The system shown is in static equilibrium when x = 0 and x2 = 0 where...

Transcribed Image Text:

The system shown is in static equilibrium when x₁ = 0 and x2 = 0 where x₁ and x₂ are the displacements of the center of mass of the uniform discs. F₁ (t) and F₂ (t) are forces always horizontal to the ground. Assume discs roll without slipping. The equations of motion for small oscillations about equilibrium are given in proper SI units (i.e., meters, radians, seconds, kg, N etc.) as: 1.5x₁200x₁100x₂ = F₁ (t) 3x2 100x₁+300x2= F₂(t) a) Using Euler-Lagrange Equations, derive the equations of motion in parametric form and determine the values of m₁, m2 and k ifr = 1m. For parts (b) and (c) assume F₁ (t) = 0 and F₂ (t) = 0. b) Calculate the natural frequencies and mode shape vectors of the system. Illustrate the mode shapes by drawing sketches for each. c) If the system is excited by such a set of initial conditions that it vibrates at the highest natural frequency only, determine x₁, at the instant when x₂ = 0.01m. The system shown is in static equilibrium when x₁ = 0 and x2 = 0 where x₁ and x₂ are the displacements of the center of mass of the uniform discs. F₁ (t) and F₂ (t) are forces always horizontal to the ground. Assume discs roll without slipping. The equations of motion for small oscillations about equilibrium are given in proper SI units (i.e., meters, radians, seconds, kg, N etc.) as: 1.5x₁200x₁100x₂ = F₁ (t) 3x2 100x₁+300x2= F₂(t) a) Using Euler-Lagrange Equations, derive the equations of motion in parametric form and determine the values of m₁, m2 and k ifr = 1m. For parts (b) and (c) assume F₁ (t) = 0 and F₂ (t) = 0. b) Calculate the natural frequencies and mode shape vectors of the system. Illustrate the mode shapes by drawing sketches for each. c) If the system is excited by such a set of initial conditions that it vibrates at the highest natural frequency only, determine x₁, at the instant when x₂ = 0.01m.

Answer rating: 100% (QA)

a Given L KE PE 12m1dx1dt2 12m2dx2dt2 12kx1x22 EulerLagrange ... View the full answer