Unit 18 Vibrationdata Force Vibration Response Spectrum

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Introduction

     SDOF systems may be subjected to an applied force Modal testing, impact or steady-state force Wind, fluid, or gas pressure Acoustic pressure field Rotating or reciprocating parts Rotating imbalance Shaft misalignment Bearings Blade passing frequencies Electromagnetic force, magnetostriction

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SDOF System, Applied Force

Vibrationdata

Governing equation of motion m  x   c   kx  f ( t ) m c = mass = viscous damping coefficient k x = stiffness = displacement of the mass f(t) = applied force 3

Rayleigh Peak Response Formula Vibrationdata

Consider a single-degree-of-freedom system with the index n. The maximum response can be estimated by the following equations.

c n  2 ln  fn T  C n  c n  0 .

5772 c n Maximum Peak  C n  n fn T ln  n is the natural frequency is the duration is the natural logarithm function is the standard deviation of the oscillator response 4

Steady-State Response to Sine Force

The natural frequency fn is fn  1 2  k m

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The normalized displacement is k x F    1 2  2 where F is the applied force magnitude   f / f n f is the applied force frequency fn is the natural frequency 5

Steady-State Response to Sine Force (cont)

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The transmitted force to ground ratio is F t F  1    2 2    2 ,   f / f n where F t is the transmitted force magnitude F is the applied force magnitude

The transmitted force ratio is the same as that for the acceleration response to base excitation.

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20 10 SDOF STEADY-STATE RESPONSE TO APPLIED SINUSOIDAL FORCE Q = 10 Q = 2 Q = 1

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1 0.1

0.01

0.1

1 FREQUENCY ( f / fn ) Control by Frequency Domain Low Freq Resonance Stiffness Damping High Freq Mass 10 7

20 10 SDOF STEADY-STATE TRANSMITTED FORCE

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Q = 10 Q = 2 Q = 1 1 0.1

0.01

0.1

1 FREQUENCY ( f / fn ) 10 8

Exercise

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vibrationdata > Miscellaneous Functions > SDOF Response: Steady-State Sine Force or Acceleration Input

Practice some sample calculations for applied force using your own parameters.

Try resonant excitation and then +/- one octave separation between the excitation and natural frequencies.

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SDOF Response to Force PSD, Miles Equation

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The overall displacement x is x RMS    8 A    1 / 2   1 m   1 / 4   1 k   3 / 4 where m k  A is the mass is the stiffness is viscous damping ratio is the amplitude of the force PSD in dimensions of [force^2 / Hz] at the natural frequency

Miles equation assumes that the PSD is white noise from 0 to infinity Hz.

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Miles Equation, Velocity & Acceleration

The overall velocity is RMS   n x RMS

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• An accelerance FRF curve is shown for a sample system in the next slide • The normalized accelerance converges to 1 as the excitation frequency becomes much larger than the natural frequency • The acceleration response would be infinitely high for a white noise force excitation which extended up to an infinitely high frequency • A Miles equation for the acceleration response to a white noise applied force cannot be derived 11

Miles Equation, Acceleration

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SDOF SYSTEM: mass= 1 kg fn = 100 Hz Damp = 0.05

100 10 1 0.1

0.01

0.001

1 10 100 EXCITATION FREQUENCY (Hz) 1000 12

SDOF Response to Force PSD, General Method

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Displacement x RMS  f n ,    1 k i N   1  1   i 2  2 1   2   i  2 F PSD ( f i )  f i ,  i  f i / f n Velocity RMS  f n ,    2  k i N   1  1   i 2  2 f i 2   2   i  2 F PSD ( f i )  f i 13

SDOF Response to Force PSD, General Method

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Acceleration  x  RMS  f n ,    4  2 k i N   1  1   i 2  2 f i 4   2   i  2 F PSD ( f i )  f i ,  i  f i / f n Transmitted Force F t RMS  f n ,    i N   1  1  1   i 2   2 2    2 i   2  i  2 F PSD ( f i )  f i 14

Force PSD

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Frequency (Hz) 10 1000 Force (lbf^2/Hz) 0.1

0.1

Duration = 60 sec

The same PSD was used for the time domain calculation in Webinar 17 .

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SDOF Example

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Apply the Force PSD on the previous slide to the SDOF system.

Duration = 60 seconds (but only affects peak value) Mass = 20 lbm, Q=10, Natural Frequency = independent variable 16

SDOF Response to Force PSD, Acceleration

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Response at 400 Hz agrees with time domain result in previous webinar unit .

fn (Hz) 100 200 400 Accel (GRMS) 0.80

1.0

1.3

vibrationdata > Power Spectral Density > Force > SDOF Response to Force PSD 17

SDOF Response to Force PSD, Transmitted Force

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Acceleration VRS

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fn (Hz) 100 200 400 Accel (GRMS) 0.80

1.0

1.3

vibrationdata > Power Spectral Density > Force > Vibration Response Spectrum (VRS) 19

Velocity VRS

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Displacement VRS

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Transmitted Force VRS

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Homework Vibrationdata

 Repeat the examples in the presentation using the Matlab scripts 23