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Analysis of the displacement amplification ratio of bridge-type flexure hinge
Sensors and Actuators A 132 (2006) 730–736
Analysis of the displacement amplification ratio of bridge-type flexure hinge Hong-Wen Ma a,∗ , Shao-Ming Yao b , Li-Quan Wang a , Zhi Zhong b a b
Harbin Engineering University, China Harbin Institute of Technology, China
Received 5 August 2005; received in revised form 28 November 2005; accepted 6 December 2005 Available online 22 May 2006
Abstract The bridge-type flexure hinge is a classic displacement amplification mechanism. The existing models of theoretic displacement amplification ratio of bridge-type flexure hinges are not perfect. This makes it very difficult to design and manufacture a satisfactory structure using these models. Kinematic theory was used to analyze the ideal displacement amplification ratio of a bridge-type flexure hinge in this paper, and the flexure hinge was regarded as a pure multi-rigid body with ideal pivots. Elastic beam theory was used to analyze the theoretic displacement amplification ratio when considering the translational and rotational stiffness of the flexure pivots. The model of theoretic displacement amplification ratio explains why the bridge-type displacement amplification mechanism has an amplification ratio extremum and where the threshold is. The finite element method was used for comparison with the mathematical model, and similar results were obtained. Finally, the finite element method was used to analyze the shape mode of the structure. The result showed that increasing the amplification ratio by decreasing the thickness of the flexure pivots led to a decrease in the mode shape frequency of the bridge-type structure. Thus, redesigning of the structure was needed to solve the problem. © 2005 Elsevier B.V. All rights reserved. Keywords: Flexure hinge; Bridge-type; Piezostack; Displacement amplification ratio; Finite element method (FEM)
1. Introduction A flexure hinge is used to amplify the displacement of a piezostack in many applications [1] because of the limited deformation range of piezostacks, which is about 10 m/cm. A levertype mechanism is a kind of classic flexure hinge amplification mechanism, but it usually has large dimensions. The Moonie-type flexure hinge was developed in 1992 [1]. The output displacement of its actuator greatly increases with cavity diameter and depth [2,3]. The rainbow-type flexure hinge is another type of amplification mechanism used with piezostacks. Its output displacement can be more than 1 mm [4–6]. The cymbal-type flexure hinge was developed in 1996 [1]. Its output displacement increases through a combination of flexural and rotational motions [7–10],
∗
Corresponding author. Tel.: +86 13936164261 E-mail address: [email protected] (H.-W. Ma).
0924-4247/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.sna.2005.12.028
and this can also be more than 1 mm. Rainbow-type and cymbaltype flexure hinges produce a high output displacement using thin beam structure, but this structure will lead to an unstable state in a high-frequency mode. The bridge-type flexure hinge amplification mechanism developed recently has a compact structure and a large displacement amplification ratio [11–15]. It can also be used in a high-frequency mode. Pokines [11] designed a bridge-type microamplification mechanism and fabricated it using LIGA. Its displacement amplification ratio was 5.48. Kim et al. [12,13] designed a 3-D bridge-type hinge mechanism to amplify the output of a piezostack’s displacement that was equivalent to a grade 2 bridge-type flexure hinge. He did not analyze the actual displacement amplification ratio of the bridge-type flexure hinge. Although he used a grade 2 amplifying mechanism [12] and optimized the structure [13], the actual displacement amplification ratio measured experimentally was hardly more than 25.
H.-W. Ma et al. / Sensors and Actuators A 132 (2006) 730–736
Shih et al. [16] derived the displacement amplification ratio of rainbow-type and Moonie-type flexure hinges using elastic beam theory. Lobontiu and Garcia [14] derived the ideal displacement amplification ratio of the bridge-type mechanism using geometric relations, but the formula is not accurate because no kinematic analysis was used. Accordingly, he derived the input and output stiffness ratio using geometric relations. Unfortunately, the displacement amplification ratio cannot be derived accurately using this method. The ideal displacement amplification ratio of bridge-type mechanism is derived using kinematic theory in this paper, and a theoretic displacement amplification ratio is derived using elastic beam theory in which the elastic deformation of the flexure pivots is considered. The model of theoretic displacement amplification ratio explains why the bridge-type mechanism has an amplification ratio extremum and where the threshold is. Finally, a finite element model is used for comparison with the mathematical model, also for the analysis of the mode shape of the structure.
2. Ideal displacement amplification ratio of bridge-type flexure hinge Fig. 1 shows five different kinds of flexure hinge amplification mechanisms: the lever-type, bridge-type, Moonie-type, rainbow-type, and cymbal-type. The bridge-type flexure hinge mechanism has both a larger displacement output and a higher frequency mode that is relatively easy to design in a compact and symmetric structure. Fig. 2 shows the ideal multi-rigid body schematic of the bridge-type mechanism, with all flexure pivots considered as ideal pivots. Only one flexure hinge arm of the mechanical model needs to be analyzed because of the symmetrical structure. Fig. 3 shows the kinematic model of arm AB. Pokines [11] derived the ideal displacement amplification ratio of the bridge-type flexure hinge using geometric relations: Ramp
sin α − sin(α − ∂α) . = cos α − cos(α − ∂α)
Fig. 1. Topology of flexure hinge amplification mechanisms.
(1)
731
Fig. 2. Schematic of bridge-type amplification mechanism with ideal pivots.
Lobontiu and Garcia [14] also derived the ideal displacement amplification ratio using geometric relations: la 2 sin2 α + i(2la cos α − ∂x)−la sin α Ramp = . (2) ∂x Eqs. (1) and (2) are not the most simplified equations because kinematic analysis has not been incorporated. The displacement amplification ratio was derived using kinematic theory in this paper. The velocities of points A and B are ⎧ ∂x ⎪ ⎨ vA = ∂t . (3) ⎪ ⎩ vB = ∂y ∂t The displacement amplification ratio is Ramp =
∂y ∂y/∂t vB = . = ∂x ∂x/∂t vA
(4)
The instantaneous center of rotation of the rigid body AB is point O, and the instantaneous speed of rotation of the rigid body is ω; the instantaneous velocities of points A and B can be expressed as vA = ω · lx . (5) vB = ω · l y From Eqs. (4) and (5), the displacement amplification ratio can be written as Ramp =
ω · ly ly = = cot α. ω · lx lx
Fig. 3. Quarter kinematic model of bridge-type amplification mechanism.
(6)
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Fig. 5. Single arm of bridge-type flexure hinge.
Fig. 6. Schematic of the mechanics of single arm of bridge-type flexure hinge.
Fig. 4. Structure of bridge-type flexure hinge.
From the above analysis, it is seen that the ideal displacement amplification ratio only depends on the angle α. In fact, if ∂α → 0 in Eq. (1) or ∂x → 0 in Eq. (2), they can be both reduced to Eq. (6) using kinematic analysis. 3. Actual bridge-type flexure hinge model based on elastic beam theory The structure of the bridge-type flexure hinge is shown in Fig. 4. The flexure hinge mechanism consists of rigid arms and flexure pivots. Corner-filleted flexure pivots are used in this amplification mechanism because they are more flexible than right-circular flexure pivots of the same size. Table 1 lists the key geometrical parameters of the structure. The structure of the bridge-type flexure hinge is symmetrical, and so only one bridge arm is needed to establish the mathematical model. Fig. 5 shows one bridge arm. The angle between line AB, passing through the pivots of the arm, and the horizontal line is α. Each of the flexure pivots can be simplified to have two types of stiffness: translational and rotational. According to kinematic analysis, the static mechanism model of the flexure hinge arm AB in Fig. 3 can be simplified as shown in Fig. 6. Horizontal
forces FA and FB are imposed on the pivots A and B. A moment 2Mα is imposed on arm AB. The bridge-type mechanism in Fig. 4 can be simplified as shown in the top structure in Fig. 7: bar AA1 and CC1 have only one degree of freedom along the x axis, and bars BB1 and DD1 have only one degree of freedom along the y axis because of the structural limitation, and so bars AA1 , BB1 , CC1 , and DD1 can all be simplified as single points A, B, C, and D. The whole bridge-type mechanism can be simplified as shown in the bottom structure in Fig. 7. The mechanical model of bridge-type mechanics can be derived by replacing the arm in Fig. 7 with that in Fig. 6. The model is shown in Fig. 8. Pivot B has only two symmetric forces imposed by two bars, and so the two symmetric forces must be horizontal balance forces. Point B on bar AB has the inverse force of pivot B. Bar AB also has two moments and a force at point A provided by pivot B, and the bar is in a state of balance, and so the value of FA must be equal to that of FB . FA and FB constitute a couple (FA = FB = F): 2Mα = Fla sin α.
Table 1 Key geometrical parameters of bridge-type flexure hinge la (mm) lx (mm) ly (mm) t (mm) b (mm)
2 0.9 13.0 0.4 6.0 Fig. 7. Simplification of bridge-type flexure hinge.
(7)
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Fig. 8. Schematic of the mechanics of bridge-type flexure hinge.
At point A on bar AB Fl = F cos α.
(8)
Then: 2Mα Fla sin α 2Kα α = = , Fl F cos α Kl l Kl la sin α α = , l 2Kα cos α
(10)
Fx = Fl l + 2Mα = F cos α · l + Fla sin α · α, (11) x = cos α · l + la sin α · α.
(12)
From Eqs. (10) and (12)
2Kα cos α x = cos α + la sin α α, Kl la sin α
(13)
and the theoretic displacement amplification ratio is Ramp
la cos α
cos α ·
2Kα cos α Kl la sin α
+ la sin α
According to elastic beam theory, each of the pivots is not an ideal pivot, and it has a compliance matrix (its inverse matrix is the stiffness matrix) [12,17]. Assuming a flexure pivot to be the six-degrees-of-freedom spring element shown in Fig. 9, the hinge compliance equation is as follows: Xh = C h F h ,
(15)
where
y la cos α · α = = 2Kα cos α x cos α Kl la sin α + la sin α Δα =
Fig. 9. Coordinate system of corner-filleted flexure pivots.
(9)
.
(14)
T X = x, y, z, α, β, γ ,
(16)
T
F = Fx , Fy , Fz , Mx , My , Mz .
(17)
In Eq. (15), the superscript h represents the pivot coordinate system, and the compliance matrix of the right-angle hinge is derived from the beam theory as follows.
⎤
⎡
l ⎢ Ebt ⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎢ ⎢ ⎢ 0 ⎢ h C =⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎣ 0
0 l + 3 3Ebt Gbt
0
4l3
0
0
0
0
4l3 l + 3Eb3 t Gbt
0
0
0
0 6l2 Ebt 3
−
6l2 Eb3 t 0
l Gk2 bt 3
0 0 −
6l2 Eb3 t 0
0
12l Eb3 t
0
0
0
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥. ⎥ 0 ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ 12l ⎦ Ebt 3 6l2 Ebt 3
(18)
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Fig. 10. Single arm of bridge-type flexure hinge with flexure pivots parallel with the axis passing through two pivots.
E is the elastic modulus, G the shear modulus of the hinge material, and k2 is a geometrical constant determined by b/t [12,17]. For a bridge-type mechanism, only the plan stiffness needs to be considered:
T X = x, y, γ , (19)
T F = Fx , Fy , Mz ,
(20) ⎤
⎡
l ⎢ Ebt ⎢ ⎢ Ch = ⎢ ⎢ 0 ⎢ ⎣ 0
0 l + 3Ebt 3 Gbt 6l2 Ebt 3 4l3
0
⎥ ⎥ ⎥ ⎥. 3 Ebt ⎥ ⎥ 12l ⎦ Ebt 3 6l2
(21)
Where the angle α in Fig. 5 is smaller or the structure in Fig. 10 is adopted, the stiffness of Kl and Kα can be derived from Eq. (21):
Kh = Ch
−1
,
(22)
⎧ Ebt ⎪ ⎪ ⎨ Kl ≈ l 3 . ⎪ Ebt ⎪ ⎩ Kα ≈ 12l
(23)
From Eqs. (14) and (23), the theoretic displacement amplification ratio when taking into account the elastic stiffness of the corner-filleted flexure pivot is Ramp =
la cos α t2
α cos α 6la cos sin α + la sin α
.
Fig. 11. Restraint and load of bridge-type flexure hinge and results of the FEM analysis.
The displacement amplification ratio obtained using the FEM is Ramp =
O . I1 + I 2
Fig. 12 shows the relationship between the displacement amplification ratio and angle α (in Fig. 2) using different equations (Eqs. (6), (24), and (25)). It is seen that with decreasing angle α, the ideal displacement amplification ratio becomes infinite. However, the theoretic displacement amplification ratio has a maximum value and α reaches a threshold value. Whereas α decreases continuously, the displacement amplification ratio decreases sharply. The threshold angle, αmax , at the maximum
(24)
It is obvious that the theoretic displacement amplification ratio is related to the thickness, t (in Fig. 9), when using a cornerfilleted flexure pivot. 4. Comparison with finite element model The finite element method (FEM) is used to analyze the static mechanics of the bridge-type mechanism, as shown in Fig. 11. Surface D is fixed. There is a displacement output on surface C when imposing a force on surfaces A and B. The results of FEM analysis are shown in Fig. 11 using Cosmos DesignStar. A tetrahedral element is used in the FEM model, and a Newton–Simpson solver is used.
(25)
Fig. 12. Displacement amplification ratio vs. angle α.
H.-W. Ma et al. / Sensors and Actuators A 132 (2006) 730–736
735
displacement amplification ratio can be derived from Eq. (14) or Eq. (24):
5. Frequency analysis of bridge-type flexure hinge
αmax = α dRamp . dα =0
The bridge-type mechanism is often used in high-frequency scanning mechanisms, and so the self-vibration frequency of the bridge-type mechanism must be analyzed. An FEM model is established to analyze its mode shape. Fig. 13 shows the first–fourth order mode shape of the bridge-type mechanism. The frequency value of the first–fifth order mode shape is shown in Table 2. It is seen in Eq. (24) that when the thickness, t, is decreased, the displacement amplification ratio increases but the first order mode shape frequency decreases quickly.
(26)
The FEM results indicate that the actual displacement amplification ratio is smaller than the theoretic value. This is mainly because the rigid arm is not an absolute rigid component and which also has and undergoes elastic deformation. When the angle α is small enough, the deformation of the arm cannot be neglected.
Fig. 13. Mode shape analysis of bridge-type flexure hinge without load.
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H.-W. Ma et al. / Sensors and Actuators A 132 (2006) 730–736
Table 2 Frequency of mode shape analysis without load t (mm)
Mode shape 1 (Hz)
Mode shape 2 (Hz)
Mode shape 3 (Hz)
Mode shape 4 (Hz)
Mode shape 5 (Hz)
0.4 0.2
201 75
409 154
602 259
711 308
1383 531
6. Conclusions The ideal displacement amplification ratio and the theoretic displacement amplification ratio are analyzed based on kinematic and elastic mechanisms. The FEM is used for a comparison with the mathematical model and for the analysis of the mode shape of the structure. From the above analysis, we arrived at some conclusions: (1) The ideal displacement amplification ratio becomes infinite when the angle between the bridge arm and the catercorner line goes to zero when the bridge-type mechanism is considered as a multi-rigid body with ideal pivots. (2) The theoretic displacement amplification ratio has an extremum when deformation at the flexure pivots is taken into account. The threshold of the extremum is related to the ratio between the translational stiffness and the rotational stiffness. (3) The actual displacement amplification ratio decreases compared with the theoretic displacement amplification ratio when the other deformation of the primary “rigid body” is considered. (4) A decrease in the thickness of the flexure pivots increases the displacement amplification ratio, but the mode shape frequency decreases. References [1] C. Niezrecki, D. Brei, S. Balakrishnan, A. Moskalik, Piezoelectric actuation: state of the art, Shock Bib. Digest. 33 (4) (2001) 269–280. [2] R.E. Newnham, A. Dogan, Q.C. Xu, Flextensional “Moonie” actuators, in: IEEE Ultrasonics Symposium, 1993, pp. 509–513. [3] F. Lalande, Z. Chaudhry, C.A. Rogers, A simplified geometrically nonlinear approach to the analysis of the Moonie actuator, IEEE Trans. Ultrasonics Ferroelectrics Freq. Control 42 (1995) 21–27. [4] G.H. Haertling, G.C. Robinson, Compositional study of PLZT rainbow ceramics for piezo actuators, in: Proceedings of the Ninth IEEE International Symposium on Applications of Ferroelectrics, 1994, pp. 313– 318. [5] J. Juuti, E. Heinonen, V.-P. Moilanen, Displacement, stiffness and load behaviour of laser-cut RAINBOW actuators, J. Eur. Ceram. Soc. 24 (2004) 1901–1904. [6] S. Chandran, V.D. Kugel, L.E. Cross, Characterization of the linear and nonlinear dynamic performance of RAINBOW actuator, in: Proceedings of the 10th IEEE International Symposium on Applications of Ferroelectrics, 1996, pp. 743–746. [7] A. Dogan, K. Uchino, Composite piezoelectric transducer with truncated conical endcaps “Cymbal”, IEEE Trans. Ultrasonics Ferroelectrics Freq. Control 44 (1997) 597–605. [8] A. Dogan, J.F. Fernandez, K. Uchino, The “CYMBAL” electromechanical actuator, in: IEEE International Symposium on Applications of Ferroelectrics, 1996, pp. 213–216.
[9] R.E. Newnham, J. Zhang, R.J. Meyer, Cymbal transducers: a review, in: Proceedings of the 12(th) IEEE International Symposium on Applications of Ferroelectrics, 2001, pp. 29–32. [10] Y. Ke, T. Guo, J. Li, A new-style, slotted-cymbal transducer with large displacement and high energy transmission, IEEE Trans. Ultrasonics Ferroelectrics Freq. Control (2004) 1171–1177. [11] B.J. Pokines, E. Garcis, A smart material microamplification mechanism fabricated using LIGA, Smart Mater. Struct. 7 (1998) 105–112. [12] J.H. Kim, S.H. Kim, Y.K. Kwak, Development of a piezoelectric actuator using a three-dimensional bridge-type hinge mechanism, Rev. Sci. Instrum. 74 (2003) 933–956. [13] J.H. Kim, S.H. Kim, Y.K. Kwak, Development and optimization of 3-D bridge-type hinge mechanisms, Sens. Actuators A 116 (2004) 530–538. [14] N. Lobontiu, E. Garcia, Analytical model of displacement amplification and stiffness optimization for a class of flexure-based compliant mechanisms, Comput. Struct. 81 (2003) 2797–2810. [15] J. Juuti, K. Kord´as, R. Lonnakko, V.-P. Moilanen, S. Lepp¨avuori, Mechanically amplified large displacement piezoelectric actuators, Sens. Actuators A 120 (2005) 225–231. [16] W.Y. Shih, W.-H. Shih, I.A. Aksay, Scaling analysis for the axial displacement and pressure of flextensional transducers, J. Am. Ceram. Soc. 80 (1997) 1073–1078. [17] Y. Koseki, T. Tanikawa, T. Arai, N. Koyachi, Kinematic analysis of translational 3-DOF micro parallel mechanism using matrix method, in: Proceedings of the 2000 IEEE/RSJ International Conference on Intelligent Robots and Systems, 2000, pp. 786–792.
Biographies Hong-Wen Ma received the BSc degree in automobile engineering in 1995 from the Jilin university of technology (JUT), China, and the BSc degree in automation engineering in 1999, the PhD degree in instrument and instrumentation in 2005 all from the Harbin institute of technology (HIT), China, where his research focused on precision positioning technique. From 2005, he has been employed by the Harbin engineering university (HEU) as associate professor. His current interests include legged robot and machine vision. Shao-Ming Yao received MS in mechanical engineering from Ship Building and Engineering Institute, in 1992. He is currently pursuing PhD in instrument science at Harbin Institute of Technology (HIT). From 1999 to 2003, he served as senior engineer in Ship Industry Group Company, where his research focused on power transmission gear. From 2003, he has been employed by HIT as associate professor in Instrument Science Department. His current research focused on opto-electromechanical integration technique and ultra-precision motion reference technique. Li-Quan Wang was born in Herbei, China in 1957. He received the PhD in control theory and control engineering from the Harbin engineering university (HEU), China. He is employed by the HEU as professor currently. His current interests include bionics, underwater robot and underwater construction equipment. Zhi Zhong was born in 1976. He received the PhD in optoelectronic information technology and instrumentation engineering, from Department of Automatic Measurement and Control Engineering, Harbin Institute of Technology, China in 2005. His research interests are currently in the field of fast ultra-precision laser heterodyne interferometric and signal processing.
Analysis of the displacement amplification ratio of bridge-type flexure hinge Hong-Wen Ma a,∗ , Shao-Ming Yao b , Li-Quan Wang a , Zhi Zhong b a b
Harbin Engineering University, China Harbin Institute of Technology, China
Received 5 August 2005; received in revised form 28 November 2005; accepted 6 December 2005 Available online 22 May 2006
Abstract The bridge-type flexure hinge is a classic displacement amplification mechanism. The existing models of theoretic displacement amplification ratio of bridge-type flexure hinges are not perfect. This makes it very difficult to design and manufacture a satisfactory structure using these models. Kinematic theory was used to analyze the ideal displacement amplification ratio of a bridge-type flexure hinge in this paper, and the flexure hinge was regarded as a pure multi-rigid body with ideal pivots. Elastic beam theory was used to analyze the theoretic displacement amplification ratio when considering the translational and rotational stiffness of the flexure pivots. The model of theoretic displacement amplification ratio explains why the bridge-type displacement amplification mechanism has an amplification ratio extremum and where the threshold is. The finite element method was used for comparison with the mathematical model, and similar results were obtained. Finally, the finite element method was used to analyze the shape mode of the structure. The result showed that increasing the amplification ratio by decreasing the thickness of the flexure pivots led to a decrease in the mode shape frequency of the bridge-type structure. Thus, redesigning of the structure was needed to solve the problem. © 2005 Elsevier B.V. All rights reserved. Keywords: Flexure hinge; Bridge-type; Piezostack; Displacement amplification ratio; Finite element method (FEM)
1. Introduction A flexure hinge is used to amplify the displacement of a piezostack in many applications [1] because of the limited deformation range of piezostacks, which is about 10 m/cm. A levertype mechanism is a kind of classic flexure hinge amplification mechanism, but it usually has large dimensions. The Moonie-type flexure hinge was developed in 1992 [1]. The output displacement of its actuator greatly increases with cavity diameter and depth [2,3]. The rainbow-type flexure hinge is another type of amplification mechanism used with piezostacks. Its output displacement can be more than 1 mm [4–6]. The cymbal-type flexure hinge was developed in 1996 [1]. Its output displacement increases through a combination of flexural and rotational motions [7–10],
∗
Corresponding author. Tel.: +86 13936164261 E-mail address: [email protected] (H.-W. Ma).
0924-4247/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.sna.2005.12.028
and this can also be more than 1 mm. Rainbow-type and cymbaltype flexure hinges produce a high output displacement using thin beam structure, but this structure will lead to an unstable state in a high-frequency mode. The bridge-type flexure hinge amplification mechanism developed recently has a compact structure and a large displacement amplification ratio [11–15]. It can also be used in a high-frequency mode. Pokines [11] designed a bridge-type microamplification mechanism and fabricated it using LIGA. Its displacement amplification ratio was 5.48. Kim et al. [12,13] designed a 3-D bridge-type hinge mechanism to amplify the output of a piezostack’s displacement that was equivalent to a grade 2 bridge-type flexure hinge. He did not analyze the actual displacement amplification ratio of the bridge-type flexure hinge. Although he used a grade 2 amplifying mechanism [12] and optimized the structure [13], the actual displacement amplification ratio measured experimentally was hardly more than 25.
H.-W. Ma et al. / Sensors and Actuators A 132 (2006) 730–736
Shih et al. [16] derived the displacement amplification ratio of rainbow-type and Moonie-type flexure hinges using elastic beam theory. Lobontiu and Garcia [14] derived the ideal displacement amplification ratio of the bridge-type mechanism using geometric relations, but the formula is not accurate because no kinematic analysis was used. Accordingly, he derived the input and output stiffness ratio using geometric relations. Unfortunately, the displacement amplification ratio cannot be derived accurately using this method. The ideal displacement amplification ratio of bridge-type mechanism is derived using kinematic theory in this paper, and a theoretic displacement amplification ratio is derived using elastic beam theory in which the elastic deformation of the flexure pivots is considered. The model of theoretic displacement amplification ratio explains why the bridge-type mechanism has an amplification ratio extremum and where the threshold is. Finally, a finite element model is used for comparison with the mathematical model, also for the analysis of the mode shape of the structure.
2. Ideal displacement amplification ratio of bridge-type flexure hinge Fig. 1 shows five different kinds of flexure hinge amplification mechanisms: the lever-type, bridge-type, Moonie-type, rainbow-type, and cymbal-type. The bridge-type flexure hinge mechanism has both a larger displacement output and a higher frequency mode that is relatively easy to design in a compact and symmetric structure. Fig. 2 shows the ideal multi-rigid body schematic of the bridge-type mechanism, with all flexure pivots considered as ideal pivots. Only one flexure hinge arm of the mechanical model needs to be analyzed because of the symmetrical structure. Fig. 3 shows the kinematic model of arm AB. Pokines [11] derived the ideal displacement amplification ratio of the bridge-type flexure hinge using geometric relations: Ramp
sin α − sin(α − ∂α) . = cos α − cos(α − ∂α)
Fig. 1. Topology of flexure hinge amplification mechanisms.
(1)
731
Fig. 2. Schematic of bridge-type amplification mechanism with ideal pivots.
Lobontiu and Garcia [14] also derived the ideal displacement amplification ratio using geometric relations: la 2 sin2 α + i(2la cos α − ∂x)−la sin α Ramp = . (2) ∂x Eqs. (1) and (2) are not the most simplified equations because kinematic analysis has not been incorporated. The displacement amplification ratio was derived using kinematic theory in this paper. The velocities of points A and B are ⎧ ∂x ⎪ ⎨ vA = ∂t . (3) ⎪ ⎩ vB = ∂y ∂t The displacement amplification ratio is Ramp =
∂y ∂y/∂t vB = . = ∂x ∂x/∂t vA
(4)
The instantaneous center of rotation of the rigid body AB is point O, and the instantaneous speed of rotation of the rigid body is ω; the instantaneous velocities of points A and B can be expressed as vA = ω · lx . (5) vB = ω · l y From Eqs. (4) and (5), the displacement amplification ratio can be written as Ramp =
ω · ly ly = = cot α. ω · lx lx
Fig. 3. Quarter kinematic model of bridge-type amplification mechanism.
(6)
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Fig. 5. Single arm of bridge-type flexure hinge.
Fig. 6. Schematic of the mechanics of single arm of bridge-type flexure hinge.
Fig. 4. Structure of bridge-type flexure hinge.
From the above analysis, it is seen that the ideal displacement amplification ratio only depends on the angle α. In fact, if ∂α → 0 in Eq. (1) or ∂x → 0 in Eq. (2), they can be both reduced to Eq. (6) using kinematic analysis. 3. Actual bridge-type flexure hinge model based on elastic beam theory The structure of the bridge-type flexure hinge is shown in Fig. 4. The flexure hinge mechanism consists of rigid arms and flexure pivots. Corner-filleted flexure pivots are used in this amplification mechanism because they are more flexible than right-circular flexure pivots of the same size. Table 1 lists the key geometrical parameters of the structure. The structure of the bridge-type flexure hinge is symmetrical, and so only one bridge arm is needed to establish the mathematical model. Fig. 5 shows one bridge arm. The angle between line AB, passing through the pivots of the arm, and the horizontal line is α. Each of the flexure pivots can be simplified to have two types of stiffness: translational and rotational. According to kinematic analysis, the static mechanism model of the flexure hinge arm AB in Fig. 3 can be simplified as shown in Fig. 6. Horizontal
forces FA and FB are imposed on the pivots A and B. A moment 2Mα is imposed on arm AB. The bridge-type mechanism in Fig. 4 can be simplified as shown in the top structure in Fig. 7: bar AA1 and CC1 have only one degree of freedom along the x axis, and bars BB1 and DD1 have only one degree of freedom along the y axis because of the structural limitation, and so bars AA1 , BB1 , CC1 , and DD1 can all be simplified as single points A, B, C, and D. The whole bridge-type mechanism can be simplified as shown in the bottom structure in Fig. 7. The mechanical model of bridge-type mechanics can be derived by replacing the arm in Fig. 7 with that in Fig. 6. The model is shown in Fig. 8. Pivot B has only two symmetric forces imposed by two bars, and so the two symmetric forces must be horizontal balance forces. Point B on bar AB has the inverse force of pivot B. Bar AB also has two moments and a force at point A provided by pivot B, and the bar is in a state of balance, and so the value of FA must be equal to that of FB . FA and FB constitute a couple (FA = FB = F): 2Mα = Fla sin α.
Table 1 Key geometrical parameters of bridge-type flexure hinge la (mm) lx (mm) ly (mm) t (mm) b (mm)
2 0.9 13.0 0.4 6.0 Fig. 7. Simplification of bridge-type flexure hinge.
(7)
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733
Fig. 8. Schematic of the mechanics of bridge-type flexure hinge.
At point A on bar AB Fl = F cos α.
(8)
Then: 2Mα Fla sin α 2Kα α = = , Fl F cos α Kl l Kl la sin α α = , l 2Kα cos α
(10)
Fx = Fl l + 2Mα = F cos α · l + Fla sin α · α, (11) x = cos α · l + la sin α · α.
(12)
From Eqs. (10) and (12)
2Kα cos α x = cos α + la sin α α, Kl la sin α
(13)
and the theoretic displacement amplification ratio is Ramp
la cos α
cos α ·
2Kα cos α Kl la sin α
+ la sin α
According to elastic beam theory, each of the pivots is not an ideal pivot, and it has a compliance matrix (its inverse matrix is the stiffness matrix) [12,17]. Assuming a flexure pivot to be the six-degrees-of-freedom spring element shown in Fig. 9, the hinge compliance equation is as follows: Xh = C h F h ,
(15)
where
y la cos α · α = = 2Kα cos α x cos α Kl la sin α + la sin α Δα =
Fig. 9. Coordinate system of corner-filleted flexure pivots.
(9)
.
(14)
T X = x, y, z, α, β, γ ,
(16)
T
F = Fx , Fy , Fz , Mx , My , Mz .
(17)
In Eq. (15), the superscript h represents the pivot coordinate system, and the compliance matrix of the right-angle hinge is derived from the beam theory as follows.
⎤
⎡
l ⎢ Ebt ⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎢ ⎢ ⎢ 0 ⎢ h C =⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎣ 0
0 l + 3 3Ebt Gbt
0
4l3
0
0
0
0
4l3 l + 3Eb3 t Gbt
0
0
0
0 6l2 Ebt 3
−
6l2 Eb3 t 0
l Gk2 bt 3
0 0 −
6l2 Eb3 t 0
0
12l Eb3 t
0
0
0
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥. ⎥ 0 ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ 12l ⎦ Ebt 3 6l2 Ebt 3
(18)
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Fig. 10. Single arm of bridge-type flexure hinge with flexure pivots parallel with the axis passing through two pivots.
E is the elastic modulus, G the shear modulus of the hinge material, and k2 is a geometrical constant determined by b/t [12,17]. For a bridge-type mechanism, only the plan stiffness needs to be considered:
T X = x, y, γ , (19)
T F = Fx , Fy , Mz ,
(20) ⎤
⎡
l ⎢ Ebt ⎢ ⎢ Ch = ⎢ ⎢ 0 ⎢ ⎣ 0
0 l + 3Ebt 3 Gbt 6l2 Ebt 3 4l3
0
⎥ ⎥ ⎥ ⎥. 3 Ebt ⎥ ⎥ 12l ⎦ Ebt 3 6l2
(21)
Where the angle α in Fig. 5 is smaller or the structure in Fig. 10 is adopted, the stiffness of Kl and Kα can be derived from Eq. (21):
Kh = Ch
−1
,
(22)
⎧ Ebt ⎪ ⎪ ⎨ Kl ≈ l 3 . ⎪ Ebt ⎪ ⎩ Kα ≈ 12l
(23)
From Eqs. (14) and (23), the theoretic displacement amplification ratio when taking into account the elastic stiffness of the corner-filleted flexure pivot is Ramp =
la cos α t2
α cos α 6la cos sin α + la sin α
.
Fig. 11. Restraint and load of bridge-type flexure hinge and results of the FEM analysis.
The displacement amplification ratio obtained using the FEM is Ramp =
O . I1 + I 2
Fig. 12 shows the relationship between the displacement amplification ratio and angle α (in Fig. 2) using different equations (Eqs. (6), (24), and (25)). It is seen that with decreasing angle α, the ideal displacement amplification ratio becomes infinite. However, the theoretic displacement amplification ratio has a maximum value and α reaches a threshold value. Whereas α decreases continuously, the displacement amplification ratio decreases sharply. The threshold angle, αmax , at the maximum
(24)
It is obvious that the theoretic displacement amplification ratio is related to the thickness, t (in Fig. 9), when using a cornerfilleted flexure pivot. 4. Comparison with finite element model The finite element method (FEM) is used to analyze the static mechanics of the bridge-type mechanism, as shown in Fig. 11. Surface D is fixed. There is a displacement output on surface C when imposing a force on surfaces A and B. The results of FEM analysis are shown in Fig. 11 using Cosmos DesignStar. A tetrahedral element is used in the FEM model, and a Newton–Simpson solver is used.
(25)
Fig. 12. Displacement amplification ratio vs. angle α.
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735
displacement amplification ratio can be derived from Eq. (14) or Eq. (24):
5. Frequency analysis of bridge-type flexure hinge
αmax = α dRamp . dα =0
The bridge-type mechanism is often used in high-frequency scanning mechanisms, and so the self-vibration frequency of the bridge-type mechanism must be analyzed. An FEM model is established to analyze its mode shape. Fig. 13 shows the first–fourth order mode shape of the bridge-type mechanism. The frequency value of the first–fifth order mode shape is shown in Table 2. It is seen in Eq. (24) that when the thickness, t, is decreased, the displacement amplification ratio increases but the first order mode shape frequency decreases quickly.
(26)
The FEM results indicate that the actual displacement amplification ratio is smaller than the theoretic value. This is mainly because the rigid arm is not an absolute rigid component and which also has and undergoes elastic deformation. When the angle α is small enough, the deformation of the arm cannot be neglected.
Fig. 13. Mode shape analysis of bridge-type flexure hinge without load.
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Table 2 Frequency of mode shape analysis without load t (mm)
Mode shape 1 (Hz)
Mode shape 2 (Hz)
Mode shape 3 (Hz)
Mode shape 4 (Hz)
Mode shape 5 (Hz)
0.4 0.2
201 75
409 154
602 259
711 308
1383 531
6. Conclusions The ideal displacement amplification ratio and the theoretic displacement amplification ratio are analyzed based on kinematic and elastic mechanisms. The FEM is used for a comparison with the mathematical model and for the analysis of the mode shape of the structure. From the above analysis, we arrived at some conclusions: (1) The ideal displacement amplification ratio becomes infinite when the angle between the bridge arm and the catercorner line goes to zero when the bridge-type mechanism is considered as a multi-rigid body with ideal pivots. (2) The theoretic displacement amplification ratio has an extremum when deformation at the flexure pivots is taken into account. The threshold of the extremum is related to the ratio between the translational stiffness and the rotational stiffness. (3) The actual displacement amplification ratio decreases compared with the theoretic displacement amplification ratio when the other deformation of the primary “rigid body” is considered. (4) A decrease in the thickness of the flexure pivots increases the displacement amplification ratio, but the mode shape frequency decreases. References [1] C. Niezrecki, D. Brei, S. Balakrishnan, A. Moskalik, Piezoelectric actuation: state of the art, Shock Bib. Digest. 33 (4) (2001) 269–280. [2] R.E. Newnham, A. Dogan, Q.C. Xu, Flextensional “Moonie” actuators, in: IEEE Ultrasonics Symposium, 1993, pp. 509–513. [3] F. Lalande, Z. Chaudhry, C.A. Rogers, A simplified geometrically nonlinear approach to the analysis of the Moonie actuator, IEEE Trans. Ultrasonics Ferroelectrics Freq. Control 42 (1995) 21–27. [4] G.H. Haertling, G.C. Robinson, Compositional study of PLZT rainbow ceramics for piezo actuators, in: Proceedings of the Ninth IEEE International Symposium on Applications of Ferroelectrics, 1994, pp. 313– 318. [5] J. Juuti, E. Heinonen, V.-P. Moilanen, Displacement, stiffness and load behaviour of laser-cut RAINBOW actuators, J. Eur. Ceram. Soc. 24 (2004) 1901–1904. [6] S. Chandran, V.D. Kugel, L.E. Cross, Characterization of the linear and nonlinear dynamic performance of RAINBOW actuator, in: Proceedings of the 10th IEEE International Symposium on Applications of Ferroelectrics, 1996, pp. 743–746. [7] A. Dogan, K. Uchino, Composite piezoelectric transducer with truncated conical endcaps “Cymbal”, IEEE Trans. Ultrasonics Ferroelectrics Freq. Control 44 (1997) 597–605. [8] A. Dogan, J.F. Fernandez, K. Uchino, The “CYMBAL” electromechanical actuator, in: IEEE International Symposium on Applications of Ferroelectrics, 1996, pp. 213–216.
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Biographies Hong-Wen Ma received the BSc degree in automobile engineering in 1995 from the Jilin university of technology (JUT), China, and the BSc degree in automation engineering in 1999, the PhD degree in instrument and instrumentation in 2005 all from the Harbin institute of technology (HIT), China, where his research focused on precision positioning technique. From 2005, he has been employed by the Harbin engineering university (HEU) as associate professor. His current interests include legged robot and machine vision. Shao-Ming Yao received MS in mechanical engineering from Ship Building and Engineering Institute, in 1992. He is currently pursuing PhD in instrument science at Harbin Institute of Technology (HIT). From 1999 to 2003, he served as senior engineer in Ship Industry Group Company, where his research focused on power transmission gear. From 2003, he has been employed by HIT as associate professor in Instrument Science Department. His current research focused on opto-electromechanical integration technique and ultra-precision motion reference technique. Li-Quan Wang was born in Herbei, China in 1957. He received the PhD in control theory and control engineering from the Harbin engineering university (HEU), China. He is employed by the HEU as professor currently. His current interests include bionics, underwater robot and underwater construction equipment. Zhi Zhong was born in 1976. He received the PhD in optoelectronic information technology and instrumentation engineering, from Department of Automatic Measurement and Control Engineering, Harbin Institute of Technology, China in 2005. His research interests are currently in the field of fast ultra-precision laser heterodyne interferometric and signal processing.