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4 D1 = ( 2 2 ) ( 2 2 2 ) 4 2 2 2( 1 1 + 2 2) ( 2 4 2 2 2 ) 2 1 2 2 2

D2 = 2 [( 2 2) ( 1 1 + 2 2 ) + 2 2 ( 2 2 + 2 ) 2 2 3 ] 2 1 2 2 2 = 1 k1 m1 c1 2m1 1 m2 m1 2 = 2 k2 m2 c2 2m2 2

(31.69)

1 =

2 =

(31.70)

Responses may also be written in the form x1 = B1 sin ( t 1) x2 = B2 sin ( t 2) (31.71)

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Here 1 and 2 are the phase angles by which the responses of masses m1 and m2, respectively, will lag behind the applied force.The response amplitudes B1 and B2 are plotted in Figs. 31.19 and 31.20, respectively. The amplitude B1 has a minimum between 1 and 2. The equations of motion for torsional systems with two degrees of freedom have the same form as Eqs. (31.60) and (31.66). The solution will also be similar and will exhibit the same characteristics as discussed earlier.

31.3.2 Multidegree-of-Freedom Systems In many applications, it is necessary to know several higher modes of a vibrating system and evaluate the vibration response. Here, the elastic system has to be treated as one with distributed mass and elasticity. This is possible for simple elements such as beams, plates, or shells of regular geometry. However, when the structural system is complex, it may be modeled as a multidegree-of-freedom discrete

FIGURE 31.19 Amplitude frequency response of the mass of a twodegree-of-freedom system subject to forced excitation.

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system by concentrating its mass and stiffness properties at a number of locations on the structure. The number of degrees of freedom of a structure is the number of independent coordinates needed to describe the configuration of the structure. In a lumped-mass model, if motion along only one direction is considered, the number of degrees of freedom is equal to the number of masses; and if motion in a plane is of interest, the number of degrees of freedom will equal twice the number of lumped masses. Holzer Method. When an undamped torsional system consisting of several disks connected by shafts vibrates freely in one of its natural frequencies, it does not need any external torque to maintain the vibration. In Holzer s method, this fact is used to calculate the natural frequencies and natural modes of a vibrating system. Figure 31.20 shows a torsional system with several disks connected by shafts. In this procedure, an initial value is assumed for the natural frequency, and a unit amplitude is specified at one end. The resulting torques and angular displacements are progressively calculated from disk to disk and carried to the other end. If the resulting torque and displacement at the other end are compatible with boundary conditions, the initial assumed value for the natural frequency is a correct natural frequency; if not, the whole procedure is repeated with another value for the natural frequency until the boundary conditions are satisfied. For a frequency and 1 = 1, the corresponding inertial torque of the first disk in Fig. 31.20 is T1 = J 1 1 = J1 2 1 This torque is transmitted to disk 2 through shaft 1; hence, T1 = J1 2 1 = k1 ( 1 2)

(31.73)

(31.74)

which relates 2 and 1. The inertial torque of the second disk is J2 2, and the sum of the inertial torques of disk 1 and disk 2 is transmitted to disk 3 through shaft 2, which gives J1 2 1 + J2 2 2 = k2( 2 3) (31.75)

Continuing this process, we see the torque at the far end is the combined inertial torques of all the disks and is given by

i = 1