Sine Vibration
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Sine Vibration
VibrationdataVibrationdataUnit 2
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VibrationdataVibrationdataSine Amplitude Metrics
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Question
Does sinusoidal vibration ever occur in rocket vehicles?
VibrationdataVibrationdata
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Space Shuttle, 4-segment booster 15 Hz
Ares-I, 5-segment booster 12 Hz
VibrationdataVibrationdataSolid Rocket Booster, Thrust Oscillation
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Main Engine Cutoff (MECO)Transient at ~120 Hz
MECO could be a high force input to spacecraft
VibrationdataVibrationdataDelta II
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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.
VibrationdataVibrationdataPegasus XL Drop Transient
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-2.5
-2.0
-1.5
-1.0
-0.5
0
0.5
1.0
1.5
2.0
2.5
0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
y=1.55*exp(-0.64*(x-0.195))Flight Data
damp = 1.0%fn = 9.9 Hz
TIME (SEC)
AC
CE
L (
G)
PEGASUS REX2 S3-5 PAYLOAD INTERFACE Z-AXIS5 TO 15 Hz BP FILTERED
VibrationdataVibrationdataPegasus XL Drop Transient Data
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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.
VibrationdataVibrationdata
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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.
VibrationdataVibrationdata
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Flight Anomaly VibrationdataVibrationdata
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.
-4
-3
-2
-1
0
1
2
3
4
87.0 87.5 88.0 88.5 89.0 89.5 90.0 90.5 91.0 91.5 92.0 92.5
TIME (SEC)
AC
CE
L (
G)
LAUNCH VEHICLECONTROL SYSTEM OSCILLATION AT STAGE 1 BURN-OUT
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Flight Accelerometer Data
-10
-5
5
10
0
44.35 44.36 44.37 44.38
DOMINANT FREQUENCY = 1600 Hz
TIME (SEC)
AC
CE
L (
G)
MTTV6 RV X-AXIS GAS GENERATOR OSCILLATION1000 Hz to 2000 Hz
VibrationdataVibrationdata
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Sine Function Example VibrationdataVibrationdata
-1.0
-0.5
0
0.5
1.0
0 0.5 1.0 1.5 2.0
TIME (SEC)
AC
CE
L (
G)
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Sine Function Bathtub Histogram VibrationdataVibrationdata
-1.5 -1 -0.5 0 0.5 1 1.50
200
400
600
800
1000
1200
1400
1600
1800
2000 Histogram
Co
un
ts
Amplitude
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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.
VibrationdataVibrationdata
x(t) = X sin (t)
v(t) = X cos (t)
a(t) = -2 X sin (t)
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Peak Sine Values VibrationdataVibrationdata
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 VibrationdataVibrationdata
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)
0.1 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
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Sine Calculation Example
peaktozeroinch039.0peak
X
G2sec/in386G5.2
])Hz25(2[1
peakX
peakX1
peakX
2
2
What is the displacement corresponding to a 2.5 G, 25 Hz oscillation?
VibrationdataVibrationdata
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Sine Amplitude VibrationdataVibrationdata
Sine vibration has the following relationships.
These equations do not apply to random vibration, however.
RMSX2peakX
peakXX2RMS
1
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SDOF System Subjected to Base Excitation VibrationdataVibrationdata
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Free Body Diagram VibrationdataVibrationdata
Summation of forces in the vertical direction
Let z = x - y. The variable z is thus the relative displacement.
Substituting the relative displacement yields
)x(yk)xy(cxm
kzzc)yzm(
ymkzzczm
y(k/m)zz(c/m)z
xmF
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Equation of Motion VibrationdataVibrationdata
By convention,
nωξ 2c/m
2nωk/m
yz2nωznω2ξz
Substituting the convention terms into equation,
is the natural frequency (rad/sec)
is the damping ratio
nω
This is a second-order, linear, non-homogenous, ordinary differential equation with constant coefficients.
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Equation of Motion (cont) VibrationdataVibrationdata
yz2nωznω2ξz
Solve for the relative displacement z using Laplace transforms.
Then, the absolute acceleration is
yzx
y could be a sine base acceleration or an arbitrary function
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yz2nωznω2ξz
A convolution integral can be used for the case where the base input is arbitrary.
d)-t(sin)-t(nexp)(Y1
=)tz(t
0 dd
2nd 1
A unit impulse response function h(t) may be defined for this homogeneous case as
where
tsin)texp(1
=h(t) dnd
Equation of Motion (cont) VibrationdataVibrationdata
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Equation of Motion (cont) VibrationdataVibrationdata
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) VibrationdataVibrationdata
2idnd
n
1idd
dn
idnd
2in
1idni
yTsinTexpT
1T2exp
yTsinT
1TcosTexp2
yTsinTexpT
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xt2exp
xtcostexp2x
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Sine Vibration Exercise 1VibrationdataVibrationdata
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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Sine Vibration Exercise 2 VibrationdataVibrationdata
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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Sine Vibration Exercise 3 VibrationdataVibrationdata
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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Sine Vibration Exercise 4 VibrationdataVibrationdata
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 VibrationdataVibrationdata
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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Sine Vibration Exercise 6 VibrationdataVibrationdata
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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