AA 242B: Mechanical Vibrations (Spring 2014)Homework #3Due May 7, 20141 Control TabA control tab of an airplane elevator is hinged about axis A in the elevator(see Figure 1). Its dynamic behavior depends on the moment of inertia IA, andthe elastic stiffness of the control linkage which can be modeled as a torsionalspring with a stiffness coefficient kθ. Since kθcannot be measured statically,a resonance test is typically used for that purpose. The elevator is held fixed,the tab is supported by a spring with a stiffness coefficient k1, and a harmonicexcitation is applied through a spring with a stiffness coefficient k2, as shown inFigure 1.1. Explain why the coefficient kθcannot be measured by a static test.2. Find the natural frequency(ies) ωnof the system described above as afunction of IAand kθ.3. Find kθas a function of the resonance frequency ωr, IA, and other pa-rameters shown in Figure 1.Figure 1: Dynamic testing of a control tab of an airplane elevator.1Figure 2: Three-dof mechanical system.2 Modal Superposition1. Use the modal superposition method to determine the free-vibrations ofthe system shown in Figure 2, given the initial conditionsu1(0) = u2(0) = u3(0) = 1˙u1(0) = ˙u2(0) = ˙u3(0) = 02. Comment on your solution.3 Vehicle DynamicsA vehicle is modeled as a rigid body AB having a mass M and a mass momentof inertia IGabout its mass center. The mass of the axles and wheels is modeledby lumped masses m, the stiffness of the springs by k1, the stiffness of the tiresby k2, and the shock absorbers are modeled by viscous dashpots, as shown inFigure 3.Use as generalized coordinates for this system the vertical displacements ofthe smaller masses and the displacements at A and B. Assume a small angle ofrotation of the rigid body AB.1. Derive the Lagrange equations of dynamic equilibrium of the system de-scribed above and write them in matrix form.4 Tuned Vibration AbsorberOne method of reducing the vibration amplitude of a single degree-of-freedomsystem subjected to a harmonic excitation is to attach a “tuned vibration asor-ber”, which is a second spring-mass system. For the resulting two degree-of-freedom system shown in Figure 4:1. Determine the equations of motion of this system.2. Let the steady-state (forced response) of the system be given byu1(t) = U1cos Ωt and u2(t) = U2cos ΩtDetermine the amplitudes U1and U2as functions of k1, k2, Ω, m1, m2,and P1.2Figure 3: Simplified dynamic model of a vehicle.33. The absorber is tuned so thatk2m2=k1m1. When the original system isexcited at resonance — that is, when Ω2=k1m1— what becomes theresponse amplitude U1?4. Letm2m1= 0.25 for a particular tuned absorber system. Plotk1U1P1andk2U2P1versus the frequency ratio r =Ωqk1m1. Comment on your plot.4Figure 4: Simplified model of a tuned vibration