**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