News 53 Tribologie + Schmierungstechnik · volume 72 · issue 5/ 2025 Introduction Stick-slip is a frequently occurring effect in which sticking and sliding alternate repeatedly on a contact surface. This unsteady movement usually results in considerable wear as well as unwanted heat and noise generation. Stick-slip is responsible, for example, for the squeaking of brakes or the squeaking of chalk on a blackboard. When playing a violin, the effect is produced in such a controlled manner that music can also be created. State of the art The stick-slip effect is usually illustrated in the literature using one of two common models. A rigid body of mass m (counter body) is pulled at a constant speed v 2 over a rigid base (base body) (Figure 1, left). A spring of stiffness c is placed between the pulling point and the base body. This spring can be perceived as a circuit of different springs, including, for example, the tangential con- GfT award Bachelorthesis 2025 Modelling the Stick-slip Effect in Matlab/ Simulink to Identify Influencing Parameters Gerrit W. Schnelle* The topic was submitted for the GfT Sponsorship Award 2025 in the category “bachelor or similar theses”. The award took place at the GfT conference in September 2025. In this paper, the stick-slip phenomenon will be discussed, taking into account the velocity-dependent friction according to Stribeck. The underlying modelling approach and its numerical implementation will be considered. In addition, different effect characteristics and a dimensionless mapping of the system behaviour will be presented. Keywords Stick-slip, Stribeck Curve, Stability Map, Stability, Frictional Dampening, Matlab, Simulink, Stable Oscillation Behaviour, Instable Oscillation Behaviour, Stick Slip Abstract * Gerrit W. Schnelle B. Sc., Orcid-ID: https: / / orcid.org/ 0009-0006-0940-2809 Universität Paderborn Chair of Design and Drive Technology Warburger Str. 100, 33098 Paderborn Figure 1: Modelling the stick-slip effect according to [Pop15] (Model 1) and Modelling the stick-slip effect according to [Sau18] (Model 2, frictional force F μ = F f , 1 counter body, 2 base bodies) News 54 Tribologie + Schmierungstechnik · volume 72 · issue 5/ 2025 tact stiffness of the base and counter body as well as other elasticities in the system. Friction μ prevails between the base body and counter body, which depends on the relative speed in the contact. Sticking occurs as long as the adhesive force F μ is greater than the spring force F c . At the beginning, spring force F c and adhesive force F μ balance each other out. The counter body remains behind the acting force for this period (see a, b remaining in position i). If the spring force F c becomes greater than the holding force F μ by tensioning the spring with continuous tension v 2 , sliding begins. During sliding (“slip”), the spring releases its stored energy and the counter body catches up (c to d, catching up from i to ii). After the slip phase, the spring force is again less than the adhesive force and the counter body remains stationary (d). Position d) corresponds to position a) and the characteristic, periodic stuttering of the body on the surface occurs. In the second model, instead of sideways movement on a rigid base, it is assumed that a treadmill moves under the counter body (Figure 1, Model 2). This allows the oscillation around a rest position x 0 to be considered. [Sau18] also introduces damping parallel to the spring. It should be noted that the relative velocity at the contact surface is now v rel = v 2 - x˙ 1 . To understand the effect, the equation of motion of the second model is considered: (1) x 1 m counter body position x˙ 1 m/ s counter body velocity ẍ 1 m/ s 2 counter body acceleration m kg counter body mass c N/ m spring stiffness (> 0) d Ns/ m viscous damping (> 0) F μ N friction force F d N viscous damping force F c N spring force In the case of sticking (v rel = 0), the spring is charged as an energy store until the spring and damper force are greater than the adhesive force, then the spring releases its energy and the mass slides. Either until it sticks to the belt again, or a smooth glide occurs in the rest position x 0 > 0. The case of quiet sliding is called “stable to the rest position”, the case of periodic adhesion “stable to the limit cycle”. In the case of smooth sliding (x˙ 1 = 0, ẍ 1 = 0), there is a permanent equilibrium of forces between spring force F c and frictional force F μ . According to C OULUMB ’s law of friction, the frictional force (tangential force) F t is the product of the normal force F N and the coefficient of friction μ: 𝑚 ∙ 𝑥̈+ 𝑑 ∙ 𝑥̇ ⏟ 𝐹 d + 𝑐 ∙ 𝑥 ⏟ 𝐹 c = 𝐹 μ (𝑣 2 − 𝑥̇) (2) F t N tangential force F N N normal force μ 1 coefficient of friction v rel m/ s relative velocity between two surfaces The frictional force is always opposite to the direction of movement, the switching function in equation (2.2) changes sign accordingly. For hydrodynamic bearings, S TRIBECK showed in 1902 the particular dependence of the coefficient of friction μ, on the relative speed v rel of the contact surfaces, with the S TRIBECK curve [Str02]. The S TRIBECK curve typically characterizes a greater static friction than dynamic friction. Modelling stick-slip The equation of motion (1) of the model (Figure 1, Model 2) was rearranged so that it can be represented in Simulink. Simulink’s solver ODE3 (B OGACKI -S HAMPINE ) with a step size of 10 -4 was used for the solution. Only linear spring stiffness and damping are considered in the model. Friction is idealized as solely dependent on relative velocity. The work is based on characteristic S TRIBECK curves using the following equation (3) and parameters (Figure 2): (3) μ(v rel ) 1 coefficient of friction a bis g 1 shaping coefficients v rel m/ s relative velocity Discussion of results The stick-slip effect can be described well using a v-x diagram, as well as an associated t-x diagram. A “normal” occurrence of the effect can be seen in Figure 3 a) with phases of adhesion marked green. In systems with the same parameterization, the placement of the counter body can determine whether stickslip occurs or not. Both solutions coexist until the start condition is defined. Five cases are defined for more precise differentiation: • local stability denotes the oscillation of the system with small disturbances around the rest position x 0 • local instability refers to the oscillation of the system with small perturbations around the rest position x 0 • global stability to the rest position refers to the system’s oscillation down to the rest position regardless of any disturbances 𝐹 t = −𝜇 ∙ 𝐹 N ∙ 𝑣 el |𝑣 el | ⏟ sign switch 𝜇(𝑣 el ) = (𝑒 ∙ r𝑏 + (1 − 𝑒 𝑐∙ r𝑑 ) ∙ ) ∙ 𝑔 News 55 Tribologie + Schmierungstechnik · volume 72 · issue 5/ 2025 • global stability to the limit cycle denotes achievement of the stick-slip effect with suitable disturbances • complete compensation describes the fragile state of permanent oscillation when friction damping and viscous damping balance each other out Figure 2: Parameters and progression of the defined S TRIBECK curve. (v min , v max ) is the low point of the curve, ( μ min / μ max )[GS1.1] is the ratio of the minimum to maximum coefficient of friction Figure 3: from a) to e): “normal” occurrence, decay, aperiodic limit case, overshoot and compensation News 56 Tribologie + Schmierungstechnik · volume 72 · issue 5/ 2025 In Figure 4 (left and right) is an example in which two different systems are each started with a small and large disturbance x. All four simulations in Figure 4 belong to the same S TRIBECK curve and were simulated at low viscous damping d. The left system is locally stable (green), as it decays for small perturbation and lies in the range dμ/ dx˙ (v 2 ) > 0. However, if the same system is started with a larger disturbance, the global stability to the limit cycle of the system is recognizable (orange). Both states can coexist. The grey dashed line shows the limit at which the system can oscillate in a fragile state (compensation). The system on the right is in the range dμ/ dx˙ (v 2 ) < 0 and is therefore locally unstable (small perturbation, yellow) and globally stable to the limit cycle (large perturbation, purple). From a linearization around the rest position [Pop15], the compensation of viscous damping and frictional damping can be traced back to a critical normal force F N,crit . This represents the limit between upswing and downswing when touching down on the rest position: (4) F N,crit N critical normal force dμ/ dx˙ s/ m gradient of the S TRIBECK curve d Ns/ m viscous damping v 2 m/ s belt speed If a normal force F N > F N,crit , the influence of friction is increased and the system becomes unstable, for F N < F N,crit it stabilises. Systems of local stability to the rest position and global oscillation to stick-slip (Figure 4, left) pass through smaller and smaller gradients dμ/ dx˙ (v 2 ) with increasing amplitude and thus inevitably pass the state of compensation. The following dimensionless numbers are taken from K EN N AKANO and S ATORU M AEGAWA [NM10]: 𝐹 N,c it = − 𝑑 𝑑𝜇 𝑑𝑥̇(𝑣 2 ) Figure 4: Local and global stability of two systems. Both systems are parameterized identically with blue S TRIBECK curve, only the belt speed varies on the left and right. left green: locally stable (positive slope) left orange: globally stable to limit cycle, oscillates to stick-slip left grey: complete compensation at the transition from stability to instability. right yellow: locally unstable, oscillates to stick-slip (negative slope) right purple: globally stable to limit cycle, oscillates to stick-slip News 57 Tribologie + Schmierungstechnik · volume 72 · issue 5/ 2025 (5) (6) ζ 1 dimensionless damping λ 1 stick-slip parameter v 2 m/ s belt speed m kg counter body mass c N/ m spring stiffness d Ns/ m viscous damping F N N normal force [NM10] assume a friction model with static friction and lower, constant sliding friction. Under this assumption, they find two asymptotes in the double logarithmic: ζ -1 λ 2 = 4 π, and ζ = 1. Both asymptotes are superimposed on the simulink results in Figure 5. 𝜆 = 𝐹 N 𝑣 2 √𝑐𝑚 𝜁 = 𝑑 2√𝑐𝑚 In future, the aim is to implement the model in dimensionless form. This reduces numerical errors and enables better-resolved images with acceptable computing times. Furthermore, a separate consideration of the tangential stiffness (depending on force, material and contact type) and other elasticities in the system, as well as an extended consideration of the S TRIBECK curve as a function of pressure, viscosity, roughness and sliding speed by means of the S CHIPPER number [Sch91] in order to take into account locally different friction coefficients, are conceivable. References [NM10] Nakano, K.; Maegawa, S.: Occurrence limit of stickslip: dimensionless analysis for fundamental design of robust-stable systems. Lubrication Science, (22)1: 2010, S. 1-18. Figure 6: Significance of the critical normal force. Figure 5: λ-ζ diagram: S TRIBECK curve #2, v 2 = 0.25 · v min yellow: stick-slip, blue: smooth glide a) Oscillation due to low friction and viscous damping b) Stick-slip because the frictional damping compensates for the viscous damping The stick-slip effect is also clearly recognizable for ζ > 1 (Figure 5, b). The simulation suggests that the viscous damping is compensated for at these points. Summary The following findings were obtained in this work: For global stability to the rest position in systems without viscous damping, local stability to the rest position is a necessary condition. In viscous damped systems, local instability can be compensated depending on the critical normal force. Simulation results of various S TRIBECK curves indicate a clear gradientand disturbance-dependent occurrence of the stick-slip effect. Particularly in operating ranges with rising S TRIBECK curves, stick-slip cannot generally be ruled out under the assumptions made (Figure 6). News 58 Tribologie + Schmierungstechnik · volume 72 · issue 5/ 2025 [Pop15] Popov, V. L.: Kontaktmechanik und Reibung. Springer Berlin Heidelberg, Berlin, Heidelberg: 2015. [Sau18] Sauer, B.: Konstruktionselemente des Maschinenbaus 2. Springer Berlin Heidelberg, Berlin, Heidelberg: 2018. [Sch91] Schipper, D. J.: Prediction of Lubrication Regimes of Concentrated Contacts. Lubrication Science, 3: 1991, S. 191-200. [Str02] Stribeck, R. H.: Die wesentlichen Eigenschaften der Gleit- und Rollenlager. Zeitschrift des Vereines Deutscher Ingenieure, 46: 1902, S. 1241-1348.