unit 18
DESCRIPTION
Unit 18. Force Vibration Response Spectrum. 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 - PowerPoint PPT PresentationTRANSCRIPT
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Vibrationdata
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Unit 18
Force Vibration Response Spectrum
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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
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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
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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.
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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
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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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Vibrationdata
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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
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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
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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.
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VibrationdataSDOF Response to Force PSD, Miles Equation
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4/3
k
14/1
m
12/1
8
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.
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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
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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
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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,
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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
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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.
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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)
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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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Vibrationdata
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SDOF Response to Force PSD, Transmitted Force
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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
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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