The effects of damping ratio  can best be seen in Figure 9-6d, which shows a 3-D plot of forced vibration amplitude as a function of both frequency ratio and damping ratio. The addition of damping reduces the amplitude of vibration at the natural frequency, but very large damping ratios are needed to keep the output amplitude less than or equal to the input amplitude. This is much more damping than is found in cam-follower systems and most machinery. About 50 to 60% of critical damping will eliminate the resonance peak. Unfortunately,

most cam follower systems have damping ratios of less than about 10% of critical. At those damping levels, the response at resonance is about five times the static response. This will create unsustainable stresses in most systems if allowed to occur.

It is obvious that we must avoid driving an underdamped system at or near its natural frequency. One result of  operation of an underdamped cam-follower system near n can be follower jump . The system of follower ass and spring can oscillate violently at its natural frequency and leave contact with the cam. When it does reestablish contact, it may do so with severe impact loads that can quickly fail the materials.

The designer has a degree of control over resonance in that the system’s mass m and stiffness k can be tailored to move its natural frequency away from any required operating frequencies. A common rule of thumb is to design the system to have a fundamental natural frequency n at least ten times the highest forcing frequency expected in service, thus keeping all operation well below the resonance point. This is often difficult to achieve in mechanical systems. One tries to achieve the largest ratio n / f possible nevertheless. As will be shown in later chapters, the high harmonic content of cam functions can require that even higher ratios than 10 are needed in some cases.

Follower Rise Time

For cams with finite jerk having a dwell after the rise of the same order of duration as the rise, Koster [2] defines a dimensionless ratio,

where T n is the period of the follower system’s natural frequency and T r is the time to complete the rise portion of the follower motion. To minimize the effects of the transient vibrations on the system, this ratio should be as small as possible and always less than 1. A small value of  is equivalent to a small ratio of f / n. The error in acceleration of the follower due to vibrations during the dwell period will be proportional to the first power of , and the error in follower position will be proportional to the third power of , for values of  < 0.5. If the dwell’s duration is about as long or longer than that of the rise, these transient vibrations will tend to die out by the end of the dwell.[2] The next fall or rise will again provide an input to the system and cause a new transient response.

Thus, the response of a cam-follower system of this type will be dominated by the recurring transient responses rather than by the forced, or steady-state, response. It is important to adhere to the fundamental law of cam design and use cam programs with finite jerk in order to minimize these residual vibrations in the follower system. Koster [2] reports that the cycloidal and 3-4-5 polynomial programs both gave low residual vibrations in the double-dwell cam. The modified sine acceleration program will also give good results. All these have finite jerk. Other functions that offer even lower residual vibration levels will be introduced in later chapters.

Some thought and observation of equation 9.1d (p. 216) will show that we would like our system members to be both light (low m ) and stiff (high k ) to get high values for n and thus low values for . Unfortunately, the lightest materials are seldom also the stiffest. Aluminum is one-third the weight of steel but is also about a third as stiff. Titanium is about half the weight of steel but also about half as stiff. Some of the exotic composite materials such as carbon fiber/epoxy offer better stiffness-to-weight ratios, but their cost is high and processing is difficult.

Note in Figure 9-6 (p. 225) that the amplitude of vibration at large frequency ratios approaches zero for forced and one for self-excited systems. So, if the system can be brought up to speed through the resonance point without damage and then kept operating at a large frequency ratio, the vibration will be minimal, especially if it is a forced system. An example of systems designed to be run this way are large self-excited devices that must run at higher speed such as electrical power generators. Their large mass creates a lower natural frequency than their required operating speeds. They are “run up” as quickly as possible through the resonance region to avoid damage from their vibrations and “run down” quickly through resonance when stopping them. They also have the advantage of long duty cycles of constant speed operation in the safe frequency region between infrequent starts and stops.