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.


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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.


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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


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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.


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?


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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


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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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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 Python script: vibrationdata.py

Miscellaneous Analysis > Generate Signal > sine

Amplitude = 1

Duration = 5 sec

Frequency = 10 Hz

Phase = 0

Sample Rate = 8000 Hz

Export Data > sine_data.txt

sine_data.txt will be used in next exercise.

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Sine Vibration Exercise 2 VibrationdataVibrationdata

Use Python script: vibrationdata.py

Select Signal Analysis > Statistics

Read Input > sine_data.txt

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 Python script: python.py

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 > Select Signal 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: vibrationdata > Sine & Damped Sine Curve-fit

to find the two dominant natural frequencies.

Enter: 10000 trials, 2 frequencies

Select Type: Damped Sine

Enter time limits: 9.5 to 9.6 seconds

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:

vibrationdata > Miscellaneous Analysis > Sine Amplitude Conversion

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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 Analysis > Sine Amplitude Conversion

Sine Vibration

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