Use MATLAB.

Ground vibrations from a crane operation, a forging press, and an air compressor are trans- mitted to a nearby milling machine and are found to be detrimental to achieving specified accuracies during precision milling operations. The ground vibrations at the locations of the crane, forging press, and air compressor are given by xe(t) = Ace sin wct, xf(t) = A respectively, sin wft, and xa(t) = Aa sin wat, where Ac =22 m, Af = 30 m, Aa = 25 m, wc 14 Hz, wf = 17 Hz, wa = 19 Hz, and c = 0.1. The ground vibrations travel at the shear wave velocity of the soil, which is equal to 300 m/s, and the amplitudes attenuate according to the relation A, Age -0.005r, where Ao is the amplitude at the source and A, is the amplitude at a distance of r ft from the source. The crane, forging press, and air compressor are located at a distance of 18 m, 24 m, and 12 m, respectively, from the mill- ing machine. The equivalent mass, stiffness, and damping ratio of the machine tool head in vertical vibration (at the location of the cutter) are experimentally determined to be 500 kg, 480 kN/m, and 0.15, respectively. The equivalent mass of the machine tool base is 1000 kg. It is proposed that an isolator for the machine tool be used, as shown in Fig. 9.55, to improve the cutting accuracies [9.2]. Design a suitable vibration isolator, consisting of a mass, spring, and damper, as shown in Fig. 9.55(b), for the milling machine such that the maximum vertical displacement of the milling cutter, relative to the horizontal surface being machined, due to ground vibration from all the three sources does not exceed 5 m peak-to-peak. Isolator = 9.2 The root mean square value of a signal x(t), Xrms, is defined as T 1/2 [lim 1 = { 1 0 7 / ^x (1) di}" dt T T k;/2 Using this definition, find the root mean square values of the displacement (xrms), velocity (ms), and acceleration (ms) corresponding to x(t) = X cos wt. 3 Ci = Xrms k;/2 Machine tool head (meq) Milling cutter Horizontal surface being machined Machine tool base (mt) Isolator (mass, m;) keq ki FIGURE 9.55 Isolation system for machine tool. meq m + m Ceq (b) Ci Xcutter Lay Xbase Xground