Page 150 - Cam Design Handbook

THB5 8/15/03 1:52 PM Page 138

138 CAM DESIGN HANDBOOK

boundary conditions x(0) = F 1 and x¢(1) = F r is expressed, not including boundary condi-
tions, using Eq. (5.25), as:

n n n
a A N j k , 2 () ()+t i b A N jk , 1 () ()+t i c A N () = F() (5.26)
t
t
jk ,
j
j
j
i
i
=
=
=
j 1 j 1 j 1
where j = 1,... , n and i = 2,... , n - 1.
These two boundary conditions are satisfied by:
n n
 AN j k , 1 () 0 () = F 1 and  AN j k , 1 () 1 () = F . (5.27)
r
j
j
=
=
j 1 j 1
5. Collect the values of initial or boundary conditions F j, (j = l,..., n) and the dif-
ferential equation at collocation points, and form the linear systems of equations, in terms
of the unknowns, A j¢ as follows (and as done earlier):
AE + AE + ... + A E = F
11 1 , 11 2 , n 1, n 1
M M M M (5.5, repeated)
AE n 1 , + A E n 2 , + ... + A E n n , = F .
n
2
1
n
Combine the equations above for the coefficients E I,k (i, j = 1, n). The first subscript iden-
tifies the initial or boundary condition or the collocation point, and the second identifies
the spline. Written, in matrix notation, the system of equations above becomes
[][ F (5.6, repeated)
EA] = [].
6. Solve the system of equations above for the coefficients, A j.
7. Evaluate the B-splines or their derivatives as needed to determine cam displacement
and vibrational responses between collocation points.
Dynamic Response of the Follower System. Regardless of the precision of manufac-
ture some departure of the actual output motion of a cam-follower system from the ideal
will occur. The departure is likely to be of most concern in demanding applications in
which the cam operates at high speed. The analysis given here addresses the vibrational
responses of the follower system as the cam drives it.
During the rise portion of the output motion response a transient vibrational response
exists. This response will be referred to as the primary vibration. This vibration may persist
into the dwell phase that follows the rise. Additionally, the transient from the rise to the
dwell may induce a response during the dwell. The departure from the ideal dwell will be
referred to as the residual vibration. The severity of the vibration is of concern to the
designer because the amplitude and intensity of the vibration produce impact, noise, wear,
and potential damage to the follower system and the cam.
For calculating vibrational responses, the deviation, r, and its derivatives relative to its
static equilibrium position are defined as follows (Wiederrich and Roth, 1975):
r = Y Y K ( K + K ) (5.28)
-
f
s
c
f
1 ()
1 ()
1 ()
r = Y - Y K ( K + K ) (5.29)
c f s f
2 ()
r 2 () = Y 2 () - Y K ( K + K ) (5.30)
c f s f
Substituting Eqs. (5.28) to (5.30) into Eq. (5.18) with normalized values and replacing d (1)
2
by d¢w and d (2) by d≤w , the following vibrational response equation can be obtained: