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Dynamic stability and bifurcation of a nonlinear in-extensional rotating shaft with internal damping

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Abstract

In this paper, stability and bifurcations in a simply supported rotating shaft are studied. The shaft is modeled as an in-extensional spinning beam with large amplitude, which includes the effects of nonlinear curvature and inertia. To include the internal damping, it is assumed that the shaft is made of a viscoelastic material. In addition, the torsional stiffness and external damping of the shaft are considered. To find the boundaries of stability, the linearized shaft model is used. The bifurcations considered here are Hopf and double zero eigenvalues. Using center manifold theory and the method of normal form, analytical expressions are obtained, which describe the behavior of the rotating shaft in the neighborhood of the bifurcations.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Kharazmi University, Mofatteh Avenue, P.O. Box 15719-14911, Tehran, Iran

    S. A. A. Hosseini

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  1. S. A. A. Hosseini
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Correspondence to S. A. A. Hosseini.

Appendices

Appendix A

Parameters ω, Λ j (j=1–4), Δ j (j=1–8) and Θ j (j=1,2) presented in Sect. 4 are defined as

$$\begin{aligned} &{\omega = \mu _{e}/\bigl(n^{2}\pi ^{2}\mu _{i}\bigr)} \\&{\varLambda _{1} = 2n^{6}\pi ^{6}\mu _{i}/\vartheta} \\&{\varLambda _{2} = n^{8}\pi ^{8}\mu _{i}^{2}\bigl(\mu _{e} + n^{4}\pi ^{4}\mu _{i}\bigr) \bigl(\mu _{e}^{2} + 2n^{4}\pi ^{4}\mu _{i}\mu _{e}}\\&{\hphantom{\varLambda _{2} =} {} - 12n^{4}\pi ^{4} + n^{8}\pi ^{8}\mu _{i}^{2}\bigr)\big/\vartheta^{3}} \\&{\varLambda _{3} = \bigl(\mu _{e}^{2} + n^{4}\pi ^{4}\mu _{e}\mu _{i} + 4n^{4}\pi ^{4}\bigr)/\vartheta} \\&{\varLambda _{4} = 2n^{10}\pi ^{10}\mu _{i}^{2}\bigl(3\mu _{e}^{2} + 6n^{4}\pi ^{4}\mu _{e}\mu _{i} + 3n^{8}\pi ^{8}\mu _{i}^{2}}\\&{\hphantom{\varLambda _{4} =} {} - 4n^{4}\pi ^{4}\bigr)\big/\vartheta ^{3}} \\&{\Delta _{1} = - \bigl[2n^{2}\pi ^{2} \bigl(n^{2}\pi ^{2}/3 - 3/8\bigr)\mu _{e}^{2} \bigl(\mu _{e} + n^{4}\pi ^{4}\mu _{i} \bigr)}\\&{\hphantom{\Delta _{1} =} {} + n^{10}\pi ^{10}\mu _{i}^{2}\bigl(5 \mu _{e} + 2n^{4}\pi ^{4}\mu _{i}\bigr) \bigr]\big/\bigl(2n^{4}\pi ^{4}\mu _{i}^{2} \vartheta \bigr)} \\&{\Delta _{2} = \bigl[7n^{6}\pi ^{6}\mu _{e}\mu _{i}^{2}\bigl(\mu _{e} + n^{4}\pi ^{4}\mu _{i}\bigr)}\\&{\hphantom{\Delta _{2} =} {} + 8 \bigl(n^{2}\pi ^{2}\bigl(n^{2}\pi ^{2}/3 - 3/8\bigr)\mu _{e}^{2} + n^{10}\pi ^{10}\mu _{i}^{2}\bigr)\bigr]}\\&{\hphantom{\Delta _{2} =} {}\big/\bigl(4n^{2}\pi ^{2} \mu _{i}^{2}\vartheta \bigr)} \\&{\Delta _{3} = - \bigl[2\varGamma \mu _{e}^{2} \bigl(\mu _{e} + n^{4}\pi ^{4}\mu _{i} \bigr) + \mu _{i}^{2}n^{10}\pi ^{10}\bigl( - 3\mu _{e}}\\&{\hphantom{\Delta _{3} =} {} + 2n^{4}\pi ^{4}\mu _{i}\bigr) \bigr]\big/\bigl(2n^{4}\pi ^{4}\mu _{i}^{2} \vartheta \bigr)} \\&{\Delta _{4} = - \bigl[n^{6}\pi ^{6}\mu _{e}\mu _{i}^{2}\bigl(\mu _{e} + n^{4}\pi ^{4}\mu _{i}\bigr)}\\&{\hphantom{\Delta _{4} =} {} - 8\bigl(\varGamma \mu _{e}^{2} + n^{10}\pi ^{10}\mu _{i}^{2}\bigr)\bigr]\big/\bigl(4n^{2}\pi ^{2} \mu _{i}^{2}\vartheta \bigr)} \\&{\Delta _{5} = \bigl[3n^{6}\pi ^{6}\mu _{e}\mu _{i}^{2}\bigl(\mu _{e} + n^{4}\pi ^{4}\mu _{i}\bigr)}\\&{\hphantom{\Delta _{5} =} {} - 8\bigl(\varGamma \mu _{e}^{2} + n^{10}\pi ^{10}\mu _{i}^{2}\bigr)\bigr]\big/\bigl(4n^{2}\pi ^{2} \mu _{i}^{2}\vartheta \bigr)} \\&{\Delta _{6} = - \bigl[2\varGamma \mu _{e}^{2} \bigl(\mu _{e} + n^{4}\pi ^{4}\mu _{i} \bigr) + n^{10}\pi ^{10}\mu _{i}^{2}\bigl( - 5\mu _{e}}\\&{\hphantom{\Delta _{6} =} {} + 2n^{4}\pi ^{4}\mu _{i}\bigr) \bigr]\big/\bigl(2n^{4}\pi ^{4}\mu _{i}^{2} \vartheta \bigr)} \\&{\Delta _{5} = - \bigl[5n^{6}\pi ^{6}\mu _{e}\mu _{i}^{2}\bigl(\mu _{e} + n^{4}\pi ^{4}\mu _{i}\bigr)}\\&{\hphantom{\Delta _{5} =} {} + 8\bigl(\varGamma \mu _{e}^{2} + n^{10}\pi ^{10}\mu _{i}^{2}\bigr)\bigr]\big/\bigl(4n^{2}\pi ^{2} \mu _{i}^{2}\vartheta \bigr)} \\&{\Delta _{8} = - \bigl[2\varGamma \mu _{e}^{2} \bigl(\mu _{e} + n^{4}\pi ^{4}\mu _{i} \bigr) }\\&{\hphantom{\Delta _{8} =} {}+ n^{10}\pi ^{10}\mu _{i}^{2}\bigl(3 \mu _{e} + 2n^{4}\pi ^{4}\mu _{i}\bigr) \bigr]\big/\bigl(2n^{4}\pi ^{4}\mu _{i}^{2} \vartheta \bigr)} \\&{\varTheta _{1} = \varGamma \mu _{e}\bigl(\mu _{e} + n^{4}\pi ^{4}\mu _{i}\bigr)\big/ \bigl(n^{2}\pi ^{2}\mu _{i}\vartheta \bigr)} \\&{\varTheta _{2} = - 2\varGamma \mu _{e}n^{2} \pi ^{2}\big/\bigl(n^{2}\pi ^{2}\mu _{i} \vartheta \bigr)} \end{aligned}$$

where

$$\vartheta = \mu _{e}^{2} + 2n^{4}\pi ^{4}\mu _{e}\mu _{i} + n^{8}\pi ^{8}\mu _{i}^{2} + 4n^{4}\pi^{4} $$

Appendix B

Parameter Ψ j (j=1–7) presented in Sect. 4 is defined as

$$\begin{aligned} &{\varPsi _{1} = 1/2(\varTheta _{2} + i\varTheta _{1})} \\&{\varPsi _{2} = 1/8\bigl[( - \Delta _{7} + \Delta _{2} + \Delta _{5} - \Delta _{4})i + \Delta _{1} - \Delta _{3}}\\&{\hphantom{\varPsi _{2} =} {} - \Delta _{6} + \Delta _{8}\bigr]} \\&{\varPsi _{3} = 1/8\bigl[( - 3\Delta _{2} + 3\Delta _{5} + \Delta _{7} - \Delta _{4})i + \Delta _{3} + 3\Delta _{6}}\\&{\hphantom{\varPsi _{3} =} {} + 3\Delta _{1} + \Delta _{8}\bigr]} \\&{\varPsi _{4} = 1/8\bigl[(3\Delta _{2} + 3\Delta _{5} + \Delta _{7} + \Delta _{4})i + \Delta _{3} - 3\Delta _{6}}\\&{\hphantom{\varPsi _{4} =} {} + 3\Delta _{1} - \Delta _{8}\bigr]} \\&{\varPsi _{5} = 8\bigl(\varOmega _{1} + \varLambda _{3}\varepsilon + \varLambda _{4}\varepsilon ^{2} \bigr)} \\&{\varPsi _{6} = 8\bigl(\varLambda _{1}\varepsilon + \varLambda _{2}\varepsilon ^{2}\bigr)} \\&{\varPsi _{7} = 1/8\bigl[( - \Delta _{7} - \Delta _{2} + \Delta _{5} + \Delta _{4})i + \Delta _{1} - \Delta _{3}}\\&{\hphantom{\varPsi _{7} =} {} + \Delta _{6} - \Delta _{8}\bigr]} \end{aligned}$$

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Hosseini, S.A.A. Dynamic stability and bifurcation of a nonlinear in-extensional rotating shaft with internal damping. Nonlinear Dyn 74, 345–358 (2013). https://doi.org/10.1007/s11071-013-0974-9

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