DOF Lead Screw Drive Model

Consider the 1-DOF lead screw drive model in Fig. 5.4. Here, for simplicity, the Coulomb friction of the translating part is neglected (i.e., Fq = 0). The equation of 

At steady-sliding we have z = 0, z = 0, and z = zq. Substituting these values in (8.14) yields 

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Our study of the lead screw drives entails systems with a single bilateral contact with friction (between lead screw threads and nut threads). As demonstrated by the example in Sect. 4.3.4, in order to study the behavior of a system in the paradoxical regions of parameters, a compliant approximation to rigid contact may be used. In Chap. 8, the limit process approach presented in [51] is utilized to determine the true motion of a 1-DOF lead screw drive model under similar paradoxical conditions. In the limit process approach, the behavior of the rigid body system is taken as that of a similar system with compliant contacts when the contact stiffness tends to infinity. Related to this topic, a discussion of the method of penalizing function can be found in Brogliato ([96], Chap. 2). Other examples include [101-103]. 

The velocity-dependent friction model used in this work is discussed in Sect. 5.1. The dynamics of a pair of meshing lead screw and nut threads is studied in Sect. 5.2. Based on the relationships derived in this section, the basic 1-DOF lead screw drive model is developed in Sect. 5.3. This model is used in Chaps. 6 and 8 to study the negative damping and kinematic constraint instability mechanisms, respectively. A model of the lead screw with antibacklash nut is presented in Sect. 5.4, and the role of preloaded nut on the increased friction is highlighted. Additional DOFs are introduced to the basic lead screw model in Sects. 5.5 to 5.8 in order to account for the flexibility of the threads, the axial flexibihty of the lead screw supports, and the rotational flexibility of the nut. These models are used in Chaps. 7 and 8 to investigate the mode coupling and the kinematic constraint instability mechanisms, respectively. Finally, in Sect. 5.9, srane remarks are made regarding the models developed in this chapter. 

Fig. 5.9 2-DOF lead screw drive model including thread compliance...

Eliminating N among (5.16), (5.17), and (5.43) yields the equations of motion for this 2-DOF lead screw drive model ... 

In Sect. 7.1, we have seen the role of friction in the two lead screw models through breaking the symmetry of the linearized system inertia, damping, and stiffness matrices. The damping and stiffness matrices of model (7.8) (i.e., 2-DOF lead screw drive model with axially compliant lead screw supports) are symmetric... 

In this chapter, the 1-DOF model of the lead screw drives developed in Sect 5.3 was used to study the instability caused by the negative gradient of the friction coefficient with respect to velocity. The local stability of the steady-sUding equilibrium point of the system was studied by examining the eigenvalues of the Jacobian matrix of the linearized system. It was shown that the steady-sliding equdibiium point of the system loses stability when the condition given by (6.9) is satisfied. 

In Sects. 5.5 and 5.6, we have introduced two 2-DOF models for the lead screw drives. In this section, the equations of motion of these models are transformed into matrix form and linearized with respect to their respective steady-sliding equilibrium point. These equations are then used in the next sections to study the local stability of the equilibrium point and the role of mode coupling instability mechanism. In this chapter, for simplicity, the coefficient of friction, n, is taken as a constant. 

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. 

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