The relative displacement u(t) of a single-storey shear building subjected to an earthquake ground motion is represented by the following second-order linear ordinary differential equation: d'u du + c + ku = ag (t) dt2 dt where m, c, and k are the mass (kg), damping constant (Ns/m), and stiffness of the structure (N/m), respectively. Meanwhile, a,(t) is the function of earthquake ground acceleration. Suppose the building with a mass of 2000 kg and supported by columns of combined stiffness of 32 x 103 N/m is subjected to earthquake with ground acceleration given by the following function: ag (t) : = 36000 cos 2t Find the equation of the displacement, u(t), given the damping is not installed to the building. Then, find how much the building is displaced fromt =30 seconds to t 60 seconds of earthquake.

C (a) Write down the equation of motion for the point particle of mass m moving in the Kepler potential U(x) = -A/x+B/x² where x is the particle displacement in m. (b) Introduce a dissipative term in the equation of motion assuming that the dissipative force acting on a particle is proportional to the partical velocity with the coefficient of proportionality v. Write down the modified equation of motion. From the dimensionless equation of motion modified by dissipation Compare to question (2 b) derive a linearised equation of motion around a stable equilibrium state. Solve it and determine what is the range of values of the free parameter B for which the particle motion is oscillatory or non-oscillatory? What is the critical value of that determines a transition from the subcritical case with the oscillatory motion to the super-critical case with the non-oscillatory motion?