Nonlinear dynamics investigation of contact force in a cam–follower system using the Lyapunov exponent parameter, power spectrum analysis, and Poincaré maps

The aim of this article is to detect the detachment between the cam and the follower using the largest Lyapunov exponent parameter, the power density function of fast Fourier transform, and Poinaré maps due to the nonlinear dynamics phenomenon of the follower. The detachment between the cam and the follower was investigated for different cam speeds and different internal distances of the follower guide from inside. This study focuses on the use of the cam–follower system with a bionic quadruped robot through a linkage mechanism. Multishock absorber (spring–damper–mass) systems at the very end of the follower were used to improve the dynamic performance and to reduce the detachment between the cam and the follower. The SolidWorks program was used in the numerical solution, while a high‐speed camera at the foreground of the OPTOTRAK 30/20 equipment was used to identify the follower position. The friction and impact were considered between the cam and the follower and between the follower and its guide. In general, when the cam and the follower are in permanent contact, there is no loss in potential energy, and no impact or restitution. The detachment between the cam and the follower increases with increasing coefficient of restitution in the presence of the impact.

and roller follower system. 5 Moreover, Yousuf used the photo-elastic technique to calculate the value of contact stress between the pear cam and the roller follower mechanism. Four different positions of the follower with respect to the cam such as 0°, 90°, 180°, and 270°at different values of the compression load are considered. 6 Pugliese et al. 7 designed an apparatus to measure the contact force using an optical interferometry sensor and a high-speed camera. They showed different conditions of the contact at low and high magnifications to detect the angular positions of the cam. Yang et al. 8

extended a transient impact
hypothesis of the contact model by considering the tangential slip between the cam and the follower. They showed that the cam and the follower maintained permanent contact when the cam rotational speed was low, while separation and oblique impact occurred at high speeds.
The signal of follower motion in the y-direction has been processed using a data acquisition technique. Yousuf used the Lyapunov exponent parameter and fast Fourier transform (FFT) analysis to detect the nonperiodic motion of the follower at the contact point for different internal dimensions of the follower guide and cam angular speeds. 9 Nonperiodic motion is examined using power spectrum analysis of FFT and a phase plane diagram at the contact point between a polydyne cam and the knife follower mechanism. 10 Yousuf and Marghitu 11 used the largest Lyapunov exponent parameter to investigate the nonperiodic motion in a globoidal cam with a roller follower system using different cam angular velocities and different internal dimensions of the follower guide. Yan and Tsay 12 solved the problem of separation between the cam and the follower by increasing the preload rate of the spring force. They proposed a mathematical model for the cam profile with a variable speed to change the follower motion by using the Bezier function. Jamali et al. 13 discussed the parameters that affect the contact problem between the cam and the flat-faced follower such as the radius of curvature, surface velocities, and applied load under a thin film of lubricants. They improved numerically the level of film thickness lubricant by taking into consideration the parabolic shape of the pressure distribution through cam depth, which reduced the detachment between the cam and the follower. Desai and Patel 14 determined the critical angular speed when the follower jumped off the cam. They determined the kinematic parameters of the dynamic force analysis at different motions for the follower such as constant velocity motion, cycloidal motion, parabolic motion, and simple harmonic motion.
Lassaad et al. 15 studied the effect of the error in the cam profile on the nonlinear dynamics of the oscillated roller follower by solving nonlinear second-order differential equations. DasGupta and Ghosh 16 used a constant pressure angle to assess jamming of the follower in its guide based on the follower guide friction.
This study focuses on the use of a cam-follower system with a bionic quadruped robot through a linkage mechanism. Multishock absorber (spring-damper-mass) systems at the very end of the follower were used to improve the dynamic performance and to reduce the detachment between the cam and the follower. This paper is organized as follows: (a) In Section 2, the analytic displacement of the follower was derived using the Newton-Euler approach in the presence of three degrees of freedom for the follower. (b) In Section 3, the experiment setup that is used in this study with a high-speed camera at the foreground of OPTOTRAK 30/20 equipment to identify the follower position and to calculate the contact force is described. (c) Section 4 describes the method of numerical simulation of the detachment between the cam and the follower through follower displacement and contact force using the SolidWorks program. (d) In Sections 5-7, the detachment between the cam and the follower is detected using the largest Lyapunov exponent parameter, the power density function of FFT, and Poincaré map. The highlights of this paper are as follows: • The detachment between the cam and the follower was detected using the largest Lyapunov exponent parameter, the power density function using FFT, and Poincaré maps due to the nonlinear dynamics phenomenon of the follower.
• Multishock absorber (spring-damper-mass) systems at the very end of the follower were used to improve the dynamic performance and to reduce the detachment height between the cam and the follower.
• Friction and impact were considered between the cam and the follower and between the follower and its guides.
• The experimental test was carried out using a high-speed camera at the foreground of the OPTOTRAK 30/20 equipment.

| EQUATIONS OF MOTION OF A THREE-DEGREE-OF-FREEDOM SYSTEM
In this paper, nonlinear dynamics occurred because of the clearance between the follower stem and its guide and due to motion of the follower with three degrees of freedom (right-left, up-down, and rotation about the z-axis). The nonlinear dynamics is tested over four values of the internal distance (ID) of the follower guide from inside (ID = 16, 17, 18, and 19). The more the ID, the more the clearance between the follower stem and its guide since the follower stem geometry and dimensions are constants. The nonlinear dynamics phenomenon is tested over high values of ID, which yields large translation and rotation (the motion of the follower stem is nonperiodic motion and chaos). Periodic motion of the follower occurs when there is no clearance between the follower and its guide or at a small value of ID The Newton-Euler approach for translational and rotational motions was applied to describe motion of the threedegree-of-freedom follower. Figure 1 shows the three degrees of freedom of a flat-faced follower stem. All springs have the same stiffness K 1 = K 2 = K 3 = K 4 = K 5 = 73.56 N/mm, while the damping coefficient has the values C 1 = C 2 = C 3 = C 4 = C 5 = 9.19 N·s/mm. The mass of the follower is assumed to be m = 0.2759 kg.
The cam-follower mechanism can be treated as a single-or a multi-degree-of-freedom system based on the motion of the follower. 17 Applying the Newton equation for rigid body dynamics mX∑Forces = 2 to the translational motion in the vertical direction where center of gravity point 2 is the reference point, one obtains After simplification, Equation (1) yields mẌC ẊK X P + + = . (2) Applying the Newton equation mX∑Forces = 1 to the translational motion in the horizontal direction on the follower stem where center of gravity point 2 is the reference point, one After simplification, Equation (3) leads to Taking the moment about the center of gravity point 2 in the follower stem structure, the Euler equation Iθ ∑Moments =̈leads to After simplification, Equation (5) can be written as The geometry of the follower stem is treated as a beam. Also, the summation of area moments of inertia is obtained around the center of gravity (black dot point 2). After applying the theory of the parallel axis theorem to transfer the area moment of inertia from point 1 to Equations (2), (4), and (6) can be written in matrix form as where  Equation (7) is applied for a small value of the clearance in which the translation and rotation are very small. The nonlinear dynamics phenomenon arises from the contact force. Equation (7) can be used to define the frequency-response matrix, called the impedance matrix, as follows 18 : where Ω is the frequency of external sinusoidal excitations, and • is the complex amplitude of • for a steady-state response with both amplitude and phase information.
The eigenvalue  ∈ λ of a damped system is complex and can be solved by substitution of i λ Ω → into Equation (8), with the determinant of the impedance matrix being zero, that is The complex amplitude of a steady-state response is calculated by taking the inverse of the impedance matrix and multiplying by the force vector as The analytical solution of the steady-state response of the follower is Note that the phase angles are different for each degree of freedom. This is the difference between a multi-DOF system and a single-DOF system. The contact force between the cam and the follower is defined in the following equation 19 : where P C is the contact force between the cam and the follower; K is the stiffness of the spring between the follower stem and the installation table; Δ is the preload spring extension; and m is the mass of the follower stem. Further, 20 where Ø 1 is the pressure angle, R b is the radius of the base circle of the cam, and X t ( ) is the product of the follower linear displacement.
The analytical solution of the follower displacement is given as since the follower displacement varies in horizontal and vertical directions.

| Numerical simulation of follower displacement
A CAD program is used to simulate the numerical model of the follower displacement. 24 In the numerical simulation, the follower has three degrees of freedom (up-down, right-left, and rotation about the z-axis). The two rollers in both sides between the wall and the mass of the spring-damper system allowed the multishock absorber systems to move up and down as shown in Figure 5   shown in Figure 7. The peak of the nonlinear response of the follower displacement was reduced to 15%, 32%, 45%, and 62% after using multishock absorber systems. The system with ID = 17 mm and N = 300 rpm was used in the simulation. The amplitude of the nonlinear response of the follower is damped using multi spring-damper-mass systems. In other words, the more the spring-damper-mass systems, the lower the amplitude of vibrations.  In this paper, the cam profile with return-dwellrise-dwell-return-dwell-rise-dwell-return-dwell-rise-dwell was selected. 25 The system with ID = 16 mm and N = 200 rpm was used in the verification of the follower displacement as shown in Figure 9 for one cycle of cam rotation.
The analytical follower displacement was obtained using Equation (13). Follower linear displacement was tracked experimentally using a high-speed camera at the foreground of the OPTOTRAK/3020 equipment. In the analytical solution of the follower displacement, the dwell stroke period of time was a straight line, which means that there was no detachment between the cam and the follower. In the numerical simulation, the dwell stroke was divided between the rise and return strokes, and the detachment was intangible between the cam and the follower. In

| Numerical analysis of contact force
The weight of the follower was found to be sufficient to maintain the contact when the follower moved with simple harmonic motion. Also, the preload spring between the installation table and the follower stem must be properly selected to maintain the contact. In general, the detachment between the cam and the follower increases when the contact force has a minimum value.

| LARGEST LYAPUNOV EXPONENT PARAMETER
The largest Lyapunov exponent is a numeric value that can have either a positive or negative sign. In the design, the largest Lyapunov exponent parameter is one, used to detect separation between the cam and the follower through the contact force. A positive value of F I G U R E 19 Follower-displacement power spectrum analysis from the experimental test F I G U R E 20 Follower-displacement power spectrum analysis from numerical simulation the Lyapunov exponent of the contact force indicates that there is a detachment between the cam and the follower (nonperiodic motion).
A negative largest Lyapunov exponent value of the contact force implies periodic motion (the cam and the follower are in continuous contact), and the contact force has a maximum value. The Wolf algorithm based on Matlab software was used to calculate the values of the largest Lyapunov exponent by monitoring the orbital divergence for the contact force. 27 Equations (14) and (15) where D is the average displacement between trajectories at t = 0, d(t) is the rate of change in the distance between nearest neighbors, d j (i) is the distance between the jth pair at (i) nearest neighbors (in mm), t is the single time series (in s), y(i) is the curve fitting of the least square method for the follower displacement data (in mm), λ is the largest Lyapunov exponent parameter, and t Δ is the time interval (in s).  29 In this paper, a local and optimal time delay was estimated based on the recommendation that the local time delay should be chosen dependent on the embedding dimensions. 30 The algorithm code of average mutual information was used to obtain the value of time delay, and the first minimum time in the average mutual information trend represents the value of the time delay. The optimal time delay was calculated from the following equation: where τ* is the local time delay, τ w is the optimal for independence of time series, and P represents the embedding dimensions.

| FAST FOURIER TRANSFORM
The power density function was used to detect the detachment between the cam and the follower since it gives six frequencies peaks along with the fundamental frequency. 31 The signal power is the most important factor for signal quality in which the noise can be measured using the signal-to-noise ratio (SNR). A maximum value of the SNR of the follower displacement signal implies no error or noise in the follower displacement. The SNR power of the follower displacement signal is measured using the dB scale since it can be either positive or negative. A negative SNR value means that the signal power is lower than the noise power. The power of the amplitude peak of the fundamental frequency and the other frequencies decreases with increasing number of samples, which means that separation is about to occur between the cam and the follower (the contact force starts approaching zero). When the frequency peaks start disappearing from the FFT diagram, the motion is quasiperiodic (contact force is approaching zero). Figures 19 and 20

CONFLICT OF INTEREST
The author declares no conflict of interest.

DATA AVAILABILITY STATEMENT
The data that support the findings of this study are available from the corresponding author upon reasonable request.