Transcript No Slide Title

Vibrationdata
Practical Application of the Rayleigh-Ritz Method
to Verify Launch Vehicle Bending Modes
By Tom Irvine
AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS
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Objective
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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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Need for Analytical Verification
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•
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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Comparison of Methods
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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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Suborbital Vehicle
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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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Vibrationdata
Mass Variation at Time Zero
LAUNCH VEHICLE MASS DATA
DENSITY ( lbm / in )
500
400
300
200
100
0
0
200
400
600
800
STATION (inch)
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Vibrationdata
Stiffness Variation
LAUNCH VEHICLE EI DATA
2
EI ( 1.0e+12 lbf in )
1.0
0.8
0.6
0.4
0.2
0
0
200
400
600
800
STATION (inch)
The EI values are very low at the vehicle’s six joints
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Rayleigh-Ritz Method
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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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Maximum Kinetic Energy T
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1 2 L
T  n
  y 2 dx
0
2

where
y
is the displacement function

is mass per length
L
is the length
n
is the natural frequency
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Maximum Potential Energy P
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2
2


1 L
d y
P
EI 
dx


2
2 0
 dx 

where
E
I
is the elastic modulus
is the area moment of inertia
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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Shear Force and Moment
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The shear force V is the integral of the inertial loading from the free end.
2 L
V   n
 y d


The moment M at x is found from the integral of the shear force.
L
M x  
V d
x

The strain energy of the beam is then found from the integral of the moment.
P
1 L
1
2


M
(
x
)
dx

2 0 Ex  I x 
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Vibrationdata
Mass Coefficients
Divide the vehicle into r segments.
The mass coefficients are determined from the kinetic
energy.
mi j 
r

q 1
 
  y  y
 x
q
i
j
q
q
q 

where
yi
is the displacement shape for the i th mode
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Vibrationdata
Stiffness Coefficients
 1

M i q M j q   x q
kij 

E I

q 1  q q
r

 
where
M i q
But each moment term
is the moment function for mode shape i at station q
M has an embedded n
2
, which is unknown.
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Vibrationdata
Modifications
ˆ
M
i
Define a modified moment
ˆ 
M
i
1
Mi
2
n
The stiffness coefficients become
.
kij 
 
n4
 1

ˆ
ˆ
  E I M i q M j q  xq

q 1  q q
r
   
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Vibrationdata
Modifications (Continued)
Define a modified stiffness matrix
ˆ 
K
1
n4
K
ˆ
K  n4 K
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Vibrationdata
Eigenvalue Problem
The modification leads to an unusual form of the generalized
eigenvalue problem.

det K  n 2 M

 0

ˆ  2 M 0
det n 4 K
n

ˆ M
det n 2 K
 0
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Rayleigh-Ritz Displacement Function
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The launch vehicle is modeled as a free-free beam.
The displacement function is assumed to be a fourth-order polynomial.
y( x )  a x 4  b x 3  c x 2  d x  e
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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Optimization Goals
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The polynomial coefficients are determined by a trial-and-error optimization
method. The optimization seeks to satisfy the following 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.
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Steps for First Mode
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1. Require that the polynomial meets the following conditions:
y(0)
y(1)
y(x)
y(x)
= 1. (aft end)
= a random number from zero to 4. (fwd end)
= zero at each of two randomly selected x values, representing nodes.
= 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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Steps for First Mode (continued)
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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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Steps for Second Mode
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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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Steps for Third Mode
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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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Vibrationdata
Rayleigh-Ritz Frequency Results
LAUNCH VEHICLE
NATURAL FREQUENCIES vs. TIME
NATURAL FREQUENCY (Hz)
60
50
40
30
20
10
0
0
10
20
30
40
50
60
TIME (SEC)
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Vibrationdata
Frequency Comparison
Natural Frequency Results, Time Zero
Mode
RR
Freq (Hz)
FEA
Freq (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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Vibrationdata
First Mode Shape
LAUNCH VEHICLE
MODE 1
TIME = 0
1.5
FEA
RR
DISPLACEMENT
1.0
0.5
0
-0.5
-1.0
-1.5
0
10
20
30
40
50
60
70
STATION (ft)
The arrows indicate the locations of two of the vehicle joints.
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Vibrationdata
Second Mode
LAUNCH VEHICLE TIME = 0
MODE 2
1.5
FEA
RR
DISPLACEMENT
1.0
0.5
0
-0.5
-1.0
-1.5
0
10
20
30
40
50
60
70
STATION (ft)
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Payload Considerations
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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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Conclusions





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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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