Mode Coupling in 3-DOF Lead Screw Model

In this section, local stability of the equilibrium point of the 3-DOF model described in Sect. 5.7 is investigated. Parameter studies and comparisons are done numerically. Similar to what was done in Sect. 7.1, the equations of motion are 

To transfer this point to the origin and present the system in state-space form, the following change of variables is applied  

To study the stability of the steady-sliding equilibrium point, the eigenvalues of the Jacobian matrix (evaluated at y = 0) are calculated. The Jacobian matrix is given by (F 0, O 0) 

In this chapter, the mode coupling instability in the lead screw drives was studied using several multi-DOF models. It was found that the necessary conditions for the mode coupling instability to occur are (a) the lead screw must be self-locking (i.e., p tan 1) and (b) the direction of the applied axial force must be the same as the direction of motion of the translating part (i.e., RQ. 0). The flutter instability boundary in the space of system parameters for the 2-DOF models of Sects. 5.5 and 5.6 was given by (7.36) and (7.49), respectively. 

As shown by the numerical simulation results of Sect. 7.4, mode coupling instability mechanism can lead to diverse range of system behaviors from simple stick-slip limit cycles to complex multiperiod or chaotic responses. 

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