ENHANCED SHAPE MEMORY ALLOY ACTUATORS

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1 ENHANCED SHAPE MEMORY ALLOY ACTUATORS M. Leary, J. Mac, M. Mazur, F. Schiavoe, A. Subic RMIT Uiversity, Budoora, Victoria 3083, Australia Phoe: , marti.leary@rmit.edu.au Abstract Shae memory alloys (SMA) rovide a oortuity to develo ultralight mechaical actuators with far suerior ecoomic ad evirometal erformace to traditioal actuators. Desite their may favorable attributes, SMA actuators are subject to a series of iheret limitatios, i articular, a relatively slow exasio rate. The authors have roosed a ovel method that eables a sigificat ehacemet of the heat trasfer rate i a cost effective ad robust maer. A advaced redictive model has bee develoed to eable the otimisatio of a virtual rototye before egagig i exerimetal validatio ad commercialisatio of this ovel techology.. Itroductio Shae memory alloys (SMA) rovide a oortuity to achieve sigificat evirometal beefit over traditioal actuators. SMA actuators are subject to a series of costraits to imlemetatio, i articular, a relatively slow exasio rate, which tyically occurs by free covective heat trasfer to the ambiet eviromet (Va Humbeeck 999; Huag 2002). Alterate coolig methods have bee roosed, icludig forced covectio with a highly coductive media (Rediiotis ad Lagoudas, 2002). This method allows very high actuatio frequecies, however the requisite hardware itroduces a sigificat cost ad weight ealty. The authors have roosed a ovel method that eables a icreased heat trasfer rate i a cost effective ad robust maer (Leary, Schiavoe ad Subic 2009). The beefits of the roosed method have bee rove exerimetally by the authors for a articular secime cofiguratio. I rearatio for commercialisatio of this techology, a advaced redictive model is required to eable the desig refiemet ad otimisatio of a virtual rototye before egagig i exerimetal validatio. This work reorts o a Fiite Differece Equatio (FDE) imlemetatio to eable greater redictive certaity i the secificatio of laggig arameters. A assessmet of FDE attributes accordig to the desig requiremets idetified the referred FDE imlemetatio. The adoted imlemetatio was develoed to eable raid simulatio of SMA arameters rior to hysical testig. These out- J. Wellitz et al. (eds.), Sustaiable Automotive Techologies 200, DOI 0.007/ _23, Sriger-Verlag Berli Heidelberg 200

2 84 M. Leary, J. Mac, M. Mazur, F. Schiavoe, A. Subic comes rovide a temlate for the imlemetatio of SMA actuators that will eable sigificat mass reductio ad icreased desig flexibility i electromechaical automotive systems. 2. The desig roblem Shae memory alloys udergo a crystallograhic trasformatio i the solid state. The trasformatio effect results i a sigificat volumetric chage which ca be utilised i the desig of ultra-light mechaical actuators. Tyically, resose time is a critical costrait to the imlemetatio of SMA actuators i automotive systems. SMA actuatio is achieved by resistive heatig which is accetably fast. However, SMA coolig is achieved by heat trasfer mechaisms that are iheretly slower tha resistive heatig (Va Humbeeck 999), ad resose time is domiated by the associated heat trasfer rate (Leary ad Schiavoe 2008). A ovel method has bee develoed by the authors to eable a icreased rate of covective heat trasfer by the additio of a laggig material, eablig otimisatio of the heat trasfer rate i a cost effective ad robust maer (Leary, Schiavoe ad Subic 2009). The method utilizes cylidrical heat trasfer media, i.e. laggig, to maiulate the heat trasfer rate (Figure ). Fig. Novel method to maiulate the heat trasfer rate, after (Leary, Schiavoe, Subic 2009). Icreasig laggig thickess results i a icreased resistace to coductive heat trasfer cocurretly with a reduced resistace to covective heat trasfer. The total resistace to heat trasfer dislays a local miima, kow as the critical radius, r cr (Holma 990). Cylidrical laggig allows heat trasfer resistace to be either icreased or decreased deedig o whether the laggig is greater-tha or less-tha the associated critical radius. Selectio of the laggig media ad dimesios allows cotrol of the critical radius to otimise the coolig rate withi the desig costraits.

3 Ehaced shae memory alloy actuators Fiite Differece Equatios The Secod Law of Thermodyamics idetifies that a temerature gradiet will result i a trasfer of heat. The associated local temerature is redicted by the oe-dimesioal heat diffusio equatio (Equatio ). T t 2 ( x, t) d T( x, t) = α () x 2 A Fiite Differece Equatio (FDE) rovides a aroximate solutio to differetial equatios such as the oe-dimesioal heat diffusio equatio, ad was alied i this work to eable robust ad raid estimatio of the ifluece of laggig arameters o SMA resose. Numerous FDE embodimets are available, each with associated trucatio error ad stability. 3. Trucatio error The itet of the FDE aroximatio is to relicate the associated differetial equatio. However, the discretizatio of the roblem results i a differece betwee these solutios, kow as the trucatio error. Trucatio error is deedat o the associated FDE method, ad ca be reduced by icreasig mesh size at the exese of comutatioal efficiecy. 3.2 Stability The FDE method origiates from a Taylor series exasio, ad cosequetly there is a ossibility for istability i results. Stability is a fuctio of the FDE embodimet ad ca be evaluated by comletig a Vo Neuma Stability Aalysis (Richtmyer ad Morto 957). Of the roosed FDE methods the followig have bee foud to be articularly effective i solvig the heat trasfer roblem:. Forward differece equatio (Exlicit) 2. Backward differece equatio (Imlicit) 3. Cetral differece equatio (Crak ad Nicholso)

4 86 M. Leary, J. Mac, M. Mazur, F. Schiavoe, A. Subic 3.3 Forward differece equatio (Exlicit) The forward differece equatio uses kow arameters at a iitial time ste to calculate values at the ext time ste. For examle, the exlicit method cosiders the temerature of three coected odes, T m-, T m, T m+, at time,, ad uses their kow values to calculate the temerature of the cetral ode, T m, at time (Equatio 2). This method is comutatioally straightforward as each equatio has oly oe ukow, ad it ca be solved without simultaeous methods. However, the forward differece equatio is ustable if the associated Fourier umber ( 2 Fo=α t / x ) is ot maitaied betwee 0 ad ½. The associated trucatio error is O[ t, x 2 ]. ( 2Fo) T + FoT T FoT + + = (2) 3.4 Backward differece equatio (Imlicit) The imlicit method of the fiite differece equatio requires that the iitial state of the system is fully uderstood. From this data a exressio for the temerature of odes i the followig time ste is created (Equatio 3). The exressio must be solved simultaeously by the iversio matrix method ad requires sigificatly higher comutatioal resources tha the exlicit method. However, the associated trucatio error is reduced ad is ideedet of mesh selectio: O[ t, x 2 ]. ( + Fo) T FoT T FoT = (3) Cetral differece equatio (Crak-Nicholso) The Crak-Nicolso equatio combies the exlicit ad imlicit methods, resultig i a average of the two, yieldig a smaller trucatio error due to a decreased sesitivity to chages withi x. The Crak-Nicolso method is as a imlicit scheme i which a series of simultaeous equatios are solved for each time ste (Equatio 4). The Crak-Nicolso method cosiders six odes whereas oly four odes were cosidered i the simle exlicit ad imlicit methods. The Crak- Nicolso method rovides high accuracy results ad is ucoditioally stable, allowig for maximum flexibility i the choice of discretizatio. (Crak ad Nicolso 947). The trucatio error is secod order accurate with both sace ad time, O[ t 2, x 2 ], illustratig the miimal effect the discretizatio of the equatio has o the FDE solutio.

5 Ehaced shae memory alloy actuators 87 ( + 2Fo) T FoT = FoT + ( 2 Fo) T FoT (4) FoT Simulatio The Crak-Nicholso method was alied to solve the oe-dimesioal heat equatio for a simle olar geometry cosistig of a 0.254mm diameter SMA wire surrouded by a aulus of laggig material (Table ). To simulate a tyical actuatio sceario, two-stages of boudary coditios were alied to simulate wire cotractio (heatig) ad exasio (coolig) resectively. Table Material roerties alied (Leary et al. 2009). Material T i [ C ] ρ [g/cm 3 ] k [W/mK] h [W/m 2 ] D mi (mm) D max (mm) SMA Laggig Shell Air 2 6 SMA actuatio was simulated by alyig a iitial temerature boudary coditio, T i, associated with the SMA actuatio temerature ad covectio to the eviromet to a system that was iitially at ambiet temerature (Table ). The SMA was assumed to reach its actuatio temerature istataeously. The laggig material temerature icreases mootoically, util the heat flow rate stabilises ad the system reaches steady-state (Figure 2). Oce steady-state coditios were achieved, the SMA wire temerature boudary coditio was removed to simulate deactivatio. I this stage the SMA wire temerature is determied by coolig to the ambiet eviromet (Figure 3). The exothermic heat trasfer of hase chage was igored (Va Humbeeck 999).

6 88 M. Leary, J. Mac, M. Mazur, F. Schiavoe, A. Subic Fig. 2 Reresetative cotractio (heatig) hase FDE solutio. Fig. 3 Reresetative exasio (coolig) hase FDE solutio.

7 Ehaced shae memory alloy actuators 89 Fig. 4 Reresetative exasio (coolig) hase FDE solutio. 5. Outcomes The roosed FDE aalysis eables raid simulatio of the erformace of a rage of SMA laggig arameters, rovidig a oortuity to redict the erformace of feasible desig arameters before buildig hysical rototyes. The roosed FDE aalysis eables the evaluatio of a secific cofiguratio. To assess a larger desig sace requires a automated Desig of Exerimets (DoE) aroach. Taguchi methods may be alied to geerate a artial factorial DoE aroach to eable automated aalysis of a large umber of feasible desig cofiguratios. To esure robust outcomes of the automated aalysis it is ecessary to esure that the steady-state coditio is reached durig the heatig hase. This ca be achieved by esurig that the root-mea-square (RMS) temerature differece betwee last two iteratios of the FDE is sufficietly small. The system erformace is the characterised by the total time required for the SMA to reach the deactivatio temerature (Table ). The cotractio ad exasio hases of the SMA simulatio were liked for a rage of feasible laggig arameters ad the deactivatio time recorded. Iitial outcomes idicate that:. Deactivatio time is miimized if aste ad shield coductivity are maximized. The desig costraits restrict the material selectios available for the shell, ad the associated coductivity is likely to be sigificatly lower tha the aste (Table ). This restrictio may be mitigated by miimizig shell thickess. 2. The redicted coolig time is comatible with exerimetal observatios ad critical radius theory ad idicates a reductio of coolig time with icreasig

8 90 M. Leary, J. Mac, M. Mazur, F. Schiavoe, A. Subic radius util a local miima is reached; additioal icrease i radius results i a icreasig coolig time. Refereces Crak, J. ad Nicolso, P. (947). A ractical Method for Numerical Evaluatio of Solutios of Partial Differetial Equatios of Heat-Coductio Tye, Proc. Cambridge Philos., vol.43, Holma, J.P., (990) Heat Trasfer, Volume 7. McGraw-Hill, New York. Leary, M., F. Schiavoe ad A. Subic. (2009) Laggig for Cotrol of Shae Memory Alloy Actuator Resose Time, Materials ad Desig, i Press. Leary, M. ad Schiavoe, F. (2008) Shae Memory Alloys for Evirometally Sustaiable Mechaical Actuatio. I Subic, A., Leary, M., ad Wellitz, J. (Eds.) Meetig the Challeges to Sustaiable Mobility, RMIT Uiversity, Melboure. Richtmyre, R.D. ad Morto, K.W., (957), Differece Methods For Iitial-Value Problems, Secod Editio, Joh Wiley & Sos, Ic, Uited States of America. Va Humbeeck, J., (999) No-medical alicatios of shae memory alloys. Materials Sciece ad Egieerig A, :

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