Problem 15.20.35. Consider the eigenvalues of the matrix 0 -k/m-c/m for the undamped (c 0) and da...
Problem 1.Consider the harmonically forced undamped oscillator described by the following ODE:mx′′+kx=F0cosωt, k >0, m >0, ω >0, F0∈R. Problem 1. Consider the harmonically forced undamped oscillator described by the following ODE: mx" + kx = Fo cos wt, k > 0, m > 0,w > 0, F0 E R. (1) a) Suppose wa #k/m. Find the general solution of the ODE ). b) Consider the initial value problem of the ODE () with initial conditions x(0) = 0 and...
Problem 2 (25 points): Consider an undamped single-degree-of-freedom system with k = 10 N/m, 41 = 10 N 92 = 8N, and m = 10 kg subjected to the harmonic force f(t) = qı sin(vt) + 92 cos(vt), v = 1 rad/ sec. Assume zero initial conditions (0) = 0 and c(0) = 0. Derive and plot the analytical solution of the displacement of the system. mm m = f(t) WWWWWWWW No friction Problem 2 Problem 3 (30 points): Using...
1. The change of position of the center of mass of a rigid body in a mechanical system is being monitored. At time t 0, when the initial conditions of the system were x = 0.1 m and x -0m/s, a step input of size 10 N began to apply to the system. The response of the system was represented by this differential equation: 2r + 110x + 500 x = 10 a) Write the order of the system, its...
Consider the forced but undamped system described by the initial value problem 3cosuwt, (0) 0, (0 2 (a) Determine the natural frequency of the unforced system (b) Find the solution (t) forw1 (c) Plot the solution x(t) versus t for w = 0.7, 0.8, and 0.9. (Feel free to use technology. MatLab, Mathematica, etc.) Describe how the response (t) changes as w varies in this interval. What happens as w takes values closer and closer to 1? Briefly explain why...
16. Assume that M is a 4 x 4 matrix with eigenvalues 11 = -1, 12 = 0, 13 = 5, y = 1. Choose the correct answer(s) (a) A basis of R* can be formed using eigenvectors of M (b) The matrix M is nonsingular c) The matrix M is diagonalizable (d) All of the above 17. Let S be a 3 x 3 symmetric matrix whose eigenvalues are 12 = 4, 13 = -1. Choose the correct answer(s)...
solve d ,e , f, g ® Consider a damped unforced mass-spring system with m 1, γ 2, and k 26. a) (2 points) Find if this system is critically damped, underdamped, or overdamped. b) (4 points) Find the position u(t) of the mass at any time t if u(0)-6 and (0) 0. c) (4 points) Find the amplitude R and the phase angle δ for this motion and express u(t) in the form: u(t)-Rcos(wt -)e d) (2 points) Sketch...
Section 7.6 Complex Eigenvalues: Problem 5 Previous Problem Problem List Next Problem (1 point) Consider the initial value problem date [10 ] x x(0) = [2] (a) Find the eigenvalues and eigenvectors for the coefficient matrix. X = * , ū = (b) Solve the initial value problem. Give your solution in real form. x(t) = Use the phase plotter pplane9.m in MATLAB to answer the following question An ellipse with clockwise orientation 1. Describe the trajectory
1) Answer the following questions for harmonic oscillator with the given parameters and initial conditions Find the specific solution without converting to a linear system Convert to a linear system Find the eigenvalues and eigenvectors of the corresponding linear system Classify the oscillator (underdamped, overdamped, critically damped, undamped) (use technology to) Sketch the direction field and phase portrait Sketch the x(t)- and v(t)-graphs of the solution a. b. c. d. e. f. A) mass m-2, spring constant k 1, damping...
Question 2 please MATH308: Differential Equattons Problems for Chapter 7.6 (Complex-Valued Eigenvalues) 1. The following ODE systems have complex eigenvalues. Find the general solution and sketch the phase plane diagrams 3 -2 1 -A x=( x, 5 -1 1 -1*.(49) mu+ku 0 (50) where u(t) is the displacement at time t of the mass from its equilibrium position (a) Let -und show that the resulting system is 1) (51) b) Find the eigenvalues of the matrix in part (a). (c)...
# 2: Consider the real symmetric matrix A= 4 1 a) What are the eigenvalues and eigenvectors. [Hint: Use wolframalpha.] b) What is the trace of A, what is the sum of the eigenvalues of A. What is a general theorem th c) The eigenvalues of A are real. What is a general theorem which assert conditions that t d) Check that the eigenvectors are real. What is a general theorem which asserts conditions th asserts equality? eigenvalues are real...