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AMERICAN INSTITUTE OF AERONAUTICS AND
ASTRONAUTICS1
VibrationdataVibrationdata
Practical Application of the Rayleigh-Ritz Method
to Verify Launch Vehicle Bending Modes
By Tom Irvine
AMERICAN INSTITUTE OF AERONAUTICS AND
ASTRONAUTICS2
VibrationdataVibrationdataObjective
Determine the natural frequencies and mode shapes of a suborbital launch vehicle via the Rayleigh-Ritz method.
Compare the Rayleigh-Ritz results with the finite element results.
The implementation of the Rayleigh-Ritz method is innovative in that it uses random number generation to determine an optimum displacement function for each mode.
AMERICAN INSTITUTE OF AERONAUTICS AND
ASTRONAUTICS3
VibrationdataVibrationdataNeed for Analytical Verification
• Launch vehicles have closed-loop guidance and control systems.
• The body-bending frequencies and mode shapes must be determined for the control system analysis.
• The FEM is used as the primary analysis method for determining the bending modes.
• The Rayleigh-Ritz method is used to verify the FEM results.
• Ideally, modal testing would also be performed for verification. Program managers often forgo modal testing out of cost and schedule considerations. This decision increases the importance of analytical verification of the bending modes.
AMERICAN INSTITUTE OF AERONAUTICS AND
ASTRONAUTICS4
VibrationdataVibrationdataComparison of Methods
The Rayleigh-Ritz Method uses a single displacement function for each mode across the entire length of the vehicle.
The FEM uses a single displacement function across each element. The elemental displacement functions are then assembled into a piecewise continuous function for the entire length.
Both methods can account for mass and stiffness variation with length.
As an aside, the FEM itself can be derived from the Rayleigh-Ritz method, but the application is different as explained above.
AMERICAN INSTITUTE OF AERONAUTICS AND
ASTRONAUTICS5
VibrationdataVibrationdataSuborbital Vehicle
A suborbital launch vehicle has three solid motor stages, a liquid trim fourth stage, a payload, and a fairing.
The vehicle length is 848 inches. The maximum diameter is 92 inches.
The total vehicle mass is 196,000 lbm at time zero.
AMERICAN INSTITUTE OF AERONAUTICS AND
ASTRONAUTICS6
VibrationdataVibrationdataMass Variation at Time Zero
0
100
200
300
400
500
0 200 400 600 800
STATION (inch)
DE
NS
ITY
( lb
m /
in )
LAUNCH VEHICLE MASS DATA
AMERICAN INSTITUTE OF AERONAUTICS AND
ASTRONAUTICS7
VibrationdataVibrationdataStiffness Variation
0
0.2
0.4
0.6
0.8
1.0
0 200 400 600 800
STATION (inch)
EI (
1.0
e+12
lbf i
n2 )
LAUNCH VEHICLE EI DATA
The EI values are very low at the vehicle’s six joints
AMERICAN INSTITUTE OF AERONAUTICS AND
ASTRONAUTICS8
VibrationdataVibrationdataRayleigh-Ritz Method
maximum kinetic energy = maximum potential energy
Note that strain energy is the potential energy for beam bending modes.
The suborbital vehicle is modeled as a beam for the Rayleigh-Ritz and FE methods.
AMERICAN INSTITUTE OF AERONAUTICS AND
ASTRONAUTICS9
VibrationdataVibrationdataMaximum Kinetic Energy T
L
0dx2y2
n2
1T
where
y is the displacement function
is mass per length
L is the length
n is the natural frequency
AMERICAN INSTITUTE OF AERONAUTICS AND
ASTRONAUTICS10
VibrationdataVibrationdataMaximum Potential Energy P
where
E is the elastic modulusI is the area moment of inertia
L
0dx
2
2dx
y2dEI
2
1P
This equation is only useful for very simple cases, because any error in the displacement function is compounded by taking the first and second derivatives.
AMERICAN INSTITUTE OF AERONAUTICS AND
ASTRONAUTICS11
VibrationdataVibrationdataShear Force and Moment
L
dy2nV
The shear force V is the integral of the inertial loading from the free end.
The moment M at x is found from the integral of the shear force.
L
xdVxM
The strain energy of the beam is then found from the integral of the moment.
L
02 dx)x(M
xIxE
1
2
1P
AMERICAN INSTITUTE OF AERONAUTICS AND
ASTRONAUTICS12
VibrationdataVibrationdataMass Coefficients
r
1qqxjyiyqjim qq
Divide the vehicle into r segments.
The mass coefficients are determined from the kinetic energy.
where
y i is the displacement shape for the i th mode
AMERICAN INSTITUTE OF AERONAUTICS AND
ASTRONAUTICS13
VibrationdataVibrationdataStiffness Coefficients
r
1qqx
qjMqiMqIqE
1jik
is the moment function for mode shape i at station q
where
qiM
But each moment term M has an embedded n2 , which is unknown.
AMERICAN INSTITUTE OF AERONAUTICS AND
ASTRONAUTICS14
VibrationdataVibrationdataModifications
iM2
n
1iM̂
Define a modified moment iM̂
.
r
1qqqjqi
4nji xM̂M̂
IE
1k
The stiffness coefficients become
AMERICAN INSTITUTE OF AERONAUTICS AND
ASTRONAUTICS15
VibrationdataVibrationdataModifications (Continued)
Define a modified stiffness matrix
K1
K̂4
n
K̂K 4n
AMERICAN INSTITUTE OF AERONAUTICS AND
ASTRONAUTICS16
VibrationdataVibrationdataEigenvalue Problem
The modification leads to an unusual form of the generalized eigenvalue problem.
0M2nK̂4
ndet
0MK̂2ndet
0M2nKdet
AMERICAN INSTITUTE OF AERONAUTICS AND
ASTRONAUTICS17
VibrationdataVibrationdataRayleigh-Ritz Displacement Function
exd2xc3xb4xa)x(y
The launch vehicle is modeled as a free-free beam.
The displacement function is assumed to be a fourth-order polynomial.
The coefficients are determined by trial-and-error using random number generation, via a computer program. The program was written in C++ by the author.
AMERICAN INSTITUTE OF AERONAUTICS AND
ASTRONAUTICS18
VibrationdataVibrationdataOptimization Goals
1. The number of nodes should be equal to the mode number plus 1.
2. The CG displacement should be zero. (free-free beam)
3. The shear force and bending moment at each end should be equal to zero.
4. The mode shapes should be orthogonal with respect to one another such that the off-diagonal terms of the mass and stiffness matrices are zero.
The polynomial coefficients are determined by a trial-and-error optimization method. The optimization seeks to satisfy the following goals:
AMERICAN INSTITUTE OF AERONAUTICS AND
ASTRONAUTICS19
VibrationdataVibrationdataSteps for First Mode
1. Require that the polynomial meets the following conditions:
y(0) = 1. (aft end)
y(1) = a random number from zero to 4. (fwd end)
y(x) = zero at each of two randomly selected x values, representing nodes.
y(x) = a negative random number as low as –3 between the nodes.
These conditions yield four equations with four unknown coefficients.
2. Use Gaussian elimination to solve for the polynomial coefficients that satisfy the conditions in step 1.
3. Add a constant to the polynomial so that the CG displacement is zero.
AMERICAN INSTITUTE OF AERONAUTICS AND
ASTRONAUTICS20
VibrationdataVibrationdataSteps for First Mode (continued)
4. Verify that the polynomial has two nodes.
5. Scale the coefficients so that the maximum value of y(x) is 1.
6. Calculate the shear and moment functions.
7. Repeat steps 1 through 6, say, one hundred thousand times.
8. The selected mode shape is the polynomial which most closely satisfies the boundary conditions of zero shear and zero bending moment at each end of the beam.
9. Determine the natural frequency using the Rayleigh method.
AMERICAN INSTITUTE OF AERONAUTICS AND
ASTRONAUTICS21
VibrationdataVibrationdataSteps for Second Mode
• The first mode shape is permanently retained for calculation of the second mode shape.
• The calculation steps for the second mode are similar to those for the first mode. The second mode, however, is required to have three nodes.
• Again, compliance with the boundary conditions is checked.
• Furthermore, an orthogonality check is performed between each candidate mode shape with respect to the first mode shape.
• The results are used to select the optimum polynomial for the second mode shape.
• Finally, the natural frequencies are calculated from the 2 x 2 generalized eigenvalue problem.
AMERICAN INSTITUTE OF AERONAUTICS AND
ASTRONAUTICS22
VibrationdataVibrationdataSteps for Third Mode
The first and second mode shapes are permanently retained for calculation of the third mode shape.
Four nodes are required for the third bending mode. Otherwise, the steps for the third bending mode are similar to those of the second mode.
The natural frequencies are calculated from the 3 x 3 generalized eigenvalue problem.
AMERICAN INSTITUTE OF AERONAUTICS AND
ASTRONAUTICS23
VibrationdataVibrationdataRayleigh-Ritz Frequency Results
0
10
20
30
40
50
60
0 10 20 30 40 50 60
TIME (SEC)
NA
TU
RA
L F
RE
QU
EN
CY
(H
z)
LAUNCH VEHICLENATURAL FREQUENCIES vs. TIME
AMERICAN INSTITUTE OF AERONAUTICS AND
ASTRONAUTICS24
VibrationdataVibrationdataFrequency Comparison
Natural Frequency Results, Time Zero
Mode RRFreq (Hz)
FEAFreq (Hz)
Difference
1 6.94 6.94 0.0%
2 15.6 15.2 2.6%
3 34.2 31.6 8.2%
FE model had 867 CBAR elements. FE software was NE/Nastran.
AMERICAN INSTITUTE OF AERONAUTICS AND
ASTRONAUTICS25
VibrationdataVibrationdataFirst Mode Shape
-1.5
-1.0
-0.5
0
0.5
1.0
1.5
0 10 20 30 40 50 60 70
FEARR
STATION (ft)
DIS
PLAC
EM
EN
TLAUNCH VEHICLE TIME = 0
MODE 1
The arrows indicate the locations of two of the vehicle joints.
AMERICAN INSTITUTE OF AERONAUTICS AND
ASTRONAUTICS26
VibrationdataVibrationdataSecond Mode
-1.5
-1.0
-0.5
0
0.5
1.0
1.5
0 10 20 30 40 50 60 70
FEARR
STATION (ft)
DIS
PLA
CE
ME
NT
LAUNCH VEHICLE TIME = 0MODE 2
AMERICAN INSTITUTE OF AERONAUTICS AND
ASTRONAUTICS27
VibrationdataVibrationdataPayload Considerations
The vehicle in the previous example had a 6000 lbm payload. The payload was considered to have sufficiently high stiffness that its natural frequency could be neglected. This is typical for suborbital vehicle payloads.
The payload mass was thus lumped into the model at the payload CG location.
A later analysis treated the payload mass with greater fidelity. Specifically, the payload mass was attached to the payload interface via a rigid-link. This required a branch in both the Rayleigh-Ritz and finite element models.
The branch technique may be the subject of a future paper.
AMERICAN INSTITUTE OF AERONAUTICS AND
ASTRONAUTICS28
VibrationdataVibrationdataConclusions
The Rayleigh-Ritz method uses a single displacement function to represent each mode shape.
The displacement function was taken as a fourth order polynomial in this report.
The polynomial coefficients were derived using random numbers.
Coefficients were selected on the basis of the optimum compliance with the boundary conditions and orthogonality requirements.
The agreement between Rayleigh-Ritz and FE methods was within 8.2% for the respective frequencies of the first three modes, at time zero.
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