Mode Coupling Instability Mechanism

In this chapter, the mode coupling instability in the lead screw drives is studied. As mentioned in Sect. 4.2, mode coupling is exclusive to multi-DOF systems and can destabilize a system even when the coefficient of friction is independent of sliding velocity. 

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In Sect. 4.2, the mode coupling instability mechanism is considered. In Chap. 3, we have seen the effect of nonconservative forces in creating circulatory systems capable of exhibiting flutter instability. Examples are presented in this section to study the flutter instability with or without friction. Material presented in this section is a prelude to Chap. 7 where we study the mode coupling instability in the lead screw drives. 

In Chap. 3, we mentioned circulatory systems which are described (after linearization) by asymmetric stiffness and/or damping coefficient matrices. Stability of these class of systems has been studied by many authors (see, e.g., [48, 49, 72, 73]). In Sect. 4.2.1 below, we give a classic example where a follower force causes flutter instability. In multi-DOF systems, friction force may act as a follower force and destroy the symmetry of the stiffness and damping matrices resulting in flutter instability known as the mode coupling instability mechanism. This mechanism was first used to explain brake squeal [7]. Ono et al. [74] and Mottershead and Chan [75] studied hard disk drive instability using a similar concept. In Sect. 4.2.2, we study the mode coupling instability mechanism in a simple 2-DOF system with friction. 

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. 

The remainder of this chapter is dedicated to the mode coupling instability mechanism and is focused on the linear system given by (7.18). In Sect. 7.3, the similarities between the two models are explored. The undamped case will be treated first and then the effect of damping is studied. 

Although the linear complex eigenvalue analysis method is useful in establishing the local stability boundaries of the equilibrium point in the system s parameter space, it does not reveal any information regarding dynamic behavior of the system. Further investigations such as numerical simulations or nonlinear analysis methods may be utilized to study the amplitude and frequency of the resulting vibrations under the mode coupling instability mechanism. 

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. 

In this chapter, using a 3-DOF model, it was shown that when mode coupling instability mechanism can affect a system, all the relevant DOFs must be included in the model. It was also shown that the compliance caused by the thread flexibility has similar effects on the stability of the system as the axial compliance in the lead screw supports. 

The other two instability conditions (8.25) and (8.26), relate to the mode coupling instability mechanism and their analysis closely follows Sect 7.2.1. Note that the situation 02 = 0 satisfies (8.26), thus (8.25) does not define an instability boundary. The inequality (8.26) gives the necessary and sufficient for the mode coupling instability. Replacing the less-than sign with an equal sign for the instability boundary and after simplifications, one finds... 

The linearized equations of motion for this system were developed in Sect. 7.1.1. and the conditions for mode coupling instability were derived in the previous chapter. Here, we will only focus on the possibility of instability due to the kinematic constraint mechanism in the undamped system. 

The relationship of physical properties to the chemical instability in these materials should provide the vital link which would, in principal, explain the explosivity of these and related materials in the condensed phase. However, this connection is difficult to make and has to be approached from all possible points of view. The crucial mechanism here for thermal behavior may be the transfer of energy via mode-mode coupling processes. A detailed study of the anharmonic dynamics of such crystals coupled with diffraction work under high pressure and variable temperature, therefore, forms the next logical step in this research. 

In the two-variable models studied for glycolytic oscillations and birhythmicity, periodic behaviour originates from a unique instability mechanism based on the autocatalytic regulation of an allosteric enzyme by its reaction product. The question arises as to what happens when two instabiUty-generating mechanisms are present and coupled within the same system can new modes of dynamic behaviour arise from such an interaction ... 

An example of such a situation was considered at the end of the preceding chapter the system with two oscillatory isozymes (fig. 3.23) contains two instability mechanisms coupled in parallel. Compared with the model based on a single product-activated enzyme, new behavioural modes may be observed, such as birhythmicity, hard excitation and multiple oscillatory domains as a function of a control parameter. The modes of dynamic behaviour in that model remain, however, limited, because it contains only two variables. For complex oscillations such as bursting or chaos to occur, it is necessary that the system contain at least three variables. 

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. 

The two linear systems given by (7.8) and (7.18) share one very important feature not all coefficient matrices are symmetric. The asymmetry, which is caused by friction, may lead to flutter instability (also known as mode coupling). The system defined by (7.8) may also lose stability due to kinematic constraint instability mechanism. ... 

Friction can cause instability in dynamical systems through three distinct mechanisms (1) negative damping, (2) kinematic constraint, and (3) mode coupling. Chapter 4 is dedicated to the introduction of these mechanisms. Illustrative examples are worked out in this chapter to demonstrate the techniques that are applied to the lead screw drives in the later chapters. 

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