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Unit 2
Sine Vibration
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Sine Amplitude Metrics
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Question
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Does sinusoidal vibration ever occur in rocket vehicles?
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Solid Rocket Booster, Thrust Oscillation
Space Shuttle, 4-segment booster
15 Hz
Ares-I, 5-segment booster
12 Hz
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Delta II
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Main Engine Cutoff (MECO)
Transient at ~120 Hz
MECO could be a high force input to
spacecraft
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Pegasus XL Drop Transient
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The Pegasus launch vehicle oscillates
as a free-free beam during the 5second drop, prior to stage 1 ignition.
The fundamental bending frequency is
9 to 10 Hz, depending on the
payload’s mass & stiffness properties.
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Vibrationdata
Pegasus XL Drop Transient Data
PEGASUS REX2 S3-5 PAYLOAD INTERFACE Z-AXIS
5 TO 15 Hz BP FILTERED
2.5
y=1.55*exp(-0.64*(x-0.195))
Flight Data
2.0
1.5
ACCEL (G)
1.0
0.5
0
-0.5
-1.0
fn = 9.9 Hz
damp = 1.0%
-1.5
-2.0
-2.5
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
TIME (SEC)
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Pogo
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Pogo is the popular name for a dynamic phenomenon that sometimes occurs during
the launch and ascent of space vehicles powered by liquid propellant rocket
engines.
The phenomenon is due to a coupling between the first longitudinal resonance of
the vehicle and the fuel flow to the rocket engines.
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Gemini Program Titan II Pogo
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Astronaut Michael Collins wrote:
The first stage of the Titan II vibrated longitudinally,
so that someone riding on it would be bounced up
and down as if on a pogo stick. The vibration was
at a relatively high frequency, about 11 cycles per
second, with an amplitude of plus or minus 5 Gs in
the worst case.
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Vibrationdata
Flight Anomaly
LAUNCH VEHICLE
CONTROL SYSTEM OSCILLATION AT STAGE 1 BURN-OUT
4
3
ACCEL (G)
2
1
0
-1
-2
-3
-4
87.0
87.5
88.0
88.5
89.0
89.5
90.0
90.5
91.0
91.5
92.0
92.5
TIME (SEC)
The flight accelerometer data was measured on a launch vehicle which shall remain anonymous. This was due to an
oscillating thrust vector control (TVC) system during the burn-out of a solid rocket motor. This created a “tail wags
dog” effect. The resulting vibration occurred throughout much of the vehicle. The oscillation frequency was 12.5 Hz
with a harmonic at 37.5 Hz.
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Flight Accelerometer Data
MTTV6 RV X-AXIS GAS GENERATOR OSCILLATION
1000 Hz to 2000 Hz
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DOMINANT FREQUENCY = 1600 Hz
ACCEL (G)
5
0
-5
-10
44.35
44.36
44.37
44.38
TIME (SEC)
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Sine Function Example
1.0
ACCEL (G)
0.5
0
-0.5
-1.0
0
0.5
1.0
1.5
2.0
TIME (SEC)
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Sine Function Bathtub Histogram
Histogram
2000
1800
1600
1400
Counts
1200
1000
800
600
400
200
0
-1.5
-1
-0.5
0
0.5
1
1.5
Amplitude
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Vibrationdata
Sine Formulas
Sine Displacement Function
The displacement x(t) is
x(t) = X sin (t)
where
X is the displacement
ω is the frequency (radians/time)
The velocity v(t) is obtained by taking the derivative.
v(t) =  X cos (t)
The acceleration a(t) is obtained by taking the derivative of the velocity.
a(t) = -2 X sin (t)
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Vibrationdata
Peak Sine Values
Peak Values Referenced to Peak Displacement
Parameter
Value
displacement
X
velocity
X
acceleration
2 X
Peak Values Referenced to Peak Acceleration
Parameter
acceleration
Value
A
velocity
A/
displacement
A/2
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Acceleration Displacement Relationship
Freq (Hz)
Displacement
(inches zero-to-peak)
0.1
9778
1
97.8
10
0.978
20
0.244
50
0.03911
100
9.78E-03
1000
9.78E-05
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Displacement
for 10 G sine Excitation
Shaker table test specifications typically have a lower frequency limit of 10 to 20 Hz to
control displacement.
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Sine Calculation Example
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What is the displacement corresponding to a 2.5 G, 25 Hz oscillation?
1 
X

X
peak 2 peak

2 
1
386
in
/
sec



X

2 .5 G 

peak [2  ( 25 Hz )] 2
G


X
 0.039 inch zero  to  peak
peak
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Vibrationdata
Sine Amplitude
Sine vibration has the following relationships.
Xpeak  2 XRMS
XRMS 
1
Xpeak
2
These equations do not apply to random vibration, however.
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SDOF System Subjected to Base Excitation
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Vibrationdata
Free Body Diagram
Summation of forces in the vertical direction
 F  m x
m x  c (y  x )  k (y  x)
Let z = x - y. The variable z is thus the relative displacement.
Substituting the relative displacement yields
m(z  y)  cz  kz
m z  cz  kz   my
z  (c/m)z  (k/m)z  y
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Equation of Motion
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By convention,
c/m  2 ξ ωn
k/m  ω n 2
ωn
is the natural frequency (rad/sec)

is the damping ratio
Substituting the convention terms into equation,
z  2ξ ωnz  ωn2z  y
This is a second-order, linear, non-homogenous, ordinary differential equation
with constant coefficients.
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Equation of Motion (cont)
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z  2ξ ωnz  ωn2z  y
y
could be a sine base acceleration or an arbitrary function
Solve for the relative displacement z using Laplace transforms.
Then, the absolute acceleration is
x  z  y
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Equation of Motion (cont)
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z  2ξ ωnz  ωn2z  y
A unit impulse response function h(t) may be defined for this homogeneous case as
h(t) =
1
exp(  n t )sin d t 
d
A convolution integral can be used for the case where the base input is arbitrary.
z( t ) = 
where
1 t 
Y() exp  n (t - )sin d ( t - )  d
d 0
d  n 1 2
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Equation of Motion (cont)
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The convolution integral is numerically inefficient to solve in its equivalent digital-series form.
Instead, use…
Smallwood, ramp invariant, digital recursive filtering relationship!
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Equation of Motion (cont)
x i 
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 2 exp  n t cosd t x i 1
 exp  2n t x i  2
  1 

 exp  n T sin d T   y i
 1  
  d T 




 1 
 sin d T  y i 1
  2 exp  n T   cos d T   


 d T 




 1 
 exp  n T sin d T  y i  2
 exp  2n T   


 d T 
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Sine Vibration Exercise 1
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Use Matlab script: vibrationdata.m
Miscellaneous Functions > Generate Signal > Begin Miscellaneous Analysis >
Select Signal > sine
Amplitude = 1
Duration = 5 sec
Frequency = 10 Hz
Phase = 0 deg
Sample Rate = 8000 Hz
Save Signal to Matlab Workspace > Output Array Name > sine_data > Save
sine_data will be used in next exercise. So keep vibrationdata opened.
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Sine Vibration Exercise 2
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Use Matlab script: vibrationdata.m
Must have sine_data available in Matlab workspace from previous exercise.
Select Analysis > Statistics > Begin Signal Analysis >
Input Array Name > sine_data > Calculate
Check Results.
RMS^2 = mean^2 + std dev^2
Kurtosis = 1.5 for pure sine vibration
Crest Factor = peak/ (std dev)
Histogram is a bathtub curve.
Experiment with different number of histogram bars.
.
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Sine Vibration Exercise 3
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Use Matlab script: vibrationdata.m
Must have sine data available in Matlab workspace
from previous exercise.
Apply sine as 1 G, 10 Hz base acceleration to SDOF
system with (fn=10 Hz, Q=10). Calculate response.
Use Smallwood algorithm (although exact solution
could be obtained via Laplace transforms).
Vibrationdata > Time History > Acceleration > Select Analysis > SDOF Response to
Base Input
This example is resonant excitation because base excitation and natural frequencies are the
same!
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Sine Vibration Exercise 4
Vibrationdata
File channel.txt is an acceleration time history that was measured during a test of an aluminum
channel beam. The beam was excited by an impulse hammer to measure the damping.
The damping was less than 1% so the signal has only a slight decay.
Use script: sinefind.m to find the two dominant natural frequencies.
Enter time limits: 9.5 to 9.6 seconds
Enter: 10000 trials, 2 frequencies
Select strategy: 2 for automatically estimate frequencies from FFT & zero-crossings
Results should be 583 & 691 Hz (rounded-off)
The difference is about 110 Hz. This is a beat frequency effect. It represents the low-frequency
amplitude modulation in the measured time history.
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Sine Vibration Exercise 5
Vibrationdata
Astronaut Michael Collins wrote:
The first stage of the Titan II vibrated longitudinally, so that someone riding on it
would be bounced up and down as if on a pogo stick. The vibration was at a
relatively high frequency, about 11 cycles per second, with an amplitude of plus
or minus 5 Gs in the worst case.
What was the corresponding displacement?
Perform hand calculation.
Then check via:
Matlab script > vibrationdata > Miscellaneous Functions > Amplitude
Conversion Utilities > Steady-state Sine Amplitude
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Sine Vibration Exercise 6
Vibrationdata
A certain shaker table has a displacement limit of 2 inch
peak-to-peak.
What is the maximum acceleration at 10 Hz?
Perform hand-calculation.
Then check with script:
vibrationdata > Miscellaneous Functions > Amplitude Conversion Utilities >
Steady-state Sine Amplitude
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