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Sine Vibration

Vibrationdata. Unit 2. Sine Vibration. Vibrationdata. Sine Amplitude Metrics. Question. Vibrationdata. Does sinusoidal vibration ever occur in rocket vehicles?. Solid Rocket Booster, Thrust Oscillation. Vibrationdata. Delta II. Vibrationdata. Main Engine Cutoff (MECO)

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Sine Vibration

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  1. Vibrationdata Unit 2 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

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

  6. Pegasus XL Drop Transient Vibrationdata 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

  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 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. Flight Anomaly Vibrationdata 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 Vibrationdata

  12. Sine Function Example Vibrationdata

  13. Sine Function Bathtub Histogram Vibrationdata

  14. Sine Formulas Vibrationdata Sine Displacement Function The displacement x(t) is The acceleration a(t) is obtained by taking the derivative of the velocity. 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) a(t) = -2 X sin (t)

  15. Peak Sine Values Vibrationdata Peak Values Referenced to Peak Displacement Peak Values Referenced to Peak Acceleration

  16. Acceleration Displacement Relationship Vibrationdata 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 What is the displacement corresponding to a 2.5 G, 25 Hz oscillation?

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

  19. SDOF System Subjected to Base Excitation Vibrationdata

  20. Free Body Diagram Vibrationdata 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. Equation of Motion Vibrationdata By convention, 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) Vibrationdata could be a sine base acceleration or an arbitrary function Solve for the relative displacement z using Laplace transforms. Then, the absolute acceleration is

  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 Python script: vibrationdata.py Miscellaneous Analysis > Generate Signal > sine Amplitude = 1 Duration = 5 sec Frequency = 10 Hz Phase = 0 Sample Rate = 8000 Hz Export Data > sine_data.txt sine_data.txt will be used in next exercise.

  27. Sine Vibration Exercise 2 Vibrationdata Use Python script: vibrationdata.py Select Signal Analysis > Statistics Read Input > sine_data.txt 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 Python script: python.py 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 > Select Signal 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: vibrationdata > Sine & Damped Sine Curve-fit to find the two dominant natural frequencies. Enter: 10000 trials, 2 frequencies Select Type: Damped Sine Enter time limits: 9.5 to 9.6 seconds 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: vibrationdata > Miscellaneous Analysis > Sine Amplitude Conversion

  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 Analysis > Sine Amplitude Conversion

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