Vibration Isolation Measurement and Simulation

Vibration IsolationMeasurement and Simulation

University of Kentucky

July 8, 2021

David HerrinUniversity of Kentucky

The Vibro-Acoustics Consortium

July 8, 2021

Overview

2

• Basics• Simulation

Method 1 Mobility MatrixMethod 2 Impedance Matrix

• MeasurementMethod 1 Direct MeasurementMethod 2 Indirect Measurement

• Correlation


July 8, 2021

Transmissibility

Machine

Foundation

𝑣

𝑣

Machine

Foundation

𝐹

𝐹

Motion Transmissibility

𝑇motion𝑣𝑣

Force Transmissibility

𝑇force𝐹𝐹

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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July 8, 2021

Force TransmissibilityInman, 2014

If 𝜉 0

𝑇1

1 𝜔𝜔

𝜔2𝜋𝑛60

Δ𝑚𝑔𝑘

𝑅 1 𝑇

𝜔𝑘 𝑚⁄

2 𝑅1 𝑅

𝑛30𝜋

𝑔 2 𝑅Δ 1 𝑅 29.9

2 𝑅Δ 1 𝑅

4

Machine

Foundation

𝐹

𝐹


July 8, 2021

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

5

𝑘𝑚𝑔

Δ


July 8, 2021

Introduction Characterization of Isolator

𝐹𝑣

𝑎 𝑎𝑎 𝑎

𝐹𝑣

The effectiveness of an isolator can be described using isolator insertion loss:

Foundation

Machine

Foundation

Machine

Rigid

Isolator

𝑣rigid

𝑣isolated

𝐼𝐿 20 ⋅ log𝑣 |rigid

𝑣 |isolated

20 ⋅ log𝑎 𝑍 𝑎 𝑎 𝑍 𝑍 𝑎 𝑍

𝑍 𝑍

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

8




• Correlation


July 8, 2021

Analysis Steps

• Static Analysis to pre-load mount (nonlinear, large deformation analysis)• Modal Analysis to find loaded/pre-stressed modes• 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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July 8, 2021

Method 1 Mobility Matrix

Reconfigure into mobility matrix

Solve model twiceSolve 1: 𝐹 1; 𝐹 0 Solve 2: 𝐹 0; 𝐹 1

𝑣𝑣

𝑏 𝑏𝑏 𝑏

𝐹𝐹

𝑏𝑣𝐹 ,

𝑏𝑣𝐹 ,

𝑏𝑣𝐹 ,

𝑏𝑣𝐹 ,

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July 8, 2021

Convert to traditional four-poles

Wu et al., 1998

Method 1 Mobility Matrix

𝑣𝑣

𝑏 𝑏𝑏 𝑏

𝐹𝐹

𝑎𝑏𝑏

𝑎 𝑏𝑏 𝑏𝑏

𝑎1𝑏

𝑎𝑏𝑏

𝐹𝑣

𝑎 𝑎𝑎 𝑎

𝐹𝑣

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July 8, 2021Dickens, 1998

Method 2 Impedance Matrix

Reconfigure into impedance matrix

Solve model twiceSolve 1: 𝐹 1 ; 𝑣 0Solve 2: 𝐹 1; 𝑣 0

𝐹𝐹

𝑐 𝑐𝑐 𝑐

𝑣𝑣

𝑐𝐹𝑣

𝑐𝐹𝑣

𝑐𝐹𝑣

𝑐𝐹𝑣

12


July 8, 2021

Convert to traditional four-poles

Method 2 Impedance Matrix

𝐹𝐹

𝑐 𝑐𝑐 𝑐

𝑣𝑣

𝑎𝑐𝑐

𝑎1𝑐

𝑎 𝑐𝑐 𝑐𝑐

𝑎𝑐𝑐

𝐹𝑣

𝑎 𝑎𝑎 𝑎

𝐹𝑣

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July 8, 2021

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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July 8, 2021

Simple Spring Properties

𝑘𝐺𝑑

8𝑛𝐷

𝜌 density of material𝐺 shear modulus of material𝑑 diameter of the spring wire𝐻 height of spring𝐷 average diameter of the spring𝑛 number of active coil turns

d

D

Spring Stiffness (Ungar, 2007)

Spring Mass

𝑚 𝜌 𝐻 𝑛𝜋𝐷𝜋𝑑

4

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July 8, 2021

Simple Relationships

𝑑

𝐷

First surge frequency

Insertion loss proportional to

𝜌 density of material𝐺 shear modulus of material𝑑 diameter of the spring wire𝐻 height of spring𝐷 average diameter of the spring𝑛 number of active coil turns

𝐼𝐿 ∝ 20 log𝜔𝑛𝐷𝐺𝑑

𝑓 ∝𝑑𝑛𝐷

𝐺𝜌

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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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July 8, 2021

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

𝐼𝐿 ∝ 20 log 𝐷

𝑓 ∝1𝐷

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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

𝐼𝐿 ∝ 20 log1𝑑

𝑓 ∝ 𝑑

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July 8, 2021

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

𝐼𝐿 ∝ 20 log 𝑛

𝑓 ∝1𝑛

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July 8, 2021

Foundation Side• 50 cm X 30 cm• 5 cm thick steel plate

Machine Side• 1 cm thick ribbed steel structure

Case 2 Isolator Between Two Structures

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Local Model of Isolator

𝐹

Response Point

Transfer Matrix Approach1. Determine impedances of foundation and

machine sides.2. Use Insertion loss equation.

23


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Sensitivity Study

• Case 1 – All Ribs• Case 2 – No Ribs • Case 3 – Remove Yellow• Case 4 – Remove Yellow and Red

24


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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

25


July 8, 2021

Overview

26




• Correlation


July 8, 2021

ISO 10846

Acoustics and vibration – 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

27


July 8, 2021

Foundation(Receiving structure)

Machine(Vibration source)

Isolator

𝑢1 𝐹1

𝐹2 𝑢2

𝑢1 𝐹1

𝐹2 𝑢2

𝐹 𝑘 𝑢 𝑘 𝑢𝐹 𝑘 𝑢 𝑘 𝑢

𝐹𝐹

𝑘 𝑘𝑘 𝑘

𝑢𝑢

𝑘 and 𝑘 indicate dynamic driving point stiffness when the output/input is blocked (𝑘 = 𝑘 at low frequencies).

𝑘 and 𝑘 indicate dynamic transfer stiffness (𝑘 𝑘 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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July 8, 2021

Foundation Dynamic Stiffness

𝑘𝐹𝑢

𝐹𝑘

1 𝑘𝑘

𝑢

when 𝑘 0.1𝑘𝐹 𝐹 𝑘 𝑢

𝑘𝐹𝑢



Isolator

𝑢1 𝐹1

𝐹2 𝑢2

𝑢1 𝐹1

𝐹2 𝑢2


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July 8, 2021

Assume 𝑘 ≫ 𝑘𝐹 𝑘 𝑢𝐹 𝑘 𝑢At low frequencies𝑘 𝑘 𝑘Complex low-frequency dynamic stiffness𝑘 𝑘 1 𝑗𝜂𝜂 tan𝜓𝑘 real part of dynamic stiffness𝜂 loss factor𝜓 phase angle of the dynamic stiffness




Isolator

𝑢1 𝐹1

𝐹2 𝑢2

𝑢1 𝐹1

𝐹2 𝑢2

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July 8, 2021

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

𝑘𝐹𝑢

Assume 𝑢 ≫ 𝑢

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Test Rig Design

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July 8, 2021

Direct Method Test Rig Design

1

23

4

5

67

8

Direct measurement

𝑘𝐹𝑢

assume 𝑢 ≫ 𝑢Schematic of test rig for Direct Method1. Dynamic excitation (shaker) 2. Static preload 3. Decoupling springs4. Excitation mass (𝑚 )5. Test element (isolator)6. Lower force distribution flange7. Force measurement (load cells)8. Rigid foundation

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July 8, 2021


Valid Frequency Range: ∆𝐿 𝐿 𝐿 20 dB

𝐿

𝐿

Accelerometers

0

10

20

30

40

50

60

70

80

10 100 1000

Acce

lera

tion

Diff

eren

ce (d

B)

Frequency (Hz)

34


July 8, 2021

Adequacy of blocking force measurement

𝑚 0.0610

10 kg


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)

35


July 8, 2021

Unwanted input vibration 1:𝐿 𝐿 15 dB


𝐿 𝐿

Accelerometers

0

5

10

15

20

25

30

35

40

45

10 100 1000

Acce

lera

tion

Diff

eren

ce (d

B)

Frequency (Hz)

36


July 8, 2021

Unwanted input vibration 2:𝐿 𝐿 0.5 dB


𝐿 𝐿

-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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July 8, 2021

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.


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July 8, 2021

ISO 10846-3 Indirect Method

Indirect measurement of force

Blocking Mass 𝑚(on isolators)

Vibration Source

Isolator

𝑢1 𝐹1

𝐹2 𝑢2

𝑢1 𝐹1

𝐹2 𝑢2

Parts1. Exciter2. Traverse3. Connecting rod4. Dynamic decoupling springs,

static preload5. Test element6. Blocking mass7. Rigid foundation

Image from ISO 10846-3

39

𝑘 ,𝐹𝑢 𝜔 𝑚

𝑢𝑢


July 8, 2021

Schematic of test rig for Indirect Method1. Dynamic excitation (shaker) 2. Static preload 3. Decoupling springs4. Excitation mass (𝑚 )5. Test element (isolator)6. Lower force distribution flange7. Blocking mass (𝑚 )8. Rigid foundation

1

2

3

5

6

7

8

4

Indirect measurement

Indirect Method Test Rig Design

40

𝑘 ,𝐹𝑢 𝜔 𝑚

𝑢𝑢


July 8, 2021

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: 𝑘 =1 N/m

41


July 8, 2021

Overview

42




• Correlation


July 8, 2021

43

𝐹𝑣

cos 𝑘𝐿 𝜌 𝑐 𝑆𝑗 sin 𝑘𝐿1

𝜌 𝑐 𝑆 𝑗 sin 𝑘𝐿 cos 𝑘𝐿𝐹𝑣

𝐹𝐹

𝜌 𝑐 𝑆𝑗 sin 𝑘𝐿

cos 𝑘𝐿 11 cos 𝑘𝐿

𝑣𝑣

Transfer Matrix

May be rearranged in Impedance Matrix form𝐾

𝑣

𝑣

𝐹

𝐹

Bies et al., 2009

1D Spring Models Transfer Matrix


July 8, 2021

𝑘 𝑘 1 𝑗𝜂

𝑚𝜌 𝜋𝑑

4 𝑛𝜋𝐷 𝐿

Longitudinal Wave Speed

Model spring as an equivalent longitudinal force element.

1D Spring Models Transfer MatrixBies and Hansen, 2009

𝑐𝐸𝜌 𝐿

𝑘 𝐿 𝑆⁄𝑚 𝐿𝑆⁄ 𝐿

𝑘𝑚

Spring Stiffness with Damping

Spring Mass

𝑘𝐺𝑑

8𝑛𝐷Spring Stiffness

44


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45

ANSYS FEM Simulation


July 8, 2021

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

46


July 8, 2021

Results Acceleration Transmissibility

1.E‐05

1.E‐04

1.E‐03

1.E‐02

1.E‐01

1.E+00

1.E+01

1.E+02

10 100

Acceleratio

n Transm

issibility

Frequency (Hz)

MeasurementFEMAnalytical

𝜏𝑎𝑎

𝑎

𝑎

47


July 8, 2021

References

48

• Inman, D. J., Engineering Vibration, Prentice Hall, 4th Edition, 2001.• Molloy, C. T., “Use of Four-Pole Parameters in Vibration Calculations,” Journal of the

Acoustical Society of America, Vol. 29, No. 7, pp.842-853, 1957.• Snowdon, J. C., “Mechanical Four-Pole Parameters and Their Application,” Journal of

Sound and Vibration, Vol. 15, No. 3, pp.307-323, 1971.• Dickens, J. D., and Norwood, C. J., “Universal Method to Measure Dynamic Performance

of Vibration Isolators under Static Load,” Journal of Sound and Vibration, Vol. 244, No. 4, pp. 685-696, 2001.

• Dickens, J. D., “Methods to Measure the Four-Pole Parameters of Vibration Isolators,” Acoustics Australia, Vol. 28, No. 1, pp. 15-21, 2000.

• Bies, D. A., Hansen, C. H., and Howard, C. Q., Engineering Noise Control, 5th Edition, CRC Press, Boca Raton, FL, 2018.

• Sun, S., Herrin, D. W., and Baker, J. R., “Determination of the Transfer Matrix of Isolators using Simulation with Applications to Determining Insertion Loss,” SAE International Journal of Materials and Manufacturing, Vol. 8, No. 3, 2015.

Documents

Vibration Isolation Measurement and Simulation