Vibration Isolation Measurement and Simulation
Transcript of Vibration Isolation Measurement and Simulation
Vibration IsolationMeasurement and Simulation
University of Kentucky
July 8, 2021
David HerrinUniversity of Kentucky
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Overview
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β’ Basicsβ’ Simulation
Method 1 Mobility MatrixMethod 2 Impedance Matrix
β’ MeasurementMethod 1 Direct MeasurementMethod 2 Indirect Measurement
β’ Correlation
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Transmissibility
Machine
Foundation
π£
π£
Machine
Foundation
πΉ
πΉ
Motion Transmissibility
πmotionπ£π£
Force Transmissibility
πforceπΉπΉ
Note: Transmissibility does not account for changes in the excitation force or motion that may occur when a more flexible isolator is used. Most models using transmissibility assume the machine and foundation to be rigid and the mass of the isolator to be negligible.
Ungar, 2007
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Force TransmissibilityInman, 2014
If π 0
π1
1 ππ
π2ππ60
Ξπππ
π 1 π
ππ πβ
2 π 1 π
π30π
π 2 π Ξ 1 π 29.9
2 π Ξ 1 π
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Machine
Foundation
πΉ
πΉ
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Design Curves
1. Identify static deflection using design curve.
2. Calculate spring stiffness.
3. Clearance between machine and foundation should be more than twice the static deflection of the spring.
Inman, 2013
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πππ
Ξ
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Introduction Characterization of Isolator
πΉπ£
π ππ π
πΉπ£
The effectiveness of an isolator can be described using isolator insertion loss:
Foundation
Machine
Foundation
Machine
Rigid
Isolator
π£rigid
π£isolated
πΌπΏ 20 β logπ£ |rigid
π£ |isolated
20 β logπ π π π π π π π
π π
ZS and ZF are the mechanical impedances at the isolator mounting point on source and foundation sides, respectively.
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Effect of Wave Propagation in Isolator
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Overview
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β’ Basicsβ’ Simulation
Method 1 Mobility MatrixMethod 2 Impedance Matrix
β’ MeasurementMethod 1 Direct MeasurementMethod 2 Indirect Measurement
β’ Correlation
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Analysis Steps
β’ Static Analysis to pre-load mount (nonlinear, large deformation analysis)β’ Modal Analysis to find loaded/pre-stressed modesβ’ Forced Response Analysis to find the transfer matrix
Static Analysis Modal AnalysisModal Superposition /
Forced Response Analysis
Boundary conditions depend upon the method used.
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Method 1 Mobility Matrix
Reconfigure into mobility matrix
Solve model twiceSolve 1: πΉ 1; πΉ 0 Solve 2: πΉ 0; πΉ 1
π£π£
π ππ π
πΉπΉ
ππ£πΉ ,
ππ£πΉ ,
ππ£πΉ ,
ππ£πΉ ,
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Convert to traditional four-poles
Wu et al., 1998
Method 1 Mobility Matrix
π£π£
π ππ π
πΉπΉ
πππ
π ππ ππ
π1π
πππ
πΉπ£
π ππ π
πΉπ£
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Method 2 Impedance Matrix
Reconfigure into impedance matrix
Solve model twiceSolve 1: πΉ 1 ; π£ 0Solve 2: πΉ 1; π£ 0
πΉπΉ
π ππ π
π£π£
ππΉπ£
ππΉπ£
ππΉπ£
ππΉπ£
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Convert to traditional four-poles
Method 2 Impedance Matrix
πΉπΉ
π ππ π
π£π£
πππ
π1π
π ππ ππ
πππ
πΉπ£
π ππ π
πΉπ£
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Difference Between Methods
Mobility Matrix Impedance Matrix
Isolator is free-free after static analysis.
Isolator is fixed on one side.
The isolator was constrained in the lateral direction in each case. The difference in boundary conditions leads to slight differences.
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Simple Spring Properties
ππΊπ
8ππ·
π density of materialπΊ shear modulus of materialπ diameter of the spring wireπ» height of springπ· average diameter of the springπ number of active coil turns
d
D
Spring Stiffness (Ungar, 2007)
Spring Mass
π π π» πππ·ππ
4
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Simple Relationships
π
π·
First surge frequency
Insertion loss proportional to
π density of materialπΊ shear modulus of materialπ diameter of the spring wireπ» height of springπ· average diameter of the springπ number of active coil turns
πΌπΏ β 20 logπππ·πΊπ
π βπππ·
πΊπ
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Case 1 Isolator Between Two Masses
10 kg mass
100 kg mass
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0
20
40
60
80
100
120
140
10 100 1000
Inse
rtion
Los
s (d
B)
Frequency
0.001
0.010
0.100
Effect of Damping
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0
20
40
60
80
100
120
10 100 1000
Inse
rtion
Los
s
Frequency
9 cm
7 cm
5 cm
3 cm
Insertion Loss Vary Spring Diameter
πΌπΏ β 20 log π·
π β1π·
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-20
0
20
40
60
80
100
120
10 100 1000
Inse
rtion
Los
s
Frequency
0.3 cm
0.4 cm
1.0 cm
1.5 cm
Insertion Loss Vary Wire Diameter
πΌπΏ β 20 log1π
π β π
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0
20
40
60
80
100
120
140
10 100 1000
Inse
rtion
Los
s (d
B)
Frequency (Hz)
3.5 Turns
7 Turns
Insertion Loss Vary Number of Turns
πΌπΏ β 20 log π
π β1π
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Foundation Sideβ’ 50 cm X 30 cmβ’ 5 cm thick steel plate
Machine Sideβ’ 1 cm thick ribbed steel structure
Case 2 Isolator Between Two Structures
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Local Model of Isolator
πΉ
Response Point
Transfer Matrix Approach1. Determine impedances of foundation and
machine sides.2. Use Insertion loss equation.
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Sensitivity Study
β’ Case 1 β All Ribsβ’ Case 2 β No Ribs β’ Case 3 β Remove Yellowβ’ Case 4 β Remove Yellow and Red
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Insertion Loss Comparison
0
20
40
60
80
100
120
1 10 100 1000
Inse
rtion
Los
s (d
B)
Frequency (Hz)
Case 1Case 2Case 3Case 4
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Overview
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β’ Basicsβ’ Simulation
Method 1 Mobility MatrixMethod 2 Impedance Matrix
β’ MeasurementMethod 1 Direct MeasurementMethod 2 Indirect Measurement
β’ Correlation
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ISO 10846
Acoustics and vibration β Laboratory measurement of vibro-acoustic transfer elements of resilient elementsPart 1 (2008): Principles and guidelinesPart 2 (2008): Direct method for determination of the dynamic stiffness of resilient supports for translator motionPart 3 (2002): Indirect method for determination of the dynamic stiffness of resilient supports for translator motion
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Foundation(Receiving structure)
Machine(Vibration source)
Isolator
π’1 πΉ1
πΉ2 π’2
π’1 πΉ1
πΉ2 π’2
πΉ π π’ π π’πΉ π π’ π π’
πΉπΉ
π ππ π
π’π’
π and π indicate dynamic driving point stiffness when the output/input is blocked (π = π at low frequencies).
π and π indicate dynamic transfer stiffness (π π if inertial forces can be neglected).
Assume1. Linearity for vibrational behavior under a static preload.2. Contact interfaces can be considered point contacts.
ISO 10846-1 General Principles
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Foundation Dynamic Stiffness
ππΉπ’
πΉπ
1 ππ
π’
when π 0.1ππΉ πΉ π π’
ππΉπ’
Foundation(Receiving structure)
Machine(Vibration source)
Isolator
π’1 πΉ1
πΉ2 π’2
π’1 πΉ1
πΉ2 π’2
ISO 10846-1 General Principles
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Assume π β« ππΉ π π’πΉ π π’At low frequenciesπ π πComplex low-frequency dynamic stiffnessπ π 1 πππ tanππ real part of dynamic stiffnessπ loss factorπ phase angle of the dynamic stiffness
ISO 10846-1 General Principles
Foundation(Receiving structure)
Machine(Vibration source)
Isolator
π’1 πΉ1
πΉ2 π’2
π’1 πΉ1
πΉ2 π’2
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ISO 10846-2 Direct Method
Schematic of typical test rig1. Static preload and dynamic excitation
(shaker)2. Moveable traverse3. Columns (guide rods, frame)4. Test element (isolator)5. Force measurement (load cells)6. Rigid foundation (Blocking mass)
Image from ISO 10846-1
ππΉπ’
Assume π’ β« π’
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Test Rig Design
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Direct Method Test Rig Design
1
23
4
5
67
8
Direct measurement
ππΉπ’
assume π’ β« π’Schematic of test rig for Direct Method1. Dynamic excitation (shaker) 2. Static preload 3. Decoupling springs4. Excitation mass (π )5. Test element (isolator)6. Lower force distribution flange7. Force measurement (load cells)8. Rigid foundation
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ISO 10846-2 Direct Method
Valid Frequency Range: βπΏ πΏ πΏ 20 dB
πΏ
πΏ
Accelerometers
0
10
20
30
40
50
60
70
80
10 100 1000
Acce
lera
tion
Diff
eren
ce (d
B)
Frequency (Hz)
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Adequacy of blocking force measurement
π 0.0610
10 kg
ISO 10846-2 Direct Method
2 is the force distribution plate (image from ISO-10846-2)
012345678910
0 200 400 600 800 1000
Crite
ria M
ass (kg)
Frequency (Hz)
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Unwanted input vibration 1:πΏ πΏ 15 dB
ISO 10846-2 Direct Method
πΏ πΏ
Accelerometers
0
5
10
15
20
25
30
35
40
45
10 100 1000
Acce
lera
tion
Diff
eren
ce (d
B)
Frequency (Hz)
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Unwanted input vibration 2:πΏ πΏ 0.5 dB
ISO 10846-2 Direct Method
πΏ πΏ
-0.5
0
0.5
1
1.5
2
10 100 1000
Acce
lera
tion
Diff
eren
ce (d
B)
Frequency (Hz)
Accelerometers
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Other Notes1. Dynamic stiffness can be averaged in 1/3 octave bands using a
minimum of 5 frequencies per 1/3 octave band.2. Results should be presented in dB with a reference of 1 N/m.3. Vibration levels should be similar to those in practice.4. Linearity check is required. Reduce input by 10 dBA to ensure that
the dynamic stiffness dB levels do not differ by more than 1.5 dB.
ISO 10846-2 Direct Method
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ISO 10846-3 Indirect Method
Indirect measurement of force
Blocking Mass π(on isolators)
Vibration Source
Isolator
π’1 πΉ1
πΉ2 π’2
π’1 πΉ1
πΉ2 π’2
Parts1. Exciter2. Traverse3. Connecting rod4. Dynamic decoupling springs,
static preload5. Test element6. Blocking mass7. Rigid foundation
Image from ISO 10846-3
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π ,πΉπ’ π π
π’π’
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Schematic of test rig for Indirect Method1. Dynamic excitation (shaker) 2. Static preload 3. Decoupling springs4. Excitation mass (π )5. Test element (isolator)6. Lower force distribution flange7. Blocking mass (π )8. Rigid foundation
1
2
3
5
6
7
8
4
Indirect measurement
Indirect Method Test Rig Design
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π ,πΉπ’ π π
π’π’
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Results Transfer Dynamic Stiffness
40
60
80
100
120
140
160
10 100 1000
Tran
sfer
Dyn
amic
Stif
fnes
s (d
B)
Frequency (Hz)
Direct MethodIndirect MethodStatic
Note: Static stiffness of the target spring is 75.72 dB (6113 N/m). Stiffness reference: π =1 N/m
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Overview
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β’ Basicsβ’ Simulation
Method 1 Mobility MatrixMethod 2 Impedance Matrix
β’ MeasurementMethod 1 Direct MeasurementMethod 2 Indirect Measurement
β’ Correlation
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πΉπ£
cos ππΏ π π ππ sin ππΏ1
π π π π sin ππΏ cos ππΏπΉπ£
πΉπΉ
π π ππ sin ππΏ
cos ππΏ 11 cos ππΏ
π£π£
Transfer Matrix
May be rearranged in Impedance Matrix formπΎ
π£
π£
πΉ
πΉ
Bies et al., 2009
1D Spring Models Transfer Matrix
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π π 1 ππ
ππ ππ
4 πππ· πΏ
Longitudinal Wave Speed
Model spring as an equivalent longitudinal force element.
1D Spring Models Transfer MatrixBies and Hansen, 2009
ππΈπ πΏ
π πΏ πβπ πΏπβ πΏ
ππ
Spring Stiffness with Damping
Spring Mass
ππΊπ
8ππ·Spring Stiffness
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ANSYS FEM Simulation
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Measurement Setup
Material Structural Steel /Young's Modulus 2.00E+11 PaShear Modulus 7.69E+10 PaNumber of Effective Coils ~4 /Material Density 7850 kg m^-3Wire Diameter 0.005 mOuter Diameter 0.05 mLength (Uncompressed) 0.075 m
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Results Acceleration Transmissibility
1.Eβ05
1.Eβ04
1.Eβ03
1.Eβ02
1.Eβ01
1.E+00
1.E+01
1.E+02
10 100
Acceleratio
n Transm
issibility
Frequency (Hz)
MeasurementFEMAnalytical
πππ
π
π
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References
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β’ Inman, D. J., Engineering Vibration, Prentice Hall, 4th Edition, 2001.β’ Molloy, C. T., βUse of Four-Pole Parameters in Vibration Calculations,β Journal of the
Acoustical Society of America, Vol. 29, No. 7, pp.842-853, 1957.β’ Snowdon, J. C., βMechanical Four-Pole Parameters and Their Application,β Journal of
Sound and Vibration, Vol. 15, No. 3, pp.307-323, 1971.β’ Dickens, J. D., and Norwood, C. J., βUniversal Method to Measure Dynamic Performance
of Vibration Isolators under Static Load,β Journal of Sound and Vibration, Vol. 244, No. 4, pp. 685-696, 2001.
β’ Dickens, J. D., βMethods to Measure the Four-Pole Parameters of Vibration Isolators,β Acoustics Australia, Vol. 28, No. 1, pp. 15-21, 2000.
β’ Bies, D. A., Hansen, C. H., and Howard, C. Q., Engineering Noise Control, 5th Edition, CRC Press, Boca Raton, FL, 2018.
β’ Sun, S., Herrin, D. W., and Baker, J. R., βDetermination of the Transfer Matrix of Isolators using Simulation with Applications to Determining Insertion Loss,β SAE International Journal of Materials and Manufacturing, Vol. 8, No. 3, 2015.