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where: wn = the natural frequency of the cam-follower system, w n = x = the damping ratio, (x = 05(Cs + Cf)/M/wn)
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When the cam is rotating at the frequency of w = wd/b the values of d(1) and d (1) (t = 1) in Eq (532) will vanish and the amplitude of the residual vibration becomes zero To assess vibrational responses, both primary and residual vibrations can be examined in a form of response spectrum for each synthesized output motion (Chen, 1981 and 1982; Mercer and Holowenko, 1958; Rees Jones and Reeves, 1978; Neklutin, 1954) The vibrational characteristic for the camshaft at various rotational speeds can be observed from the response spectrum even though the output motion is synthesized for the camshaft at a constant speed, wd Example Application
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Example 8: Synthesis of Cam Motion with Nonrigid Follower An example application is provided here to illustrate the mechanics of applying the procedure described earlier A DRD motion is synthesized using the spline interpolation procedure The dynamic behavior of the follower system is then investigated and the cam motion is found using the spline collocation procedure In preparing this example a case was selected from the literature for which a cam had been designed using an optimized polynomial (Peisakh, 1966; Chen, 1981) The solution obtained using the optimized polynomial gives a convenient basis for evaluating the performance of the methods described here, at least for one case The results shown were re ned iteratively by trying a total of ten different combinations of spline parameters and sets of constraints Space limitations prevent the presentation of all the iterations; however, it should be clear that the procedures being used allow great freedom in specifying motion constraints and can easily evaluate the effects of changing these constraints and of altering the spline parameters In this example, each step of the process is illustrated even though some of the operations can be easily reduced to an algorithm and eliminated as a concern to the
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designer For instance, selection of the collocation points is readily automated Such operations are described in the example, however, for the sake of thoroughness
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Synthesis of the Output Motion In this illustration the basic goal is to satisfy the ten basic constraints at t = 0 and at t = 1 in the list of constraints that follows These constraints are the primary design constraints The two additional constraints at t = 05 were added in the iterative process to re ne the motion somewhat Since the degree of the spline curve is k - 1, splines of order k = 6 are required for S(4) to be continuous It is necessary to achieve this continuity if the constraints on S(4) are to be satis ed, as they must be if continuous cam acceleration is to be obtained The output motion that is synthesized using spline functions is shown in Figures 524 to 528 and compared to the motion produced by the optimized polynomial Collocation Solution of the Differential Equations To solve the differential Eqs (518) and (531) for the cam displacements and for vibrational responses, the normalized time domain (t) is divided into sixteen equal elements (e = 16) and in each element four Gaussian points (p = 4) are located Both the number of elements used and the choice of
1 Spline (k=10) Optimized Polydyne
Displacement of output motion
0 0 5 Normalized time 1
FIGURE 524 Normalized displacements of output motions for spline (k = 10) and optimized polydyne in Example 8
CAM MOTION SYNTHESIS USING SPLINE FUNCTIONS
3 Spline (k = 10) Optimized Polydyne
Velocity of output motion
0 0 5 Normalized time 1
FIGURE 525 Normalized velocities of output motions for spline (k = 10) and optimized polydyne in Example 8
a uniform mesh for the elements are arbitrary Based on the equal division, the rst element is located on the normalized time subdomain [0, 1/16], the second element on the subdomain [1/16, 2/16], and so on through the sixteenth element on the subdomain [15/16, 1] The four Gaussian points x = 033992104 and 086113631 of the fourth Legendre polynomial de ned on [-1, 1] are scaled to t = ((b - a)x + b + a)/2 on the subdomain [a, b] for each element (Stroud and Secrest, 1986) For example, the rst four Gaussian points located on the subdomain, [0, 1/16], of the rst element are 000433949, 002062747, 004187253, and 005816051 For Eq (531), there are 66 constraints the two end conditions and the sixty-four values for the right sides of the differential equations to be satis ed at the Gaussian points Sixty-six B-splines of order six are used to formulate the system of linear equations in Eq (56) The knot sequence used to evaluate the splines for the collocation process is established by repeating four knots at each mesh point, t = 1/16, 2/16, 3/16, , and 15/16 At the end points, t = 0 and t = 1, the knots are repeated six times as necessary because of the order of the spliones (k = 6) After Eq (56) is solved for the unknown coef cients, the solution of Eq (531) is readily found for any point in the interval 0 t 1 enabling the dynamic behavior of the follower system to be determined
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