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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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When the rise is d and the rise angle is b, we can obtain displacement, velocity, and acceleration curves as follows: y( d , b ,q ) = d y* (q b ) d v( d , b ,q ) = v * (q b ) b d a( d , b ,q ) = 2 a* (q b ) b (1359)
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where v = dy/dq and a = dv/dq Note that y*(h) = y(1,1,h) To use the Fourier transform, we de ne v*(h) = 0 for h < 0 and h > 1 Likewise, v(d,b,q) = 0 for q < 0 and q > b The Fourier transforms of v*(h) and v(d,b,q) are de ned as
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V* (l ) = v * (h) e - ilh dh
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V ( d , b , l ) = v( d , b ,q ) e - ilq dq
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(1360)
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In a similar manner, Y*(l), A*(l), Y(d,b,l), and A(d,b,l) are de ned as the Fourier transforms of y*(h), a*(h), y(d,b,q), and a(d,b,q) respectively, although for Y*(l) and Y(d,b,q), the upper limit of the transform integral must extend to + The analog of Eq (1359) in the transform plane is as follows: Y ( d , b , l ) = dbY* (bl ) V ( d , b , l ) = dV* (bl ) d A( d , b , l ) = A* (bl ) b (1361)
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Note that Y*(l) = Y(1,1,l) The dynamic error Z is de ned as the difference between the actual and the static follower responses Let k = wn/w, where wn is the natural frequency of a one degree-offreedom elastic model of the cam-follower system and w is the rotational speed of the cam Neglecting damping, the governing equation of Z is assumed to be Z + k 2 Z = a( d , b ,q ) (1362)
The absence of a damping term in Eq 1362, required for the method developed here, is usually conservative The amplitude of the residual vibration induced by the rise curve is then R( k ) = 1 A( d , b , k ) k (1363a)
Because v is de ned to be zero for q < 0 and q > b, Eq (1363a) becomes R( k ) = V ( d , b , k ) (1363b)
The residual response spectrum is thus the Fourier spectrum of the velocity curve v(d,b,q) The area of an admissible velocity curve between q = 0 and b is equal to the follower rise d Convolution, h(q), of two functions f(q) and g(q) is de ned as h(q ) = f * g = In the transform plane, H (l ) = F(l ) G(l ) (1364b)
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