Term Project Proposal

Topic: Eigen values in vibrations

Heather Case

Problem Description:  A beam of length L is fixed at both ends and is 
experiencing longitudinal vibrations.  The beam is made of aluminum.  
V(x,t) is the longitudinal displacement at a given position and time.  Figure 
1is a picture of the proposed problem.









Boundary Conditions: x(0)=0, X(L)=0
Initial Conditions: V(x,0)=0
L=2m
Background:  This is a classic eigenvalue problem. This problem can also be 
solved through the use of the Rayleigh- Ritz method and the Rayleigh 
Energy method.  These methods are typically used when the calculation of 
eigenvalues is extremely difficult.  Both methods are design to find the 
natural frequency of a system without finding the eigenvalues.  These 
methods are considered estimates and not extremely accurate.  

I propose to solve for the eigenvalues of the system using the methods 
developed in class.  The "exact" solution will also be calculated using the 
standard assumptions.  The problem will also be solved with the Rayleigh-
Ritz and Rayleigh’s energy method procedures.  The results from these 
calculation methods will be compared and the accuracy of each method 
determined.  Plots of the natural frequencies from each computation will be 
shown.  This problem will be repeated with a couple different beam lengths 
to see the effects on the accuracy of the given methods.  An initial condition 
may also be added to determine the effects on the solution.  The location of 
the mass on the beam can be moved to determine the effects on the accuracy 
of the results.  The resolution of the problem will be 2 and then changed to 3 
to again determine the effects on accuracy.


References
1 Elements of Vibrational Analysis, Leonard Meirovitch, 1986 McGraw-
Hill Inc.

2 Numerical Analysis, R. Burden and J. Faires, 1997 Brooks/Cole 
Publishing Company