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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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Presentation Transcript
slide1
Vibrationdata

Unit 2

Sine Vibration

slide2
Vibrationdata

Sine Amplitude Metrics

question
Question

Vibrationdata

Does sinusoidal vibration ever occur in rocket vehicles?

delta ii
Delta II

Vibrationdata

Main Engine Cutoff (MECO)

Transient at ~120 Hz

MECO could be a high force input to spacecraft

pegasus xl drop transient
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.

slide8
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.

gemini program titan ii pogo
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.

flight anomaly
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.

sine formulas
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)

peak sine values
Peak Sine Values

Vibrationdata

Peak Values Referenced to Peak Displacement

Peak Values Referenced to Peak Acceleration

acceleration displacement relationship
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.

sine calculation example
Sine Calculation Example

Vibrationdata

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

sine amplitude
Sine Amplitude

Vibrationdata

Sine vibration has the following relationships.

These equations do not apply to random vibration, however.

free body diagram
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

equation of motion
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.

equation of motion cont
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

equation of motion cont1
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

equation of motion cont2
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!

sine vibration exercise 1
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.

sine vibration exercise 2
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.

.

sine vibration exercise 3
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!

sine vibration exercise 4
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.

sine vibration exercise 5
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

sine vibration exercise 6
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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