Vibrations of cam-follower system

Livija Cveticanin

University of Novi Sad

21000, Novi Sad, Trg D., Obradovica, 6, Serbia

 

Many studies on the cam mechanisms concern the problem of vibrations. As machine speed increases the problem of vibrations of the cam mechanism has the more significant importance. The vibration level has the influence on the wear rate, noise level and service life of the cam actuated machines and devices and also to the precision operation of machines. Because of that it is important to understand the cause of vibrations and provide means to control or to minimize unwanted vibrations so that desirable system response characteristics may be predicted and obtained.

The cam mechanism may be modeled as a three mass system (leading element, cam and follower) with three degrees of freedom (displacement of the leading element, cam and follower). Due to complexity of such a model, usually, the mechanism is divided into two systems: leading element-camshaft-cam system and cam-follower system which are considered as one-degree-of-freedom systems.

The most of the papers consider the vibrations of the cam-follower system which is assumed as an oscillator with a mass and a spring. The return system of the follower contains a spring and a damper. The oscillator is excited with the function of shape of the cam which depends on the angle of rotation of the cam. This basic model is extended including the Columb friction at the rocker arm pivot and the Hertzian contact between the follower and cam. To reduce the sensitivity of the follower motion on the parameter variations the optimal design methods for cam curve are developed. Unfortunately, the results are obtained by omitting the influence of the camshaft and leading element.

For the second type of models the follower is assumed as a rigid body. The mathematical model of the system is a parametrically excited differential equation which is simplified and transformed into a differential equation with constant parameters. In these considerations the elastic properties of the follower are neglected.

In this paper the vibrations of the complete cam-mechanism with translator motion of the cam are considered. The motion of the follower as a function of cam curve and parameters of the cam-mechanism are determined. The special attention is given to analysis of the cam velocity, damping properties of the camshaft and mass ratio of the follower and cam. As an example the vibrations of the cam mechanism with polynomial cam-curve are investigated.

The mechanism consists of a leading element, an elastic cam-shaft, a heavy cam and an elastic follower. The leading force and the force of the follower act. If the shaft which connects the leading system and the cam is rigid, the model is a system with one degree of freedom. The generalized coordinate is the displacement of the cam. The cam has a profile which causes the follower to move in certain manner. The differential equation of motion is the second order, non-linear and with time variable coefficients one. For some parameters of the cam mechanism the differential equation of motion is with small parameters. Then the approximate analytic solution of the equation is obtained.

For the small velocity of cam motion in comparison the other parameters (approximately 0.5) the strong non-linear differential equation with slowly varying parameters is solved using the elliptic-Krylov-Bogolubov method. Analyzing the obtained solution it is obvious that the amplitude of vibrations depend on the velocity of cam motion and the relation between the masses of the follower and the cam. For higher velocity of cam motion the accuracy of follower motion is smaller. It is recommended the mass of the cam to be decreased in comparison to the mass of the follower. The amplitude variation depends on the cam profile, too.

If the relation between the masses of the follower and of the cam is small the differential equation of motion can be considered as an equation with small non-linearities. Adopting the Krylov-Bogolyubov method for the differential equation with strong damping term and small non-linearities the approximate analytic solution is obtained. The analytical approximate solution is compared with the theoretical motion which corresponds to a rigid cam mechanism. Based on the obtained solution the vibration properties of the cam mechanism and the motion of the follower are analyzed. It is concluded that the follower motion directly depends on the cam curve. The usual cam form is a polynomial deflection function.

The vibration properties of the system depend on the parameters of the mechanism.

The damping properties of the cam-shaft have a significant influence on the vibrations of the system. For higher values of the damping coefficient of the connecting shaft the vibrations of the mechanism are smaller and the motion of the follower differs only a bit from the projected theoretical one. Namely, for higher damping coefficient the motion of the follower is much more accurate than for smaller damping coefficient.

The mass ratio between the follower and the cam influences the follower motion. With increase of the follower mass in comparison to the mass of the cam the vibrations of the follower decrease. For high accuracy of the follower motion the mass of the follower have to be increased or the mass of the cam decreased.

The slower the cam motion the vibrations of motion are smaller. The increase of the cam velocity has an unacceptable influence on the follower motion.

Due to the fact that the working period of the cam is short, the introduced approximations in the analytical solving procedures are available. The obtained analytical solutions are in good agreement with the numerical results.

cveticanin@uns.ns.ac.yu