Consider the following shaftflywheel system that is modeled

Consider the following shaft-flywheel system that is modeled as a second order rotating system with equation of motion J theta (t) + b_r theta (t) + k_r theta (t) = tau (t) where J is the moment of inertia, b_r is the rotational damping coefficient and k_r is the rotational stiffness coefficient. The input (t) is an external torque input. Suppose that J = 5 kg m^2. Compute the values of the system parameters k_r and b_r such that the following conditions are satisfied The system is critically damped The system reaches a steady-state response equal to = 0.5 rad for a unit step torque input.

Solution

the given equation is in the form of ax^2+bx+c=0

if the system is critically damped then the equation will have real roots which are repeated

b^2=4ac is the condition for critical damping

repeated roots are -b/2a ,-b/2a

by substituting the given values and by solving the polynomial we will get the required values