Page 370 - Cam Design Handbook

THB12 9/19/03 7:34 PM Page 358

358 CAM DESIGN HANDBOOK

Models with a relatively rigid camshaft are the most general. Models with an elastic
camshaft are an occasional requirement. Models with an elastic camshaft are coupled to
the follower resulting in nonlinear second-order differential equations, since the cam
movement on the shaft is not a direct function of time.
Multi-degree-of-freedom systems are shown, but note that they are shown to be com-
plicated and difficult to apply, time-consuming, and thus costly. It has been found that the
one-degree-of-freedom model is satisfactory as long as the excitations of the second and
higher modes are much less than those near the first mode, which is usually true. The rep-
resentation of a cam-follower mechanism by means of a single-degree-of-freedom model
having a natural frequency equal to the fundamental natural frequency of the actual system
is satisfactory. The fundamental natural frequency dominates the transient follower
responses, which are treated in Chap. 13. Most researchers involved with the dynamics of
cam-follower systems use linear one-degree-of-freedom spring-mass models with either
open or closed-track cams for constraint of the follower. The closed-track cam is pre-
dominantly utilized for production machinery, and the open-track cam (spring force
loaded) is specifically used for automotive valve gear systems. Nonlinear stiffness models
have rarely been used, since the nonlinear component of the stiffness is usually relatively
small. In addition, nonlinearities caused by passive parameters are not addressed in this
book. These nonlinearities are damping of various kinds (e.g., Coulomb friction, quadratic
damping, stiction, and combinations), backlash, and tolerances in mechanical components.
The nonlinear models must be solved numerically with Runge-Kutta and Adams Bulirsch-
Stoer algorithms. The Runge-Kutta Methods are the most popular; for other choices. Chen
(1982), Chen and Polvanich (1975a,b), and Koster (1970) have employed computer sim-
ulations using models of complex cam systems including multiple nonlinear component
effects.
The following are the commercially available computer programs:
TK Solver, Universal Technical Services
1220 Rock Street
Rockford, Ill. 61101
www.uts.com
Mathcad, Mathsoft, Inc.
101 Main Street
Cambridge, Mass. 02142
www.mathsoft.com
MATLAB/SIMULINK, The Mathworks, Inc.
2W Prime Parkway
Natick, Mass. 01760
www.mathworks.com

12.2 SYSTEM VIBRATIONS

This section looks at the practical and theoretical sources of vibration that may occur in
the design and study of high-speed cam-follower mechanisms. This information is item-
ized to enhance the knowledge of the theoretical simplified modeling that is used. Although
simplified models are invaluable, the designer should not be “trapped” totally by conclu-
sions that models reveal. One should always maintain freedom of thought in the process
of designing high-speed machinery. All vibrations are not always what the model reveals