Effect of Different Material Properties on the Nonlinear Dynamics Phenomenon of Cam-Follower Mechanism

Abstract

The effect of different material properties for both radial cam and translated roller follower on the nonlinear dynamics phenomenon is investigated at different cam speeds. The equations of the dynamic motion of the follower movement are derived based on the conception of mechanical vibration theory. In this chapter, we investigate the nonlinear dynamics behavior by detecting it using phase-plane diagram and Poincare’ map at different material properties and different cam speeds of the cam. The follower is moved with three degrees of freedom inside its guides and the cam is spinning about a fixed pivot. The nonlinear response of the follower is calculated using SolidWorks program at different cam speeds, different guides’ clearances, and different material properties. The nylon, acrylic, and aluminum greasy and steel dry and greasy material properties are examined for nonlinear dynamics behavior of cam-follower mechanism.

1. Introduction

The radial cam and translated roller follower mechanism can be found in windshield wiper of the car in which the oscillating motion of the wiper is controlled by the rotary motion of the cam. This kind of cam has been selected based on the irregular profile in which it has two noses and two flanks. At high speeds of the cam, the follower will jump off the cam and introduce chaotic motion at the locations of the tip of the noses. The application of the proposed mechanism is in controlling the opening and closing of valves in internal combustion engine. Nonlinear dynamics phenomenon has happened in the presence of the follower guide’s clearance. The mathematical model of transient impact, separation, and contact is described by Yang et al. using an oblique impact in cam-follower system [1]. They explained that at low speed for the cam and without the use of coefficient of restitution, the cam, and the follower are in permanent contact. Due to the nonlinear dynamics phenomenon, the largest Lyapunov exponent, the power spectrum of Fast Fourier Transform (FFT), Poincare’ map are discussed by Yousuf to investigate the detachment between the cam and the follower. The response of the follower is calculated at different coefficients of restitution, different cam speeds, and different follower Guides’ clearaances [2]. By taking into consideration the coefficient of restitution, Sundar et al. analyzed the model of nonlinear contact damping and contact stiffness of single degree of freedom system [3]. Moreover, the same group in Ref. [3] discussed the effect of rolling and sliding contact on the nonlinear dynamics phenomenon [4]. Osorio et al. studied the bifurcation of corner impact at variable cam speeds since the detachment occurred between the cam and the follower [5]. Li and Du used the model of coefficient of restitution to control and analyze the bifurcation diagram of collision constrained vibrational system. On the other hand, phase-plane diagram has been built between the displacement and velocity of the follower due to the energy dissipation outside the phase-plane envelope, while the Poincare’ map [6] shows the single and group points of black dots. The aim of this chapter is to study the effect of different contact material properties on the nonlinear dynamics phenomenon for radial cam and translated roller follower system at different cam speeds and different follower guides’ clearances.

2. Phase-plane diagram and Poincare’ map of nonlinear dynamics phenomenon

Four follower guides’ clearances (C = 16·10-3, 17·10-3, 18·10-3, and 19·10-3 mm) at different cam speeds are used in SolidWorks program. The spring between the follower and the installation table works as a secondary force actuator. The total follower displacement is shown in the following eq. [7]:

$${\mathrm{X}}_{\mathrm{c}}\left(\mathrm{t}\right)={\mathrm{e}}^{-\mathrm{\beta t}}\left({\mathrm{C}}_1\mathrm{Sin}\left(\sqrt{{\mathrm{\omega }}_{\mathrm{n}}^2-{\mathrm{\beta }}^2}\mathrm{t}\right)+{\mathrm{C}}_2\mathrm{Cos}\left(\sqrt{{\mathrm{\omega }}_{\mathrm{n}}^2-{\mathrm{\beta }}^2}\mathrm{t}\right)\right)$$

$$+\frac{{\mathrm{X}}_{\mathrm{st}}}{\mathrm{H}}\left[1-\left(\frac{{\mathrm{\Omega }}^2}{{\mathrm{\omega }}_{\mathrm{n}}^2}\right)\right]\mathrm{Sin}\left(\mathrm{\Omega t}\right)-\frac{{\mathrm{X}}_{\mathrm{st}}}{\mathrm{H}}\frac{2\mathrm{\beta \Omega }}{{\mathrm{\omega }}_{\mathrm{n}}^2}\mathrm{Cos}\left(\mathrm{\Omega t}\right)$$E1

Where:

$$\frac{\mathrm{c}}{\mathrm{m}}=2\mathrm{\beta }$$E2

$$\frac{\mathrm{k}}{\mathrm{m}}={\mathrm{\omega }}_{\mathrm{n}}^2$$E3

$$\mathrm{P}=\frac{\mathrm{F}}{\mathrm{m}}$$E4

$$\mathrm{H}={\left[1-\left(\frac{{\mathrm{\Omega }}^2}{{\mathrm{\omega }}_{\mathrm{n}}^2}\right)\right]}^2+\frac{4{\mathrm{\beta }}^2{\mathrm{\Omega }}^2}{{\mathrm{\omega }}_{\mathrm{n}}^4}$$E5

$${\mathrm{X}}_{\mathrm{st}}=\frac{\mathrm{P}}{\mathrm{m}{\mathrm{\omega }}_{\mathrm{n}}^2}$$E6

$${\mathrm{C}}_1=\frac{\mathrm{\Omega }{\mathrm{X}}_{\mathrm{st}}\left[\frac{{\mathrm{\Omega }}^2}{{\mathrm{\omega }}_{\mathrm{n}}^2}+\frac{2{\mathrm{\beta }}^2}{{\mathrm{\omega }}_{\mathrm{n}}^2}-1\right]}{\mathrm{H}\sqrt{{\mathrm{\omega }}_{\mathrm{n}}^2-{\mathrm{\beta }}^2}}$$E7

$${\mathrm{C}}_2=\frac{2{\mathrm{X}}_{\mathrm{st}}\mathrm{\beta \Omega }}{\mathrm{H}{\mathrm{\omega }}_{\mathrm{n}}^2}$$E8

The preload spring is included with the contact force as in below [8] in Eqs. (9) and (10):

$${\mathrm{P}}_{\mathrm{c}}=\frac{1}{\mathrm{Cos}\left(\mathrm{\varphi }\right)}\left[\mathrm{K}\left(\mathrm{\Delta }+{\mathrm{X}}_{\mathrm{c}}\left(\mathrm{t}\right)\right)-\mathrm{K}{\mathrm{X}}_{\mathrm{c}}\left(\mathrm{t}\right)-\mathrm{m}{\ddot{\mathrm{X}}}_{\mathrm{c}}\left(\mathrm{t}\right)\right]$$E9

$$tan\left(\mathrm{\varphi }\right)=\frac{{\dot{\mathrm{X}}}_{\mathrm{c}}\left(\mathrm{t}\right)}{{\mathrm{X}}_{\mathrm{c}}\left(\mathrm{t}\right)+{\mathrm{R}}_{\mathrm{b}}^2}$$E10

In which $\mathrm{K}\left(\mathrm{\Delta }+{\mathrm{X}}_{\mathrm{c}}\left(\mathrm{t}\right)\right)$ is the preload spring.

SolidWorks program already has the library of material properties since both the radial cam and the translated roller follower are assumed to have the same material properties at the contact point. Different material properties for the radial cam and translated roller follower such as steel greasy, steel dry, aluminum greasy, aluminum dry, nylon, and acrylic are considered in the contact model in SolidWorks program. In the future study, the different material properties at the contact point between the cam and the follower are taken into account. In other meaning both the cam and the follower have different material properties. Figure 1 shows the mechanism of cam-follower with its dimensions.

Figure 1.

Cam-follower system with its dimensions.

Phase-plane diagram shows how the attractor of the follower displacement-velocity grows or shrinks over the time at different contact conditions. Phase-plane diagram is another proof of chaotic motion alongside with Poincare’ map. The broken lines in the upper and lower surfaces in the phase-plane diagram show the effect of impact in one cycle of the cam rotation for the given follower displacement and follower velocity. The broken lines increased with the increasing of cam speeds and with the increasing of follower guides’ clearances. When the orbit of the follower displacement-velocity in state space domain forms a closed cycle, it signifies periodic motion. When the attractor of the follower displacement-velocity diverges with no limit of spiral cycles, it indicates non-periodic and chaotic motion [9]. Figure 2 shows the phase-plane mapping when the contact condition is aluminum greasy for follower guide’s clearance (16.10–3 mm) at different cam speeds. The cross-linking of the follower displacement-velocity orbits increases as the cam speeds increase starting from follower displacement (30 to 50 mm) as indicated in Figure 2b–d. The follower motion variation is increased with the increasing of follower velocity since there will be an energy dissipation outside the envelope of phase-plane diagram. The system in Figure 2a shows the quasi-periodic motion for the follower displacement since the follower starts double impact and detachment at follower displacement (30 mm) and the follower comes back to the cam at follower displacement (25 mm). The motion of the follower in Figure 2b–d shows the non-periodic motion for the follower displacement. Figure 3 shows the phase-plane mapping when the contact condition is steel greasy for follower guide’s clearance (17.10-3 mm) at different cam speeds. The multi and double impacts have occurred when the follower displacement is between (22 and 31 mm) as indicated in Figure 3a and b at (N = 100 and 300 rpm) respectively. The cross-linking of the follower displacement is increased with the increasing of cam speeds and follower guides’ clearances since there will be an energy dissipation outside the envelope of phase-plane diagram. The non-periodic motion of the follower displacement is shown in Figure 3c and d when the follower starts multi-impacts since the follower variation is increased with the increasing of cam speeds. The Poincare’ maps investigate the chaotic motion of the follower due to multi-impact in one cycle of the cam rotation [10]. Figure 4 shows the Poincare’ maps when the contact condition is nylon for follower guide’s clearance (16.10-3 mm). Poincare’ map represents that the follower motion reached the steady state (periodic motion) as shown in Figure 4a based on the single dot in Poincare’ map for the given displacement and velocity. Figure 4b reflects the periodic motion of the follower displacement at (55 and 59.5 mm) while the non-periodic motion is occurred when the black dots are stationed around one area inside Poincare’ map. The more black dots in Poincare’ maps, the more detachment heights between the cam and the follower. The non-periodic motion of the follower displacement is indicated in Figure 4d. The quasi-periodic motion of the follower displacement is shown in Figure 5a and b at follower displacement (27 mm), and (30.5 to 31 mm). The non-periodic motion of the follower displacement is shown in Figure 5d. The black dots are increased with the increasing of cam speeds and with the increasing of follower guides’ clearances. SolidWorks software is used to calculate the follower displacement and follower velocity at different contact conditions, different cam speeds, and different follower guides’ clearances. Follower guide clearances (C = 16·10-3, 17·10-3, 18·10-3, and 19·10-3 mm) are used in the simulation while the cam is rotating at constant speed (100, 300, 500, and 700 rpm). Figure 6 shows the mapping of follower displacement when the contact condition is nylon at different cam speeds and follower guide’s clearance (17.10-3 mm). The cam and the follower are in permanent contact as shown in Figure 6a in which it indicates periodic motion. The quasi-periodic motion of the follower displacement is shown in Figure 6b. The detachment of the follower is shown in Figure 6c in which it indicates non-periodic motion and chaos as shown in Figure 6d. Figure 7 shows the mapping of follower displacement when the contact condition is steel greasy at different cam speeds and follower guide’s clearance (17.10-3 mm). The cam and the follower are in permanent contact as shown in Figure 7a and b which indicates periodic motion for the follower displacement. The quasi-periodic motion of the follower displacement is shown in Figure 7c in which the detachment is occurred between the cam and the follower. Figure 7d shows the non-periodic motion and chaos for the follower displacement due to the high speed of the cam. Figure 6d indicates non-periodic motion and chaos over all the periods of time while in Figure 7d the motion of the follower displacement is divided between periodic and non-periodic motion and chaos over some periods of time. It can be concluded from all the contact conditions that the non-periodic motion starts at (N = 300 rpm) except for the contact condition of nylon material properties where the non-periodic motion starts sometimes earlier after (N = 100 rpm). All the contact conditions have periodic motion at (N = 100 rpm). Figure 8 shows the follower displacement mapping when the contact condition is steel greasy for follower guide’s clearance (17.10-3 mm) while Figure 9 shows the follower displacement mapping for follower guide’s clearance (19.10-3 mm) at different contact conditions. Figure 10 shows the follower displacement mapping when the contact condition is aluminum greasy at different guides’ clearances.

Figure 2.

Phase-plane mapping when the contact condition is aluminum greasy for follower guide’s clearance (16.10-3 mm).

Figure 3.

Phase-plane mapping when the contact condition is steel greasy for follower guide’s clearance (17.10-3 mm).

Figure 4.

Poincare’ map when the contact condition is nylon for follower guide’s clearance (16.10-3 mm).

Figure 5.

Poincare’ map when the contact condition is steel greasy for follower guide’s clearance (17.10-3 mm).

Figure 6.

Follower displacement mapping when the contact condition is nylon for follower guide’s clearance (17.10-3 mm).

Figure 7.

Follower displacement mapping when the contact condition is steel greasy for follower guide’s clearance (17.10-3 mm).

Figure 8.

Follower displacement mapping when the contact condition is steel greasy for follower guide’s clearance (17.10-3 mm).

Figure 9.

Follower displacement mapping for follower guide’s clearance (19.10-3 mm) at different contact conditions.

Figure 10.

Follower displacement mapping when the contact condition is aluminum greasy at different guides’ clearances.

3. Conclusions

Phase-plane diagram and Poincare map are used to detect the nonlinear dynamics phenomenon in radial cam and flat-faced follower mechanism at different cam speeds and different contact condition of the material properties. As stated, the level of chaos of steel greasy material in nonlinear dynamics behavior is more than the level of chaos of nylon material properties due to the contact between the cam and the follower. The peak of rise stroke is increased for steel dry and greasy while it decreases for nylon and acrylic material properties. The dwell stroke is varied and increased with the increasing of guide’s clearance of aluminum greasy material properties. The level of chaos of nonlinear dynamics of steel greasy is more than the level of chaos of aluminum greasy since this level is increased with the increasing of cam speeds.

Nomenclatures

${\mathrm{X}}_{\mathrm{c}}\left(\mathrm{t}\right)\mathit{,}{\dot{\mathrm{X}}}_{\mathrm{c}}\left(\mathrm{t}\right),{\ddot{\mathrm{X}}}_{\mathrm{c}}\left(\mathrm{t}\right)$	Linear displacement, velocity, and acceleration of the roller follower, mm, mm/s, mm/s^2.
m	Mass of follower stem, kg.
Ω	Cam angular velocity, rad/s.
${\mathrm{X}}_{\mathrm{st}}$	Static deflection, mm.
${\mathrm{\omega }}_{\mathrm{n}}^2$	Follower angular velocity, rad/s.
t	Time, s.
F	External force on the follower, N.
c	Damping coefficient, N·s/mm.
${\mathrm{P}}_{\mathrm{c}}$	Contact force between the cam and the follower, N.
K	Spring stiffness, N/mm.
${\mathrm{R}}_{\mathrm{b}}$	Radius of base circle of the cam, mm.
Δ	Preload deflection before the cam starts spinning, mm.
ϕ	Pressure angle, angle.