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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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Sine Vibration

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Sine vibration

Vibrationdata

Unit 2

Sine Vibration


Sine vibration

Vibrationdata

Sine Amplitude Metrics


Question

Question

Vibrationdata

Does sinusoidal vibration ever occur in rocket vehicles?


Solid rocket booster thrust oscillation

Solid Rocket Booster, Thrust Oscillation

Vibrationdata


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.


Pegasus xl drop transient data

Pegasus XL Drop Transient Data

Vibrationdata


Sine vibration

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.


Flight accelerometer data

Flight Accelerometer Data

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Sine function example

Sine Function Example

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Sine function bathtub histogram

Sine Function Bathtub Histogram

Vibrationdata


Sine formulas

Sine Formulas

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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)

  • 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.


    Sdof system subjected to base excitation

    SDOF System Subjected to Base Excitation

    Vibrationdata


    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!


    Equation of motion cont3

    Equation of Motion (cont)

    Vibrationdata


    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 = 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.


    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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