dyn. 20, 131–158 (1999)MathSciNetzbMATHCrossRefGoogle Scholar

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Bamadev Sahoo
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1.Department Mechanical EngineeringIIITBhubaneswarIndia

Nonlinear dynamics of a viscoelastic beam traveling with pulsating speed, variable axial tension under two-frequency parametric excitations and internal resonance

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