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## 13.6 USE OF THE CONVOLUTION OPERATOR TO REDUCE RESIDUAL VIBRATIONS

### 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 profiles of increasing continuity. For example, starting with a uniform rise curve, one sequence of modifications 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).

### Symbols

a = cam curve acceleration

a*( ?) = acceleration curve for any lift curve with unit rise and unit rise angle

A = Fourier transform of a

A*( ?) = Fourier transform of a*( ?)

d = cam rise

k = ? n/ ?

R = amplitude of the residual vibrations induced by the rise curve

v = cam curve velocity

v*( ?) = velocity curve for any lift curve with unit rise and unit rise angle

V = Fourier transform of v

V*( ?)...

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