Nonlinear dynamics of system with combined rolling–sliding contact and clearance

Rolling–sliding contacts are found in a variety of systems, such as gears, drum brakes, and tire pavements. Such systems inherently have multiple nonlinearities such as kinematic, contact, and friction nonlinearities. Further, most of these systems have clearance between components which causes excessive vibration and noise. The combination of clearance with other nonlinearities makes the system dynamically interesting. It is important to understand the dynamic behavior of such systems under various operating conditions. Hence, this article focuses on theoretically investigating the transient and steady-state responses of a cam–follower system with rolling–sliding contact and clearance as an exemplary case. A contact mechanics-based model for the same has been developed for this purpose. The transient behavior of the system is examined based on energy and time-varying frequency contents. The domain of attractors, frequency response plots, and phase portraits are deployed to analyze the effect of initial conditions and excitation speed on the steady-state behavior, quantitatively and qualitatively. The steady-state solutions were classified into various branches based on periodicity and phase portraits. In addition, parametric analyses of the effects of loading, damping, and friction on the response have been conducted. Finally, the system is examined for start-up and shut-down characteristics with a constant rate of change in excitation frequency.

characteristics of rolling-sliding contact [2][3][4][5][6]. For example, O'donoghue et al. [7] published experimental results and empirical relations between the parameters like load, speed, viscosity, and surface finish on coefficient of friction using disk machines. Similarly, the effect of various operating parameters on wear and friction was studied by Ramalho et al. [8] using a roller-onroller model. Further, the effect of wet and dry conditions on the transient traction characteristics was studied by Baek et al. [9] using the twin-disk rolling-sliding frictional machine.
The nonlinear behavior of systems with clearance nonlinearity has been studied using analytical, numerical, and experimental approaches in the literature. For instance, Kahraman et al. [10] examined the nonlinear frequency response characteristics of a spur gear pair with backlash using a simplified formulation. They emphasized the need for searching a large domain of initial conditions to obtain all possible solutions at a given frequency. Steady-state solutions were obtained using numerical simulation and the harmonic balance method (HBM), which is a popularly used semianalytical technique [11][12][13] for systems with clearance nonlinearity. Later, Zheng et al. developed an analytical-FEM based framework to study nonlinear dynamics of high-speed spur gears [14]. Experimental investigations were conducted by Kahraman et al. [15] on a four-square gear dynamics test rig to study sub-harmonic resonance and jump discontinuity using the frequency response plots. Further vibro-impacts in a torsional system with clearance were studied and verified by Xia et al. [16] using vehicle driveline experiments. Cam-follower system with an oscillating follower and rotating cam [17] is a classical example of a system which includes both the combined rollingsliding contact and the clearance nonlinearity. Since system properties such as geometry, contact, excitation, and clearance can be easily controlled, it is commonly used to analyze the nonlinear dynamics [18,19]. Researchers [20,21] have quantified the effect of individual nonlinearity on the dynamic response of the system theoretically. However, nonlinear phenomena such as bifurcations and chaos were investigated by Alzate et al. [18,22] using experiments and simplified models, while some have focused on the stability aspects of the cam-follower system [23,24]. Yousuf et al. [25] studied the system's dynamics with follower's guidance clearance employing a combination of experimental and simulation methods, and more recently he documented the dynamics of polydyne cam [26]. Researchers have made attempt to evaluate the effect of bearing clearance on the cam mechanism [27].

Gaps in literature and objectives
In essence, there have been extensive studies focused on the dynamics of rolling-sliding contacts, camfollower systems, and systems with clearance nonlinearity. However, there has not been much attention on the transient behavior of systems with clearance nonlinearity. Further, response of cam-follower system with clearance nonlinearity under various conditions has not been well documented. Consequently, there is a strong need to systematically analyze the behavior of cam-follower system with rolling-sliding contact and clearance nonlinearity. Accordingly, the objectives of this article are: (i) to develop a mathematical formulation for a cam-follower system incorporating the clearance nonlinearity and with effective control of the system parameters; (ii) to study the transient behavior of the system using instantaneous energy and frequency responses; (iii) to investigate the steady-state response quantitatively (using frequency response) and qualitatively (using domain of attractors, and phase portraits) for various operating conditions; and (iv) to analyze the start-up and shut-down characteristics.

Development of mathematical model
The cam-follower system under investigation (as shown in Fig. 1) is similar to the one used in literature [17], with the following differences. First, the current system has a torsional spring at pivot P that provides a restoring torque to the follower instead of a translational spring used in literature. Unlike the previous system, the cam profile is perfectly circular (whose eccentricity is denoted by φ), and there is no dowel pin attached to the follower in the current system. Finally, the angular displacements of cam (θ ) and follower (α) are measured positive in the counter-clockwise direction with respect toê x . Refer to Sundar et. al. [17] for a detailed description of the system, and its kinematic formulations. The loss of contact between cam and follower introduces the clearance nonlinearity to the system. The effects of kinematic nonlinearity are minimal [21], and friction nonlinearity can be ignored due to negligible moment arm (further discussed in Sect. 5.2). The system's dynamics is majorly governed by the clearance nonlinearity.
The following were the major assumptions that helped in the development of the mathematical model: (1) the stiffness (κ) of torsional spring is linear; (2) the follower is rigid (due to high flexural modulus and low bending moment); (3) the pivots P and E have friction-less bearings; (4) Hertzian contact is used for modeling the mechanics of impact (as it is more appropriate than concept of coefficient of restitution [21]). Wherein, a line contact with linear contact stiffness of K and linear viscous contact damping coefficient C is assumed between the cam and follower. Note that the contact constraint between the cam and follower is non-holonomic. However, since the motion of cam and follower is restricted to be on the same plane (ê x −ê y , as shown in Fig. 1) the relative motion at the contact is also along theê x −ê y plane. In other words, there is no motion along theê z direction. Hence, the non-holonomic constraint of the rolling-sliding contact is accounted in the formulation. Adopting an approach developed earlier [21,28], the relative position vector of contact point O c on the cam is given by − − → Q O c = ψ iî + ψ jĵ , where ψ i and ψ j are local coordinates. Closed-form solutions for ψ i (t) and ψ j (t) and their time-derivatives are given by he following equations: along theê x andê y , respectively. χ 0 is the distance of contact point O b from pivot P along follower length and is constant [21].
− → P E is also constant.

Governing equation of motion
By performing the moment balance about P, the equation of motion is, here the contact normal force F N is given by, The friction force F f (t) is given by μF N (t) sgn(v r (t)), and Here, m b and I b denote the mass and mass moment of inertia of follower about P, respectively. Please refer Appendix A for the detailed nomenclature. Equation 5 is solved piece-wise to avoid the discontinuity arising out of clearance while integrating numerically, which helps in accurate estimation of the system response (α(t),α(t)).
A reference state at α = θ = 0 • is defined as shown in Fig. 2 to specify the nominal clearance (backlash, b) for the system. The backlash (b) is the gap between cam and follower in this reference state. Going forward, response of the system is characterized using a new variable q(t), which represents the relative displacement between the cam and follower at the contact points alongê y .
Here, (t) = φsin(θ (t)) denotes the instantaneous displacement of O c alongê y , which eventually is the motion input to the follower. Note that, the positive value of (t) implies the positive displacement of cam center G alongê y . Further, as per definition q is zero in the reference state. For normalization, considering small angle approximation for α(t) and ignoring the variation in χ(t) (compound to its mean value) and also neglecting friction force, Eq. 5 can be written in terms of relative displacement q(t) as, Eqution 9, which is a simplified version of Eq. 5, has the excitation force terms clubbed in to mean (F m ) and alternating components (F a ). The mean force is primarily controlled by the preload (α p ), while the amplitude of alternating force is determined by φ. The following normalized parameters are used to describe the system: Normalized mean load,F m = F m /(b · K ); Normalized alternating load,F a = φ/b; Load ratio,F =F m /F a ; Normalized cam speed,¯ c = c /ω n ; Normalized relative displacement,q = q/b; Normalized relative velocity,q =q/(b · ω n ) where, ω n = κ/χ 2 +K I b /χ 2 is the natural frequency of the system in contact. Other key variables used for the purpose of this study are based on [17,28] and their values are listed in Table. 1. Note that Eq. 9 would be used to define the system using a set of non-dimensional parameters (F m ,F,¯ c , ζ ); however, response of the system was obtained using the original equation of motion (Eq. 5) and then post-processed to obtain q(t) for the entire study.

Transient response of the system
The system can be defined by the set of parameters consisting operating conditions [F m ,F, ζ, μ,¯ c ] and initial conditions [q 0 ,q 0 ]. A typical responseq(t) of the system obtained withF m = 0.1,F = 0.5, ζ = 0.02, μ = 0 and¯ c = 0.9 and initial conditions (q 0 ,q 0 ) = (1, −1) is shown in Fig. 3. It was observed that the system's response initially has some transience (till about 0.03 s) which eventually dies out and the system attains a steady state. To study the behavior of a system in transience, it is essential to demarcate the transient and steady-state phases of the system's response. This section addresses the methods to quantify the duration of transience and analyze the transient response of the system.

Estimating the duration of transience
Duration of transient response can be quantified by estimating the time needed for the system to attain steady state (t ss ). Since the system reaches the steady-state asymptotically, a couple of methods have been proposed to logically estimate t ss for a given set of parameters and operating conditions.

Method 1: By tracking the periodic state of the system
The variables (q,q) completely define the instantaneous state of system, and for the steady-state, they should repeat after a specific time period, T . Mathematically, they follow repeatability condition for t ≥ t ss as: Here, T should be an integer multiple of the time period of cam (T c ), which is used to estimate t ss . After obtaining the response of the system for an extended duration, the iterative process of evaluating t ss starts with an initial guess t guess = t end . The repeatability condition (eq. 10) was checked for different values of t guess as it was decreased. The least value of t guess which satisfies (within allowable tolerances) eq. 10 was taken as t ss .
Though this method is computationally efficient, the error associated with the numerical integration of eq. 5 can cause some inaccuracy in the evaluation of the state variables. This problem can be minimized by using smaller tolerances, which, however, demands higher computational power.

Method 2: Moving window method
Another iterative method to estimate t ss compares a time window of the response to a known steadystate response. Once the system response till steadystate is attained and the time period of oscillation T was extracted. Steady-state signal for one period was considered as a reference and compared with signal from the moving window of same duration. The window was moved backward in time, starting with W indow(t guess ) = [t end − T, t end ] and reducing the t guess until the mean of response is within 5% of the mean of the reference signal. This method is more robust compared to the first method as it is based on a global indicator (average over a time period) as opposed to a local indicator (instantaneous value). However, since the method is based on comparison of signals, it has a higher computational cost. Figure 4 shows the estimation oft ss = t ss /T c for variousq(0) using both the methods which follow a similar trend.

Effect of initial conditions on the t ss
Variation of t ss with initial condition is shown in Fig.  5. As observed, no clear trend or relation between t ss and the initial displacement (q 0 ) could be found (Fig.  5a). However, t ss increased with the increase in magnitude of initial velocity magnitude, |q 0 | (Fig. 5b). As the system gets higher initial kinetic energy, the transient phase gets extended.

Time-frequency analysis of the transient response
The time-frequency characteristics of the transient responses were studied using the help of a short-time Fourier transform ofq. Instantaneous frequency distribution of the response was observed from the spectrograms ( Fig. 6) for¯ c = 0.6 with various initial conditions. It was inferred that, all the types of solution had time-varying DC component and few alternating components. Although it is obvious that different initial conditions cause different transience (as noted by the frequency contents and amplitudes of the spectrogram), it was interesting to note that some of these different transient phases lead to the same steady-state response. For instance, Fig. 6b and c has different transience with same steady-state response evident from the same constant harmonic amplitudes in the later part of the spectrogram. For example, system response has a clear discontinuity between the transient and steady-state phases (as seen in Fig. 6a) and eventually has a constant harmonic amplitude of about 0.7 at¯ c = 1. However, Fig. 6b, c has overlap of the two phases and eventually attained the same steady-state amplitude of 0.7. In the final plot 6d, a different steady-state solution was obtained altogether with distinguished higher amplitude at the normalized frequency of 0.5, indicating that steady-state solution is of period 2, along with associated different transience. It is also noteworthy from the spectrograms that all the transient responses have very high harmonic amplitude initially about the frequency of 0.5.

Energy variation during transience
The initial conditions determine the initial energy of the system. The time history of the system's total energy (kinetic+potential) at¯ c = 0.6 is depicted in Fig. 7, from which the role of initial energy can be comprehended. Ten different initial conditions resulted in only two distinct steady-state energy levels corresponding to period-1 and period-2 solutions. It can further be inferred that the steady-state energy of the system does not have any correlation with the initial energy. In other words, the response which has a lower initial energy can have a higher steady-state energy level and vice versa.

Role of initial conditions
The steady-state responses were observed qualitatively at each¯ c , using a set of initial conditions (q 0 ,q 0 ), obtained by combination of 21 equally distributed values forq 0 andq 0 within [-2,2]. This set of 21×21 initial conditions broadly results in a couple of steady-state response types, namely impacting and non-impacting. However, the impacting type of response can be further classified based on its periodicity (n). Figure 8 graphically presents the domains of attractors corresponding to each type of solution atF m = 0.1,F = 0.5, μ = 0, and ζ = 0.02 for different¯ c . At lower excitation frequencies, such as at¯ c = 0.3 all initial conditions led to the same solution as the follower always remained in contact (n = 1) with the cam (Fig. 8a). However, upon increasing the excitation frequency to¯ c = 0.4, period-1 impacting (n = 1) solutions were also found to coexist with period-1 non-impacting solutions (Fig.  8b). Upon further increasing the cam speed (¯ c ) to 0.5, the number of initial conditions leading to the impacttype response increased, indicating the increase in system's affinity toward the impacting response. Finally, at higher¯ c = 0.9, only impact-type solutions with higher periodicities (n = 2 & n = 3) were observed.
Here, cam completes two and three rotations, respectively, within every impact of follower. This suggests that periodicity of the solutions increases with¯ c . It can be easily concluded that the initial conditions play a critical role in determining the type of steady-state response at each cam speed for a given set of operating conditions.

Frequency responses
Steady-state response is quantitatively analyzed by plotting zero-to-peak amplitude ofq (q z ) vs.¯ c for various initial conditions on a frequency response plot (Fig. 9). As the instantaneous gap, at the contact point between the cam and the follower, is given by b − q(t), the ordinate of Fig. 9 indicates the zero-to-peak value of the gap. As the excitation frequency¯ c was increased from 0.05 to 1.5 in the steps of 0.05, all the responses were observed to be periodic (T = nT c ). The solutions corresponding to each periodicity followed a continuous trend (bounded by certain values¯ c ), hereafter called as a 'branch' of the solutions or responses (marked by the same color in the figure). In addition, all the solutions corresponding to a particular branch have similar topological phase portraits. For instance, the first branch of solutions (represented by blue in the figure) at lower cam speeds (¯ c = 0.05 to 0.55) is characterized by period-1 non-impacting responses. The phase portraits (q vsq) for these type of solutions were observed to be elliptical, denoted as type-1 (Fig. 10a, b). Note that the value ofq in these phase portraits is always greater than one, indicating that system is always in contact. Also, as¯ c increases from 0.3 to 0.4, indentation between cam & follower increases which in-turn leads to increase in the amplitude ofq(t); thus,q z increases as seen from Fig. 9 for this branch of solution.
The second branch (given in red) of solutions observed from (¯ c = 0.35 to 0.75) has period-1 solution, but with follower loosing contact with the cam and having higher amplitude compared to the previ-  ous branch. Type-2 phase portraits were observed (10c) for this (n = 1 impacting) branch. The left end (noncontact side) of the phase portraits has a sharp edge (smaller radius of curvature) compared to right end due to the loss of contact. However, with increase in c theq z of the solution reduced and the left end of the phase portrait became less sharper (Fig. 10d). The loss of contact between the cam and the follower can also visualized in terms of a bifurcation diagram as shown in Fig. 11 which compares the amplitude of the vertical oscillation of contact point of the cam with that of the follower as¯ c increases. At lower cam speeds (¯ c ≤ 0.3), both the amplitudes are essentially the same (contact phase); however, they start to bifurcate as the cam speeds increase.
Period-2 responses (n = 2 impacting, yellow color branch in the figure) were observed from¯ c = 0.6 to 1. For this branch of solution, follower impacts once in every two cycles of cam rotation. In this region, type-3 phase portraits were observed (10e, 10f). Here, the sharp left edge of the phase portraits slowly built up into a small inner loop as¯ c was increased (10f). This inner loop is formed because, while the follower is not in contact with the cam, the cam completes one extra rotation. This causes a significant reduction in the amplitude of q ( (t) term in Eq. 8).
Next branch of solutions (purple color in the figure) was observed in the range of¯ c = 0.85 to 1.25, corresponding to period-3 (n = 3 impacting) responses. In this case, the follower impacts once for every three rotations of the cam. The phase portraits (type-4) asso- Fig. 7 Time history of the total energy of the system for different initial conditions at¯ c = 0.6 (F m = 0.1,F = 0.5, ζ = 0.02, μ = 0) ciated with these solutions were dumbbell-like shaped (10g), with a small dent on the top and bottom which are formed due to two extra periods during the non-contact duration. These dents became more pronounced as thē c was increased (Fig. 10h). Period-4 solutions were observed from¯ c = 1.15 to 1.5 as the next branch (green color in figure). Here, follower impacts once in every four cam rotations. Phase portraits for this n = 4 impacting branch were similar to a combination of types 3 & 4, and these are called type-5 phase portraits (10i). This behavior of two dents and a small internal loop can be explained by three extra periods available during loss of contact regime. Finally, period-5 impacting responses were obtained for¯ c >1.4, and here the phase portraits were double dumbbell-shaped (type-6). Follower in this case impacts once in five rotations of cam. Four dents were observed on the phase portraits due to the extra four periods spent out of contact (Fig. 10j).
For each branch of the solutions corresponding to impacting type (type-2 to type-6),q z decreased as¯ c increased. This is physically explained by the reduction of T c with an increase in¯ c , which means the follower has to return to impact with the cam quickly and thus cannot afford a larger amplitude. With increase in c , if it is infeasible to retain the periodicity, the system moves to the solution with the next higher period. However, not all solutions could be categorized under a certain branch in the frequency response plot. These solutions generally had higher periodicity as compared to the other solutions observed at a given¯ c . Further, their phase portraits have multiple inner loops (Fig.  10k & 10l). Thus, these higher period responses with multiple inner loops are termed as 'outlier' solutions (marked in black).
It is to be noted that efforts were made to characterize all the steady-state solutions observed for¯ c ≤ 1.5 (within the range of parameters considered for the study) based on their periodicity and phase portraits. Solutions with periodicity as high as 12 were observed in certain cases. In addition, in this range excitation speed, solutions with n=1 represent one cam rotation per impact, while solutions with n>1 represent more than one cam rotations per impact (sub-harmonic response). However, responses with more than one impact for per rotation of the cam (super-harmonic response) were not observed. However, upon investigating response at much higher cam speeds, nonperiodic chaotic solutions were observed for some cases. For instance, Fig. 12 shows time history and phase portrait of a non-periodic chaotic response at c = 2 . The remaining studies are performed for 0 ≤¯ c ≤ 1.5 only and thus only solutions with periodicity up to 12 were observed.
Discussion on largest Lyapunov exponent: The largest Lyapunov exponent (LLE) of the system is calculated for various¯ c within 1.5 (as given in Appendix C) to check for the possibility of any chaotic solutions. As LLEs for all the cases are negative, it can be con- cluded that only non-chaotic solutions exist within this speed range.

Affinity toward a steady-state solution
Affinity in this context is defined as the percentage of initial condition (total 21 2 of them) that would converge to a particular steady-state solution for a given¯ c . This is indicated by the number close to the solution in Fig. 9. It was observed in every given branch that, the affinity is highest near the middle of the branch and it gradually decreases toward both ends similar to normal distribution (Fig. 13). Further, the affinity for most outlier solutions was much lower than those which fall under certain branches, which shows the tendency of the system to have simpler solutions with lower periodicity.

Harmonic analysis
The frequency response of the system (q) was plotted based on its harmonic contents (Fig. 14). The fundamental harmonic of all the branches had significant amplitude, thus indicating that all the impacting steadystate responses irrespective of their periodicity have an appreciable first harmonic component. This also indicates a considerable contribution of the cam motion (in terms of φ and¯ c ) to q(t). In the 2 nd and 3 rd harmonics, the period-1 impacting response has a higher amplitude compared to other branches. The maximum amplitude for each individual branch decreases as with super harmonics (14a-c).
The amplitude corresponding to the nth sub-harmonic ( c / = n) is denoted by q a1/n . In the sub-harmonic amplitudes of q a1/2 , n = 2&4 branches have higher amplitudes, while the amplitudes corresponding to other branches are negligible (Fig. 15a). The outlier solutions whose periods are integral multiple of 2 also have significant sub-harmonic amplitude here. Along the same lines, at q a1/3 , the period-3 branch has higher amplitude, while the amplitudes of other branch amplitudes are insignificant. However, some period-6 (outlier) solutions were also seen (Fig. 15b). A similar trend was observed for q a1/4 ,q a1/5 sub-harmonics, where period-4 and period-5 branches have higher amplitudes, respectively (Fig. 14), compared to other branches. In general, it can be inferred that solutions with a period n and its integer multiple have significant sub-harmonic (1/n) amplitude. Since frequency response withq z portraited the nonlinear behavior of the system well, it was used in further analyses.   Figure 16 gives the frequency response of the system withF = 2 while retaining other parameters as constant to study the effect decrease in the alternating load. It was observed that a decrease inF a caused a reduction inq z at each¯ c , although the phase portraits remained qualitatively unchanged. The period-1 no-impact branch extended till¯ c = 0.75, as compared to the system withF = 0.5. Moreover, all the branches with impacting steady-state solutions started at marginally higher¯ c , and extended significantly, thus increasing the overall span of excitation frequency. Eventually, the period-5 solutions did not appear till The effect of varyingF in the frequency response was further analyzed in Fig. 17. ForF = 0.5, period-1 to period-5 solutions were observed within¯ c ≤ 1.5. However, forF = 1, solutions up to period-4 were observed within the same speed range. This further reduced to up to period-3 solutions forF = 4, and finally, only period-1 solutions could be observed upon increasingF to 10. It was also observed that the amplitude ofq z and the number of outlier solutions gradually reduced with an increase inF. Next, the region of overlap between the period-1 non-impacting solutions and period-1 impacting solutions was observed. Interestingly forF=10, the period-1 no-impact branch reappears at¯ = [1.1, 1.5], after the period-1 impacting branch. In general, it could be inferred that the system had a tendency toward solutions of lower period asF was increased and eventually started having a linear system-like behavior at sufficiently highF. The physical reason behind these characteristics is that as F a decreases, the external energy imparted to the follower decreases. Thus, either the amplitude decreases or higher excitation frequency is needed to obtain a certain periodic solution.
Another effect of loading was investigated by changing theF m while keeping other parameters constant (Fig. 18). AtF m = 0.05; period-1 to period-4 solutions along with some outlier solutions were observed for a given range of¯ c . The overlap region (between period-1 non-impacting and period-1 impacting solutions) was¯ c = [0.45, 0.6]. AsF m was increased to 0.1, only solutions up to period-3 and fewer outlier solutions were noticed. The region of overlap was c = [0.6, 0.75]. Similarly, for higherF m values, the higher period solutions cease to exist and the overlap region moved to higher speed ranges. For higher mean loads (F m ≥ 0.2), period-1 non-impacting solutions reappeared at higher excitation frequencies. In essence, these characteristics of the system can be associated with fact that the increase in mean load reduces the tendency of the system to lose contact.
The following observations were made by comparing the frequency responses of the system with an increase in the damping ratio (ζ = 2.5%, 5%, 7.5%& 10%). First, the solutions with higher periodicity occur only at higher¯ c (Refer Fig. 21). Next, the maximum amplitude of each branch of solution gradually decreases. The number of outlier solutions (occurring between the branches quickly vanishes). Further, the region of overlap shifts toward higher¯ c , while the span of the overlap decreases. The rationale behind these observations is, increase in damping ratio causes the system to lose more energy at every impact. Hence, the amplitude of oscillations is lower, and hence, solutions with lower periodicity are preferred.
In summary, the system qualitatively had similar behavior for the reduction in F a , increase in F m , and increase in ζ as noticed from the following: a) Reduction inq z ; b) Preference of the system toward solu-  tions with lower periodicity; d) Increase in the upper frequency limit of a given branch; and e) Shift of the frequency response analogous to that of a linear system.

Effect of friction
The contribution of the contact friction to the system dynamics was evaluated by accounting for the frictional force. Figure 19 shows the system's response when μ = 0.2. The amplitude of all the solutions was the same as that of the system without friction. In fact, even the phase portraits for both systems were similar. It is to be noted that the system without friction had three more solutions (at¯ c = 0.8, 0.85 & 1.5); however, their affinity values were very low (7%,8%,1%, respectively). Thus, friction had a negligible effect, which can be explained by the small moment arm (w b /2) of the friction force about the pivot in Eq. 5. Since almost all significant solutions with friction could be predicted without friction, it was neglected in most of the discussions in this article.

Steady speed sweep
Even though the analyses in previous sections considered the steady-state response at a given¯ c , many reallife systems reach excitation frequency only slowly. In other words, their frequency of excitation continuously changes during the operation. A steady speed sweep of the cam has been carried out to study the start-up and shut-down phenomena. The dynamics of the system was simulated for¯ c increasing at a constant rate of 0.1 per second till it reached 1.5. This result was overlaid (in maroon color) on the steady-state result obtained previously (Sect. Fig. 20a. During this steady sweep, the system had negligible duration of initial transience (≈ 0.02 s). For lower¯ c , the response closely followed the period-1 non-impact branch. Around¯ c = 0.57, the response jumped to period-1 impacting branch and kept following it till¯ c = 0.75. Then, the response jumped to the next higher period branch. The same trend continues as the excitation frequency increases till the system has period-4 solutions at¯ c = 1.5 as seen in the figure. The system follows the steady-state branches; however, at the end of each branch, a small diversion is observed (like an auxiliary branch), where the solutions had double the periodicity of the previous branch. Further, in some cases, this auxiliary branch passes over the nearest outlier solution. It was interesting to note that frequency at which the jump from one branch to other happens matched with the highest affinity point of the target branch. Similarly, the effect of decreasing the cam speed at a constant rate is depicted in Fig. 20b (shown by magenta color). Initial¯ c of 1.5 was reduced to 0 in over 15 seconds. After a very small initial transience, the solution started following the period-4 branch. As opposed to the start-up case, with the reduction in¯ c , the solution overshot the branch considerably before jumping to period-3 branch at¯ c = 1.06. The system then followed the period-3 and overshot. The same trend was followed till¯ c was reduced till 0. The jump to lower period branch happens when the affinity of the target branch is the highest (similar to the start-up analysis). It is interesting to note that the rate of change of the cam speed (˙ c ) had a negligible effect on the start-up and shut-down characteristics. Although initial conditions were very critical in steady-state response, they do not have any role in the start-up and shut-down responses.

Conclusion
In this work, a cam-follower system was formulated and normalized to control operating parameters effectively. A couple of methods to evaluate the duration of transient phase were proposed. Time-frequency analysis and quantification of average energy were performed to characterize the transient behavior. It was found that multiple transiences with different initial conditions and energy can lead to the same steady-state solution. The effect of initial conditions on the steadystate response was then studied based on domains of attractors. Frequency responses were analyzed in detail and the responses were categorized into multiple branches based on periodicity and similarity of phaseportrait topology. The sub-and super-harmonic amplitudes of all the branches of solutions were also analyzed. Further, the effects of parameters such as load, damping, and friction on the steady-state response were then analyzed. Finally, the system's start-up and shutdown characteristics were obtained, while the jump was found to be closely related to the affinity value of the solution.
The major contributions from this study are: (a) characterization of the transient response of the system based on instantaneous energy and spectrograms; (b) qualitative classification of the steady-state solutions into various branches based on the topology of the phase portrait along with physical explanation; c) quantitative analysis of the harmonic contents of the steady-state solutions using frequency response plots; d) estimation of the constant speed start-up and shutdown characteristics & relating the point of transition to the affinity of the solution. The outcome of the study would be used to understand the systems with clearance nonlinearity. As a follow-up work, semi-analytical solutions to the system can be sought. Further, this study can be extended to gears and brake systems with similar contact mechanics but with multiple clearances.