System Dynamics
3rd Edition
ISBN: 9780073398068
Author: III William J. Palm
Publisher: MCG
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Chapter 2, Problem 2.46P
Use the Laplace transform to solve the following problem:
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Chapter 2 Solutions
System Dynamics
Ch. 2 - Prob. 2.1PCh. 2 - Solve each of the following problems by direct...Ch. 2 - Solve each of the following problems by separation...Ch. 2 - Prob. 2.4PCh. 2 - Prob. 2.5PCh. 2 - Obtain the Laplace transform of the following...Ch. 2 - Obtain the Laplace transform of the function shown...Ch. 2 - Prob. 2.8PCh. 2 - Prob. 2.9PCh. 2 - Prob. 2.10P
Ch. 2 - Prob. 2.11PCh. 2 - Obtain the inverse Laplace transform xt for each...Ch. 2 - Solve the following problems: 5x=7tx0=3...Ch. 2 - Solve the following: 5x+7x=0x0=4 5x+7x=15x0=0...Ch. 2 - Solve the following problems: x+10x+21x=0x0=4x0=3...Ch. 2 - Solve the following problems: x+7x+10x=20x0=5x0=3...Ch. 2 - Solve the following problems: 3x+30x+63x=5x0=x0=0...Ch. 2 - Solve the following problems where x0=x0=0 ....Ch. 2 - Invert the following transforms: 6ss+5 4s+3s+8...Ch. 2 - Invert the following transforms: 3s+2s2s+10...Ch. 2 - Prob. 2.21PCh. 2 - Compare the LCD method with equation (2.4.4) for...Ch. 2 - Prob. 2.23PCh. 2 - Prob. 2.24PCh. 2 - (a) Prove that the second-order system whose...Ch. 2 - For each of the following models, compute the time...Ch. 2 - Prob. 2.27PCh. 2 - Prob. 2.28PCh. 2 - Prob. 2.29PCh. 2 - If applicable, compute , , n , and d for the...Ch. 2 - Prob. 2.31PCh. 2 - For each of the following equations, determine the...Ch. 2 - Prob. 2.33PCh. 2 - Obtain the transfer functions Xs/Fs and Ys/Fs for...Ch. 2 - a. Obtain the transfer functions Xs/Fs and Ys/Fs...Ch. 2 - Prob. 2.36PCh. 2 - Solve the following problems for xt . Compare the...Ch. 2 - Prob. 2.38PCh. 2 - Prob. 2.39PCh. 2 - Prob. 2.40PCh. 2 - Determine the general form of the solution of the...Ch. 2 - a. Use the Laplace transform to obtain the form of...Ch. 2 - Prob. 2.43PCh. 2 - Prob. 2.44PCh. 2 - Obtain the inverse transform in the form xt=Asint+...Ch. 2 - Use the Laplace transform to solve the following...Ch. 2 - Express the oscillatory part of the solution of...Ch. 2 - Prob. 2.48PCh. 2 - Prob. 2.49PCh. 2 - Prob. 2.50PCh. 2 - Prob. 2.51PCh. 2 - Prob. 2.52PCh. 2 - Prob. 2.53PCh. 2 - 2.54 The Taylor series expansion for tan t...Ch. 2 - 2.55 Derive the initial value theorem:
Ch. 2 - Prob. 2.56PCh. 2 - Prob. 2.57PCh. 2 - Use MATLAB to obtain the inverse transform of the...Ch. 2 - Use MATLAB to obtain the inverse transform of the...Ch. 2 - Use MATLAB to solve for and plot the unit-step...Ch. 2 - Use MATLAB to solve for and plot the unit-impulse...Ch. 2 - Use MATLAB to solve for and plot the impulse...Ch. 2 - Use MATLAB to solve for and plot the response of...
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- Problem 2 (55 pts). We now consider the FEM solution of Problem 1.(a) [5pts] Briefly describe the 4 steps necessary to obtain the approximate solution of thatBVP using the Galerkin FEM. Use the minimum amount of math necessary to supportyour explanations.(b) [20pts] Derive the weak form of the BVP.(c) [10pts] Assuming a mesh of two equal elements and linear shape functions, sketch byhand how you expect the FEM solution to look like. Also sketch the analytical solutionfor comparison. In your sketch, identify the nodal degrees of freedom that the FEMsolution seeks to find.(d) [10pts] By analogy with the elastic rod problem and heat conduction problem considered in class, write down the stiffness matrix and force vector for each of the twoelements considered in (c).(e) [10pts] Assemble the global system of equations, and verbally explain how to solve it.arrow_forwardAn aluminum rod of length L = 1m has mass density ρ = 2700 kgm3 andYoung’s modulus E = 70GPa. The rod is fixed at both ends. The exactnatural eigenfrequencies of the rod are ωexactn =πnLqEρfor n=1,2,3,. . . .1. What is the minimum number of linear elements necessary todetermine the fundamental frequency ω1 of the system? Discretizethe rod in that many elements of equal length, assemble the globalsystem of equations KU = ω2MU, and find the fundamentalfrequency ω1. Compute the relative error e1 = (ω1 − ωexact1)/ωexact1.Sketch the fundamental mode of vibration.arrow_forwardProblem 1 (65 pts, suggested time 50 mins). An elastic string of constant line tension1T is pinned at x = 0 and x = L. A constant distributed vertical force per unit length p(with units N/m) is applied to the string. Under this force, the string deflects by an amountv(x) from its undeformed (horizontal) state, as shown in the figure below.The PDE describing mechanical equilibrium for the string isddx Tdvdx− p = 0 . (1)(a) [5pts] Identify the BCs for the string and identify their type (essential/natural). Writedown the strong-form BVP for the string, including PDE and BCs.(b) [10pts] Find the analytical solution of the BVP in (a). Compute the exact deflectionof the midpoint v(L/2).(c) [15pts] Derive the weak-form BVP.(d) [5pts] What is the minimum number of linear elements necessary to compute the deflection of the midpoint?(e) [15pts] Write down the element stiffness matrix and the element force vector for eachelement.arrow_forward
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