アナログ計算機を用いた衝突振動のシミュレーション

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Impacting vibration, that is a mechanical vibration accompanied with impact, is usually complicated and can hardly be analyzed, owing to extreme nonlinearity of the impact. Then, a general method of analog computer simulation applicable to impacting vibration of a multi-degree-of-freedom system has been investigated. In the collinear impact of two bodies A and B (Fig. 1 (a)), the relation between the gap (x_A-x_B) and the contact force f_i is given by Fig. 1 (b) or Eq. 1. In Eq. 1, the negative gap represents local deformation at the impact point. The stiffness of the local deformation s_n is assumed to be constant. It can be considered that a virtual elastic component of nonlinear stiffness s_N is always connected between the impact points of two bodies. In the impedance analogy, the reciprocal of s_N corresponds to a nonlinear capacitance C_N. By use of this C_N, an electrical equivalent circuit of a vibration system with impact is given in Fig. 2, where f_<Ak> and f_<Bk> are the external forces applied to the mechanical terminals A_k and B_k, respectively, and x^^^. _A and x^^^. _B are the velocities of the impact points A_1 and B_1, respectively. The nth mode equivalent circuit inside the black box in Fig. 2 can be given as Fig. 3 (a) or (b). A computing element simulating the C_N, which has an ideal characteristic as shown in Fig. 4 (b), can be constructed as Fig. 4 (a) provided that D_1 is an ideal diode. The actually obtained characteristic, however, is unsatisfactory for the exact simulation as shown by crosses (×) in Fig. 5. Then, a clipping circuit shown in Fig. 7 is added in cascade to the output terminal of Fig. 4 (a). Thus, such an improved characteristic as shown by dotts (・) in Fig. 5 is obtained. This element is named an 'impact-element' and represented by the symbol shown in Fig. 9. A setup diagram for analog computer simulation of the nth mode is shown in Fig. 10 (a) and it is represented by diagramatic symbol as in Fig. 10 (b). A block diagram for simulating the whole vibration system represented by the equivalent circuit shown in Fig. 2 is set up as in Fig. 8 by use of the symbols of Fig. 9 and Fig. 10 (b). By the above-mentioned method, simulation of some impacting vibrations has been done. Simulated waveforms obtained for models given in Fig. 11 and 13 are shown in Fig. 12 and 14, respectively. These results indicate that the simulation method mentioned here is available and effective for simple analyses of impacting vibrations.
社団法人日本音響学会の論文
1975-08-01

アナログ計算機を用いた衝突振動のシミュレーション

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