Question

Problem 1 (Harmonic Oscillators) A mass-damper-spring system is a simple harmonic oscillator whose dynamics is governed by the equation of motion where m is the mass, c is the damping coefficient of the damper, k is the stiffness of the spring, F is the net force applied on the mass, and x is the displacement of the mass from its equilibrium point. In this problem, we focus on a mass-damper-spring system with m = 1 kg, c-4 kg/s, k-3 N/m, and F 0 N with two measurements in time that x(t 0s) 1 m and x(t-1 s)-(3-e/(2) m. 1) Verify that the exact solution of Eq. (1) is given by 2

Answer 1

Given that A mass - damper - spring system is simple harmonic oscillator, whose dynamics are governed by the following equation of motion. M d^2 x/dt^2 + c dx/dt + KX = F m rightarrow mass c rightarrow damping coefficient of the damper k rightarrow stiffness of the spring F rightarrow net force X rightarrow displacement of mass From eqn (a) m = 1, c = 4, k = 3, F = 0 d^2 x/dt^2 + 4 dx/dt + 3x = 0 (D^2 + 4 D + 3) X = 0 since D = d/dt D^2 + 4 D + 3 = 0 (D + 3) (D + 1) = 0 \r\n D = -1, -3 so x (-1) = c_1 e^-t + c_2 e^- 3t x (0) = 1 x (1) = (3 . e^-2)/2 c Rightarrow (3/2 - 1/2e^-2)/e x (1) = 3/2 e^-1 - 1/2 e^-3 x (0) = c_1 + c_2 = 1 x