Question

A spring-mass system has a spring constant of 3 N/m. A mass of 2 kg is attached to the spring, and the motion takes place in a viscous fluid that offers a resistance numerically equal to the magnitude of the instantaneous velocity. If the system is driven by an external force of (27 cos 3t − 18 sin 3t) N, determine the steady state response. Express your answer in the form R cos(ωt − δ). (Let u(t) be the displacement of the mass from its equilibrium position at time t.) uss(t) = Incorrect: Your answer is incorrect.

A spring-mass system has a spring constant of 3 N/m. A mass of 2 kg is attached to the spring, and the motion takes place in a viscous fluid that offers a resistance numerically equal to the magnitude of the instantaneous velocity. If the system is driven by an external force of (27 cos 3t - 18 sin 3) N, determine the steady state response. Express your answer in the form R cos(t ). (Let u(t) be the displacement of the mass from its equilibrium position at time t.) Additional Materials eBook

Answer 1

Given data the spring constant of spring mass system is 3N/m. the mass of spring m = 2kg the external force is , (27 cos 3t - 18 sin 3t) N here, 2y'' + y' + 3y = 27 cos (3t) - 18 sin (3t) the general solution is the sum of a complementary function the auxiliary equation is 2r^2 + r + 3 = 0 this has two complex roots, (a + bi) and (a - bi) where, a = -1/4 , b = 1/4 squareroot 23 so, the complementary function is exp(at){A cos(bt) + B sin(bt)} \r\n since a < 0 and the amplitude of sin and cos is limited to plusminus 1 this part of the solution decays with time and will not contribute to the ultimate state. Here, the ultimate state is determined by PI. The PI takes the form is , Y = A COS (3T) + B sin (3t) now, subbing the derivative equation gives, 2y'' + y' + 3y = R.H.S 2{ -9A cos(3t)-9B sin (3t) } + {-3A sin(3t)+ 3B cos(3t)} + 3{ A cos(3t) + B sin (3t)} = RHS here, equating coefficients of sin and cos, -18A + 3B + 3A = 27 -18B - 3A + 3B = -18 By solving , we get A= -3/2 , B = 3/2 so, the PI is 3/2 (-cos(3t) + sin(3t)) this is the ultimate state as t rightarrow infinity