**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]:

$$\mathbf{X\_{c}(t)} = \mathbf{e^{-\beta t}} \left( \mathbf{C\_{1}Sim} \left( \sqrt{\alpha\_{\mathrm{n}}^{2} - \beta^{2}} \,\mathrm{t} \right) + \mathbf{C\_{2}Cons} \left( \sqrt{\alpha\_{\mathrm{n}}^{2} - \beta^{2}} \,\mathrm{t} \right) \right)$$

$$+ \frac{\mathbf{X\_{st}}}{\mathrm{H}} \left[ \mathbf{1} - \left( \frac{\mathfrak{Q}^{2}}{\alpha\_{\mathrm{n}}^{2}} \right) \right] \mathrm{Sim}(\mathfrak{\Omega t}) - \frac{\mathbf{X\_{st}}}{\mathrm{H}} \,\frac{2 \beta \mathfrak{Q}}{\alpha\_{\mathrm{n}}^{2}} \mathrm{Cov}(\mathfrak{\Omega t}) \tag{1}$$

Where:

$$\frac{\text{c}}{\text{m}} = \text{2} \mathfrak{P} \tag{2}$$

$$\frac{\mathbf{k}}{\mathbf{m}} = \alpha\_{\mathbf{n}}^{2} \tag{3}$$

$$\mathbf{P} = \frac{\mathbf{F}}{\mathbf{m}}\tag{4}$$

$$\mathbf{H} = \left[\mathbf{1} - \left(\frac{\Omega^2}{\alpha\_\mathbf{n}^2}\right)\right]^2 + \frac{4\beta^2\Omega^2}{\alpha\_\mathbf{n}^4} \tag{5}$$

$$\mathbf{X\_{st}} = \frac{\mathbf{P}}{\mathbf{m}\mathbf{o\_n^2}} \tag{6}$$

*Effect of Different Material Properties on the Nonlinear Dynamics Phenomenon… DOI: http://dx.doi.org/10.5772/intechopen.112795*

**Figure 1.** *Cam-follower system with its dimensions.*

*Chaos Monitoring in Dynamic Systems – Analysis and Applications*

$$\mathbf{C\_1} = \frac{\mathbf{Q} \mathbf{X\_{st}} \left[ \frac{\Omega^2}{\alpha\_\mathbf{n}^2} + \frac{2\mathfrak{P}^2}{\alpha\_\mathbf{n}^2} - \mathbf{1} \right]}{\mathbf{H} \sqrt{\alpha\_\mathbf{n}^2 - \mathfrak{P}^2}} \tag{7}$$

$$\mathbf{C}\_{2} = \frac{2\mathbf{X}\_{\text{st}} \| \Omega}{\text{H} \mathbf{o}\_{\text{n}}^{2}} \tag{8}$$

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

$$\mathbf{P\_c = \frac{\mathbf{1}}{\mathrm{Cov}(\phi)} \left[ \mathbf{K} (\Delta + \mathbf{X\_c(t)}) - \mathbf{K} \mathbf{X\_c(t)} - \mathbf{m} \ddot{\mathbf{X\_c(t)}} \right] \tag{9}$$

$$\tan\left(\phi\right) = \frac{\dot{\mathbf{X}}\_{\text{c}}(\mathbf{t})}{\mathbf{X}\_{\text{c}}(\mathbf{t}) + \mathbf{R}\_{\text{b}}^{2}}\tag{10}$$

In which Kð Þ Δ þ Xcð Þt 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

*Effect of Different Material Properties on the Nonlinear Dynamics Phenomenon… DOI: http://dx.doi.org/10.5772/intechopen.112795*

**Figure 4.** *Poincare' map when the contact condition is nylon for follower guide's clearance (16.10-3 mm).*

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 camfollower with its dimensions.

Phase-plane diagram shows how the attractor of the follower displacementvelocity 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
