Vibrationdata

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Unit 18

Force Vibration Response Spectrum

Vibrationdata

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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

VibrationdataSDOF System, Applied Force

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m = mass

c = viscous damping coefficient

k = stiffness

x = displacement of the mass

f(t) = applied force

)t(fkxxcxm

Governing equation of motion

VibrationdataRayleigh Peak Response Formula

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Tfnln2nc

nc

5772.0ncnC

nnC Maximum Peak

fn is the natural frequency

T is the duration

ln is the natural logarithm functionis the standard deviation of the oscillator responsen

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

VibrationdataSteady-State Response to Sine Force

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222 21

1

F

xk

The normalized displacement is

nf/f

The natural frequency fn is

1 kfn

2 m

f is the applied force frequency

fn is the natural frequency

where F is the applied force magnitude

VibrationdataSteady-State Response to Sine Force (cont)

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The transmitted force to ground ratio is

222

2t

21

21

F

F

where

Ft is the transmitted force magnitude

F is the applied force magnitude

nf/f,

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

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0.01

0.1

1

10

20

0.1 1 10

Q = 10Q = 2Q = 1

FREQUENCY ( f / fn )

DIS

PL

AC

EM

EN

T M

AG

NIT

UD

E [

k x

/ F

]

SDOF STEADY-STATE RESPONSE TO APPLIED SINUSOIDAL FORCE

Low Freq Resonance High Freq

Stiffness Damping Mass

Control by Frequency Domain

Vibrationdata

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0.01

0.1

1

10

20

0.1 1 10

Q = 10Q = 2Q = 1

FREQUENCY ( f / fn )

TR

AN

S F

OR

CE

MA

G

| F

t / F

|SDOF STEADY-STATE TRANSMITTED FORCE

VibrationdataExercise

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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.

VibrationdataSDOF Response to Force PSD, Miles Equation

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4/3

k

14/1

m

12/1

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Ax RMS

m is the mass

k is the stiffness

is viscous damping ratio

Ais the amplitude of the force PSD in dimensions of [force^2 / Hz] at the natural frequency

The overall displacement x is

where

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

VibrationdataMiles Equation, Velocity & Acceleration

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RMSRMS xx nThe overall velocity is

• 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

Vibrationdata

Miles Equation, Acceleration

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0.001

0.01

0.1

1

10

100

1 10 100 1000

EXCITATION FREQUENCY (Hz)

AC

CE

LE

RA

NC

E (

m

/se

c2

/ N

)

ACCELERANCE MAGNITUDE ( ACCELERATION / FORCE )SDOF SYSTEM: mass= 1 kg fn = 100 Hz Damp = 0.05

VibrationdataSDOF Response to Force PSD, General Method

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Displacement

Velocity

N

1iiiPSD

2i

22i

nRMS f)f(F

21

1

k

1,fx

N

1iiiPSD

2i

22i

2i

nRMS f)f(F

21

f

k

2,fx

nii f/f,

VibrationdataSDOF Response to Force PSD, General Method

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Acceleration

Transmitted Force

N

1iiiPSD

2i

22i

2i

nRMS f)f(F

21

21,ftF

nii f/f,

N

1iiiPSD

2i

22i

4i

2

nRMS f)f(F

21

f

k

4,fx

Vibrationdata

Force PSD

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Frequency (Hz)

Force (lbf^2/Hz)

10 0.1

1000 0.1

Duration = 60 sec

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

VibrationdataSDOF Example

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Mass = 20 lbm, Q=10,

Natural Frequency = independent variable

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

Duration = 60 seconds (but only affects peak value)

Vibrationdata

SDOF Response to Force PSD, Acceleration

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

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

fn (Hz)

Accel (GRMS)

100 0.80

200 1.0

400 1.3

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SDOF Response to Force PSD, Transmitted Force

Vibrationdata

Acceleration VRS

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

fn (Hz)

Accel (GRMS)

100 0.80

200 1.0

400 1.3

Vibrationdata

Velocity VRS

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Vibrationdata

Displacement VRS

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Vibrationdata

Transmitted Force VRS

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VibrationdataHomework

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Repeat the examples in the presentation using the Matlab scripts