Frequency Island and Nonlinear Vibrating Systems

Abstract

Piecewise linear vibration isolator system is one of the development to introduce dual rate stiffness and damping. There are several vibrating behavior if the system is designed properly. Analytical treatment of the system determines some difficulties such as jump. This investigation indicates that such strong nonlinear systems have a new phenomenon called Frequency Island in their frequency response plot. Frequency Island is a possible isolated frequency response that the vibrating system may jump into the island and stays there until the excitation frequency moves out of the range of the island.

In this student existence, appearance, growing and disappearing of frequency island will be studied and examined. Frequency Island corresponds to large amplitude vibration for certain range of system parameters and considered as a dangerous phenomena in real system. As a result, understanding its appearance will help designers and engineers to design the system to avoid Frequency Island.

Appendix

Parameters of the analytical equation to predict the frequency response of a piecewise linear suspension system

$$ Z_{1} = \uppi^{2} \left( {A^{2} - 1} \right) $$

$$ \begin{aligned} Z_{2} & = \uppi A\left[ {\upomega_{3}^{2} \left( {4\updelta \,\cos \left( {\sin^{ - 1} (\frac{\updelta }{A})} \right) - A\,\sin \left( {2\sin^{ - 1} (\frac{\updelta }{A})} \right) - \uppi A + 2A\,\sin^{ - 1} (\frac{\updelta }{A})} \right)} \right] \\ & \quad + \, 2A^{2} \left[ {\left( {\upxi_{1} \upomega_{1} + \upxi_{3} \upomega_{3} } \right)\left( {\uppi - \sin \left( {2\sin^{ - 1} (\frac{\updelta }{A})} \right) - 2\sin^{ - 1} (\frac{\updelta }{A})} \right) } \right. \\ & \quad +\, \left. {\upxi_{1} \upomega_{1} \left( {\sin \left( {2\sin^{ - 1} (\frac{\updelta }{A})} \right) + 2\sin^{ - 1} (\frac{\updelta }{A})} \right)} \right]^{2} - \upomega_{1}^{2} \uppi^{2} \\ \end{aligned} $$

$$ \begin{aligned} Z_{3} & = 4\upomega_{1}^{2} A^{2} \left[ {\upxi_{1} \upomega_{1} \left( {\sin (2\sin^{ - 1} \left( {\frac{\updelta }{A}} \right)) + 2\sin^{ - 1} \left( {\frac{\updelta }{A}} \right)} \right)} \right. \\ & \quad +\, \left. {\left( {\upxi_{1} \upomega_{1} + \upxi_{3} \upomega_{3} } \right)\left( {\uppi - \sin (2\sin^{ - 1} \left( {\frac{\updelta }{A}} \right)) - 2\sin^{ - 1} \left( {\frac{\updelta }{A}} \right)} \right)} \right]^{2} + \upomega_{1}^{4} \uppi^{2} \\ & \quad +\, \left( { - \upomega_{3}^{2} A\,\sin (2\sin^{ - 1} \left( {\frac{\updelta }{A}} \right)) - \upomega_{3}^{2} \uppi A + 2\upomega_{3}^{2} \sin^{ - 1} \left( {\frac{\updelta }{A}} \right)A + 4\upomega_{3}^{2} \updelta \,\cos (\sin^{ - 1} \left( {\frac{\updelta }{A}} \right))} \right)^{2} \\ \end{aligned} $$

$$ Z_{6} = 2A\left( {\upomega_{1}^{2} + \upsigma } \right)\left[ {\left( {\upxi_{1} \upomega_{1} + \upxi_{3} \upomega_{3} } \right)\left( {2\sin^{ - 1} \left( {\frac{\updelta }{A}} \right) - \uppi - \sin (\sin^{ - 1} \left( {\frac{\updelta }{A}} \right)} \right)} \right] $$

$$ Z_{7} = 2A\left( {\upomega_{1}^{2} + \upsigma } \right)\upxi_{1} \upomega_{1} \left( {2\,\sin^{ - 1} \left( {\frac{\updelta }{A}} \right) + \sin (\sin^{ - 1} \left( {\frac{\updelta }{A}} \right))} \right) $$

$$ Z_{8} = \uppi A\upsigma + \upomega_{3}^{2} A\left( {2\sin^{ - 1} \left( {\frac{\updelta }{A}} \right) - \uppi - \sin (2\sin^{ - 1} \left( {\frac{\updelta }{A}} \right))} \right) + 4\updelta \upomega_{3}^{2} \cos (\sin^{ - 1} \left( {\frac{\updelta }{A}} \right) $$

$$ \begin{aligned} Z_{10} & = 4f_{3}^{4} \upxi_{3}^{4} Z_{15}^{4} + 32\uppi^{2} f_{1} \upxi_{1}^{{}} f_{3}^{3} \upxi_{3}^{3} Z_{15}^{3} + 4\uppi \upxi_{3}^{2} f_{3}^{2} Z_{15}^{2} \left( {f_{3}^{2} Z_{16} + 4\uppi^{3} f_{1}^{2} \left( {6\uppi^{4} \upxi_{1}^{2} - 1} \right)} \right) \\ & \quad + \, 16\uppi^{3} \upxi_{1}^{{}} \upxi_{3}^{{}} f_{1} f_{3}^{{}} Z_{15}^{2} \left( {f_{3}^{2} - 4\uppi^{3} f_{1}^{2} \left( {1 - 2\upxi_{1}^{2} } \right)} \right) + 16\uppi^{5} \upxi_{1}^{2} f_{1}^{2} f_{3}^{2} Z_{16}^{{}} \\ \end{aligned} $$

$$ Z_{11} = 64\uppi^{3} \updelta \sqrt {1 - \frac{{\updelta^{2} }}{{A^{2} }}} \left( {\upxi_{3}^{2} f_{3}^{4} Z_{15}^{2} + 4\uppi^{2} \upxi_{1}^{{}} \upxi_{3}^{{}} f_{1} f_{3}^{3} Z_{15} + 4\uppi^{4} \upxi_{1}^{2} f_{1}^{2} f_{3}^{2} } \right) $$

$$ Z_{12} = \uppi^{2} f_{3}^{4} Z_{15}^{2} - 8\uppi^{5} f_{1}^{2} f_{3}^{2} Z_{15} + 16\uppi^{8} f_{1}^{4} \begin{array}{*{20}c} {} & {} \\ \end{array} Z_{13} = 32\uppi^{4} \updelta \sqrt {1 - \frac{{\updelta^{2} }}{{A^{2} }}} f_{3}^{2} \left( {f_{3}^{2} Z_{15}^{2} - 8\uppi^{3} f_{1}^{2} } \right) $$

$$ Z_{14} = 256\uppi^{6} f_{3}^{4} \updelta^{4} \left( {1 - \frac{1}{{A^{2} }}} \right)\begin{array}{*{20}c} {} & {} \\ \end{array} Z_{15} = - 2\uppi \,\sin \left( {2\sin^{ - 1} (\frac{\updelta }{A})} \right) + 2\uppi^{2} - 4\uppi \left( {\sin^{ - 1} (\frac{\updelta }{A})} \right)^{2} $$

$$ Z_{16} = - 4\sin \left( {2\sin^{ - 1} (\frac{\updelta }{A})} \right) - 8\uppi^{3} + 8\uppi^{2} \left( {\sin^{ - 1} (\frac{\updelta }{A})} \right)^{2} $$