# Minimal Model for Sprag-Slip Oscillation as Catastrophe-Type Behavior
## 1 Introduction  
Friction-induced vibration is usually known as self-excited vibration by friction producing unwanted phenomena such as vibration, noise, and fatigue. Owing to its frequent and serious problems in many applications such as automotive brakes, machine joints, and tools, enormous research for the source of instability in the friction-engaged system has been conducted over the decades [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17]. In the literature, the minimal model has been widely used to explain the dynamic instability induced by the negative friction-slope and mode-coupling instability [10; 11; 12; 13]. In a general approach, a stability analysis based on linear eigenvalue analysis was performed and stability boundaries in the parameter space were derived for the equilibrium of the system. Then, their nonlinear stick-slip oscillations at the unstable steady-sliding equilibrium have been demonstrated by solving discontinuous differential equations using several different friction models such as the smoothing [10] and switching [11] methods. In this scenario, the determination of the dynamic instability in the parameter space is valid when steady-sliding exists, and the resulting nonlinear motion in this space can be reasonably found. However, in the parameter space for which steady-sliding does not exist, the above analysis is no more applicable, and the true nonlinear response cannot be examined. Therefore, the determination of the parameter space for which the steady-sliding does not exist is critical for the so-called \"sprag-slip oscillation\". Sprag-type instability and its realistic interpretation were of particular interest in this study. Its dynamic behavior has not been vigorously investigated due to its unbounded and discontinuous nature. The term \"sprag\" was firstly named by [PERSON] in his article [1] by using two simple examples: One, the inclined rigid rod hinged at one end under sliding and can have infinite contact forces, and two, the mass attached by an inclined spring and may experience an unbounded steady-sliding equilibrium (equivalent to the infinite contact forces) for a certain geometric condition. Later, this kinematic constrained instability was investigated in the beam-on-disk system by [PERSON] et al. [14] and the pin-disc model by [PERSON] et al. [15]. In the beam-on-belt configuration, [PERSON] et al. [16] extended the spragtheory by assuming that the sprag condition can be, in another way, characterized as the non-existence of the steady-sliding equilibrium, which is presumably equivalent to the lift-off of the sliding beam at the contact. [PERSON] et al. [17] also investigated the sprag conditions of the several sliding mass-spring systems and found that, in the parametric space, the unbounded spragging force corresponds to the boundary of the regime for the non-existence of the steady-sliding equilibrium. In other words, the sprag condition in [1] is shown to be the condition for the onset of loss-of-contact in [16; 17]. However, it is still unclear that the spray-type oscillation truly exists in the friction-sliding system. The infinite contact force or the loss of contact at sliding is quite ambiguous from the engineering point of view. Honestly, the sprag-slip oscillation is not based on the mathematical principles, but the hypothesis that a lacking steady-sliding presumably leads to stick-slip oscillation. The infinite contact force is generated in a specific region in the parametric space by a linearized equilibrium. [17] Kinematic and dynamic analysis includes initial condition and static equilibrium position in open and closed loop mechanisms. Hence, it is very important to find the equilibrium position. In dynamical system, steady-sliding equilibrium can be represented by a nonlinear equation. Its nonlinear equation derives a solution with abrupt changes that are difficult to predict in a parametric space.