VIBRATION AND CONTROL OF VIBRATION 31.19

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VIBRATION AND CONTROL OF VIBRATION

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FIGURE 31.17 Short-circuit excitation form.

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31.3 SYSTEMS WITH SEVERAL DEGREES OF FREEDOM

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Quite often, a single-degree-of-freedom system model does not sufficiently describe the system vibrational behavior. When it is necessary to obtain information regarding the higher natural frequencies of the system, the system must be modeled as a multidegree-of-freedom system. Before discussing a system with several degrees of freedom, we present a system with two degrees of freedom, to give sufficient insight into the interaction between the degrees of freedom of the system. Such interaction can also be used to advantage in controlling the vibration.

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31.3.1 System with Two Degrees of Freedom Free Vibration. A system with two degrees of freedom is shown in Fig. 31.18. It consists of masses m1 and m2, stiffness coefficients k1 and k2, and damping coefficients c1 and c2. The equations of motion are m1 1 + (c1 + c2)x 1 + (k 1 + k2 )x1 c2x 2 k2x2 = 0 x m2 x2 + c2 x 2 + k2 x2 c2 x 1 k2 x1 = 0 (31.60)

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VIBRATION AND CONTROL OF VIBRATION 31.20

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LOAD CAPABILITY CONSIDERATIONS

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FIGURE 31.18 Two-degree-of-freedom system.

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Assuming a solution of the type x1 = Ae st and substituting into Eqs. (31.60) yield [m1s2 + (c1 + c2)s + k1 + k2]A (c2s + k2)B = 0 (k2 + c2 s)A + (m2 s2 + c2 s + k2 )B = 0 Combining Eqs. (31.62), we obtain the frequency equation [m1s2 + (c1 + c2 )s + k1 + k2 ] (m2 s 2 + c2 s + k2 ) (c2 s + k2 )2 = 0 (31.63) (31.62) x2 = Be st (31.61)

This is a fourth-degree polynomial in s, and it has four roots; hence, the complete solution will consist of four constants which can be determined from the four initial conditions x1, x2, x 1, and x 2. If damping is less than critical, oscillatory motion occurs, and all four roots of Eq. (31.63) are complex with negative real parts, in the form s1,2 = n1 ip1 So the complete solution is x1 = exp ( n1t) (A1 cos p1t + A2 sin p1t) + exp ( n2t) (B1 cos p2t + B2 sin p2t) x2 = exp ( n1t) (A 1 cos p1t + A2 sin p1t) + exp ( n2t) (B cos p2 t + B 2 sin p2t) 1 Since the amplitude ratio A/B is determined by Eq. (31.62), there are only four independent constants in Eq. (31.65) which are determined by the initial conditions of the system. Forced Vibration. Quite often an auxiliary spring-mass-damper system is added to the main system to reduce the vibration of the main system. The secondary system is (31.65) s3,4 = n2 ip2 (31.64)

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VIBRATION AND CONTROL OF VIBRATION 31.21

VIBRATION AND CONTROL OF VIBRATION

called a dynamic absorber. Since in such cases the force acts on the main system only, consider a force P sin t acting on the primary mass m. Referring to Fig. 31.18, we see that the equations of motion are m1 1 + (c1 + c2 )x 1 + (k1 + k2 )x1 c2 x 2 k2 x2 = P sin t x m2 2 + c2 x 2 + k2 x2 c2 x 1 k2 x 1 = 0 x Assuming a solution of the type x1 = A1 cos t + A2 sin t P/k1 (31.67) x2 = A3 cos t + A4 sin t P/k1 and substituting into Eqs. (31.66), we find that the Ai are given as A1 = 2 [2D1 2 2 D2( 2 2)] 1 2 D2 + D2 1 2 2 [D1( 2 2) + 2D2 2 2] 1 2 D 21 + D 2 2 (31.68) A3 = (2D1 2 2 D2 ) D2 + D2 1 2

2 1 2 2

(31.66)

A2 =