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(a) No damping
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(b) Damping 10%
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Primary response
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Harmonic curve Residual response
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Cycloidal curve
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FIGURE 131 Vibratory response characteristics of cam-follower for simple harmonic and cycloidal input, Hrones (1948) Mitchell (1950)
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132 RESPONSE IN FREQUENCY DOMAIN
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As speed increases in a cam-follower system, a rapid exchange of energy occurs within the system and the noise of operation also increases This sudden energy shift takes place between the elastic members and the masses during operation The energy shift is also visible as high follower vibration This action is termed mechanical shock and its related system response a shock response A shock is de ned as the physical manifestation of the transfer of mechanical energy from one body to another during an extremely short interval of time (see 9) The shock response spectrum or the response spectrum is one of the two most commonly used methods of analyzing mechanical shock The other method is the Fourier spectrum analysis In both cases, the time history of the transient is converted into an amplitude versus frequency picture, or spectrum Neklutin (1954) was the rst to employ this technique in the study of cam-driven systems Thus, a valuable method of expressing the dynamic response of a cam-follower system is to obtain the dynamic response spectra (DRS) of the cam s excitations A DRS is de ned as a plot of individual peak-acceleration responses of a multitude of single-degree-offreedom, mass-spring systems subject to a particular input transient The ordinate is usually acceleration, or some normalized expression relating to acceleration, while the abscissa is in terms of the system natural frequency, or the ratio of pulse duration to the system natural period Damping is a parameter, and if possible, its values should be stated; otherwise, it is usually assumed to be zero To illustrate the DRS, we start with a given input pulse and carry out a mathematical computation to obtain the response of a single-degree-of-freedom linear system subject to that input It is best to rst compute the follower acceleration as a function of time Then nd the maximum follower acceleration and plot it on a graph versus the fundamental period of the one DOF system This provides one point on the diagram By holding the damping of the system constant and varying the system s natural frequency by changing
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the stiffness of the spring, the calculations are repeated to de ne a curve called a response spectrum We can calculate these curves over a range of damping ratios to obtain a set of curves, each applicable for a different damping ratio The normalized displacement, velocity, and acceleration DRSs are de ned, respectively, as SND = SNV xm ym w n xm = ym
2 w n xm ym
SNA =
(131)
Nomenclature for these equations and the gures: wn = cam-follower system natural frequency xm = maximum displacement of output ym = cam maximum displacement ym = cam maximum velocity m = cam maximum acceleration y T1/TN = ratio of pulse width to system natural period Figure 132 shows the acceleration DRS of a modi ed trapezoidal cam with 5 percent critical damping Curves with other damping values can be added if so desired The curves peak at several values of wn From the spectra a designer can ascertain how systems of different periods relate to speci c inputs Also, it is found that damping has a greater effect on the residual vibration than on the primary response In establishing the acceptable damping factor it is usually conservative to use a value near the low end of the choices 1321 Example For illustration, consider a cam-follower system used in a high-speed automatic machine modeled with a single DOF, Chen (1982) The follower is actuated by a dwell-rise-dwell cam with a modi ed trapezoidal pro le The peak input acceleration is 500 m/sec2, and the duration of the excitation is 0015 sec If the follower linkage of the system is such that it has a natural frequency of 100 Hz, what will be the peak acceleration response of the follower during the lift stroke and during the dwell period Let us assume the damping factor is 005 1322 Solution With reference to Fig 132 with 5 percent damping and at time ratio T1/TN = 0015 100 = 15, the primary acceleration ampli cation is 298, and the residual acceleration amplication is 188 Therefore, the primary acceleration response will be 298 times the input or 1490 m/sec2, and the residual acceleration response will be 188 times the input or 940 m/sec2 If the effective mass of the follower is 40 N, the corresponding inertial load of the follower is 6075 N due to primary vibration and 3833 N due to residual vibration Next, a quantitative comparison (Chen, 1981) of the dynamic characteristics of various types of dwell-rise-dwell cam pro les will be shown Based on a single-DOF model, a
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