T1-23 8.0    3-D OBJECT DRIVEN BY VIBRATORY FORCE AND TORQUES Figure 37 shows an object with its C.G. at C, mounted on 4 flexible mounts and acted upon by a disturbing harmonic force F_y in the y-direction (vertical) and/or by torques, T_x, T_y and T_z acting singly or in combination about the x, y and z axes, which are principal inertia axes passing through the C.G. (point C). The four mounts are symmetrically disposed relative to the C.G., their location defined by distances b x , b y and b z from the axes, as shown. The mass moments of inertia about the principal inertia axes are I x , I y and I z , respectively. As a result of the external force and torques, the object motion is (a) a displacement of C.G., maximum values of which are denoted by translational motions of the C.G. (x, y, z) and (b) rotations of the object (from equilibrium) about the coordinate axes (q x , q y^, qz). These displacements are generally small relative to the major dimensions of the object. Let: M = mass of object (W/g where W is weight of the object, g = 386 in/sec 2 = 9.8 m/sec 2 ); ^ky = total vertical stiffness of the four supports in lb./in. or N/m; i.e., 4 times the stiffness of each support if all four supports are identical ^ks = total horizontal or shear stiffness of the four supports; i.e., 4 times the horizontal stiffness of each support, if all supports are identical and for each support k x = k z = k s , lb./in. or N/m; 3 = angular frequency of sinusoidally applied force and torques (rad/sec) Damping is assumed to be negligible. 8.1    Displacement of the Object Due to F y only: y =                                                                                                                                                 (31) Due to T z only: x =                                                                                                                                                 (32) qz =                                                                                                                                                 (33) C x y Fy Tx Tz bz by bx q z q x ky/4 ky/4 ky/4 ky/4 Ty C (x, y, z) y z C z x q y F_y ________ k_y – M3² T_zb_yk_s _____________________________________ I_zM3⁴ – 3² (I_zk_s + k_yb_x² M + k_sb_y² M) + k_yk_sb_x² T_z(k_s – M3²) _____________________________________ I_zM3⁴ – 3² (I_zk_s + k_yb_x² M + k_sb_y² M) + k_yk_sb_x² Figure 37    Solid Body on Vibration Isolators