On the Stability and Long-Term Behaviour of Structural Systems Excited by Nonideal Power Sources

Abstract

This chapter investigates the problem of an unbalanced motor attached to a fixed frame by means of a nonlinear spring and a linear damper. The proposed mathematical model is simple enough to allow for an analytical treatment of the equations, while sufficiently complex to preserve the main nonlinear phenomena that can be observed in real unbalanced rotating machinery. The primary focus is on the bidirectional interaction that in general exists between the excitation provided by the motor and the response of the vibrating structure. By combining various mathematical tools (Averaging, Singular Perturbation Theory, classification of Hopf bifurcations, Poincaré-Bendixson Theorem), the long-term behaviour of the system is investigated in detail. The analytical results are verified numerically. It should be noted that the study presented in this Chapter was originally published in [1, 2].

Appendix

This section provides the expressions of parameters ${f}_{ij}$ and ${g}_{ij}$ in Eq. (80). These are simply the coefficients of the nonlinear terms of system (79), which result when system (73) is transformed according to change of variables (76),

$${f}_{20}=-\frac{3\rho aR}{4{c}_{2}}\left(3{c}_{1}^{2}+{c}_{2}^{2}\right)-2{c}_{1}\left(\frac{2\xi }{aR}+\frac{3}{4}\rho {a}^{2}\right)$$

(A.1)

$${f}_{02}=-\frac{9\rho aR{\omega }_{0}^{2}}{4{c}_{2}}$$

(A.2)

$${f}_{11}=-\frac{3}{4}\rho {\omega }_{0}a\left(a+3R\frac{{c}_{1}}{{c}_{2}}\right)-\frac{2\xi {\omega }_{0}}{aR}$$

(A.3)

$${f}_{30}=\frac{9\rho {c}_{1}}{4{c}_{2}}\left({c}_{1}^{2}+{c}_{2}^{2}\right)$$

(A.4)

$${f}_{03}=\frac{9\rho {\omega }_{0}^{3}}{4{c}_{2}}$$

(A.5)

$${f}_{21}=\frac{3\rho {\omega }_{0}}{4{c}_{2}}\left(3{c}_{1}^{2}+{c}_{2}^{2}\right)$$

(A.6)

$${f}_{12}=\frac{9{c}_{1}\rho {\omega }_{0}^{2}}{4{c}_{2}}$$

(A.7)

$${g}_{20}=\frac{\left({c}_{1}^{2}+{c}_{2}^{2}\right)}{{\omega }_{0}}\left[\frac{9\rho a}{4}\left(R\frac{{c}_{1}}{{c}_{2}}+a\right)+\frac{4\xi }{aR}\right]$$

(A.8)

$${g}_{02}=\frac{3\rho {\omega }_{0}a}{4}\left(a+3R\frac{{c}_{1}}{{c}_{2}}\right)$$

(A.9)

$${g}_{11}=\frac{3}{2}\rho {a}^{2}{c}_{1}+\frac{3\rho aR}{4{c}_{2}}\left(3{c}_{1}^{2}+{c}_{2}^{2}\right)+\frac{2\xi {c}_{1}}{aR}$$

(A.10)

$${g}_{30}=-\frac{9\rho }{4{\omega }_{0}{c}_{2}}{\left({c}_{1}^{2}+{c}_{2}^{2}\right)}^{2}$$

(A.11)

$${g}_{03}=-\frac{9{c}_{1}\rho {\omega }_{0}^{2}}{4{c}_{2}}$$

(A.12)

$${g}_{21}=-\frac{9{c}_{1}\rho }{4{c}_{2}}\left({c}_{1}^{2}+{c}_{2}^{2}\right)$$

(A.13)

$${g}_{12}=-\frac{3\rho {\omega }_{0}}{4{c}_{2}}\left(3{c}_{1}^{2}+{c}_{2}^{2}\right),$$

(A.14)

where ${a}_{eq}$ and ${R}_{eq}$ have been shortly written as $a$ and $R$, respectively.