# Minimal Model for Sprag-Slip Oscillation as Catastrophe-Type Behavior
## 2 Materials and Methods  
A simple compliant model was proposed, as shown in Figure 1. The \(k_{x}\) and \(k_{y}\) represent system stiffness fixed to ground as opposed that the inclined stiffness and \(k\) was applied for representing kinematic variation of system stiffness during sliding. The contact stiffness \(k_{c}\) was applied for allowing the downward acceleration. It should be noted that the inclined stiffness \(k\) may produce the infinite spragging force for the constant inclined angle [17]. In the present work, the inclined angle \(\theta\) was allowed varying for preventing the sprag force from becoming infinite and providing the nonlinear perspective on spragging, as shown in Figure 2.  
Figure 1: Description of a spring-mass model. The inclined angle \(\theta\) is variable. Figure 2: Force components and displacements, (**a**) force components at current state, (**b**) definition of deformation. The equations of motion for this nonlinear spring-mass model are given by:  
\[m\ddot{x}=\mu N-k\Delta\cos\theta-k_{x}x \tag{1}\]  
\[m\ddot{y}=N-F_{L}-k\Delta\sin\theta-k_{y}y \tag{2}\]  
where the stretch/compression in the inclined spring is obtained from (Figure 2b):  
\[\Delta=L_{o}-L \tag{3}\]  
and where the relationship between the original length \(L_{o}\) and the deformed length \(L\) is given by:  
\[L\cos\theta=L_{o}\cos\theta_{o}-x \tag{4}\]  
\[L\sin\theta=L_{o}\sin\theta_{o}-y \tag{5}\]  
\[L=\sqrt{\left(L_{o}\cos\theta-x\right)^{2}+\left(L_{o}\sin\theta-y\right)^{2}} \tag{6}\]  
It is important to note that the contact normal load \(N\) in Equations (1) and (2) is a nonsmooth function defined as:  
\[N=\left\{ \begin{array}{ccc}-k_{c}y,&if&y<0\\ 0,&if&y\geq 0\end{array} \right. \tag{7}\]  
The equations of motion, Equations (1)-(2), can be rewritten in the dimensionless form by introducing the new time scale and system parameters:  
\[x^{\prime\prime}+\frac{\sigma}{L}(1-L)(\cos\theta_{o}-x)+\alpha x-\mu n=0 \tag{8}\]  
\[y^{\prime\prime}+\frac{\sigma}{L}(1-L)(\sin\theta_{o}-y)+\beta y-n=0 \tag{9}\]  
where \(\tau=t\sqrt{k_{c}/m}\), \((\bullet)^{\prime}=d(\bullet)/d\tau\), \(x\to x/L_{o}\), \(y\to y/L_{o}\), \(L\to L/L_{o}\), \(\sigma=k/k_{c}\), \(\alpha=k_{x}/k_{c}\), \(\beta=k_{y}/k_{c}\), \(\kappa=F_{L}/\left(k_{c}L_{o}\right)\), and \(n=N/k_{c}\). For a convenient mathematical form, \(\mu\) is set to be a regularized friction coefficient and \(n\) is expressed in a mathematical function form such that:  
\[\mu=\mu_{o}\big{[}1-\exp\big{(}-d\big{|}V-\dot{x}\big{|}\big{)}\big{]}sgn(V- \dot{x}) \tag{10}\]  
\[n=\frac{1}{2}\big{(}\big{|}y\big{|}-y\big{)} \tag{11}\]  
where \(d\) is the control parameter in the creep regime (Figure 3). Figure 3: Function curve, (**a**) regularized friction coefficient, (**b**) contact normal load. The system equilibriums can be found from Equations (8) and (9) when \(x^{\prime\prime}=x^{\prime}=0\) and \(y^{\prime\prime}=y^{\prime}=0\). The corresponding equilibrium equations become:  
\[F_{x}=\frac{\sigma}{L}(1-L)(\cos\theta_{o}-x)+\alpha x-\mu_{o}n=0 \tag{12}\]  
\[F_{y}=\frac{\sigma}{L}(1-L)(\sin\theta_{o}-y)+\beta y-n=0 \tag{13}\]  
Here the two set of equilibrium equations are formulated for the contact and noncontact conditions. For the contact condition, \(n>0\), the equilibrium equations are rewritten as:  
\[\sigma(1-L)\cos\theta+\alpha(\cos\theta_{o}-L\cos\theta)+\mu_{o}(\sin\theta_{ o}-L\sin\theta)=0 \tag{14}\]  
\[\sigma(1-L)\sin\theta+\beta(\sin\theta_{o}-L\sin\theta)+(\sin\theta_{o}-L\sin \theta)=-\kappa \tag{15}\]  
It can be also expressed in the matrix form:  
\[\mathbf{D}\Bigg{\{} \begin{array}{c}\cos\theta\\ \sin\theta\end{array} \Bigg{\}}=\mathbf{b} \tag{16}\]  
where  
\[\mathbf{D}=\sigma\mathbf{I}+L\mathbf{A} \tag{17}\]  
\[\mathbf{A}=\left[ \begin{array}{cc}-(\sigma+\alpha)&-\mu\\ 0&-(\sigma+\beta+1)\end{array} \right] \tag{18}\]  
\[\mathbf{b}=\left\{ \begin{array}{c}-\alpha\cos\theta_{o}-\mu\sin\theta_{o}\\ -(\beta+1)\sin\theta_{o}-\kappa\end{array}