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1 Sine Vibration Vibrationdata Unit 2

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2 Vibrationdata Sine Amplitude Metrics

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

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4 Space Shuttle, 4-segment booster15 Hz Ares-I, 5-segment booster12 Hz Vibrationdata Solid Rocket Booster, Thrust Oscillation

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

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6 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. Vibrationdata Pegasus XL Drop Transient

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7 Vibrationdata Pegasus XL Drop Transient Data

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8 Pogo 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. Vibrationdata

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9 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. Vibrationdata

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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.

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11 Flight Accelerometer Data Vibrationdata

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12 Sine Function Example Vibrationdata

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13 Sine Function Bathtub Histogram Vibrationdata

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14 Sine Formulas The acceleration a(t) is obtained by taking the derivative of the velocity. Sine Displacement Function The displacement x(t) is where X is the displacement ω is the frequency (radians/time) The velocity v(t) is obtained by taking the derivative. Vibrationdata x(t) = X sin ( t) v(t) = X cos ( t) a(t) = - 2 X sin ( t)

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15 Peak Sine Values Vibrationdata Peak Values Referenced to Peak Displacement ParameterValue displacementX velocity X X acceleration 2 X Peak Values Referenced to Peak Acceleration ParameterValue accelerationA velocity A/ displacement A/ 2

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16 Acceleration Displacement Relationship Vibrationdata Shaker table test specifications typically have a lower frequency limit of 10 to 20 Hz to control displacement. Freq (Hz) Displacement (inches zero-to-peak) 0.19778 197.8 100.978 200.244 500.03911 1009.78E-03 10009.78E-05 Displacement for 10 G sine Excitation

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

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

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19 SDOF System Subjected to Base Excitation Vibrationdata

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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

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

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

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

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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!

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25 Equation of Motion (cont) Vibrationdata

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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.

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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..

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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!

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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: 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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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

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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

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