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Unformatted text preview: ME563 – Fall 2011
Purdue University
West Lafayette, IN Homework Set No. 7
Assignment date: Friday, October 7
Due date: Friday, October 14
Please attach this cover sheet to your completed homework assignment. Name
PUID Problem 7.1
Problem 7.2
Problem 7.3 TOTAL Homework 7.1
Consider the three systems shown below: (a) longitudinal motion of a homogeneous rod, (b) transverse motion of a homogeneous string, and (c) torsional motion of a homogeneous shaft. Derive
the boundary conditions for each of the three systems. For each x = L boundary condition, you
must draw an appropriate free body diagram consistent with the sign conventions for corresponding
problem. You will not receive credit for your work without correct free body diagrams. Homework 7.2
Choose one of three systems described in Problem No. 7.1. For the system of your choice:
1. Derive the characteristic equation. Write this characteristic equation in terms of appropriate
nondimensional stiﬀness and mass parameters.
2. Make a sketch of the characteristic function for numerical values for the nondimensional
stiﬀness and mass parameters of your choice.
3. Based on your sketch above, provide lower and upper bounds for the ﬁrst four natural frequencies of the system. Homework 7.3
Consider the transverse motion w(x, t) for the thin bending beam shown below. Both ends of the
beam are clamped to ground.
1. Derive the characteristic equation.
2. Make a sketch of the characteristic function.
3. Based on your sketch above, provide lower and upper bounds for the ﬁrst four natural frequencies of the system.
4. Determine numerical values for the ﬁrst four natural frequencies using a numerical solver such
as the Matlab function “fsolve”.
5. Determine the ﬁrst four modal functions for the problem. Make a sketch of these modal
functions vs. x. ...
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 Fall '09

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