Damped free vibrations can be modelled by a block of mass m that is attached to a spring and a dashpot as shown. k ww Engineering Mathematics and Numerical Analysis 2023-2024 M25752 - Coursework 2 m Figure Q10 - Damped free vibration system From Newton's second law of motion, the displacement x of the mass as a function of time can be determined by solving the differential equation (a) c = 3 (N sec)/m. (b) c = 50 (N sec)/m. X 펴 dx d²x m +c+kx = 0 dt² dt Page | 5 where k is the spring constant and c is the damping coefficient of the dashpot. If the mass is displaced from its equilibrium position and then released, it will start oscillating back and forth. The nature of the oscillations depends on the size of the mass and the values of k and c. For the system shown in the figure, m = 10 kg and k = 28 N/m. At time t = 0 the mass is displaced to x = 0.18 m and then released from rest. Derive expressions for the displacement x and the velocity v of the mass, as a function of time. Consider the following two cases: [6 marks] [6 marks] Using a symbolic and a numerical solution in each case, plot the position x and the velocity v versus time comparing both solutions (two curves per plot). For case (a) take 0 ≤ t ≤ 20 sec, and for case (b) take 0 ≤ t ≤ 10 sec.

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Damped free vibrations can be modelled by a block of mass m that is attached to a spring and a dashpot as shown. k ww Engineering Mathematics and Numerical Analysis 2023-2024 M25752 - Coursework 2 m Figure Q10 - Damped free vibration system From Newton's second law of motion, the displacement x of the mass as a functio