HOMEWORK 7_W13(1) - DREXEL UNIVERSITY Department of...

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Unformatted text preview: DREXEL UNIVERSITY Department of Mechanical Engineering & Mechanics Applied Engineering Analytical & Numerical Methods II MEM 592 - Winter 2013 HOMEWORK #7: Due Thursday, March 7 @ 630pm 1. [30 points] Consider the one-dimensional (1-D) heat equation, which arises in the study of heat conduction in solids as well as in a variety of diffusion phenomena ,0 , 0. (1) 0, 0. (2) . (3) Assume the following boundary conditions, 0, , Given also is that the bar initially has the temperature profile ,0 ,0 Provide the general form of the solution. 2. [30 points] The longitudinal vibrations of an elastic bar of length 2 in the -direction are modeled by the 1-D wave equation ,0 2, 0 and free at the other i) If the bar is fastened in one end, 0. (4) , the following conditions are recovered , 0, 0, Compute the solution , that corresponds to initial displacement ii) Compute the solution , up to 0. (5) ,0 and initial velocity equal to zero. 3 that satisfies the initial displacement ,0 ,0 2. 3. [40 points] The fourth-order PDE (6) , is used to model the vertical ( -direction) deflection bending loads, where beam cross-sectional area, / ( of a uniform elastic beam extending along the elastic modulus of the beam material, axis which is subject to moment of inertia of the beam cross section, = density of the beam material). i) Derive a set of ordinary differential equations for the spatial and temporal parts of the solution. Provide a general form of the solution to (6). ii) The solution to (6) which corresponds to zero initial velocity and which also satisfies the boundary conditions 0, 0, , , 0, ∑ 0, 0, , 0 (7) is Find and write the form of the solution defined in (8) up to ,0 cos . (8) 3 in the case that it also satisfies the initial condition . (9) ...
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  • Spring '14
  • Velocity, Partial differential equation, Applied Engineering Analytical & Numerical Methods II, initial displacement, uniform elastic beam

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