p1 RJM 16/10/02EG1C2 Engineering Maths: Matrix Algebra Tutorial 6 Matrix Algebra - Tutorial 6 32.Find the eigenvalues and eigenvectors of the following matrices (note the eigenvalues are integers): Suppose the eigenvectors of A are denoted  1 and  2 and let X = [  1  2 ]. Find X -1 A X. Comment. When evaluating the eigenvectors of B, show that the three equations represented by (B- I)x = 0 are linearly dependent.

p2 RJM 16/10/02EG1C2 Engineering Maths: Matrix Algebra Tutorial 6 33.The Stochastic Matrix equation below shows how the values of R, C and I change over time. R,C and I are percentages. Use eigenvalue techniques to show that their values will stabilise if this equation is repeatedly applied and hence find their steady value.

p3 RJM 16/10/02EG1C2 Engineering Maths: Matrix Algebra Tutorial 6 In the rotational system above, the angular position of the mass is , its angular velocity is  = d  /dt, and it can be shown that -k  -F  = Jd  /dt Express these equations as two state equations in  and  if J = 2kgm 2, F = 6Nm per rad/s and k = 4Nm per rad. Find the general response of  and  and the particular response if at time 0,  = 0 rad/s and  = 1 rad. 34.

p4 RJM 16/10/02EG1C2 Engineering Maths: Matrix Algebra Tutorial The above circuit can be modelled by equations Use eigenvalue techniques to find the general response of v 2 and i 1 when L = 4H, R = 8  and C = 0.25F, and the particular response if at time t = 0, i 1 = 0 and v 2 = 2V.

p5 RJM 16/10/02EG1C2 Engineering Maths: Matrix Algebra Tutorial 6 36.Consider the following, being a permanent magnet armature controlled d.c. motor in a feedback loop with controller C. The command input is 0. Let  be the motor position and  its speed. Question continued..

p6 RJM 16/10/02EG1C2 Engineering Maths: Matrix Algebra Tutorial 6 Examining the block diagram, the equations describing the system are  = d  /dt and Td  /dt = -C  - . Express the above in terms of the state variables  and  and find the complex eigenvalues and eigenvectors to the system if T = 0.5s and C = 5. Hence write down the general solution to the system. Find the particular solution if at time t = 0,  = 0 rad/s and  = 3 rad.

p7 RJM 16/10/02EG1C2 Engineering Maths: Matrix Algebra Tutorial 6 Answers

p8 RJM 16/10/02EG1C2 Engineering Maths: Matrix Algebra Tutorial 6

p9 RJM 16/10/02EG1C2 Engineering Maths: Matrix Algebra Tutorial 6