Mechanical and Control Characteristics of Simple and Multilayer Nano- and Micro-Scale Piezo Motors

Transcription

1 ISS X, Russian ngineering Research,, Vo., o., pp Aerton ress, Inc.,. Origina Russian Text S.M. Afonin,, pubished in Vestnik Mashinostroeniya,, o., pp.. Mechanica and Contro Characteristics of Simpe and Mutiayer ano- and Micro-Scae iezo Motors S. M. Afonin Moscow State Institute of ectronic Technoogy (Technica niversity) Abstract Statistica and dynamic characteristics of simpe and mutiayer nano- and micro-scae piezo motors with ongitudina and transverse piezo effects are determined. DOI:./S68798X57 The use of piezo motors (piezo actuators) is promising in nanotechnoogy, nanobioogy, microeectronics, and astronomy for precision aignment and compensation of gravitationa and therma deformation. The range of motion may be increased to tens of microns by means of mutiayer (composite) piezo motors, which are very promising eectromagnetoeastic nano- and micromotors, ensuring precision within tens of microns at a bandwidth of around Hz [ 6]. In contro systems for the nano- and microdeformation of mutiayer piezo motors, strict constraints are imposed on the range of dispacement, the rigidity, and the precision of the nano- and micro-scae piezo motors. It is important to ensure precision of such contro systems in parae and coded contro. This entais determining the corresponding mechanica characteristics of the piezo motors. The use of mutiayer piezo motors with coded contro permits effective eectromechanica digita anaog conversion for nano- and microdispacements proportiona to the contro code. ano- and micro-scae piezo motors permit precise positioning in nanotechnoogy and microeectronics. Such piezo motors are characterized by a dispacement range from a few nm to tens of microns; sensitivity ess than nm/v; a oad capacity up to ; power up to W at the output shaft; and bandwidth of tens of Hz. The use of simpe and mutiayer piezo motors for nano- and micromanipuators permits precision aignment in microeectronics, nanotechnoogy, astronomy, and adaptive optics [ 6]. In static conditions with no oad, the dispacement range of a mutiayer piezo motor is n times that of a simpe piezo motor, where n is the number of piezo ayers in the motor. Depending on the manufacturing technoogy, the mutiayer piezo motor may take the form of a composite piezoconverter consisting of severa easticay compressed piezo pates; a composite piezoconverter consisting of piezo pates bound by siver paste; a composite piezoconverter consisting of easticay reinforced stacks of pates; a composite piezoconverter consisting of cemented pates; or a mutiayer piezoconverter with ayers appied by thick- or thinfim technoogy. Since the imiting destructive compressive stress in industria TsTS or ZT piezoceramics is an order of magnitude greater than the imiting destructive tensie stress, on average, preiminary compression of the mutiayer piezo motor using a spring or a membrane is expedient, in order to increase its strength. The preiminary compressive force must exceed the maximum tensie oad, so as to ensure working compression of the mutiayer piezo motor [4, 5]. The assemby of mutiayer piezo motors for nano- and microdispacements is as foows: a mutiayer piezoconverter, aready compressed so as to seect the gaps between the piezo pates, is camped by a preiminary deformed eastic eement (pin, spring, or membrane). The compressive force is around 5 Ma. The force seects the gaps, and the dependence of the deformation on the externa force is inear. The basic externa oad parameter of the piezo motor is its rigidity: the ratio of the eastic reaction force (in response to the oad) and the deformation. The structura parameters of a simpe piezo motor with ongitudina and transverse piezo effects are seected on the basis of the oad rigidity (Figs. a and a). ractica use of mutiayer piezo motors requires knowedge of their basic characteristics: the piezomodui, the eastic piabiities, and the compressive strength. In determining the static characteristics of the piezo motors, the equations of the inverse piezo effect and the mechanica oad are taken into account [4 6]. ano- and microscae piezo motors operate on the basis of the inverse piezo effect: dispacement is the resut of deformation of the piezo motor under the action of an externa eectric votage. We now consider the mechanica and contro characteristics of simpe and mutiayer piezo motors. In cassica eectric drives, the static characteristics of the eectric motor in steady operation are divided into mechanica ω(m) and contro ω() components, where ω is the shaft speed, M is the oad torque, and is the contro votage. Anaogousy, for nano- and 65

2 66 AFOI C Δ F Δ F S T, Ma Δ, μm 4 6 F, h Fig.. Mechanica characteristics of simpe piezo motor with ongitudina piezo effect and transverse piezo effect. C Correspondingy, the maximum working force F max aong axis of the simpe piezo motor when Δ is Fig.. Kinematic diagrams of simpe nano- and microscae piezo motors with ongitudina piezo effect and transverse piezo effect. micro-scae piezo motors, static characteristics of the form S(T) or Δ(F) wi be caed mechanica characteristics, where S is the strain, T is the mechanica stress, Δ is the reative dispacement, and F is the externa force, whie characteristics of the form S() or Δ() wi be caed contro characteristics, where is the eectric fied strength. The mechanica and contro characteristics are measured on a MM-5 press at working oads, with mechanica stress up to 5 Ma in the piezo motor. The static mechanica characteristic of the piezo motor is the equation of the inverse piezo effect with a ongitudina piezo effect and votage contro, in the form [ 6] S p d + s T. () Here S ξ/ is the strain of the piezo motor aong axis (Fig. a); ξ is the dispacement; n is the ength of the mutiayer piezo motor; n is the number of piezo pates; d is the piezomoduus; / is the eectric fied strength aong axis ; is the eectrode votage in the piezo motor; s / u is the eastic piabiity aong axis at const, with contro from the votage source; u is the Young s moduus of the piezoeectric ceramic when const; T F/S is the mechanica stress in the piezo motor aong axis ; F is the externa force; and S is the cross-sectiona area of the piezo motor. In statics, the maximum dispacement Δ max aong axis of the simpe piezo motor when F is Δ max d. F max Δ max S /( s ) d S /( s ). We now consider a simpe piezo motor (piezo pate) with a ongitudina piezo effect (Fig. a): piezomoduus d 4 m/v; suppy votage V; thickness of piezo pate.6 m; eastic piabiity s.5 m /; radius R p 7.5 m; cross-sectiona area S.77 4 m. In that case, Δ max nm; F max.8 k; the corresponding static mechanica characteristics of the form S (T ) are shown in Fig. a, with 6 (), 4 (), and () kv/m. From q. (), we obtain the equation of the static mechanica characteristic for a simpe piezo motor with a ongitudina piezo effect, in the case of votage contro Δ d s F/S d F/C or Δ Δ max ( F/F max ), where is the thickness of the piezo motor; C S /( s ) is its rigidity. Correspondingy, when const, we obtain the equation Δ(F) of the mechanica characteristic and, when F const, we obtain the equation Δ() of the piezo motor s contro characteristic. The static mechanica characteristic of the piezo motor is the equation of the inverse piezo effect [ 6]. With a transverse piezo effect, in the case of votage contro S d + s T. () In statics, to find the maximum dispacement Δ max aong axis of the simpe piezo motor (Fig. b), when F, we use the formua d h/. Δ max RSSIA GIRIG RSARCH Vo. o.

3 MCHAICAL AD COTROL CHARACTRISTICS 67 C Δ F Δ F h C Δ, μm F, k Fig. 4. Mechanica characteristics of mutiayer piezo motor with ongitudina piezo effect in the case of parae contro: () ; () ; () V. The maximum working force aong axis of the simpe piezo motor when Δ is F max d S /( s ). We now consider a simpe piezo motor with a transverse piezo effect (Fig. b), when the eectric fied strength is aigned with axis and the deformation with axis : piezomoduus d m/v; suppy votage V; thickness of piezo pate.6 m; eastic piabiity s.5 m /; height h m; width b m; cross-sectiona area S 6 6 m. In that case, Δ max μm; F max 5 ; the corresponding static mechanica characteristics of the form Δ(F) are shown in Fig. b, with (), (), and () V. The discrepancy between the cacuation resuts and experimenta data is no more than 5%. From q. (), we obtain the equation of the static mechanica characteristic for a simpe piezo motor with a transverse piezo effect, in the case of votage contro or Fig.. Kinematic diagrams of mutiayer nano- and microscae piezo motors with ongitudina piezo effect and transverse piezo effect, in the case of parae contro. Δ d h/ s F Fh/S d h/ F/C Δ Δ max ( F/F max ), where h is the ength of the piezo motor (piezo pate); h C S /( s ) is its rigidity. We now consider the mechanica and contro characteristics of mutiayer piezo motors with ongitudina and transverse piezo effects in the case of parae votage contro. It foows from the compression diagrams in the region of the working forces for piezo motors of different design that, regardess of the design and contro of the composite piezo motor, the best means of increas- ing its rigidity is preiminary compression at a pressure greater than the mechanica stress used to seect the gaps and fatten microprojections. In statics, the maximum dispacement aong axis of a mutiayer piezo motors when F (Fig. a) is Δ max d n, and its maximum working force aong axis when Δ is F mx Δ max S /( s ) d S /( s ). Hence, from q. (), the equation for the static mechanica characteristic of a mutiayer piezo motor with a ongitudina piezo effect, in the case of parae votage contro, takes the form or Δ d n s F/S d n F/C Δ Δ max ( F/F max ), () where F max d S /( s ); n is the ength of the mutiayer piezo motor (n is the number of piezo pates); C S /( s ) is the rigidity of the piezo motor. Thus, when const, we obtain the equation Δ(F) of the mechanica characteristics; when F const, we have the equation Δ() for a votage-controed piezo motor. Consider a mutiayer piezo motor consisting of bound piezo pates, with a ongitudina piezo effect (Fig. a): piezomoduus d 4 m/v; suppy votage V; eastic piabiity s m /; thickness of piezo pates.6 m; diameter of piezo pates D 5 m; number of piezo pates n 5. The motor is characterized by parae contro, when the piezo pates are mechanicay connected in series but eectricay connected in parae. In Fig. 4, curve corresponds to the mechanica characteristic of the mutiayer piezo motor with Δ max 6 μm and F max.8 k. The equation for the inverse piezo effect when T const permits the cacuation of the contro char- RSSIA GIRIG RSARCH Vo. o.

4 68 AFOI S S.5 5, V/m. Fig. 5. Contro characteristics of mutiayer piezo motor with ongitudina piezo effect in the case of parae contro. C a + C e Δ C a + C e C h Δ C acteristics (Fig. 5) for a mutiayer piezo motor with parae votage contro and a ongitudina piezo effect, in the form of a composite piezoconverter consisting of eight - piezo assembies, with piezomoduus d 4 m/v and eastic piabiity s 7.5 m /. In Fig. 5, curves correspond to T, 6.7, and.4 Ma, respectivey. Anaogousy, we obtain the static mechanica characteristic for a mutiayer piezo motor with a transverse piezo effect, in the case of parae votage contro, from q. () Δ d / s F/S d / F/C, (4) where nh is the ength of the mutiayer piezo motor (h is the ength of the piezo pate); C S /( s ) is its rigidity. quivaenty Δ Δ max ( F/F max ), where Δ max d nh/, F max d S /( s ). From q. (4), it foows that, besides the piezomoduus, the eastic piabiity is an important characteristic of the piezo motor. To obtain a more rigid mutiayer piezo motor, we need to minimize the eastic piabiity. The basic externa oad parameter of the piezo motor is its rigidity, which provides the basis for the seection of the structura parameters. Hence, taking account of q. (), we may write the dispacement of the mutiayer piezo motor in the case of parae votage contro and a ongitudina piezo effect (Fig. 6a) in the form ξ d n F/C ; F F + C a ξ + C e ξ; F σ a S, where C S /( s ) is its rigidity; F is the externa force; F is the initia compressive force appied by the eastic eement; σ a is the mechanica stress in initia Fig. 6. Kinematic diagrams of reinforced mutiayer nanoand micro-scae piezo motors with ongitudina piezo effect and transverse piezo effect, in the case of parae contro and an eastic oad. reinforcement; C a is the rigidity of the reinforcing eement; C e is the oad rigidity. Hence, the contro characteristic describing the dispacement of the reinforced mutiayer piezo motor in the case of parae votage contro and a ongitudina piezo effect takes the form ξ d n σ a s ( C a + C e )/C ( d σ a s ) ( C a + C e )/C Since the mechanica stress in initia reinforcement wi be constant for each individua piezo motor, the dispacement ξ as a function of the initia reinforcement wi be constant, regardess of the votage suppied to the motor. Hence, the contro characteristic for a mutiayer piezo motor with a ongitudina piezo effect under the action of an externa oad wi be (Fig. 7) Δ max , + ( C a + C e )/C (5) where Δ max d n is the ampitude of the dispacement of the mutiayer piezo motor prior to reinforcement; is the ampitude of the eectrode votage. The contro characteristics in Fig. 7 are shown for a piezo motor consisting of gued piezo pates, with parae votage contro: piezomoduus d 4 m/v; suppy votage V; number of piezo pates n 5; C a + C e () and.c (). Anaogousy, if we take account of q. (4), we obtain the contro characteristic corresponding to the dispacement of a reinforced mutiayer piezo motor in RSSIA GIRIG RSARCH Vo. o.

5 MCHAICAL AD COTROL CHARACTRISTICS 69 Δ, μm, V C Δ F Δ a a a F a a C a Fig. 7. Contro characteristics of reinforced mutiayer nano- and micro-scae piezo motor with ongitudina piezo effect in the case of parae contro and an eastic oad. Fig. 8. Kinematic diagrams of sectiona mutiayer nanoand micro-scae piezo motors with ongitudina piezo effect and transverse piezo effect, in the case of coded contro. the case of parae votage contro and a transverse piezo effect ξ d ( /) σ a s ( C a + C e )/C s where C S /( ) is its rigidity. ( d σ a s ) , + ( C a + C e )/C The contro characteristic for a reinforced mutiayer piezo motor with a transverse piezo effect under the action of an externa oad takes the form Δ max Δ , (6) + ( C a + C e )/C where Δ max d (/). quations () (6) correspond to the static mechanica and contro characteristics of mutiayer piezo motors and permit the seection of their parameters as a function of the externa oad in nano- and micromanipuators. The dispacement range of the piezo motor is proportiona to the piezomoduus and the suppy votage. ffective use of the working dispacement range with an eastic inertia oad requires tota oad rigidity in the range < C a + C e <.C (). Increase in rigidity reduces the time constant T (). The error of the cacuated characteristics with respect to the experimenta data is 5%. A sectiona mutiayer piezo motor characterized by a ongitudina piezo effect and coded contro of the sections may be used in the automatic contro system of nano- and micromanipuators with digita anaog conversion [6]. The mutiayer piezo motor is divided into sections, with n k piezo pates in section k. The sections are mechanicay connected in series and insuated. Correspondingy, the piezo pates (piezo ayers) in each section are eectricay connected in parae but mechanicay connected in series. Consider coded contro with eectromechanica digita anaog conversion for piezoeectric mutiayer sectiona piezo motors with ongitudina and transverse piezo effects that produce nano- and microdispacement proportiona to the contro code. We wi simuate the static and dynamic characteristics of a mutiayer sectiona piezo motor as an eectromechanica system with point parameters, at ow resistance (R ). With a ongitudina piezo effect, we find that the number of piezo pates in the section n k k ; the ength of section k is k k, where,,, ; is the number of sections. The tota ength of the mutiayer sectiona piezo motor is (Fig. 8a) k ( ). In static conditions, the maximum dispacement of the mutiayer sectiona piezo motor with a ongitudina piezo effect is Δ max d ( ) d n, where n k is the number of piezo pates in the sectiona piezo motor. The corresponding dispacement of this motor when binary code is suppied to the input takes the form Δ a k Δ k, where a k, are the vaues of the binary code. RSSIA GIRIG RSARCH Vo. o.

6 7 AFOI Δ, μm Hence We now consider the mechanica and contro characteristics of mutiayer piezo motors with a ongitudina piezo effect in the case of coded contro. In this case, the tota deformation of the mutiayer piezo motor consists of the deformation of the individua sections of the piezo pates when votage is appied and the deformation of the whoe mutiayer piezo motor under the action of externa force. From q. (), we obtain the static mechanica characteristic for a mutiayer piezo motor with a ongitudina piezo effect, in the case of coded votage contro (Fig. 9) where C S /( ). quivaenty (7) where Δ max d a k k ; F max d a k k S /( s ) or Δ a k d k d a k k Δ d a k k s F/S d a k k F/C, s Δ Δ max ( F/F max ),.8. F, k Fig. 9. Mechanica characteristics of sectiona mutiayer piezo motor with ongitudina piezo effect in the case of coded contro. Δ max d n S ; F max d n S S /( s ). C a + C e Δ C a + C e a a C a Here n S a k is the number of piezo ayers k of the mutiayer piezo motor connected to the votage source. The mechanica characteristics in Fig. 9 correspond to a mutiayer sectiona piezo motor with a ongitudina piezo effect: piezomoduus d 4 m/v; suppy votage V; eastic piabiity s 4 m /; thickness of piezo pates.6 m; diameter of piezo pates D 5 m; number of piezo pates n 5; number of sections 4. The motor is characterized by coded contro, with a, a, a, a 4 (); a, a, a, a 4 (); and a, a, a, a 4 (). From q. (5), we obtain the contro characteristic describing the static dispacement of a mutiayer sectiona piezo motor in the case of a ongitudina piezo effect and an eastic oad (Fig. a) Δ d a k k ( C a + C e )/C (8) A sectiona piezo motor (Fig. 8b) with a transverse piezo effect may take the form of a monoithic piezo motor with separate sectiona eectrodes or a mutiayer sectiona piezo motor with sections of ength k k, where,,,. The ength of the sectiona piezo motor with a transverse piezo effect is k Fig.. Kinematic diagrams of reinforced sectiona mutiayer nano- and micro-scae piezo motors with ongitudina piezo effect and transverse piezo effect, in the case of coded contro and an eastic oad. ( ). Δ a a C a RSSIA GIRIG RSARCH Vo. o.

7 MCHAICAL AD COTROL CHARACTRISTICS 7 In statics, the dispacement of the sectiona piezo motor with a transverse piezo effect when votage is suppied to section k takes the form (Fig. 8b) Δ k d k /. The maximum static dispacement of the sectiona piezo motor with a transverse piezo effect is Δ max d ( ) /. The static dispacement of the sectiona piezo motor with a transverse piezo effect when binary code is suppied to the input is Δ ( d /) a k k. We now consider the mechanica and contro characteristics of mutiayer piezo motors with a transverse piezo effect, in the case of coded contro. In that case, the tota deformation of the mutiayer sectiona piezo motor is equa to the sum of the deformation of the individua sections with votage suppy and the tota deformation under the externa force. From q. (4), we obtain the static mechanica characteristic for a mutiayer piezo motor with a transverse piezo effect, in the case of coded votage contro Δ ( d /) a k k s F/S where C S /( ). quivaenty ( d /) a k k F/C, s Δ Δ max ( F/F max ), (9) where Δ max (d /)( a k k ) ; F max (d /)( a k k ) S /( s ). From q. (6), we obtain the static contro characteristic for the dispacement of a sectiona piezo motor with a transverse piezo effect under the action of an eastic oad (Fig. b) ( d /) a k k Δ () + ( C a + C e )/C. The static dispacement of a mutiayer piezo motor with ongitudina or transverse piezo effects under an eastic oad takes the form Δ k c, () where k c is the gain. With parae contro according to qs. (5) and (6) k c d n ( C a + C e )/C with a ongitudina piezo effect; d / ( C a + C e )/C with a transverse piezo effect. With coded contro according to qs. (8) and () k c d a k k ( C a + C e )/C with a ongitudina piezo effect; ( d /) a k k ( C a + C e )/C with a transverse piezo effect. We now consider the dynamic characteristics of a mutiayer piezo motor, regarded as an eectromechanica system with distributed or point parameters. The dynamic characteristics of the mutiayer piezo motor are cacuated by soution of the wave equation and the equation of the ongitudina piezo effect, with zero initia conditions and the corresponding boundary conditions. The cacuation of the piezo motor is based on a wave equation describing the propagation of a wave in a ong ine with damping but without distortion [ 6] ξ α ξ α ξ ( c ) t c t where ξ(x, t) is the dispacement of the piezo motor s cross section; x is the coordinate; t is the time; c is the veocity of sound when const; α is the damping coefficient. With one attached end of the piezo motor (say, at x ), we obtain ξ(x, t). With an eastic inertia oad at x, we obtain an equation for the forces at the motor s other end T S where M is the mass being dispaced. By Lapace transformation ξ, x M ξ ( C t a + C e )ξ, Ξ( x, p) L{ ξ( x, t) } ξ( x, t)e pt dt, RSSIA GIRIG RSARCH Vo. o.

8 7 AFOI we obtain the transfer function W(p) of a mutiayer piezo motor with a ongitudina piezo effect, parae contro, and an eastic inertia oad, regarded as an eectromechanica system with distributed parameters ( p) d n , Mp /C + coth[ ( p/c + α) ] ( p/c + α) +( C a + C e )/C where Ξ(p) is the Lapace transform of the dispacement of the end of the motor aong axis ; (p) is the Lapace transform of the votage at the pates of the mutiayer piezo motor, with zero initia conditions. Hence, we obtain the static dispacement Δ ξ( ) of a reinforced mutiayer piezo motor with a ongitudina piezo effect and parae contro, in steady conditions, under the action of a votage (t) (t) and an eastic inertia oad ξ ( ) im ξ() t t p im ---- p p d n im p ( p/c + α)/ tanh[ ( p/c α + α) ] + ( C a + C e )/C d n ( C a + C e )/C Then, after series expansion in terms of hyperboic cotangents (retaining the first two terms), we may write the transfer function W(p) of a mutiayer piezo motor with a ongitudina piezo effect, parae contro, and an eastic inertia oad, at working frequencies in the range < ω <.c /, when m M (m is the mass of the mutiayer piezo motor) in the foowing form, if we regard the motor as an eectromechanica system with point parameters ( p) d n [ + ( C a + C e )/C ]( T p + T ξ p + ) M Here T and ξ C a + C e + C α C are, respectivey, the time constant and damping coefficient of the osciatory com- c M( C a + C e + C ) ponent in a mutiayer piezo motor with a ongitudina piezo effect and an eastic inertia oad. Correspondingy, if we take account of the resistance R of the externa circuit and the capacitance C of the piezo pate in the mutiayer sectiona piezo motor, we obtain its transfer function in the case of a ongitudina piezo effect ( p) d n [ + ( C a + C e )/C ]( RnC p + ) ( T p + T ξ p + ) The transfer function W(p) of a mutiayer piezo motor with a transverse piezo effect, parae contro, and an eastic inertia oad at working frequencies in the range < ω <.c /, when m M (m is the mass of the mutiayer piezo motor) may be written in the foowing form, if we regard the motor as an eectromechanica system with point parameters ( p) d / [ + ( C a + C e )/C ]( T p + T ξ p + ) Here T M and ξ C a + C e + C α C are, respectivey, the time constant and damping coefficient of the osciatory c M( C a + C e + C ) component in a mutiayer piezo motor with a transverse piezo effect and an eastic inertia oad. Correspondingy, the transfer function W(p) of a sectiona mutiayer piezo motor with a ongitudina piezo effect, coded contro, and an eastic inertia oad, when m M may be written in the foowing form, if we regard the motor as an eectromechanica system with point parameters [7, 8] ( p) d a k k , [ + ( C a + C e )/C ]( T p + T ξ p + ) where C a, C e are the rigidity vaues of the reinforcing eement and the oad; C S /( s ) is the rigidity of the sectiona mutiayer piezo motor with a ongitudi- RSSIA GIRIG RSARCH Vo. o.

9 MCHAICAL AD COTROL CHARACTRISTICS 7 na piezo effect; and T M , ξ C a + C e + C α C are, respectivey, the time constant and damping coefficient of the osciatory com- c M( C a + C e + C ) ponent in this case. The transfer function W(p) of a sectiona mutiayer piezo motor with a ongitudina piezo effect, coded contro, and an eastic inertia oad, when m M, takes the form where C S /( s ) is the rigidity of this piezo motor with a transverse piezo effect. Taking account of q. (), we may write the transient characteristic of a mutiayer piezo motor with an eastic inertia oad, regarded as an eectromechanica system with point parameters, in the foowing form, for sma resistance (R ) where h(t) is the normaized transient characteristic of the mutiayer piezo motor; ξ is its steady dispacement. With parae contro With coded contro ( p) d / a k k , [ + ( C a + C e )/C ]( T p + T ξ p + ) ξ() t k c h() t ξ h(), t ξ d n ( C a + C e )/C with a ongitudina piezo effect; d ( /) ( C a + C e )/C with a transverse piezo effect. ξ d a k k ( C a + C e )/C with a ongitudina piezo effect; ( d /) a k k ( C a + C e )/C with a transverse piezo effect; ξ, μm 4 The standardized transient characteristic of the mutiayer piezo motor takes the form ξ ( ) T ( ) h() t e sin( ω ( ) t + ϕ ( ) ), ξ ( ) M where T () , ξ () C e + C ( ) α C ; ( ) c M( C e + C ( ) ) ξ ( ) arctan ξ ( ) 4 t, s Fig.. Dynamic characteristics of a mutiayer piezo motor with a ongitudina piezo effect in the case of parae contro and an eastic inertia oad. ξ ω () ; ( ) ϕ () Here subscript corresponds to a ongitudina piezo effect and subscript to a transverse piezo effect. Correspondingy, for a mutiayer piezo motor made of TsTS-9 piezoceramic, with a parae contro and a stepped input function (ampitude 5 V), we obtain ξ μm, ξ., and T.9 ms and the transient characteristic in Fig.. The mechanica and contro characteristics obtained here for mutiayer piezo motors with parae and coded votage contro permit the cacuation of the static and dynamic operating conditions of simpe and mutiayer piezo motors as a function of the externa oad in nano- and micromanipuators and their physica and geometric parameters. T ( ) RSSIA GIRIG RSARCH Vo. o.

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