Question

Consider the mass M subject to periodic forcing P(t) A sin wt where A 0.3 and e is a small parameter. The mass is attached to a spring with stiffness k and dashpot with damping coefficientc to model the stiffness and damping of the structure. Resting atop the idealized structure is vibration damper consisting of a mass ma, spring ka, and dashpot ca, as shown in Figure 1. The goal is to make the appropriate choice of the parameters ma, kaca such that the amplitude of oscillations of the mass M is less than 0.03 m. For consistency, make use of the following notation; an y(t) ca x(t) al P(t) Figure 1. Schematic diagram of the idealized model for the mass M and vibration absorber Let M - 1kg, c0.5N/(m/s), k 3 N/m, m(0.05+0.1k0.2ca)kg and V3. The mass of the vibration absorber increases with ka and ca Questions 1. Derive the equations of motion for the system. 2. Derive the associated transfer functions. 3. CASE I: Consider the special case with ca -0.How will you choose ka to completely eliminate the oscillations of the structure? What range of values of ka will achieve the goal of keeping the amplitude of oscillations of M to less than 0.05 m. CASE II: Add damping Ca-0.003 N/(m/s) to the vibration absorber in CASE I. What happens when i) wa 0.5ii) wa-, and iii) wa2n? CASE II: Suppose you were in competition with another team to design the vibration absorber. It would be nice to have a design that can meet the goal of the amplitude being less than 0.03 m even when the forcing has an amplitude larger than 0.3. What is the highest value of A such that your design can no longer keep the amplitude of oscillations of the block to less than 0.03m. 4. 5.

Answer 1

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