Lyapunov Exponent for a Globoidal Cam With a Roller Follower Mechanism Using Wolf Algorithm

A globoidal cam and roller follower system is analyzed and discussed for non-periodic motion of the follower.  Wolf algorithm is used to calculate the largest Lyapunov exponent. The globoidal  cam with roller follower mechanism was extensively used in paper processing machines and automatic assembly lines. The dynamic analysis has been done by simulating the follower displacement in the y-direction at uniform cam angular velocities. Many attempts have been conducted to study the chaos phenomenon in cam-follower mechanism to keep the cam and follower in permanent contact. The nonlinear dynamic tool of computer algorithm is used to determine the time delay and the embedding dimension. The first minimum of the Average Mutual Information (AMI) algorithm is chosen to calculate the time delay. The global dimensions has been selected when the total percentage of neighboring trajectories reaches zero by using Global False Nearest Neighbors (GFNN)  algorithm. The impact between the cam, follower and the two guides is taken into consideration based on the contact parameters. The contact between the cam, follower and the two guides is occurred due to the impulse and momentum phenomenon. The effect of the dimension of the follower guide on the follower is considered at distinct angular velocity of the cam. The largest Lyapunov exponent is compared for distinct follower guide’s dimension such as (16, 17, 18, and 19 mm). Positive value of Lyapunov exponent indicates a non-periodic motion and chaos for the system. The numerical simulation has been done using SolidWorks software. In simulation process, the follower with three degrees of freedom is considered in which it has translation in (x,y) directions and rotation about (z-axis). Follower movement is processed experimentally through an infrared 3-D camera device with a high precision optical sensor. Phase plane diagram is used to explain the variation in follower motion. The phase-plane diagram is different with the increasing of follower guide’s dimension and cam angular velocity. Power spectrum analysis based on Fast Fourier Transform (FFT) is conduct to illustrate the non-periodic motion of the follower. The follower motion is non-periodic when the orbit of phase-plane diagram diverges with no limit of spiral cycles. The frequencies' peaks are disappeared gradually with the increasing of cam angular velocities and follower guides dimension. In general, all the values of largest Lyapunov exponent parameter is positive so that gives an idea to non-periodic motion. Largest Lyapunov exponent parameter is increased with the increasing of follower guide’s dimension and cam angular velocity.