Lucas S - wideband sonic and ultrasonic technology for ...

1

Experimental and computational modelling of vibration performance of

ultrasonic tools for manufacturing applications

Dr Margaret Lucas and Andrea Cardoni

Department of Mechanical Engineering, University of Glasgow

UIA Symposium, 2002

2

Objectives

• Study the causes of component failure in simple and complex ultrasoniccutting systems by finite element (FE) modelling.

• Investigate geometric modifications of the ultrasonic components inorder to reduce stress.

• Characterise the vibration behaviour of ultrasonic components by FEand modal analysis.

• Propose geometric modifications to improve vibration performance.

• Characterise the nonlinear responses of ultrasonic systems.

• Illustrate the theory of Nonlinear Cancellation Coupling.

• Propose strategies to reduce effects of nonlinear behaviour and toreduce nonlinearities.

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Customised Ultrasonic Horns

Piezoelectric transducer

Single-slotted block horn

Half wavelength blade

System exciter

Three-bladed cutting head

Ultrasonic Systems and Components

One wavelength blade

One and a half wavelength Bar horn

Double-slotted block horn

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High Gain Blade

• In a cylindrical horn the longitudinal nodecorresponds to the highest stressed section.

• The maximum displacement slope coincideswith the highest stressed section of the horn.

• The highest stress occurs at the steep section reduction.

• If the node is close to the highest stress section,the maximum stress is greater.

Bar Horn of Constant Section

Normalised Stress and Displacement

Highest stressed sectionNodal plane

ABSOLUTE DISTANCE (m)

Max.displacement

slope

0.00 0.02 0.04 0.06

0.0

0.5

1.0

Stress

Displ.

ABSOLUTE DISTANCE (m)

Max.displacement

slope

0.00 0.02 0.04 0.06

Nodal plane ≡ highest stressed section

0.0

- 1.0

1.0

Stress

Displ.

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Tapered cutting knife

Model no 7

Model no 6

Model no 4

Model no 3

Model no 2

Model no 1

Step Redesign

Reducing Maximum Stress by using Alternative Blade Geometries

Cutting knife

Model no 5 0

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6 7Blade

No

rmal

ised

Str

ess

Maximum Stress

- 46 %

Predicted Maximum Stress

• Redesign of blade step profile reduces stress.

• In particular, tapering the cutting knife provides significant stress reduction.

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Highest stress Failure plane

Highest stress

Node AntinodeAntinode

Node

Antinode

Stress Distribution in the Central Blade of a Cutting Head

• Block horn and blades are tuned at thesame frequency (35 kHz).

• The highest stress occurs after the bladestep.

• Blade node and highest stressed sectionare very close.

• Investigate the effect of shifting the bladenode backwards into the thicker bladesection to reduce stress at the failurelocation.

Block horn

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Shifting the Node by System Detuning

DISP

Block Horn

Stud

Piezoelectric Transducer Block Horn

Frequency IncreaseFrequency Decrease

ABSOLUTE DISTANCE (m)

STRESS

0.00 0.02 0.04 0.06

0.00 0.02 0.04 0.06

1.0

0.5

0.0

1.0

0.5

0.0• The node moves backwards from the highest

stressed section, when the block horn is tuned at a lower frequency.

• The highest stressed section of the blade is at the same location for each block hornmodification, however the maximum stress is reduced as the node moves backwards.

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Experimental Modal Analysis

• Modal analysis characterises a structure in

terms of its natural frequencies, damping

values and mode shapes.

• A 3D laser Doppler vibrometer (LDV), signal

analyser and modal analysis software (LMS)

allows the modal parameters to be extracted.

•Modal analysis can be used in conjunction

with FE models to improve model predictions

and assist in redesign.

• The experimental set-up can be used to

measure frequency response functions

(FRFs) or characterise the nonlinear

response.

Modal analysis using a 3D LDV

3D LDVTransducer

8 ChannelSignal Analyser

FunctionGenerator

Signal Amplifier ComputerComputer(LMS)

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High stress

(FEA) (EMA)

(kHz)

Operatingmode

• The central blade vibrates in a pure longitudinalmode, but the outer blades are characterised bylongitudinal and flexural responses.

• Participation of flexural responses in theoperating mode increases the stress in the outerblades.

• Block horn design must focus on constrainingfexural responses.

Frequency Response Functions and Modal Properties

• The FRF illustrates the modal densityaround the operating frequency of athree bladed cutting head.

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Castellating the outer columns restricts

flexural motion of the outer blades in the

longitudinal mode.

A longer central column also removes

flexural responses in the longitudinal

mode.

Block Horn Redesign Strategies

Block with castellated outer columns

(FE) (EMA)

Castellations

Block horn with longer central column

(FE) (EMA)

Central Column

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Nonlinear Effects

Modal interactions in nonlinear systems can arise when the system is harmonically excited in the vicinity of a natural frequency. In particular, if special relationships (combination resonances) between two or more linear modes and the excitation frequency exist, the system response contributes more modes. Effects of combination resonances are high noise level, component fatigue and poor operating performance.

Underneath are two ultrasonic cutting systems which are prone to these effects due to nonlinear behaviour.

Single-Blade Cutting System Three-Blade Cutting System

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Combination Resonances in a Single-Blade Cutting System

lω1ω 2ω

Combination I: 21 ωωω +≈l

432 ωωω +≈ll2ω4ω

3ω

Measured blade mode shapes

Combination II:

• System driven at 35.29 kHz

• System driven at 43.1 kHz

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Combination ICombination II

For the single blade system, mode combination IIhas a lower threshold and wider unstable region

Stability Regions

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0 10.0K 20.0K 30.0K 40.0K 50.0K0

1.0

Hz

Mag

nitu

de, m

/s

ω1 = 11437 Hz

Combination resonance: ωL ≈ ω1 + ω2

ω2 = 23675 Hz

ω1

ω2

ωL

• System driven at 35.11 kHz

0 10.0K 20.0K 30.0K 40.0K 50.0K0

1.0K

2.0K

3.0K

4.0K

5.0K

6.0K

7.0K

Hz

Mag

nitu

de, V

/ V

Combination Resonances in Three-Blade Cutting System I

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0 10.0K 20.0K 30.0K 40.0K 50.0K0

50.0

100.0

150.0

200.0

250.0

300.0

Hz

Mag

nitu

de, V

/ V

Combination Resonances in Three-Blade Cutting System II

0 10.0K 20.0K 30.0K 40.0K 50.0K0

1.0

2.0

3.0

4.0

5.0

Hz

Mag

nitu

de, V

/ V

5.0

• System driven at 34.945 kHz

A single-slotted horn cleans the response spectrum

0 10.0K 20.0K 30.0K 40.0K 50.0K0

1.0

Hz

Mag

nitu

de, m

/s

ω1

ω2

ωL

16

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

34950 35000 35050 35100 35150 35200 35250 35300

Frequency (Hz)R

esp

on

se (

m/s

)

30 V (Up)

30 V (Down)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

35000 35050 35100 35150 35200 35250 35300 35350

Frequency (Hz)

Res

po

nse

(m

/s)

30 V (Up)

30 V (Down)

5 V (Up)

5 V (Down)Instabilityregion

Thermal Effect on the Transducer Characteristic

Frequency sweep carried out oncold transducer

Frequency sweep carried out onhot transducer

The transducer shows a clear softening characteristic highlighted by the jump phenomenon and a wide unstable region

The transducer still shows a softening characteristic, but no jump phenomenonor unstable region

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Softening cubic stiffness

Hardening cubic stiffness

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0 0.5 1 1.5 2 2.5 3 3.5Ω/ωe1

x 1

Nonlinear Cancellation Coupling Analytical Theory

Equations of Motion

Nonlinear response

k2k1 m2

x2

c2

h2

FoCos Ω t

m1

x1

c1

h1

3 31 1 1 2 1 2 2 1 2 1 2 2 1 1 2 2 1( ) ( ) ( ) ( ) om x c c x c x k k x k x h x h x x F Cos t+ + − + + − + + − = Ω&& & &

32 2 2 2 2 1 2 2 2 1 2 2 1( ) 0m x c x c x k x k x h x x+ − + − − − =&& & &

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0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

34950 35000 35050 35100 35150 35200 35250 35300

Frequency (Hz)

Res

po

nse

(m

/s)

30 V (Up)

30 V (Down)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

35050 35100 35150 35200 35250 35300 35350 35400 35450

Frequency (Hz)

Resp

on

se (

m/s

)

30 V (Up)

30 V ( Down)

Transducer

The transducer exhibits a softening characteristic The transducer-bar horn system exhibits a slightly hardening characteristic

Transducer and 1.5 wavelength bar horn system

Effect of Tuned Bar on Response Characteristic(Case I: Excitation 30 V)

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Effect of Tuned Bar on Response Characteristic(Case II: Excitation 50 V)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

32300 32325 32350 32375 32400 32425 32450 32475 32500 32525 32550

Frequency (Hz)

Resp

on

se (

m/s

)

50 V ( Up)

50 V (Down)

The transducer exhibits a clear softening responsecharacterised by the jump phenomenon and an unstable region.

The transducer-bar horn system still shows a softening characteristic, however no unstable region is detected.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

35200 35250 35300 35350 35400 35450 35500 35550

Frequency (Hz)

Res

po

nse

(m

/s)

50 V (Up)

50 V (Down)

Transducer Transducer -1.5 wavelength bar horn system

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Effect of Blade on System Response Characteristic

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

34950 35000 35050 35100 35150 35200 35250 35300

Frequency (Hz)

Res

po

nse

(m

/s)

30 V (Up)

30 V (Down)

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

35100 35150 35200 35250 35300 35350 35400 35450

Frequency (Hz)

Resp

on

se (

m/s

)

30 V (Up)

30 V ( Down)

Transducer -half wavelength blade systemTransducer

(Case I: Half-wavelength Blade; Excitation 30 V)

The transducer exhibits a softening characteristic The transducer-blade system exhibits a clear softening response characterised by the jump phenomenon and a wide unstable region.

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Effect of Blade on System Response Characteristic(Case II: One wavelength Blade; Excitation 30 V)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

34950 35000 35050 35100 35150 35200 35250 35300

Frequency (Hz)

Res

po

nse

(m

/s)

30 V (Up)

30 V (Down)

Transducer Transducer -half wavelength blade system

The transducer-blade system shows a near linear response characteristic

0.000

0.500

1.000

1.500

2.000

2.500

3.000

3.500

4.000

35820 35840 35860 35880 35900 35920 35940 35960 35980 36000

Frequency (Hz)

Resp

on

se (

m/s

)

30 V (Up)

30 V( Down)

Transducer-one wavelength blade system

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0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

35150 35200 35250 35300 35350 35400 35450 35500

Frequency (Hz)

Res

po

nse

(m

/s)

30 V (Up)

30 V (Down)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

35150 35200 35250 35300 35350 35400 35450

Frequency (Hz)

Res

po

nse

(m

/s)

30 V (Up)

30 V (Down)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

35150 35200 35250 35300 35350 35400 35450 35500

Frequency (Hz)

Res

po

nse

(m

/s)

30 V (Up)

30 V (Down)

Threaded stud fully-screwed into the blade-base

Threaded stud half-screwed into the blade-base

Threaded stud fully-screwed into the transducer-base

Investigation of Stud Configuration for Nonlinear Cancellation Coupling

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Conclusions

• FE analysis is an effective numerical method for design andanalysis of ultrasonic tools.

• Strategies have been proposed to reduce stress at blade failurelocations by geometric modifications (by blade profile andblock geometries).

• The nonlinear behaviour of ultrasonic cutting systems has beencharacterised experimentally.

• Strategies to reduce the effects of nonlinear reponses bycleaning the response spectrum have been proposed.

• Strategies to reduce nonlinearity by Nonlinear CancellationCoupling have been proposed.