# Rocket Science and Technology

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Rocket Science and Technology

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90232

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-
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-
8855

Longitudinal Body Bending Dynamics

05 August

2013

By C. P. Hoult

Introduction

Whenever nonrigid (flexible) body effects are considered
such as propellant
sloshing, axial vibrations or body bending
Lagrange's equation
s come into play. In this
memo, the free
-
free behavior of a uniform non
-
rolling beam is first estimated. The
resulting mode shapes are then used in a Rayleigh
-
Ritz approximation to find the natural
frequencies and mode shapes of a non
-
uniform beam. The
results are implemented in an
Excel

code named FLEXIT.xls.

Nomenclature

_______Mnemonic_____________________Definition___________________________

EI

Bending stiffness, lb
-
in
2

EI

Average bending stiffness, lb
-
in
2

L

Beam length, in

Mass per unit length, sl/in

Average mass per unit length, sl/in

N

Number of normal modes used to approximate the beam

displacement,

)
(
t
q
i

Amplitude of the ith normal mode, ft

n

Natural frequency of the nth mode, red/sec

)
(
x
W
n

Normalized shape of the nth mode

T

Kinetic energy

U

Potential ene
rgy

t

Time, sec

x

Distance aft of the nose tip, in

y

Beam lateral displacement, in

)
(
x

Normal mode shape

Uniform Beam

The solution for the bending vibrations of a uniform, non
-
spinning beam are
based on

the material in r
ef. (2), pp.67
-
80:

EI
a

2

(1)

where the beam
-
average properties are denoted by an over bar:

2

L
dx
x
EI
L
EI
0
)
(
1

(2a)

and

L
dx
x
L
0
)
(
1

(2b)

The characteristic equation is

1
)
cosh(
)
cos(

L
a
L
a
n
n

(3)

This has roots

2
)
1
2
(
....
2
7
,
2
5
,
0

n
L
a
n

(4
a
)

More exactly,

....
5
.
3
,
4997527
.
2
,
50561873
.
1
,
0

L
a
n

(4b)

The mode shapes are

x
a
x
a
x
a
x
a
L
a
L
a
L
a
L
a
x
W
n
n
n
n
n
n
n
n
n

cos
cosh
sin
sinh
sin
sinh
cosh
cos
)
(

(5)

3

First Three Bending Mode Shapes
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
0
0.2
0.4
0.6
0.8
1
Distance from Nose Tip / Body
Length
Amplitude
Fundamental
First Harmonic
Second Harmonic

These are the first three mode shapes for a uniform, free
-
free, non
-
rolling beam as
developed from eq's. (4b) and (5).

The source is FLEXIT.xls which
was also used to find
the natural frequencies:

Mode

Natural Frequency,

Fundamental

277.452

First Harmonic

764.806

Second

Harmonic

1499.317

Lagrange's Equations

To estimate the frequencies and mode shapes of a non
-
uniform

beam requires the
solution of Lagrange's equations using the Rayleigh
-
Ritz method.

Start by assuming the
beam displacement is approximated by

)
(
)
(
1
x
W
q
x
y
i
N
i
i

.

(6)

Then, Lagrange's equations are

0

i
i
q
U
q
T
dt
d

(7)

4

The system kinetic energy is

dx
x
y
x
T
L

0
2
)
(
)
(
2
1

(8)

where

the beam displacement is
assumed to be represented by a sum of
N

mode shapes
satisfying the boundary conditions...the Rayleigh
-
Ritz technique:

)
(
)
(
1
x
W
q
t
y
x
y
i
N
i
i

(9)

Next, assume simple harmonic motion so that

)
exp(
t
j
A
q
i
i

(10)

and

)
(
)
(
1
x
W
q
j
x
y
i
i
N
l

,

and

(11
a
)

)
(
)
(
1
2
x
W
q
x
y
i
i
N
i

.

(11b)

W
ith this, the kinetic energy becomes

dx
x
W
q
x
T
L
N
i
i
i
2
0
1
)
(
)
(
2
1

,

(1
2
)

dx
x
W
x
W
q
x
q
T
i
L
N
l
l
l
i
)
(
)
(
)
(
0
1

, or

(1
3
)

dx
x
W
x
W
q
x
q
T
dt
d
i
L
N
l
l
i
)
(
)
(
)
(
0
1
1

. (1
4
)

Substituting from eq.(11
b
) into eq. (14) gives

dx
x
W
x
W
q
x
q
T
dt
d
i
N
l
l
l
L
i
)
(
)
(
)
(
1
2
0

.

(15)

Similarly, the potential strain energy is found from

)
(
"
)
(
1
2
2
x
W
q
x
y
x
y
i
N
i
i

(1
6
)

5

Therefore,

dx
x
W
q
x
EI
q
q
U
L
N
i
i
i
i
i
2
0
1
)
(
"
)
(
2
1

, or

dx
x
W
x
W
q
x
EI
i
L
N
l
l
l
)
(
"
)
(
"
)
(
0
1

(1
7
)

Substituting eq's.
(
15) and (17) into eq. (7) results in

dx
x
W
x
W
q
x
EI
dx
x
W
x
W
q
x
i
L
N
l
l
l
i
N
l
l
l
L
)
(
"
)
(
"
)
(
)
(
)
(
)
(
0
1
1
2
0

(18)

Then, there will be
N
equations of the form

0
)
(
"
)
(
"
)
(
)
(
)
(
)
(
0
2
1

dx
x
W
x
W
x
EI
x
W
x
W
x
q
L
i
l
i
l
N
l
l

(19)

Equations (19) can be easily recast in matrix form. Let the integrals be called
il
J

. Then,

0
}
{
2

l
ilst
ilma
q
J
J

,

(20
a
)

where

,
)
(
)
(
)
(
0
dx
x
W
x
W
x
J
i
L
l
ilma

and

L
i
l
ilst
dx
x
W
x
W
x
EI
J
0
)
(
"
)
(
"
)
(
. (20b)

It can be shown
(1)

that t
he only nontrival solution to eq. (20
a
) occurs when

takes on a
value
n

which causes

0
det
2

ilst
ilma
J
J

(21)

Otherwise, the
0
}
{

l
q
.

Then,

0
}
{
2
1

q
I
J
J
ilst
ilma

(22)

Expanding eq.(2
2
) leads to the characteristic equ
ation for this problem.
If three
assumed modes are used, i
t is a cubic polynomial in
2

.

6

as a function of
2

. The observed approximate zero crossings are then iteratively
refined.

Normal Mode Shapes

The normal mode shapes are derived from

eq
's
.

(6) and

(19). Suppose that

took on one of its solution values, say the kth natural frequency. Then, for three
assumed modes,
a normal mode looks like

)
(
)
(
)
(
)
(
3
1
3
2
1
2
1
x
W
q
q
x
W
q
q
x
W
x

.

(23)

since all that can be determined from unforced oscillations are the relative amplitudes of
the contributions from the various assumed mode shapes.

Equation (19
) can be expanded
to give

0
13
2
13
1
3
12
2
12
1
2
11
2
11

st
ma
st
ma
st
ma
J
J
q
q
J
J
q
q
J
J

,

0
23
2
23
1
3
22
2
22
1
2
21
2
21

st
ma
st
ma
st
ma
J
J
q
q
J
J
q
q
J
J

, and

(24)

0
33
2
33
1
3
32
2
32
1
2
31
2
31

st
ma
st
ma
st
ma
J
J
q
q
J
J
q
q
J
J

.

By implication,

must take on one of the values
n

for there to be a solution at all.
Then
,

any two
, say the first two,

of the three

equations
above can be solved for
1
2
q
q

and
1
3
q
q
:

12
23
13
22
13
21
11
23
1
2
A
A
A
A
A
A
A
A
q
q

and
12
23
13
22
22
11
21
12
1
3
A
A
A
A
A
A
A
A
q
q

where
ijst
ijma
ij
J
J
A

2

. (25)

The mode shape is governed by the choice of
n

.

.

FLEXIT Implementation

Nonuniform beam

properties

are assumed to vary linearly with x across
a

segment. In general

such a beam is
composed of segm
ents in which
)
(
x

and
)
(
x
EI

are
defined at the segment end points, and
, neither
)
(
x

nor
)
(
x
EI

are continuous across
segment boundaries. Trapezoidal integration is used

to evaluate

the integrals in eq. (20b)
.
The desired granularity (number of integration steps
InS

along the body) is input. The
number of steps used for an individual segment is based on

7

INT[
InS
L
L
BODY
SEG
]+1. This ensure
s that at least the desired number of steps are used.

The normal modes
, eq.(5) and their second derivatives
,

are generated by

external
functions:

)
sin(
)
sinh(
)
sin(
)
sinh(
)
cosh(
)
cos(
)
(
"
x
a
x
a
L
a
L
a
L
a
L
a
a
x
W
n
n
n
n
n
n
n
n

)
cos(
)
cosh(
x
a
x
a
a
n
n
n

(2
6
)

Note that
n

appearing in eq's. (5) and (2
6
) above are the natural frequencies for a
uniform beam as given by eq. (4b).

It is customary to normalize the natural mode shapes. There are many ways to do
this. In FLE
XIT they are normalized to have the same nose tip value (2) as the uniform
beam modes
.

FLEXIT uses a simple bisection algorithm to find the roots of the characteristic
equation. This requires the user to make a rough guess on these roots, and then set up
per
and lower bounds for the bisection iteration. There are two ways to do this. First, use the
roots for an roughly equivalent uniform beam as a guide. The second is to plot the
determinant, eq. (21), vs
.

the square of the frequency. The zero crossing
s of this plot are
the roots being sought.

The following table compares the uniform beam results for the
corresponding nonuniform case. The rocket analyzed is the Space Vector Aries,
Minuteman II second stage converted into a sounding rocket.

As can be
seen, the
uniform beam results can be useful to guide the search for roots to the characteristic
equation.

Mode

Fundamental
natural

fr
equency,

First Harmonic
natural frequency,

Second Harmonic
natural frequency,

Nonuniform
B
e
am

25
7.632

585.471

1523.704

Uniform
Beam

277.452

764.806

1499.317

Since our major practical interest is in the fundamental

mode
, only the first three
uniform beam modes are carried in the calculation
, and only the first two mode shapes
are estimated and pl
otted
.

References

1. F. B
.

Hildebrand, "Methods of Applied Mathematics", Prentice
-
Hall, Inc.,Englewood
Cliffs, N. J., 1956.

2. R.L. Bisplinghoff, H. Ashley, and R. L. Halfman, "Aeroelasticity", Addison
-
Wesley
Publishing Company, Cambridge, 1955