A trolley of mass 'm' fixed onto a damper with a damping constant 'b' and onto serially connected springs having stiffness coefficients of 'ki' and 'k' respectively and is moving along a smooth surface under free-vibrational motion with an initial excitation of u(t). The x(t) defines as the displacement of the trolley. b fisi m Figure A1.1. a) Determine the governing differential equation of motion for the spring-mass-damper system shown in Figure A1.1. [06 marks] b) If x(t)= K.et, then determine the respective quadratic equation in terms of X. (K & R) [03 marks] c) Determine the roots of the equation mentioned in part (b), in terms of m, b and ki and k2. [02 marks] d) State the condition for critical damping w.r.t the equation in part (b). Determine the damping ratio ''. [04 marks] e) Assuming that the said system adheres to the standard oscillatory model, deduce damping ratio ' in terms of natural frequency . [03 marks] f) Express the roots of the quadratic equation in part (b) in terms of and con. [02 marks] g) Detern ine the roots of the quadratic equation in part (b) for an underdamped scenario. [03 marks] h) If b=0.75 Nsm, m= 0.5 kg & k = 80Nm, determine the damping ratio '' for the given system. [02 marks]