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1361 Introduction This section presents a systematic method of using the convolution operator to generate cam lift curves that minimize the residual vibration after a lift event of given displacement and duration The method is a mathematical transformation that can be applied to any suitable motion to produce families of cam pro les of increasing continuity For example, starting with a uniform rise curve, one sequence of modi cations leads to a family of curves whose members include parabolic and various nonelementary rise curves Another sequence starting with a simple harmonic rise curve leads to a family whose members include cycloidal and various nonelementary rise curves An alternate application of this approach enables us to design cam curves that produce extremely low residual vibration over a range of operating speeds This method was originated by Gupta and Wiederrich (1983)

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1362 Symbols a = cam curve acceleration a*(h) = acceleration curve for any lift curve with unit rise and unit rise angle A = Fourier transform of a A*(l) = Fourier transform of a*(h) d = cam rise k = wn/w R = amplitude of the residual vibrations induced by the rise curve v = cam curve velocity

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v*(h) = velocity curve for any lift curve with unit rise and unit rise angle V = Fourier transform of v V*(l) = Fourier transform of v*(h) w = an admissible cam velocity curve W = Fourier transform of admissible cam velocity curve w y = cam curve displacement y*(h) = displacement curve for any cam lift curve with unit rise and unit rise angle Y = Fourier transform of y Y*(l) = Fourier transform of y*(h) Z = dynamic error of the follower response b = cam rise angle l = parameter for Fourier transforms of functions q = cam angle of rotation w = cam rotational speed wn = natural frequency of the one degree-of-freedom elastic follower model

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1363 Basic Theory Convolution is a basic operation performed on two functions A convolution operation in the function plane corresponds to a simple multiplication of the two functions in the transform plane (Laplace, Fourier, etc) Gupta and Wiederrich (1983) showed that convolution can be useful in the design of cam motions for linear, moderately damped cam-follower systems Any rise or return cam lift curve can be modi ed to give a new lift curve of the required rise and duration with signi cantly reduced residual follower vibration Minimizing the residual vibration is of primary importance in many applications, particularly those with long periods of dwell Knowing the residual response of the motion, one can use the principle of superposition to predict the contribution of that motion on the steady state vibration response of the system, or at least establish an upper bound on that contribution The residual response is thus a valuable indicator of the overall dynamic quality of a motion Let y*(h), v*(h), and a*(h) represent displacement, velocity, and acceleration for any cam lift curve with unit rise and unit rise angle It may be noted that v* = dv * dv * and a* = dh dh

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