Question

PARTS: a-c

Problem 1 (40 points) The rotational system shown in this diagram has a single torque input, T(t), and a single angular displacement output, (t). Also shown are the shaft polar moment of inertia, J, torsional spring constant, k, and rotational viscous damping coefficient, c. From the law of rotation (sum of the moments equal to polar moment of inertia times angular acceleration), we obtain: jä(t) + c)(t) + ko(t) = T(t). This ordinary differential equation is second-order, linear time-invariant. Using the approach presented at the lecture (Week 5) for state space: (a) Obtain the “state” differential equations for xi(t) = 0(t) and x2(t) = º(t) = či(t). (b) Derive the coefficient matrices A, B, C, and D. (c) Find the eigenvalues of the A-matrix. (d) Define the condition(s) for BIBO stability of this system. (e) For k= 20 N-m/rad, c=8 N-m-s, and J= 2 kg-m”, can we say that the system is BIBO stable? T(1)

Answer 1

The governing differential equation is given to be

-J0+20+ k =T

(1) - - في =ة.-

Defining the following state variable

(1) = 0;r(2) = 0 =r(1)

Using the state variable, we can reduce the second order differential equation (1) to two first order differential equations which can be represented in a matrix form as follows

d/1 dt (x(2) 0 1-4 11/21) + OT - x(2)

Therefore,

$\small A=\begin{bmatrix} 0 &1 \\ -\frac{k}{J} &-\frac{c}{J} \end{bmatrix}; B=\begin{Bmatrix} 0\\\frac{1}{J} \end{Bmatrix}$

The output vector can be written as

Therefore,

2 = 0 il r(2)

$\small C=\begin{bmatrix} 1 &0 \\ 0& 1 \end{bmatrix}; D= \begin{Bmatrix} 0\\0 \end{Bmatrix}$

The eigenvalues of matrix A can be found by solving the following equation

det|A-X1=0det 1-1 - - 1 +1)]

-:12+ft+' = 0

Therefore, the eigenvalues are

-..--+ V(5)? - -4-(5):- 4