Ritz Solution Techniques

1
Ritz Solution Techniques
ME6443, Dr. A.A. Ferri
Consider a general elastic rod with attached
springs, dampers, and masses and with applied
axial forces
u(x,t)
Axial displacement
2
Kinetic energy of the elastic rod
could be r(x)A(x)
Total kinetic energy
Potential energy of the elastic rod
could be E(x)A(x)
Total potential energy
3
Virtual Work of a single axial point force
Virtual work associated with a distributed force
per unit length f(x,t)
Total virtual work of applied forces
4
Expand displacement field as a finite summation
of Ritz functions (also called basis functions)
multiplied by time-varying generalized coordinates
.
This has the effect of turning an
infinite-dimensional problem (governed by partial
differential equations PDEs) into a set of N
ordinary differential equations (ODEs).
Substitute Ritz expansion into kinetic energy
expression
5
Collecting terms, we get
where
Similarly, inserting the Ritz expansion into the
potential energy expression
6
Collecting terms, we get
where
7
Substituting the Ritz expansion into the virtual
work yields
where
Substitute T, V, and Qj into Lagranges equations
Yields N coupled, second-order, ordinary
differential equations
8
t, s
Example
x
L
Fixed-free string has exact solution for natural
frequencies and eigenfunctions
i 1, 2, ...
9
Kinetic energy
Ritz expansion
where
10
Need to choose y functions that satisfy the
geometric boundary conditions of the problem
For this problem, we will choose simple
polynomials (monomials)
i 1, 2, ..., N
Yields
i,j 1,N
11
Potential energy
12
N1
Lagranges equations yield
Natural frequency
Exact answer
10.3 too high
13
N2
Lagranges equations yield
Let
Thus, for non-trivial solutions, we must have
characteristic equation
14
N2, continued
or
Define
0.3754 too high
20.3806 too high
15
Estimate eigenfunctions from eigenvectors
N2, continued
Take top row, and set c11 1
This gives the ratio of the amount of y1 to y2 in
the first vibration mode
Compare with the exact first eigenfunction
Note the amplitude is unimportant, only the
shape
16
Estimate eigenfunctions from eigenvectors
N2, continued
Take top row, and set c12 1
This gives the ratio of the amount of y1 to y2 in
the 2nd vibration mode
Compare with the exact 2nd eigenfunction
17
Error in the position of node
18
Example Non-uniform axial rod
Ritz expansion
Need to choose y functions that satisfy the
geometric boundary conditions of the problem
19
choose
Note that y(0) y(L) 0
Kinetic energy
20
Mass terms
Introduce variable
21
Potential energy
22
After applying Lagranges equations, get a system
of linear equations
Define
23
N 4
mhat 0.7500 -0.0901 -0.0000 -0.0072
-0.0901 0.7500 -0.0973 -0.0000 -0.0000
-0.0973 0.7500 -0.0993 -0.0072
-0.0000 -0.0993 0.7500
khat 7.4022 -2.2222 -0.0000 -0.6044
-2.2222 29.6088 -6.2400 -0.0000 -0.0000
-6.2400 66.6198 -12.2449 -0.6044
-0.0000 -12.2449 118.4353
natfreqs 3.1233 6.2746 9.4198
12.5790
times
24
PHI 1.1612 0.0698 -0.0075 -0.0066
0.0718 1.1686 -0.0769 -0.0090 0.0076
0.0776 -1.1700 -0.0789 0.0064 0.0075
-0.0775 -1.1626
Now, need to convert eigenvectors PHI into
continuous, eigenfunctions
Define nth eigenfunction
For example,
25
First three modes
26
What if the rod was not tapered?
mhat 0.5000 -0.0000 0.0000 -0.0000
-0.0000 0.5000 0.0000 0.0000 0.0000
0.0000 0.5000 -0.0000 -0.0000
0.0000 -0.0000 0.5000
khat 4.9348 -0.0000 -0.0000 -0.0000
-0.0000 19.7392 0.0000 0.0000 -0.0000
0.0000 44.4132 0.0000 -0.0000
0.0000 0.0000 78.9568
Note that the mass and stiffness matrices are now
diagonal. This implies that the equations of
motion are uncoupled.
This happens because the Ritz functions are the
exact eigenfunctions for this problem.