Kinematic Constraint Instability

In a study of the effect of friction on the existence and uniqueness of the solutions of the equation of motion of dynamical systems, Dupont [19] considered a 1-DOF model of a lead screw system. He investigated the situations under which no solution existed and clearly identified one of the sources of instability in the lead screw systems i.e., the kinematic constraint instability mechanism. For the selflocking screws, he found that there is a certain limiting ratio between the lead screw moment of inertia (rotating part) and the mass of the translating part, beyond which no solution exists. 

Finally, in Sect. 4.3, we turn our attention to the kinematic constraint instability. This section begins by the introduction of the Painleve s paradoxes which play an important role in the kinematic constraint instability mechanism. The self-locking property which is another consequence of MctiOTi in the rigid body dynamics is also discussed in this section. This effect has a prominent presence in the study of the lead screws in Chaps. 7 and 8. The concepts presented here form the basis for the study of the kinematic constraint instability in the lead screws in Chap. 8. 

In Sects. 4.3.1 and 4.3.2, we study the classic Painleve s example and derive the conditions for the occurrence of the paradoxes. In Sect. 4.3.3, the concept of self-locking is introduced which is closely related to the kinematic constraint instability mechanism. In the rigid body systems, this phenomenon is sometimes known as jamming or wedging [97]. As we will see later on, the self-locking is an important aspect of the study of the dynamics of the lead screws. In Sect. 4.3.4, a simple model of a vibratory system is analyzed where the kinematic constraint mechanism leads to instability. In the study of disc brake systems, similar instability mechanism is sometimes referred to as sprag-slip vibration [7]. Some further references are given in Sect. 3.3.5. 

Fig. 4.19 Simple model to demonstrate kinematic constraint instability...

Further References on the Kinematic Constraint Instability Mechanism... 

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. 

Violation of this inequality also leads to instability, which is known as the kinematic constraint instability mechanism. This instability mechanism is the subject of Chap. 8. 

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. ... 

See Sect. 8.8.1 for the analysis of kinematic constraint instability in this system. 

The third and final instability mechanism in the lead screw drives is the kinematic constraint. In Sect. 4.3, Painleve s paradoxes were introduced and - through simple examples - it was shown that under the conditions of the paradoxes, the rigid body equations of motion of a system with frictional contact do not have a bounded solution or the solution is not unique. We have also discussed the relationship between Painleve s paradoxes and the kinematic constraint instability mechanism. 

It will be shown by numerical examples in Sect. 8.5 below that when Fq > 0 and Co < 0 (i.e., negative effective damping), the instability may or may not lead to stick-slip vibrations. In contrast to this case, when Fq < 0 (i.e., Painleve s paradox), the instability is accompanied by stick-slip vibration and impulsive forces. Section 8.6 is dedicated to the study of the kinematic constraint instability and the resulting vibrations. 

Fig. 8.4 Unstable steady sliding equilibrium point due to damping kinematic constraint instability mechanism g > tan/l, Wo > 0, Q > 0, and tan/lcx > c...

Expectedly, the instability conditions given by (8.21) are the same as the necessary conditions for the Painleve s paradoxes discussed in Sect. 8.1. Limiting our study to the case of Q > 0 for simplicity. Fig. 8.7 shows that the phase plane of the system is divided by N = 0 and 6 = 0 lines into four regions. In these regions, the system s equation has either no solution or two solutions when kinematic constraint instability is active (i.e., conditions of (8.21) are satisfied). Based on the discussions in Sect. 8.2, the following conclusions are drawn for the behavior of the lead screw model ... 

As mentioned earlier, damping does not affect the stability of the 1-DOF model when the kinematic constraint instability mechanism is active. However, damping has a considerable effect on the behavior of the nonlinear system. Figure 8.11 shows phase plots of the 1-DOF model with three levels of lead screw support damping. For each of these three simulation results, the line A(m, w) = 0 is also drawn. The onset of lead screw seizure is the point where the trajectory reaches this line. Note that from (8.17), the line A(w, u) =0 is given by... 

The linear eigenvalue analysis of Sect. 8.6 showed that when Co, Fq > 0 the origin is stable. More specifically, when the kinematic constraint instability conditions given by (8.21) are not satisfied and Co > 0, the trivial equilibrium point of the system is asymptotically stable. However, there can be situations where the region of attraction of the stable equilibrium point is quite small, leading to instabilities even when conditions of (8.21) do not hold. 

Kinematic Constraint Instability in Multi-DOF System Models... 

Inequality (8.27) is the condition for the occurrence of Painleve s paradoxes or the kinematic constraint instability. 

Lemma 8.2. For the 2-DOF model of Sect. 5.6, the mode coupling and the kinematic constraint instability regions have no overlap in the parameter space. 

Fig. 8.15 Stability of the 2-DOF syston as support stiffness and coefficient of friction are varied, where m = 15 kg, mi = 11.6 kg, and RSI > 0. Slanted lines hatched area mode coupling instability region Horizontal lines hatched area primary kinematic constraint instability region...

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