Sine Vibration

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Unit 2

• Sine Vibration

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Sine Amplitude Metrics
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Question
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• Does sinusoidal vibration ever occur in rocket
  vehicles?

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Solid Rocket Booster, Thrust Oscillation
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Space Shuttle, 4-segment booster 15 Hz
Ares-I, 5-segment booster 12 Hz
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Delta II
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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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Pegasus XL Drop Transient
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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 payloads mass stiffness
  properties.

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Pegasus XL Drop Transient Data
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Pogo
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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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Gemini Program Titan II Pogo
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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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Flight Anomaly
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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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Flight Accelerometer Data
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Sine Function Example
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Sine Function Bathtub Histogram
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Sine Formulas
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Sine Displacement Function The displacement
x(t) is
x(t) X sin (?t)
where X is the displacement ? is
the frequency (radians/time) The velocity v(t)
is obtained by taking the derivative.
v(t) ? X cos (?t)

• The acceleration a(t) is obtained by taking the
  derivative of the velocity.

a(t) -?2 X sin (?t)
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Peak Sine Values
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Peak Values Referenced to Peak Displacement
Parameter Value
displacement X
velocity ? X
acceleration ?2 X
Peak Values Referenced to Peak Acceleration
Parameter Value
acceleration A
velocity A/?
displacement A/?2
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Acceleration Displacement Relationship
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Freq (Hz) Displacement (inches zero-to-peak)
0.1 9778
1 97.8
10 0.978
20 0.244
50 0.03911
100 9.78E-03
1000 9.78E-05
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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Sine Calculation Example
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• What is the displacement corresponding to a 2.5
  G, 25 Hz oscillation?

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Sine Amplitude
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Sine vibration has the following
relationships. These equations do not
apply to random vibration, however.
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SDOF System Subjected to Base Excitation
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Free Body Diagram
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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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Equation of Motion
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By convention,
is the natural frequency (rad/sec)
? is the damping ratio
Substituting the convention terms into equation,
This is a second-order, linear, non-homogenous,
ordinary differential equation with constant
coefficients.
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Equation of Motion (cont)
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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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Equation of Motion (cont)
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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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Equation of Motion (cont)
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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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Equation of Motion (cont)
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Sine Vibration Exercise 1
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Use Matlab script vibrationdata.m Miscellaneous
Functions gt Generate Signal gt Begin
Miscellaneous Analysis gt Select Signal gt
sine Amplitude 1 Duration 5 sec Frequency
10 Hz Phase 0 deg Sample Rate 8000
Hz Save Signal to Matlab Workspace gt Output
Array Name gt sine_data gt Save sine_data will
be used in next exercise. So keep vibrationdata
opened.
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Sine Vibration Exercise 2
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Use Matlab script vibrationdata.m Must have
sine_data available in Matlab workspace from
previous exercise. Select Analysis gt Statistics
gt Begin Signal Analysis gt Input Array Name gt
sine_data gt Calculate Check Results. RMS2
mean2 std dev2 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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Sine Vibration Exercise 3
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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 (fn10 Hz,
Q10). Calculate response. Use Smallwood
algorithm (although exact solution could be
obtained via Laplace transforms).
Vibrationdata gt Time History gt Acceleration gt
Select Analysis gt SDOF Response to Base Input
This example is resonant excitation because
base excitation and natural frequencies are the
same!
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Sine Vibration Exercise 4
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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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Sine Vibration Exercise 5
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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 gt vibrationdata gt Miscellaneous
  Functions gt Amplitude Conversion Utilities gt
  Steady-state Sine Amplitude

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Sine Vibration Exercise 6
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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 gt Miscellaneous
Functions gt Amplitude Conversion Utilities gt
Steady-state Sine Amplitude