1. Unit 2
Vibrationdata
Sine Vibration

2. Vibrationdata
Sine Amplitude Metrics

3. Question Vibrationdata
Does sinusoidal vibration ever occur in rocket vehicles?

4. Solid Rocket Booster, Thrust Oscillation
Vibrationdata
Solid Rocket Booster, Thrust Oscillation
Space Shuttle, 4-segment booster
15 Hz
Ares-I, 5-segment booster
12 Hz

5. Vibrationdata Delta II Main Engine Cutoff (MECO) Transient at ~120 Hz
MECO could be a high force input to spacecraft

6. Pegasus XL Drop Transient
Vibrationdata
Pegasus XL Drop Transient
The Pegasus launch vehicle oscillates as a free-free beam during the 5-second drop, prior to stage 1 ignition.
The fundamental bending frequency is 9 to 10 Hz, depending on the payload’s mass & stiffness properties.

7. Pegasus XL Drop Transient Data
Vibrationdata
Pegasus XL Drop Transient Data

8. Pogo
Vibrationdata
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.

9. Gemini Program Titan II Pogo
Vibrationdata
Gemini Program Titan II Pogo
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.

10. Vibrationdata Flight Anomaly
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.

11. Flight Accelerometer Data
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12. Vibrationdata
Sine Function Example

13. Sine Function Bathtub Histogram
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Sine Function Bathtub Histogram

14. Vibrationdata Sine Formulas v(t) =  X cos (t) a(t) = -2 X sin (t)
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)

15. 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
Value
acceleration A
velocity
A/
displacement
A/2

16. Acceleration Displacement Relationship
Vibrationdata
Freq (Hz)
Displacement (inches zero-to-peak)
0.1
9778 1
97.8 10
0.978 20
0.244 50
100
9.78E-03
1000
9.78E-05
Displacement
for 10 G sine Excitation
Shaker table test specifications typically have a lower frequency limit of 10 to 20 Hz to control displacement.

17. Sine Calculation Example
Vibrationdata
Sine Calculation Example
What is the displacement corresponding to a 2.5 G, 25 Hz oscillation?

18. Vibrationdata Sine Amplitude
Sine vibration has the following relationships.
These equations do not apply to random vibration, however.

19. SDOF System Subjected to Base Excitation
Vibrationdata

20. Vibrationdata Free Body Diagram
Summation of forces in the vertical direction
Let z = x - y. The variable z is thus the relative displacement.
Substituting the relative displacement yields

21. Vibrationdata Equation of Motion  is the natural frequency (rad/sec)
By convention,
is the natural frequency (rad/sec) 
is the damping ratio
Substituting the convention terms into equation,
This is a second-order, linear, non-homogenous, ordinary differential equation with constant coefficients.

22. Equation of Motion (cont)
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Solve for the relative displacement z using Laplace transforms.
Then, the absolute acceleration is
could be a sine base acceleration or an arbitrary function

23. Equation of Motion (cont)
Vibrationdata
A unit impulse response function h(t) may be defined for this homogeneous case as
A convolution integral can be used for the case where the base input is arbitrary.
where

24. Equation of Motion (cont)
Vibrationdata
The convolution integral is numerically inefficient to solve in its equivalent digital-series form.
Instead, use…
Smallwood, ramp invariant, digital recursive filtering relationship!

25. Equation of Motion (cont)
Vibrationdata

26. Sine Vibration Exercise 1
Vibrationdata
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.

27. Sine Vibration Exercise 2
Vibrationdata
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. .

28. Sine Vibration Exercise 3
Vibrationdata
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!

29. 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: 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.

30. 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

31. 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