Eigenvalues Application: Gyroscope The differential equation describing the moti

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Eigenvalues Application: Gyroscope The differential equation describing the motion of a gyroscope with a single degree of freedom (one that can only move about on a single axis) can be simplified to: theta(t) is the gyroscope's output angle, J is the moment of inertia, B is the damping coeffiecient, and k is the spring constant of the gyro, The gyroscope's input angle is to. Using a second differential equation, write this system of two equations as The procedure is almost identical to the procedure in problem 1. What is A in terms of B, k, and J? Solve for the eigenvalues of A, in terms of B, k, and J. Assuming B, k, and J are always positive, are the real parts of the eigenvalues of A always positive or negative? Based on your previous answer, is the homogeneous system of equations for the gyro stable and/or attractive? Let B=1, k=75, and J=0.02. Using your earlier equation for the eigenvalues A, find numeric values for the eigenvalues of A. Given the matrix A = Write your name, the assignment name, and the date at the top of the first page of each assignment. Staple multiple pages together. Number all problems and put them in order in your assignment. Please write clearly and draw a box around numerical answers. You will not get credit if your work is difficult to follow or the answer in a box is not easily seen. You will not get full credit if these directions aren't followed. For full credit show all work!!! Eigenvalues of Systems of Equations Application: Series RLC Circuit, Natural, or Transient Response (Remember EE280, maybe not) Consider a series RLC circuit, with a resistor R, inductor L, and capacitor C in series. The same current i(t) flows through R, L, and C. The switch S1 is initially closed and S2 is initially open allowing the circuit to fully charge. At t=0 the switch S1 opens and S2 closes as shown above. Solving the homogenous linear equation for i(t) allows us to observe the natural response of the series RLC circuit, which is the response of the circuit with no voltage source attached. From Kirchoff's Voltage Law, the sum of the voltages around the loop must equal zero; differentiating this equation with respect to time to remove the integral from v_c gives: This single 2 order ODE can be converted to two first-order ODEs by using the 2 ^nd equation i_2(t) = di(t)/dt a) Write the system of 2 linear 1^st-order ODEs resulting from substituting i_2(t) = di(t)/dt into the series RLC equation. The other equation is i2(t) = di(t)/dt. Your system will be i=Ai. Your 2 unknown variables will be in the vector i = [i(t) i_2(t)]. Substitute as needed so that each equation has one and only one variable differentiated, i.e. one equation has di(t)/dt, and one has di_2(t)/dt. Write I in terms of R, L, and C. The equation is V = Ai. Your mission is to populate the matrices. Solve for the eigenvalues A, and A2 of A, again in terms of R, L, and C (quadratic e q) Based on the eigenvalues that you found, is this system of equations stable? Explain why. Assume R, L, and C are all positive and non-zero. What must R be, in terms of L and C, for the eigenvalues of A to be all real? Find the eigenvalues of A if R=50 Ohm, L=200mH, and C=500pF. Make sure to account for the units properly! Also, solve for the eigenvectors of A. Choose x, in each eigenvector to be 1. g) Write an equation for the general solution of i(t) when R=50 Ohm, L=200mH, and C=500 muF.

Explanation / Answer

3. a. Eigen values of A

A=[-5 2 ; 2 -2]

eig(A)

b. Eigen vectors

A=[-5 2 ; 2 -2]

[V,D] = eig(A)

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