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Effects of Modal Interactions on Vibration Performance in Ultrasonic Cutting
Effects of Modal Interactions on Vibration Performance in Ultrasonic Cutting
’
M. Lucas’, A Cardmi’, F.C.N. Lim’, M.P. Cwtmell’ Department of Mechanical Engnearing, Unkersrty of Glasgow, Glasgrm, UK Submitted by J A . McGeough (I), Ednbrrgh, UK
Abstmd The effects of modal interactions and nonlinear response characteristics can harrper implementation of hi& prmer u h s o n i c technologes, due to the resulting modal cwpling, increased &ess, audble noise levels and poor m b o l of the operating response. These dfferent adverse responses are illu&ated by chmcterising the v i h t i o n b e h w i w r of singleblade and m b b l a d e ultrasonic cutting systems. This paper proposes desigr &ateges to eliminate the effects of modal interactions, by foeusing on reducing the number of vibation modes, and to reduce the effeets of nonlinear responses, by serial coupling of tuned cmponents with appropriate cubic softening and hardening response chmcteristics. Keywords: U h s o n i c cutting, Nonlinear responses, Vibration
1 INTRODUCTION The application of high power u h s o n i c s to manufacturing processes has been an established area of research fw several decades. However, many systems, particularly in continuous operation, sufer frm m p o n e n t kilures and noise. Ultrasonic bannsducers exceed their linear response threshold in hi& prmer applimtions [ l ] , leadng to unpredictable v i h t i o n b e h w i w r of the attiached tuned cmponents. This hampers the exploitation of ultrasonic systems that h w e c m p l e x geometries and are chmcterised by many modes, as the consequences of the nonlinearrty of the transducer are dffiwlt to incwpwate at the desigr stage. Consequentiy, the investigation of a method focused on reducing the effects of modal interactions and nonlinearities is nrm VMto provide a n w boost in the design of more reliable ultrasonic systems.
In this paper, the response characteristics of a singe blade and mubblade ultrasonic cutting device are investigated. The single blade system, shrmn in Figure 1(a), consists of a hi&-gain halCwwdength cutting blade tuned to a longtudinal mode at a nominal 35 kHr. The mubblade cutting device (Figure l@)) is also tuned to a longtudnal mode at a n m i n a l 35 lblq and consists of three halCwwelength h i g h g i n blades attached to a halCwwdength block h m . The research repwted in this paper is based on two fundamental premises; fir*, that there exists an oppwtunlty to mitigate the effects of the nonlinear characteristic of the ~ M S ~ U C W ,which is consistentiy reported f w a range of transducers as being of a cubic soitening nature [1,4, if the attached components and threaded studs have an opposing hardening characteristic; secondly, that a desigr strategy focused on minimising the number of modal frequencies below the tuned frequency CM reduce the passibillty of exciting modal interactions and allow improved control of multiple frequency responses.
Both of these premises assume that the nonlinear responses of u h s o n i c cutting systems are characteristic of a Dufhng asallatw. Therefwe, in suppwt of the experimental investigation, a preliminary theoretical investigation has bean carried out. A simple coupled oscillatw is proposed, consisting of two singe
degeeoCfreedom subsystems in which the principal nonlinewities are a hardening and soitening cubic stiffness effect respectively. The idea is to investigate the effects on the nonlinear response of v q i n g one of the cubic stiffness parameters. Althw& this thewy is not w s t m i s e d for M ultrasonic device, the results can provide insi&t into the measurements and validation of the predctions can be confirmed in a general sense.
Figure 1: (a) Singeblade and (b) multi-blade cutting head showing amplitude distribution of the tuned mode
2 MOOALlNTERACTlONS Modal interactions in pametrically excited systems can =cur if special relationships (combination resonances w internal resonances) between one w mwe modal frequencies and the excitation frequency exist. For systems chmcterised by many modes, such as a multC blade ultrasonic cutting device, external excitation of the tuned mode m y excite one or two other modes through internal resmance. Each of these modes m y excite further modes through internal resonance and the result is a frequency cascading effect [3]. Evidence suggests that systems tend to leak energy into lrmer frequency modes, with the consequence fw ultrasonic s w e m s being that energy is leaked into audible modal frequencies and that the response of a tuned, longitudinal mode, cutting device indudes coupled flexural and twsional mode responses. The immediate consequence is that less energy excites the operating mode and &ess is r i s e d due to the addtional flexural responses, causing failures in the cutting blades.
W r . The excitation was wept over a range of 300 Hr
Figure 2: (a) FRF f r m Sblade cutting head, (b) cmbination resonance, (c) m b i n a t i o n rBsonance with modulation
and the swept-sine measurement was repeated for 5 V increments of the excitation voltage. In Figure 2@) it is observed that at 15 V excitiahon, when the weepdown frequency reaches 35.2 W q two responses we excited in two lrmer frequency modes. These modes, conespondng to a bending mode of the assembly and a torsional mode of the blades, occur at 11.6 W r (fd and 23.6 W r (Q,respectively and satisfy the combination resonance, f, S' ,f + fi. M e n the excitation frequency was further decreased to 35.15 k k the response became enriched with sidebands wound all three modal frequencies, indicating a modulated response as shown in Figure 2(c). The moduliatjon frequency is present in the lrmer end of the spectrum at 750 k,at the first system natural frequency. Other combination resonanms CM be measured by driving the system in a nontuned mode. For example, the sbbilty dagams f w two combination rBsonances measured on a single blade cutting system we shown in Figure 3. The system was &iven at its tuned frequency at 35.3 k k , and subsequentty at 43 k k , a frequency conespondng to a h i a e r longtudinal mode of the assembly. A sweep of the excitation frequency revealed that a dfferent cmbination rBsonance was excited at a different =citation threshdd in each case. The excitation level versus frequency plot in Figure 3 shows the insbbilty regons inside which the combination resonances participate within the response. The regon of the second combination resonance is wider than the firsf and the threshdd at which such modal interactions appear is h e r . It is clear that cmbination resonances we c m m o n to singleblade and multLblade cutting s w e m s . A simple approach to r e b i g r i n g a system =citing a combination resonance is to uncwple the tuned mode frwn the other modes involved in the cwnbination resonance by making small gemetry modifications to provide the required modal frequency shiits to remove the internal resmance condtion. This d d n i n g &ategy only works in systems characterised by a few modes of v i h t i o n belrm the tuned frequency. For systems with many modes, detuning of a resonance alters the modal behaviwr of the system such that it favours the excitation of one or more other modal interactions. Small geometry modifications of the blade in sinde blade systems hiwe proved successful in eliminatjng combination resonanms. The approach fw m b b l a d e systems has been to reduce the number of modes, fw example by reducing the wavelength block horn and blades con&uetion, to a halCwavelength system as shrmn in Figure 4. This multLblade system has less than half the number of modal frequencies below the tuned frequency than the devim in Figure l @ ) and results in -more. sbaightfoMrd conbo~of combination resonances.
Figure 3: Stability r a m s for a singleblade system
2.1 Cornblnatlon rwonances Cmbination resonances CM arise when a system is harmonically excited in the vicinity of a natural frequency. If a relationship exists between two w mwe linear modes and the excitation frequency in the fwm of a cmbination r6sonance, the system response contributes more modes [3]. Figure 2(a) is a frequency response function (FRF) measured on M wter blade of the threeblade cutting head using a 3D laser vibmeber. The modal interactions w e identified frwn a slow frequency weep over a n m r m frequency band a r w n d the tuned longitudnal mode frequency at 35.1
Figure 4: HalCwavelength threeblade wtting system showing amplitude di&ibrtion ofthe longtudnal mode
3 NONLINEAR CHARACTERISTICS Nonllnear behindour. to some extent, Is present In mechanlcal and structural systems due to Inherent sources of nonllnearltles exlstlng even In slmple systems. Nonllnearttles are responslble for a varlety of eWects that are absent In inear systems. such as the lump phenomenon, natural frequency shlRlng, frequency modulatlons and chaotlc motlons [3,4]. Many systems excited In vlbratlm, M l c h exhlbit llnear behimdour at low levels of excitatlon, become nonllnear at hlgher levels of excitatlon. Hlgh power uitrasonlc systems, d e s l g e d to resonate In a tuned mode of vlbratlon at a kw uitrasonlc frequency, typify such systems and are prone to exhlbit adverse responses durlng operatlon that resuit from nonllnearltles. Nonllnear behidour of uitrasonlc systems Is responslble for energy leakage Into nontuned modes leadlng to an uncontrollable process performance often accompanled by mlse and component fallures [2]. 3.1
Transducer nonllneartty
The nonllnear bahiwlour of uitnsonlc systems Is strongly attributable to the plezoeledrlc transducer behimdour [I], slnce the nonllnear threshold of its constituent ceramlcs 1s reached at low excitatlon. The nonllnear charaderlstlc of a typlcal Industrlal 35 kl+ uitrasonlc transducer, obtalned from swpt-slne measurements, 1s shown In Flgure 5. +30V
M)
&30V
(om*1)
- - e50VM) Flgure 6: Frequency response curves of transducer and slngleblade cuttlng system wlth threaded stud (a) fully ltted Into bladebase, @) haK-lltted Into bladebase, (c) fullylltted Into transducer-base
3.2 Nonllnear behavlour of cuttlng Iyztemz
Flgure 5: Llnear and nonllnear response of a tnnsducer at two dit7erent excitatlon levels Flgure 5 shows the frequency response of the transducer f x e In the tuned longitudlnal mode at two excitatlon levels. The curves were obtalned by s w e p l n g the excitatlon frequency lowards and backwards. A slow s w e p rate w s adopted In order to record only steady transducer responses at each frequency Increment. From the response curves it Is posslble to Identify the Ilnear threshold, h thls case 30 V, and to charaderlse the nonllnearlly. The response at 50 V shows the jump phenomenon, typlcal of a DulAng osclllator, M e r e a j u m p u p In the response curve In the sweepup measurement and a j u n p d o w n In the s w e p down curve creates an enclosed unstable reglon. The response curve bends to the left, Illustratlng a softenlng The cublc softenlng response charaderlstlc. charaderlstlc Is typlcal of a range of measured transducers, tuned to frequencles from 20 Ibh to 70 Ibh, wlth the llnear threshold and the degree of softenlng belng unlque to each transducer. The measurements hlghllght the Importance of understandlng the transducer behimdour In the deslgn of uitrasonlc systems, because the transducer plays such a predomlnant role In determlnlng the response charaderlstlcs of the uitrasonlc system.
Manlpulidlng the nonllnear response of the cuttlng systems w m Investlgated by measurlng the response of dit7erent transducer plus tuned component conflguratlons. By condudlng a serles d swept-slne measurements. it was found that some components tended to llnearlse the response of the system, whereas other transducerkomponert conflguratlons tended to Increase the softenlng charaderlstlc. F # Instance. a haK-welength hlghgaln blade attached to a transducer tended to further soften the response, resuttlng In a soRer system wlth a lower Ilnear threshold. Thls meant that the behimdour of a s l n g l e b l d e cuttlng devlce of thls type was hlghiy nonllnear at operatlng excitatlon levels. A wavelength lowr-galn blade, on the other hand, tended to reduce the softenlng charaderlstk, resuttlng In a slngleblade cuttlng system that was more Ilnear and had a much hlgher llnwr threshold. Knowledge galned from these measurements all^ a known transducer to be serlaliy coupled to tuned components wHh the resuit that the tuned system operates M h a more Ilnear response and a hlgh llnear threshold. The nonllnear charaderlstlc could also be manlpulated by aiterlng the positbnlng of the threaded studs 1olnlng the transducer and blade. Flgure 6 shows response measurements for three dit7erent stud positlons jolnlng a transducer and a Mghgaln blade. For thls system, positlonlng the stud so that it Is fuliylltted Into the transducer base can slgnltlcantly reduce the w#th of the unstable reglon. Slmllarly, the ]olnt tlghtness also Mads the Adth of the unstable reglon.
3.3 Theoretlcal Approach In order to validate the approach of manipulating the nonlinear characteristic of senallycoupled components, using components of alternate hardening and softening characteristics, an analysis is presented for two coupled Duffingtype oscillators (Figure 7) involving linear viscous damping (ci2), and sbffness characteristics modelled by linear and cubic terms ( k j 2 and h, respectively). hr is a cubic hardening spring of (+h x f $ J and h2 is a cubic softening spring of (-h2 x2 ). A harmonic force is applied to the first subsystem. The governing drfferential equations for this hypothetical test-bed system are derived and the method of muttiple scales is used to solve the equations to a second order of approximation [5].
Figure 7: Two degree-of-freedom Duffing Oscillator 0 OlE
0 014
0 012
1
0 01
Further supporting evidence that the transducer behaviour is typical of a Dufiing oscillator is presented in Figures 9 and 10. Figure 9 is the predicted bifurcation of the excitation acceleration when the excrtation frequency is at the first mode eigenvalue of the system in Figure 7. This phenomenon was investigated experimentally by measuring the response from the bansducer
face
of
two
drfferent 35
kHz
tuned
bansducers over a range of excitation levels and the results are presented in Figure 10. Atthough the measured response from the bansducer face cannot indicate muRiple response solutions, the measurements in Figure 10 seem to exhibit a bifurcation point at higher excitation levels for each transducer. 4 CONCLUSIONS Research has shown that combination resonances are common in ulbasonic c m n g systems and that minimising the number of modes in the system and eliminating internal resonances by modrfying cutting blade geometries can mitigate the effects of such modal interactions. Nonlinearities in high power ulbasonic systems mainly stem from the transducer behaviour. A novel method for influencing the nonlinear behaviour of a system is proposed, which is based on coupling components with opposrtely signed cubic nonlinearrty. This strategy is supported by a hypothetical, but relevant case of two coupled Dufiing oscillators. Practical application of this method to the case of an utkasonic cutting system proved capable of reducing the effects of the nonlinearities within the system.
0 ODE 0 006 0 004
0 002 0
0
06
h2 (Multipleof
1
"Qim.12
2 6
36
h,)
I + 0.083 rn 0.042 A 0.0167
0.008 ]
Figure 8: Frequency response for variation of the softening characteristic h2 For the hardening system, the nonlinear effect changes significantly when a further nonlinear subsystem is coupled to it. As the softening strffness h2 is decreased, with the rest of the variables constant it tends to bend the backbane of the nonlinear response characteristic to the right, to a more linear-like response as shown in Figure 8.
Figure 9: Excitation level plotted against response showing predicted bifurcation for h2=0.0167hl [5]
Figure 10: Measured transducer response indicating bifurcation 5
REFERENCES Aurelle, N., Guyomar, D., Richard, C., Gonnard, P., Eyraud, L., 1996, Nonlinear Behaviour of an Uttrasonic Transducer, Uttrasonics, 3420521 1. Lucas, M., Petzing, J.N., Graham, G., 1998, Experimental Characterisation of Nonlinear Int. Conf. Vibration in Utkasonic Tools, Ilm Experimental Mechanics, Oxford (UK), 945949. Cartmell, M.P., 1990, Inboduction to Linear, Paramebic and Nonlinear Vibrations, Chapman and Hall. Anderson, T.J., Balachandran, B., Nayfeh, A.H., 1992, Observations of Nonlinear Interactions in a Flexible Cantilever Beam, Proc. A I M , 16781685. Lim, F.C.N., Cartmell, M.P., Cardoni, A,, Lucas, M., 2003, A Preliminary Investigation into Opbmising the Response of Vibrating systems used for Ulbasonic C m n g , Journal of Sound and Vibration (under review).
’
M. Lucas’, A Cardmi’, F.C.N. Lim’, M.P. Cwtmell’ Department of Mechanical Engnearing, Unkersrty of Glasgow, Glasgrm, UK Submitted by J A . McGeough (I), Ednbrrgh, UK
Abstmd The effects of modal interactions and nonlinear response characteristics can harrper implementation of hi& prmer u h s o n i c technologes, due to the resulting modal cwpling, increased &ess, audble noise levels and poor m b o l of the operating response. These dfferent adverse responses are illu&ated by chmcterising the v i h t i o n b e h w i w r of singleblade and m b b l a d e ultrasonic cutting systems. This paper proposes desigr &ateges to eliminate the effects of modal interactions, by foeusing on reducing the number of vibation modes, and to reduce the effeets of nonlinear responses, by serial coupling of tuned cmponents with appropriate cubic softening and hardening response chmcteristics. Keywords: U h s o n i c cutting, Nonlinear responses, Vibration
1 INTRODUCTION The application of high power u h s o n i c s to manufacturing processes has been an established area of research fw several decades. However, many systems, particularly in continuous operation, sufer frm m p o n e n t kilures and noise. Ultrasonic bannsducers exceed their linear response threshold in hi& prmer applimtions [ l ] , leadng to unpredictable v i h t i o n b e h w i w r of the attiached tuned cmponents. This hampers the exploitation of ultrasonic systems that h w e c m p l e x geometries and are chmcterised by many modes, as the consequences of the nonlinearrty of the transducer are dffiwlt to incwpwate at the desigr stage. Consequentiy, the investigation of a method focused on reducing the effects of modal interactions and nonlinearities is nrm VMto provide a n w boost in the design of more reliable ultrasonic systems.
In this paper, the response characteristics of a singe blade and mubblade ultrasonic cutting device are investigated. The single blade system, shrmn in Figure 1(a), consists of a hi&-gain halCwwdength cutting blade tuned to a longtudinal mode at a nominal 35 kHr. The mubblade cutting device (Figure l@)) is also tuned to a longtudnal mode at a n m i n a l 35 lblq and consists of three halCwwelength h i g h g i n blades attached to a halCwwdength block h m . The research repwted in this paper is based on two fundamental premises; fir*, that there exists an oppwtunlty to mitigate the effects of the nonlinear characteristic of the ~ M S ~ U C W ,which is consistentiy reported f w a range of transducers as being of a cubic soitening nature [1,4, if the attached components and threaded studs have an opposing hardening characteristic; secondly, that a desigr strategy focused on minimising the number of modal frequencies below the tuned frequency CM reduce the passibillty of exciting modal interactions and allow improved control of multiple frequency responses.
Both of these premises assume that the nonlinear responses of u h s o n i c cutting systems are characteristic of a Dufhng asallatw. Therefwe, in suppwt of the experimental investigation, a preliminary theoretical investigation has bean carried out. A simple coupled oscillatw is proposed, consisting of two singe
degeeoCfreedom subsystems in which the principal nonlinewities are a hardening and soitening cubic stiffness effect respectively. The idea is to investigate the effects on the nonlinear response of v q i n g one of the cubic stiffness parameters. Althw& this thewy is not w s t m i s e d for M ultrasonic device, the results can provide insi&t into the measurements and validation of the predctions can be confirmed in a general sense.
Figure 1: (a) Singeblade and (b) multi-blade cutting head showing amplitude distribution of the tuned mode
2 MOOALlNTERACTlONS Modal interactions in pametrically excited systems can =cur if special relationships (combination resonances w internal resonances) between one w mwe modal frequencies and the excitation frequency exist. For systems chmcterised by many modes, such as a multC blade ultrasonic cutting device, external excitation of the tuned mode m y excite one or two other modes through internal resmance. Each of these modes m y excite further modes through internal resonance and the result is a frequency cascading effect [3]. Evidence suggests that systems tend to leak energy into lrmer frequency modes, with the consequence fw ultrasonic s w e m s being that energy is leaked into audible modal frequencies and that the response of a tuned, longitudinal mode, cutting device indudes coupled flexural and twsional mode responses. The immediate consequence is that less energy excites the operating mode and &ess is r i s e d due to the addtional flexural responses, causing failures in the cutting blades.
W r . The excitation was wept over a range of 300 Hr
Figure 2: (a) FRF f r m Sblade cutting head, (b) cmbination resonance, (c) m b i n a t i o n rBsonance with modulation
and the swept-sine measurement was repeated for 5 V increments of the excitation voltage. In Figure 2@) it is observed that at 15 V excitiahon, when the weepdown frequency reaches 35.2 W q two responses we excited in two lrmer frequency modes. These modes, conespondng to a bending mode of the assembly and a torsional mode of the blades, occur at 11.6 W r (fd and 23.6 W r (Q,respectively and satisfy the combination resonance, f, S' ,f + fi. M e n the excitation frequency was further decreased to 35.15 k k the response became enriched with sidebands wound all three modal frequencies, indicating a modulated response as shown in Figure 2(c). The moduliatjon frequency is present in the lrmer end of the spectrum at 750 k,at the first system natural frequency. Other combination resonanms CM be measured by driving the system in a nontuned mode. For example, the sbbilty dagams f w two combination rBsonances measured on a single blade cutting system we shown in Figure 3. The system was &iven at its tuned frequency at 35.3 k k , and subsequentty at 43 k k , a frequency conespondng to a h i a e r longtudinal mode of the assembly. A sweep of the excitation frequency revealed that a dfferent cmbination rBsonance was excited at a different =citation threshdd in each case. The excitation level versus frequency plot in Figure 3 shows the insbbilty regons inside which the combination resonances participate within the response. The regon of the second combination resonance is wider than the firsf and the threshdd at which such modal interactions appear is h e r . It is clear that cmbination resonances we c m m o n to singleblade and multLblade cutting s w e m s . A simple approach to r e b i g r i n g a system =citing a combination resonance is to uncwple the tuned mode frwn the other modes involved in the cwnbination resonance by making small gemetry modifications to provide the required modal frequency shiits to remove the internal resmance condtion. This d d n i n g &ategy only works in systems characterised by a few modes of v i h t i o n belrm the tuned frequency. For systems with many modes, detuning of a resonance alters the modal behaviwr of the system such that it favours the excitation of one or more other modal interactions. Small geometry modifications of the blade in sinde blade systems hiwe proved successful in eliminatjng combination resonanms. The approach fw m b b l a d e systems has been to reduce the number of modes, fw example by reducing the wavelength block horn and blades con&uetion, to a halCwavelength system as shrmn in Figure 4. This multLblade system has less than half the number of modal frequencies below the tuned frequency than the devim in Figure l @ ) and results in -more. sbaightfoMrd conbo~of combination resonances.
Figure 3: Stability r a m s for a singleblade system
2.1 Cornblnatlon rwonances Cmbination resonances CM arise when a system is harmonically excited in the vicinity of a natural frequency. If a relationship exists between two w mwe linear modes and the excitation frequency in the fwm of a cmbination r6sonance, the system response contributes more modes [3]. Figure 2(a) is a frequency response function (FRF) measured on M wter blade of the threeblade cutting head using a 3D laser vibmeber. The modal interactions w e identified frwn a slow frequency weep over a n m r m frequency band a r w n d the tuned longitudnal mode frequency at 35.1
Figure 4: HalCwavelength threeblade wtting system showing amplitude di&ibrtion ofthe longtudnal mode
3 NONLINEAR CHARACTERISTICS Nonllnear behindour. to some extent, Is present In mechanlcal and structural systems due to Inherent sources of nonllnearltles exlstlng even In slmple systems. Nonllnearttles are responslble for a varlety of eWects that are absent In inear systems. such as the lump phenomenon, natural frequency shlRlng, frequency modulatlons and chaotlc motlons [3,4]. Many systems excited In vlbratlm, M l c h exhlbit llnear behimdour at low levels of excitatlon, become nonllnear at hlgher levels of excitatlon. Hlgh power uitrasonlc systems, d e s l g e d to resonate In a tuned mode of vlbratlon at a kw uitrasonlc frequency, typify such systems and are prone to exhlbit adverse responses durlng operatlon that resuit from nonllnearltles. Nonllnear behidour of uitrasonlc systems Is responslble for energy leakage Into nontuned modes leadlng to an uncontrollable process performance often accompanled by mlse and component fallures [2]. 3.1
Transducer nonllneartty
The nonllnear bahiwlour of uitnsonlc systems Is strongly attributable to the plezoeledrlc transducer behimdour [I], slnce the nonllnear threshold of its constituent ceramlcs 1s reached at low excitatlon. The nonllnear charaderlstlc of a typlcal Industrlal 35 kl+ uitrasonlc transducer, obtalned from swpt-slne measurements, 1s shown In Flgure 5. +30V
M)
&30V
(om*1)
- - e50VM) Flgure 6: Frequency response curves of transducer and slngleblade cuttlng system wlth threaded stud (a) fully ltted Into bladebase, @) haK-lltted Into bladebase, (c) fullylltted Into transducer-base
3.2 Nonllnear behavlour of cuttlng Iyztemz
Flgure 5: Llnear and nonllnear response of a tnnsducer at two dit7erent excitatlon levels Flgure 5 shows the frequency response of the transducer f x e In the tuned longitudlnal mode at two excitatlon levels. The curves were obtalned by s w e p l n g the excitatlon frequency lowards and backwards. A slow s w e p rate w s adopted In order to record only steady transducer responses at each frequency Increment. From the response curves it Is posslble to Identify the Ilnear threshold, h thls case 30 V, and to charaderlse the nonllnearlly. The response at 50 V shows the jump phenomenon, typlcal of a DulAng osclllator, M e r e a j u m p u p In the response curve In the sweepup measurement and a j u n p d o w n In the s w e p down curve creates an enclosed unstable reglon. The response curve bends to the left, Illustratlng a softenlng The cublc softenlng response charaderlstlc. charaderlstlc Is typlcal of a range of measured transducers, tuned to frequencles from 20 Ibh to 70 Ibh, wlth the llnear threshold and the degree of softenlng belng unlque to each transducer. The measurements hlghllght the Importance of understandlng the transducer behimdour In the deslgn of uitrasonlc systems, because the transducer plays such a predomlnant role In determlnlng the response charaderlstlcs of the uitrasonlc system.
Manlpulidlng the nonllnear response of the cuttlng systems w m Investlgated by measurlng the response of dit7erent transducer plus tuned component conflguratlons. By condudlng a serles d swept-slne measurements. it was found that some components tended to llnearlse the response of the system, whereas other transducerkomponert conflguratlons tended to Increase the softenlng charaderlstlc. F # Instance. a haK-welength hlghgaln blade attached to a transducer tended to further soften the response, resuttlng In a soRer system wlth a lower Ilnear threshold. Thls meant that the behimdour of a s l n g l e b l d e cuttlng devlce of thls type was hlghiy nonllnear at operatlng excitatlon levels. A wavelength lowr-galn blade, on the other hand, tended to reduce the softenlng charaderlstk, resuttlng In a slngleblade cuttlng system that was more Ilnear and had a much hlgher llnwr threshold. Knowledge galned from these measurements all^ a known transducer to be serlaliy coupled to tuned components wHh the resuit that the tuned system operates M h a more Ilnear response and a hlgh llnear threshold. The nonllnear charaderlstlc could also be manlpulated by aiterlng the positbnlng of the threaded studs 1olnlng the transducer and blade. Flgure 6 shows response measurements for three dit7erent stud positlons jolnlng a transducer and a Mghgaln blade. For thls system, positlonlng the stud so that it Is fuliylltted Into the transducer base can slgnltlcantly reduce the w#th of the unstable reglon. Slmllarly, the ]olnt tlghtness also Mads the Adth of the unstable reglon.
3.3 Theoretlcal Approach In order to validate the approach of manipulating the nonlinear characteristic of senallycoupled components, using components of alternate hardening and softening characteristics, an analysis is presented for two coupled Duffingtype oscillators (Figure 7) involving linear viscous damping (ci2), and sbffness characteristics modelled by linear and cubic terms ( k j 2 and h, respectively). hr is a cubic hardening spring of (+h x f $ J and h2 is a cubic softening spring of (-h2 x2 ). A harmonic force is applied to the first subsystem. The governing drfferential equations for this hypothetical test-bed system are derived and the method of muttiple scales is used to solve the equations to a second order of approximation [5].
Figure 7: Two degree-of-freedom Duffing Oscillator 0 OlE
0 014
0 012
1
0 01
Further supporting evidence that the transducer behaviour is typical of a Dufiing oscillator is presented in Figures 9 and 10. Figure 9 is the predicted bifurcation of the excitation acceleration when the excrtation frequency is at the first mode eigenvalue of the system in Figure 7. This phenomenon was investigated experimentally by measuring the response from the bansducer
face
of
two
drfferent 35
kHz
tuned
bansducers over a range of excitation levels and the results are presented in Figure 10. Atthough the measured response from the bansducer face cannot indicate muRiple response solutions, the measurements in Figure 10 seem to exhibit a bifurcation point at higher excitation levels for each transducer. 4 CONCLUSIONS Research has shown that combination resonances are common in ulbasonic c m n g systems and that minimising the number of modes in the system and eliminating internal resonances by modrfying cutting blade geometries can mitigate the effects of such modal interactions. Nonlinearities in high power ulbasonic systems mainly stem from the transducer behaviour. A novel method for influencing the nonlinear behaviour of a system is proposed, which is based on coupling components with opposrtely signed cubic nonlinearrty. This strategy is supported by a hypothetical, but relevant case of two coupled Dufiing oscillators. Practical application of this method to the case of an utkasonic cutting system proved capable of reducing the effects of the nonlinearities within the system.
0 ODE 0 006 0 004
0 002 0
0
06
h2 (Multipleof
1
"Qim.12
2 6
36
h,)
I + 0.083 rn 0.042 A 0.0167
0.008 ]
Figure 8: Frequency response for variation of the softening characteristic h2 For the hardening system, the nonlinear effect changes significantly when a further nonlinear subsystem is coupled to it. As the softening strffness h2 is decreased, with the rest of the variables constant it tends to bend the backbane of the nonlinear response characteristic to the right, to a more linear-like response as shown in Figure 8.
Figure 9: Excitation level plotted against response showing predicted bifurcation for h2=0.0167hl [5]
Figure 10: Measured transducer response indicating bifurcation 5
REFERENCES Aurelle, N., Guyomar, D., Richard, C., Gonnard, P., Eyraud, L., 1996, Nonlinear Behaviour of an Uttrasonic Transducer, Uttrasonics, 3420521 1. Lucas, M., Petzing, J.N., Graham, G., 1998, Experimental Characterisation of Nonlinear Int. Conf. Vibration in Utkasonic Tools, Ilm Experimental Mechanics, Oxford (UK), 945949. Cartmell, M.P., 1990, Inboduction to Linear, Paramebic and Nonlinear Vibrations, Chapman and Hall. Anderson, T.J., Balachandran, B., Nayfeh, A.H., 1992, Observations of Nonlinear Interactions in a Flexible Cantilever Beam, Proc. A I M , 16781685. Lim, F.C.N., Cartmell, M.P., Cardoni, A,, Lucas, M., 2003, A Preliminary Investigation into Opbmising the Response of Vibrating systems used for Ulbasonic C m n g , Journal of Sound and Vibration (under review).