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CAM DESIGN HANDBOOK
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As a result of the machine systems being old Generally the older the machine is, the more vibrations, noise, and lack of smoothness of operation occur As result of periodic change in power input voltage of the electrical motor or prime mover This has occurred in the performance of a high-speed textile machine in which the function had transient energy malfunction Due to the inclusion of belts and chains in the torsional drive This may produce large backlash and high compliancy in the system, seriously reducing the effective performance Note that in cases where the vibrating system cannot be modi ed satisfactorily, one may avoid resonance effects by not operating the machine at excitation frequencies at which resonances are excited This may be accomplished by speed controls on a machine or by prescribing limitations on use of the system, eg, red lines on engine tachometers to show operating speeds to be avoided
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123 CAM-FOLLOWER DYNAMICS RIGID CAMSHAFT
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1231 Single-Degree-of-Freedom System Modeling, as discussed in Chap 11, transforms the system into a set of tenable mathematical equations that describe the system in suf cient detail for the accuracy required Various tools of analysis are at the disposal of the designer-analyst and range in complexity from classical linear analysis for single-degree-of-freedom systems to complex multipledegree-of-freedom nonlinear computer programs In this section we present a single-degree-of-freedom system (DOF) with a rigid camshaft The cam mechanism consists of a camshaft, cam, follower train including one or more connecting links, and springs terminating in a load mass or force For lumped systems, linkages and springs are divided into two or more ideal mass points interconnected with weightless springs and dampers Springs The spring force generally follows the law
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where terms under the S sign are nonlinear Dampers Damping takes the general form C( x ) = Cx x r -1 0 C( x ) is Coulomb damping For r = 1 C( x ) is viscous damping 2 C( x ) is quadratic damping Stiction is accounted for as the breakaway force, , in one element i sliding along another element j Fst = 0 for xi x j for xi x j = at breakaway
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CAM SYSTEM DYNAMICS ANALYSIS
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Damping value ranges from 4 to 8 percent in most cam-and-follower systems; it is the transient response of the follower that is of basic concern Steady-state vibration in most systems rarely occurs, because in practice the angular velocity of the camshaft is low in comparison with the natural frequency of the follower system We assume that the vibration damps out during the dwell period and does not carry over to the next motion cycle The model is the one-degree-of-freedom system, which lumps the follower train and mass load into a single equivalent mass with equivalent springs and dampers connected to the closed-track cam, as shown in Fig 121 Let m = mass, lb b = damping coef cient, lb/in/sec k = spring rate, lb/in x = follower displacement, in x = follower velocity, in/sec = follower acceleration, in/sec2 x y = cam displacement, in y = cam velocity, in/sec = cam acceleration, in/sec2 y Utilizing Newton s second law, we see that mx + b( x - y) + k ( x - y) = 0 The vibration form of Eq (121) is m( x - y) + b( x - y) + k ( x - y) = - my (122) (121)
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FIGURE 121 Single-degree-of-freedom system (closed-track cam)
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Substituting z = x - y yields mz + bz + kz = - my (123)
which places in evidence the prominent role of cam function acceleration y on the vibrating system, similar to a system response to a shock input at its foundation The general solution to Eq (123) consists of the complementary or transient solution plus the particular solution The complementary solution is the solution to the homogenous equation mz + bz + kz = 0 2 z + 2zw n z + w n = 0 where w n = ( k m)
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