**2. Numerical simulation**

Follower displacement is calculated using SolidWorks software [9]. The follower moved with three degrees of freedom. Four values of the follower guide's from inside (I.D. = 16, 17, 18, 19 mm) at different cam speeds are used. The follower with the offset (O = 20, 30, 40, 50 mm) are chosen. The impact coefficient of restitution with the values (0.2, 0.3, and 0.4) is considered in the calculation of nonlinear response of the follower in the presence of follower offset. Cam follower mechanism is shown in **Figure 1**.

**Figure 1.** *Polydyne cam with an offset flat-faced follower.*

*Nonlinear Dynamics Phenomenon in a Polydyne Cam with an Offset Flat Faced Follower… DOI: http://dx.doi.org/10.5772/intechopen.106179*

**Figure 2.** *Nonlinear response mapping when the follower offsets to the right (O = 10 mm).*

The chaotic phenomenon in cam follower system is increased with the increasing of impact coefficient of restitution in which the impact will happen due the loss in potential energy of the follower and due to the increase in follower guide clearance value. **Figures 2** and **3** show the mapping of nonlinear response of the follower at

### **Figure 3.**

*Nonlinear response mapping when the follower offsets to the left (O = 10 mm).*

different cam speeds, different follower guides' clearances, and different impact coefficient of restitution when the follower offsets to the right and left respectively (O = 10 mm). The nonlinear response of the follower is periodic as shown in **Figure 2a** and both the cam and the follower are in permanent contact. The follower lost the contact with the cam at time (t = 13.58 s) and (t = 15.99 s) at detachment height

*Nonlinear Dynamics Phenomenon in a Polydyne Cam with an Offset Flat Faced Follower… DOI: http://dx.doi.org/10.5772/intechopen.106179*

(26.98 mm) and (27.43 mm) respectively. Due to the coefficient of restitution, the follower keep bouncing from the cam from (t = 0.36 s) to (t = 5.658 s) while the follower will regain energy and keep permanent contact with the cam for the period from (t = 9.208 s) to (t = 10.11 s) which is having a periodic motion as illustrated in **Figure 2b**. The chaotic motion is shown in **Figure 2c**–**f** which increased with the increasing of follower guides' clearances, cam speeds, and coefficient of restitution. There is an intangible impact when the coefficient of restitution (0.2) and the dissipation in potential energy is occurred due to sliding while the contact is still valid between the cam and the follower, as shown in **Figure 3a**. The periodic and chaotic motion is together shown in **Figure 3b** and **c**. The periodic motion is shown from the period (t = 6.1 s) to (t = 10.26 s) and from the period (t = 14.14 s) to (t = 19.55 s) as shown in **Figure 3b** while the periodic motion begins from the period (t = 1.264 s) to (t = 3.808 s) as shown in **Figure 3c**. The chaotic motion is shown in **Figure 3d**–**f**.

**Figure 4.** *Bifurcation diagram against cam speeds.*

**Figure 5.** *Bifurcation diagram when the follower offsets to the left (O = 50 mm).*
