. A vibrating mass is found to be oscillating with an amplitude that is too large. To reduce this amplitude, an auxiliary system is added as shown in Figure 8.52. This problem generally occurs when...

. A vibrating mass is found to be oscillating with an amplitude that is too large. To reduce this amplitude, an auxiliary system is added as shown in Figure 8.52. This problem generally occurs when the forcing frequency is too close to the natural frequency of the primary syste The auxiliary system acts as a damped vibration absorber. (a) Derive the equation of motion of the system plus absorber and then solve to determine what values of absorber mass m, sti¤ness k2, and damping c must be selected in order to minimize the vibration amplitude of primary mass M. Assume that m = 0:25M and F(t) = F0 cos !t. (b) Suppose that we can accept a design where k2=m is selected so that neither natural frequency of the combined system, !1 and !2, is closer than 5% to the driving frequency !. Devise the system that achieves this criterion.