# Minimal Model for Sprag-Slip Oscillation as Catastrophe-Type Behavior
## 3 Results  
For the preliminary investigation, \(\theta_{o}\) was chosen to be a control parameter and the other system parameters were set to be \(\mu_{o}=0.5\), \(\kappa=0.007\), \(\alpha=0.007\), \(\sigma=0.1\), and \(\beta=0\) for clearly showing the catastrophic behavior. The equilibrium curve and its stability types can be determined by solving Equations (21), (22), and (27) numerically with respect to \(\theta_{o}\). The equilibrium set of the contact and noncontact conditions can be obtained separately, as shown in Figure 4a,b and Figure 4c,d, respectively. The fixed points under contact (\(y_{eq}<0\)) are the steady-sliding equilibriums essential in the friction-induced vibration problems. However, the fixed points under noncontact condition (\(y_{eq}\geq 0\)) represent the static equilibrium in the lift-off state that was not of interest in this analysis. The dynamic behavior of the system can be investigated by the stability at the sliding deformation with respect to the control parameter, as illustrated in Figure 4a. In this example, three steady-sliding equilibriums exist for the lower and upper range of \(\theta_{o}\) where one is geometrically unstable and the others are stable. It is important to note that, for the constant inclined angle case [17], the steady-sliding disappeared under the so-called sprag condition. This constant angle approximation will be called the \"linear sprag model\". The corresponding dynamic behavior can be hardly analyzed due to its discontinuous characteristics. Therefore, the spragging force should be interpreted by the proposed nonlinear model. In this example, the critical value for the existence of the sprag condition in the linear sprag model was around \(\mu_{o}\equiv\mu_{cr}=0.58\). The steady-sliding equilibrium of the linear sprag model always exists in the bounded manner for \(\mu_{o}<\mu_{cr}\) as shown in Figure 5a,b. In contrast, Figure 5c,d indicates that the equilibrium of the linear sprag model goes to infinity at the certain angles for \(\mu_{o}>\mu_{cr}\). Equivalently, the spragging force increases in infinity on the contact. However, if the inclined angle is allowed varying, the real roots of the steady-sliding always exist in the bounded one or three real numbers. Figure 5 demonstrates that the equilibrium solution of the linear sprag model can approximate the one of the nonlinear model for the lower and upper range of \(\theta_{o}\), but it is not valid in between. Figure 6a illustrates a discontinuous jump from the lower to the upper stable equilibrium if \(\theta_{o}\) increases from zero. This jump is called catastrophe, which results in large static deformation in the sliding direction. In the reverse direction, as decreasing \(\theta_{o}\) there was no jump, but there was increase of the static deformation. This interpretation differs from the conclusion of the previous spray model [16; 17], i.e., that there are the critical angles leading to the infinite spragging force and the spray condition for the nonsteady-sliding equilibrium in between, as shown in Figure 6b. It implies that the spray condition may not be the correct interpretation for the so-called spray-slip oscillation presumably induced by spragging force. The above results indicate that the term \"sprag\" can be replaced by the catastrophic characteristics with changes in system parameters such as the initial inclined angle \(\theta_{o}\). In the sliding oscillation, the discontinuous jump can still dominates the oscillatory behavior. Figure 7a illustrates that the oscillation started from the zero initial position, \(x=0\) and \(y=0\) for the angle \(\theta_{o}=49^{\circ}\), and reached the stable limit cycle. As the angle varied, the amplitude of oscillation drastically increased near the value of catastrophe, as shown in Figure 7b. It implies that the \"sprag-slip oscillation\" was initiated from the catastrophic behavior rather than the infinite spragging force.