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Sine Vibration

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Unit 2

Sine Vibration

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Sine Amplitude Metrics

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Does sinusoidal vibration ever occur in rocket vehicles?

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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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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.

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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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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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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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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)

v(t) = X cos (t)

a(t) = -2 X sin (t)

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Peak Values Referenced to Peak Displacement

Peak Values Referenced to Peak Acceleration

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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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What is the displacement corresponding to a 2.5 G, 25 Hz oscillation?

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Sine vibration has the following relationships.

These equations do not apply to random vibration, however.

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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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By convention,

Substituting the convention terms into equation,

This is a second-order, linear, non-homogenous, ordinary differential equation with constant coefficients.

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could be a sine base acceleration or an arbitrary function

Solve for the relative displacement z using Laplace transforms.

Then, the absolute acceleration is

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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

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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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Use Matlab script: vibrationdata.m

Miscellaneous Functions > Generate Signal > Begin Miscellaneous Analysis >

Select Signal > sine

Amplitude = 1Duration = 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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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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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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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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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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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