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.

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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.

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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.

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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.

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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

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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

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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.

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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

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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.

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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

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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

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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.

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VibrationdataVibrationdataModifications

iM2

n

1iM̂

Define a modified moment iM̂

.

r

1qqqjqi

qq

4nji xM̂M̂

IE

1k

The stiffness coefficients become

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VibrationdataVibrationdataModifications (Continued)

Define a modified stiffness matrix

K1

K̂4

n

K̂K 4n

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VibrationdataVibrationdataEigenvalue Problem

The modification leads to an unusual form of the generalized eigenvalue problem.

0M2nK̂4

ndet

0MK̂2ndet

0M2nKdet

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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.

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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:

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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.

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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.

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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.

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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.

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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

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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.

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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.

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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

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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.

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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.