- Email: [email protected]
A novel technique for position analysis of planar compliant mechanisms
Mechanism and Machine Theory
Mechanism and Machine Theory 40 (2005) 1224–1239
www.elsevier.com/locate/mechmt
A novel technique for position analysis of planar compliant mechanisms S. Venanzi a, P. Giesen b, V. Parenti-Castelli a
a,*
DIEM—Department of Mechanical Engineering, University of Bologna, Viale Risorgimento 2, 40136 Bologna, Italy b TNO–TPD, Institute of Applied Physics, DOI—Division Optical Instrumentation, Stieltjesweg 1, 2600 AD Delft, the Netherlands Received 14 July 2004; received in revised form 31 December 2004; accepted 24 January 2005 Available online 14 March 2005
Abstract An iterative technique to perform the non-linear position analysis of planar compliant mechanisms is presented. The technique makes it possible to find the position and orientation (pose) of each link of a mechanism whose input link deflection is assigned. Unlike most papers, it does not rely on the finiteelement method. Its innovative part is in the fact that large deflections are considered, and the position analysis is solved without resorting to the linear approximation of small deflections. The technique is highly suitable for rigid-link mechanisms with compliant kinematic pairs: mathematical models for such pairs are used, and a modular approach for the study of compliant mechanisms is obtained. This modular approach reduces the time needed for the mechanism modelling, as different compliant pairs can be quickly embedded into any mechanism. Furthermore, using dedicated models for such pairs highly increases the computational efficiency. In the last part of the paper, the technique is applied to a four-bar mechanism. 2005 Elsevier Ltd. All rights reserved. Keywords: Compliant mechanisms; Position analysis; Flexures; Large deflections
*
Corresponding author. Tel.: +39 51 209 3459; fax: +39 51 209 3446. E-mail address: [email protected] (V. Parenti-Castelli).
0094-114X/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2005.01.009
S. Venanzi et al. / Mechanism and Machine Theory 40 (2005) 1224–1239
1225
1. Introduction The definition compliant mechanisms concerns those mechanisms which gain their mobility from the deflection of flexible elements rather than from kinematic pairs; if the flexible elements were considered as rigid bodies, such mechanisms would become structures and would thus have no mobility. Compliant mechanisms are becoming more and more common for a number of reasons. With respect to ordinary mechanisms, they allow for a cost reduction as no assembly is needed and the manufacturing process can often be simplified. Furthermore, they need reduced maintenance because no wear occurs and no lubrication has to be provided. This highly increases the mechanism reliability, and makes them suitable for working in difficult environments such as vacuum or high temperatures. On the other hand, compliant mechanisms present a reduced mobility, since the material strength poses a limit to deflection that cannot be overcome. Another disadvantage is the need for a driving force/torque to control the mechanism configuration and to deform its flexible parts; such a force/torque is needed even when no resistant loads act on the mechanism. Because of these peculiar features, compliant mechanisms are often intended as complementary to ordinary mechanisms, rather than alternative. In precision engineering—that is, in applications such as optics or metrology, where the positioning accuracy is by far the most relevant requirement—compliant mechanisms are the only acceptable solution. In most cases, they take the form of ordinary mechanisms; only, the kinematic pairs are replaced by flexible elements—flexures from here on. Flexures work without clearance or hysteresis, and their behavior is highly predictable. Unfortunately, a new problem arises: the lack of tools to analyze such mechanisms. Much theory has been developed for the analysis of ordinary mechanisms; on the contrary, a systematic approach dealing with compliant mechanisms has not been presented yet. In [1], a partial answer to the problem is given by the pseudo-rigid-body model. The model is used to reduce the study of compliant mechanisms to that of ordinary mechanisms, and can be very useful in mechanism design. However, it cannot cope with compliant mechanisms for precision engineering, as the assumptions and approximations behind the model are not acceptable when the requirements on accuracy become too strict. The alternative to [1] is the non-linear finite-element (FE) analysis of mechanisms, commonly used by designers. The FE analysis can be simplified by using the ‘‘chain calculation’’ [2–5]: the links are divided into small elements, and for each element loads and displacements are studied via the stiffness matrix. The main difference between the chain calculation and ordinary FE techniques lies in the fact that in the chain calculation each element is dealt with separately from the others, and the results are then assembled kinematically: no large stiffness matrices have to be assembled, and a higher computational efficiency is achieved. Both ordinary FE techniques and the chain calculation have to be performed iteratively to meet the imposed boundary conditions. The FE approach is time-consuming both in the modelling and in the computation, and not suitable to quickly compare different designs. As a result, most designers rely on a few standard designs whose accuracy has already been tested, like the leaf-spring mechanism or the doublecompound mechanism [6–9]. In this paper, an iterative technique to analyze compliant mechanisms is presented. The technique deals with planar mechanisms containing flexures. It can be used to perform the position analysis of a mechanism once the input variables are given (that is, when the input link deflection is prescribed), and provides both the exact pose of each link and the driving torques/forces needed
1226
S. Venanzi et al. / Mechanism and Machine Theory 40 (2005) 1224–1239
to achieve that configuration. Unlike previous methods, flexures are considered as whole parts undergoing large deflections, and are not divided into smaller elements. This allows for an efficient solution and, most of all, for a modular approach: flexures can be considered as effective hinges, and any mechanism with such hinges can be designed and analyzed. In the paper, some common flexure designs are listed and modelled. After its general description, the technique is applied to a planar four-bar linkage.
2. Technique outline The position analysis of ordinary mechanisms can be a challenging task because of its intrinsic non-linearity. The problem is even more complex for compliant mechanisms: because of the flexible elements, the kinematic relations used for ordinary mechanisms are not sufficient, and both equilibrium and elasticity equations have to be added. The solution of the problem has thus to satisfy kinematics, equilibrium and elasticity at the same time. Some techniques exist for structures where small displacements only are considered, and the linear beam theory can be used. However, the linear beam theory cannot express the large deflections typical of flexures. It is then necessary to use non-linear analysis. In this paper, non-linear analysis is used with the following assumptions: • Geometric non-linearity is considered. Geometric non-linearity refers to the fact that deflections are large, and have to be accounted for when considering the equilibrium configuration. • Material non-linearity is not considered. All materials are assumed to obey Hookes law, and effects like plasticity, non-linear elasticity, hyperelasticity and creep are neglected. • Strain is assumed to be small, so that it does not cause significant changes in geometrical features like section areas or beam thicknesses. It must be noted that small strain does not imply small deflections. In this paper, the three problems mentioned before—kinematics, equilibrium, and elasticity— are faced separately. The first problem is solved by a kinematic analysis. An ideal hinge is introduced between one of the extremities of each flexure and the adjacent rigid link (see Fig. 1). In this way, the relative rotation between the flexure and its adjacent rigid link becomes possible, and the congruence condition on rotations is broken. The compliant mechanism is thus replaced by an ordinary mechanism, whose mobility is given by the ideal hinges, and whose flexures do not deflect. If
Fig. 1. Ideal hinge.
S. Venanzi et al. / Mechanism and Machine Theory 40 (2005) 1224–1239
1227
the input variables for this new mechanism are given, the position analysis can be performed as for ordinary mechanisms; the position and orientation (pose) of each link are obtained. The second problem is solved by a static analysis. The static analysis is performed on the mechanism with ideal hinges, and provides the reaction force in each ideal hinge caused by the external loads acting on the mechanism. It depends on the link pose, and therefore on the kinematic analysis of the previous step. The third problem is solved by a deflection analysis, the most innovative point in the new technique. It considers each flexure in its whole, and uses a differential equation describing its large displacements. This differential equation relates the deflected shape of a flexure with the load acting on it. When working in a plane, the deflected shape of the ith flexure can be described by a three-component vector, [Xi, Yi, Hi], where Xi and Yi are linear displacements of one end section, and Hi is an angular displacement of the same section. Analogously, the load acting on the ith flexure can be expressed by a three-component vector, [Fxi, Fyi, Mi], where Fxi and Fyi are forces on one end section, while Mi is a moment on the same section. The elasticity theory relates the two vectors; in general, given three arbitrary components among the six Xi, Yi, Hi, Fxi, Fyi, Mi, it is possible to determine the remaining three. A numerical procedure has been developed, allowing the determination of the components Xi, Yi and Mi when the components Fxi, Fyi and Hi are given. Such a procedure is detailed in the following section. The choice of the components to be determined depends on the two previous steps: the two force components Fxi and Fyi have been determined by the static analysis, while the relative rotation Hi is the same as the rotation in the ideal hinge. Imposing the same value for the flexure rotation Hi and for the ideal hinge has the meaning of re-introducing the congruence condition on rotation, which had been broken when performing the kinematic analysis. The deflection analysis provides the components Xi and Yi, and thus a deflected shape for the ith flexure; it also provides Mi, bending moment acting on the same flexure. These values can be used to perform further kinematic and static analyses of the mechanism. The new kinematic analysis is performed on the mechanism with ideal hinges. But this time, flexures are deflected (and their deflected shape is expressed by the values Xi and Yi, previously obtained). The new pose of each link is evaluated. The new static analysis is performed on the mechanism with ideal hinges. With respect to the previous static analysis, however, the link pose has been updated; moreover, the bending moments Mi have to be considered. The results of the last kinematic and static analyses can be used for a new deflection analysis; in this way, a loop is built. The loop is iterated until all variables converge. The iterative procedure can be represented by the block diagram in Fig. 2. The next section will describe the mathematics used to perform the deflection analysis; the last section will illustrate how the iterative procedure can be applied to a well-known mechanism, the four-bar linkage.
3. Deflection analysis There are no standard procedures to analyze large displacements of flexures. In this paper, a mathematical procedure dealing with flexures has been developed and implemented. The
1228
S. Venanzi et al. / Mechanism and Machine Theory 40 (2005) 1224–1239
Fig. 2. Iterative procedure.
procedure is based on the Bernoulli–Euler equation, solved without the usual hypothesis of small deflections. A closed-form solution of the procedure would require elliptic integrals [3]; because of the complexity of such analytical approach, a numerical solution has been preferred. The numerical solution strongly depends on the boundary conditions and on the flexure design. In this paper, four common flexure designs are analyzed. 3.1. Leaf spring The most intuitive flexure is the simple leaf spring. According to the linear beam theory, it constrains one degree of freedom in the plane, the relative displacement along the axial direction, and allows a displacement in the shear direction and a rotation. However, axial displacement, shear displacement and rotation exist at the same time when large deflections are considered. Their value depends on the load acting on the leaf spring; an extensive study on the leaf spring, modelled by means of the pseudo-rigid-body model, can be found in [10]. In this paper the pseudo-rigid-body model is not used. On the contrary, the leaf spring is studied using Bernoulli–Euler equation, which states dhðsÞ MðsÞ ¼ ds k
ð1Þ
where s is a linear coordinate along the leaf spring, h(s) is the section rotation, M(s) is the bending moment acting on the section and k is the bending stiffness of the leaf spring. Narrow leaf springs can be considered like beams, so that their bending stiffness is given by E Æ I (E is the material elastic modulus, I is the area moment of inertia of the cross section). Wide leaf springs can be considered like plates, so that their bending stiffness is given by E Æ I/(1 m2), where m is the Poisson ratio of the material [11].
S. Venanzi et al. / Mechanism and Machine Theory 40 (2005) 1224–1239
1229
Fig. 3. Leaf spring.
The bending moment on a section can be expressed as MðsÞ ¼ M þ Fy ðX xðsÞÞ Fx ðY yðsÞÞ
ð2Þ
where M is the moment acting on the free end of the leaf spring, [Fx, Fy] is the force acting on the same end, [X, Y] are the center section coordinates of the free end, and [x(s), y(s)] are the cartesian coordinates of the section where the bending moment is evaluated (see Fig. 3). Furthermore, the cartesian coordinates of the section and its rotation are linked by the relation dxðsÞ=ds ¼ cos hðsÞ
ð3Þ
dyðsÞ=ds ¼ sin hðsÞ
Combining Eqs. (1)–(3), the large deflection of a leaf spring can be mathematically formulated by a first order differential equation 3 3 2 2 ½M þ Fy ðX xðsÞÞ Fx ðY yðsÞÞ=k hðsÞ d6 7 7 6 ð4Þ cosðhðsÞÞ 5 4 xðsÞ 5 ¼ 4 ds sinðhðsÞÞ yðsÞ subject to the boundary 3 2 3 2 0 hð0Þ 7 6 7 6 4 xð0Þ 5 ¼ 4 0 5; yð0Þ
0
conditions 3 2 3 2 H hðLÞ 7 6 7 6 4 xðLÞ 5 ¼ 4 X 5 yðLÞ
ð5Þ
Y
where H is the rotation of the leaf-spring end. In other words, the large deflection of the leaf spring can be expressed by a three-component function which has to satisfy the first order differential equation (4). Boundary conditions are applied to both bounds of the independent variable, the linear coordinate s, that varies between 0 and L, length of the leaf spring. In Eqs. (4) and (5) six parameters appear: M, moment on one end of the leaf spring; Fx and Fy, components of the force on the same end; H, rotation of the end section; and X and Y, expressing the center position of the end section. The problem can be solved numerically as an ordinary boundary-condition problem [12]. Three of the above listed parameters can be found, provided the other three are given.
1230
S. Venanzi et al. / Mechanism and Machine Theory 40 (2005) 1224–1239
The accuracy of Eq. (4) can be increased by adding the effects of the axial force acting in each section. The axial force is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Fy 2 2 ð6Þ N ðsÞ ¼ Fx þ Fy cos arctan hðsÞ Fx Eq. (4) then becomes 3 3 2 2 ½M þ FyðX xðsÞÞ FxðY yðsÞÞ=k hðsÞ d6 7 7 6 cosðhðsÞÞ 5ð1 þ N ðsÞ=EAÞ 4 xðsÞ 5 ¼ 4 ds sinðhðsÞÞ yðsÞ
ð7Þ
where A is the cross section area. A numerical procedure was developed in MATLAB. The procedure implements Eqs. (7) and (5), and solves the boundary value problem using standard numerical techniques. The procedure receives H, Fx and Fy as inputs, and provides M, X and Y as outputs. 3.2. Cross hinge Two leaf springs can be used at the same time to improve the flexure performance. The result of the assembly—which is shown in Fig. 4 and will be called cross hinge from here on—is another kind of flexure. Such a flexure has the advantages of being stiff to axial and shear forces, and to be compliant to rotation only. Its performances can be further increased by using three leaf springs, two narrow external ones and one wide internal one (see Fig. 5); in this way, the hinge has a higher torsional stiffness. To a first order approximation, the cross hinge behaves like a perfect hinge. Because of these interesting properties, it has been widely studied [13–16]. The large-deflection model developed for the leaf spring has been adapted to the cross hinge. A six-component function replaces the three-component function: three components for the first leaf spring, three for the second one. The differential equation to be satisfied is not reported, as it looks exactly like Eq. (7). However, 12 parameters instead of six appear: M1 and M2 are the moment at the end of the 1st and 2nd leaf spring respectively, [Fx1, Fy1] and [Fx2, Fy2] are the end forces, H1
Fig. 4. Cross hinge.
S. Venanzi et al. / Mechanism and Machine Theory 40 (2005) 1224–1239
1231
Fig. 5. Cross hinge.
Fig. 6. Cross hinge.
and H2 are the final section rotations, [X1, Y1] and [X2, Y2] identify the final section positions (see Fig. 6). The boundary conditions have to be slightly modified to express the link between the two leaf springs. The values of the function on the boundary are 3 3 2 3 2 3 2 2 h1 ðLÞ 0 H1 h1 ð0Þ 6 x ð0Þ 7 6 0 7 6 x ðLÞ 7 6 X 7 7 6 7 6 1 7 6 17 6 1 7 7 6 7 6 7 6 6 6 y 1 ð0Þ 7 6 0 7 6 y 1 ðLÞ 7 6 Y 1 7 7 7 6 7 6 7 6 6 ð8Þ 6 h ð0Þ 7 ¼ 6 0 7; 6 h ðLÞ 7 ¼ 6 H 7 7 6 27 6 2 7 6 7 6 2 7 7 6 7 6 7 6 6 4 x2 ð0Þ 5 4 0 5 4 x2 ðLÞ 5 4 X 2 5 y 2 ð0Þ
0
y 2 ðLÞ
Y2
Six more conditions have to be added. Three come from the fact that the deflected shape of flexures has to satisfy kinematic conditions:
1232
S. Venanzi et al. / Mechanism and Machine Theory 40 (2005) 1224–1239
H1 ¼ H2 ¼ H ðX 1 X 2 Þ sin a þ ðY 1 þ Y 2 Þ cos a ¼ L cos a sin H
ð9Þ
ðX 1 þ X 2 Þ cos a þ ðY 2 Y 1 Þ sin a ¼ L cos aðcos H þ 1Þ and three from the fact that it has to satisfy equilibrium: ðFx1 þ Fx2 Þ sin a þ ðFy 1 Fy 2 Þ cos a ¼ Fx ðFx2 Fx1 Þ cos a þ ðFy 1 þ Fy 2 Þ sin a ¼ Fy 1 ½ðX 2 2
þ
þ X 1 LÞ cos a þ ðY 2 Y 1 Þ sin a ½ðFx1 þ Fx2 Þ sin a ðFy 1 þ Fy 2 Þ cos a 1 ½ðX 2 2
ð10Þ
X 1 Þ sin a ðY 2 þ Y 1 Þ cos a ½ðFy 1 Fy 2 Þ sin a þ ðFx1 Fx2 Þ cos a
þ M1 þ M2 ¼ M In Eq. (10), M is the moment, [Fx, Fy] is the force acting on the hinge movable part, and a is a geometrical feature of the flexure—see Fig. 6. The equations were implemented in MATLAB and used to perform the deflection analysis. Similarly to the previous case, the numerical procedure receives H, Fx and Fy as inputs, and provides as output the moment M and the position [X, Y] of the point to which the force [Fx, Fy] is applied. 3.3. Notch hinge Another common design for flexures is that reported in Fig. 7, the so called notch hinge [17–20]. This design has the advantage of concentrating the strain in its central section, so that its kinematic behavior more closely resembles that of a theoretical hinge. This advantage is paid for in terms of maximum deflection, as the stress in the central section reaches high values even with limited rotations. Different shapes have been proposed for the notch hinge; a thorough comparison about this can be found in [21], where linear deflections only are considered. In this paper, the more traditional design—that with circular notches—is studied by means of Eqs. (7) and (5). To make these equation suitable, the hinge cross section moment of inertia I and area A cannot be considered constant, but have to be replaced by functions involving the linear coordinate s, the flexure width, its minimum thickness, and its curvature (see Fig. 7). Such an approach has already been adopted by other authors to evaluate the flexure stiffness [17,21], whereas a more sophisticated modelling requires FE techniques [19]. A numerical model similar to that for
Fig. 7. Notch hinge.
S. Venanzi et al. / Mechanism and Machine Theory 40 (2005) 1224–1239
1233
Fig. 8. Haberland hinge.
the leaf spring and the cross hinge has been developed and implemented. Eq. (7) has been used, with k replaced by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 , 12 ð11Þ k ¼ E b t þ 2R ð4R2 ð2s LÞ2 Þ with L hinge length, b hinge width, t hinge minimum thickness, R hinge curvature. A further factor 1/(1 m2) is to be added for plates. 3.4. Haberland hinge The last kind of flexure studied and implemented is the one reported in Fig. 8; the design is similar to the cross hinge, but the two leaf springs are connected in their central section. This design was first proposed by R. Haberland [22], and grants a higher accuracy than the cross hinge. The flexure is however stiffer to rotation than the cross hinge, and allows smaller rotations. This flexure has been studied as the union of four leaf springs, all connected to the central point; for the sake of brevity, the differential equation and the boundary conditions are not reported. A numerical model similar to the previous ones was developed and implemented in MATLAB.
4. Numerical example In this section, the technique is applied to a four-bar linkage. The technique is described in detail, and the iteration is shown step by step. The studied mechanism is shown in Fig. 9. The length of each link is reported in Table 1 (all lengths are in mm). The links are interconnected by leaf springs, whose dimensions are reported in Table 2. All leaf springs are assumed to behave like beams, with a bending stiffness k = EI. All links are assumed to be infinitely stiff, in the sense that their deformation can be ignored with respect to the deflection in the leaf springs. The whole mechanism is made of aluminium (elastic modulus E = 69 GPa). The reference configuration in which all flexures are not deflected is reported in Fig. 9, with the crank angle a0 = p/2. The orientation of the leaf springs in that configuration are those reported in Fig. 9. All angles in the
1234
S. Venanzi et al. / Mechanism and Machine Theory 40 (2005) 1224–1239
Fig. 9. Compliant four-bar linkage.
Table 1 Mechanism dimensions Link
Length (mm)
Crank, L1 Coupler, L2 Rocker, L3 Ground, L4
20 60 60 80
Table 2 Leaf spring dimensions Leaf spring
Length (mm)
Width (mm)
Thickness (mm)
1 2 3 4
l1 = 10 l2 = 10 l3 = 15 l4 = 15
b1 = 5 b2 = 5 b3 = 5 b4 = 5
t1 = 0.1 t2 = 0.1 t3 = 0.1 t4 = 0.1
Table 3 Angles—non-deflected configuration Angle
Name
Value (rad)
Crank angle Coupler angle Rocker angle
a0 b0 c0
1.570796 0.774981 1.996102
not-deflected configuration are reported in Table 3. The only external load acting on the mechanism is a horizontal force from right to left applied in the middle of the rocker, FR = 0.5 N. The crank angle a is assumed as input variable to control the mechanism; the orientation of coupler and rocker is expressed by the two angles b and c (see Fig. 9). Let a clockwise rotation
S. Venanzi et al. / Mechanism and Machine Theory 40 (2005) 1224–1239
1235
of 0.2 rad be given to the crank. The technique provides the pose of each link in the mechanism after such a rotation, and the moment that has to be applied to the crank to achieve it. It will now be applied step by step. 4.1. Step 1—guess values A guess value has to be assumed for the deflected shape of flexures and for the bending moment acting on them. The undeflected shape of flexures can be assumed, therefore X 1 ¼ l1
Y1 ¼ 0
M1 ¼ 0
X 2 ¼ l2
Y2 ¼ 0
M2 ¼ 0
X 3 ¼ l3
Y3 ¼ 0
M3 ¼ 0
X 4 ¼ l4
Y4 ¼ 0
M4 ¼ 0
ð12Þ
4.2. Step 2—kinematic analysis The ideal hinges are introduced to obtain an ideal rigid-link mechanism—see Fig. 10. Based on the guess values for flexures, the kinematic analysis of the ideal mechanism can be performed. The loop closure equations are 8 X 1 cos a0 Y 1 sin a0 þ L1 cos a þ X 2 cos a Y 2 sin a þ L2 cos b > > > > < ¼ L4 þ X 4 cos c Y 4 sin c þ L3 cos c þ X 3 cos c Y 3 sin c 0 0 ð13Þ > sin a þ Y cos a þ L sin a þ X sin a þ Y cos a þ L sin b X > 1 0 1 0 1 2 2 2 > > : ¼ þX 4 sin c0 þ Y 4 cos c0 þ L3 sin c þ X 3 sin c þ Y 3 cos c In the two equations (13), only two unknowns appear, b and c, which can be easily determined: b ¼ 0:826851 rad
ð14Þ
c ¼ 1:942198 rad
Fig. 10. Ideal rigid-link mechanism.
1236
S. Venanzi et al. / Mechanism and Machine Theory 40 (2005) 1224–1239
With this result, it is possible to evaluate the rotation in each ideal hinge: • the first hinge rotates from a0 to a, therefore H1 = a a0 = 0.200 rad; • the angle between crank and coupler changes from p a0 + b0 to p a + b, therefore H2 = b b0 a + a0 = 0.252 rad; • the angle between rocker and coupler becomes c b from c0 b0, therefore H3 = c c0 b + b0 = 0.106 rad; • the rocker angle passes from c0 to c, therefore H4 = c c0 = 0.054 rad. 4.3. Step 3—static analysis After the kinematic analysis, the static analysis has to be performed. Three equilibrium equations can be written for the crank, the coupler and the rocker: 8 Fx1 þ Fx2 ¼ 0 > > > Fy þ Fy ¼ 0 > > 1 2 > > > Fy ðL1 cos a þ X 2 cos a Y 2 sin aÞ Fx2 ðL1 sin a þ X 2 sin a þ Y 2 cos aÞþ > 2 > > > > M 1 þ M 2 þ M ¼ 0 > > > > > Fx2 Fx3 ¼ 0 < Fy 2 Fy 3 ¼ 0 ð15Þ > > > L cos b þ Fx L sin b M M ¼ 0 Fy 3 2 2 3 > 3 2 > > > þ Fx þ F ¼ 0 Fx > 4 3 R > > > > Fy 4 þ Fy 3 ¼ 0 > > > > > Fy ðL cos c þ X 3 cos c Y 3 sin cÞ Fx3 ðL3 sin c þ X 3 sin c þ Y 3 cos cÞþ > : 3 3 M 4 þ M 3 12 F R L3 sin c ¼ 0 where [Fxi, Fyi] is the force acting on the ith hinge, Mi is the moment acting on that hinge (null at this first step), M is the external moment to apply to the crank, FR is the resistant force on the rocker (see Fig. 11). The system above is a linear system of nine equations in the nine unknowns Fxi, Fyi, i = 1, . . ., 4, and M. The solution of the system is Fx1 ¼ 0:084 N
Fy 1 ¼ 0:092 N
Fx2 ¼ 0:084 N
Fy 2 ¼ 0:092 N
Fx3 ¼ 0:084 N
Fy 3 ¼ 0:092 N
Fx4 ¼ 0:216 N
Fy 4 ¼ 0:092 N
ð16Þ
M ¼ 1:933 N mm 4.4. Step 4—deflection analysis The values Hi, Fxi and Fyi, obtained in the previous analysis, can be used in Eqs. (7) and (5), to obtain new estimates for Xi, Yi and Mi. For each flexure, the force [Fx, Fy] has to be projected in the proper direction to obtain the axial and shear components.
S. Venanzi et al. / Mechanism and Machine Theory 40 (2005) 1224–1239
1237
Fig. 11. Static analysis.
H1 ¼ 0:200 Fx1 ¼ 0:092 Fy 1 ¼ 0:084 ) M 1 ¼ 0:943 X 1 ¼ 9:953 Y 1 ¼ 0:774 H2 ¼ 0:252 Fx2 ¼ 0:107 Fy 2 ¼ 0:064 ) M 2 ¼ 0:307 X 2 ¼ 9:862 Y 2 ¼ 1:482 H3 ¼ 0:106 Fx3 ¼ 0:055 Fy 3 ¼ 0:112 ) M 3 ¼ 1:040 X 3 ¼ 14:983 Y 3 ¼ 0:282
ð17Þ
H4 ¼ 0:054 Fx4 ¼ 0:172 Fy 4 ¼ 0:159 ) M 4 ¼ 1:215 X 4 ¼ 14:952 Y 4 ¼ 0:997 After this step, the procedure goes back to step 2—to be performed with the new values for Xi, Yi and Mi. 4.5. Final results After few iterations, the procedure converges towards a solution. The procedure has been tested with several planar mechanisms, different both in topology and in their geometrical features. The only convergence problems were found when buckling occurred in the flexures: since buckling is intrinsically connected with instability, it is possible to state that such convergence problems do not depend on the model. The numerical value of the two angles b and c after six iterations is b ¼ 0:817134 rad c ¼ 1:947244 rad
ð18Þ
whereas the correction from the 5th to the 6th step is 2.5e06 and 2.7e06 rad respectively. The counterclockwise moment to be applied to the crank to obtain the prescribed rotation is 4.023 N mm. The deflected shape of the mechanism is reported in Fig. 12. The same analysis was performed with non-linear FE software. The results obtained are b ¼ 0:817624 rad c ¼ 1:948497 rad
ð19Þ
There is a good match between the two models, as the difference is 0.5 and 1.3 mrad respectively.
1238
S. Venanzi et al. / Mechanism and Machine Theory 40 (2005) 1224–1239
Fig. 12. Deflected shape of the mechanism obtained with the proposed technique.
5. Conclusions In this paper, a new technique has been presented. The technique performs the position analysis of planar compliant mechanisms: it receives as input the mechanism deflection, and provides as output the actual pose of each link, the deflected shape of each flexible element, and the forces/moments needed to deflect the mechanism. Unlike other techniques, it does not rely on linear approximations and small deflections, but considers large elastic deflections for flexible elements. The technique is particularly suitable for mechanisms with compliant pairs (flexures), because it allows for a modular approach to their description. The paper describes some common flexures and mathematical models are provided for them. These hinge models can be used either as self-standing units, or as modules to study more complex mechanisms: they can be embedded as independent parts into any mechanism, so that the mechanism modelling becomes much quicker. Furthermore, using dedicated models to study flexures considerably reduces the computational time. In the last part of the paper, the technique is applied to a four-bar linkage. For this mechanism, the technique provides both the deflected shape of the entire mechanism and the moment to apply to deflect it to the desired shape. The technique shows a good match when compared with nonlinear finite-element analysis.
Acknowledgments The authors gratefully acknowledge the financial support of the Italian MIUR and CNR.
References [1] L.L. Howell, Compliant Mechanisms, John Wiley & Sons, 2001. [2] R.E. Miller, Numerical analysis of a generalized plane elastica, International Journal for Numerical Methods in Engineering 15 (1980) 325–332. [3] K. Mattiasson, Numerical results from large deflection beam and frame problems analysed by means of elliptic integrals, International Journal for Numerical Methods in Engineering 17 (1981) 145–153. [4] B.A. Coulter, R.E. Miller, Numerical analysis of a generalized plane elastica with non-linear material behavior, International Journal for Numerical Methods in Engineering 26 (1988) 617–630.
S. Venanzi et al. / Mechanism and Machine Theory 40 (2005) 1224–1239
1239
[5] T.C. Hill, A. Midha, A graphical, user-driven Newton Raphson technique for use in the analysis and design of compliant mechanisms, ASME Journal of Mechanical Design 112 (1990) 123–130. [6] G.H. Neugebauer, Designing springs for parallel motion, Machine Design 8 (1980) 119–120. [7] S. Smith, D. Chetwynd, Foundations of Ultraprecision Mechanism Design, Gordon & Breach, 1992. [8] J. van Eijk, On the Design of Plate-spring Mechanisms, Delft University of Technology, 1985. [9] R.V. Jones, Instruments and Experiences, Papers on Measurement and Design, John Wiley & Sons, 1988. [10] C. Kimball, L.W. Tsai, Modeling of flexural beams subjected to arbitrary end loads, ASME Journal of Mechanical Design 124 (2002) 223–235. [11] Timoshenko, Theory of Plates and Shells, McGraw-Hill, 1989. [12] G. Lindfield, J. Penny, Numerical Methods Using Matlab, Ellis Horwood Limited, 1995. [13] J.A. Haringx, The cross-spring pivot as a constructional element, Applied Scientific Research, A 1 (4) (1949) 313– 332. [14] W.H. Wittrick, The theory of symmetrical crossed flexure pivots, Australian Journal of Scientific Research, A 1 (2) (1948) 121–134. [15] S. Zelenika, F. De Bona, Analytical and experimental characterisation of high-precision flexural pivots subjected to lateral loads, Precision Engineering 26 (2002) 381–388. [16] B.D. Jensen, L.L. Howell, The modeling of cross-axis flexural pivots, Mechanism and Machine Theory 37 (5) (2002) 461–476. [17] J.M. Paros, L. Weisbord, How to design flexure hinges, Machine Design 11 (1965) 151–156. [18] S. Zhang, E.D. Fasse, A finite-element-based method to determine the spatial stiffness properties of a notch hinge, ASME Journal of Mechanical Design 123 (2001) 141–147. [19] G. Xu, L. Qu, Some analytical problems of high performances flexure hinge and micro-motion stage design, in: Proceedings of The IEEE International Conference on Industrial Technology, 1996, pp. 771–775. [20] J.W. Ryu, D.G. Gweon, Error analysis of a flexure hinge mechanism induced by machining imperfection, Precision Engineering 21 (2/3) (1997) 83–89. [21] N. Lobontiu, J.S.N. Paine, E. Garcia, M. Goldfarb, Design of symmetric conic-section flexure hinges based on closed-form compliance equations, Mechanism and Machine Theory 37 (2002) 477–498. [22] German Patents 2626800 and 2653427, and related US Patent 4261211.
Mechanism and Machine Theory 40 (2005) 1224–1239
www.elsevier.com/locate/mechmt
A novel technique for position analysis of planar compliant mechanisms S. Venanzi a, P. Giesen b, V. Parenti-Castelli a
a,*
DIEM—Department of Mechanical Engineering, University of Bologna, Viale Risorgimento 2, 40136 Bologna, Italy b TNO–TPD, Institute of Applied Physics, DOI—Division Optical Instrumentation, Stieltjesweg 1, 2600 AD Delft, the Netherlands Received 14 July 2004; received in revised form 31 December 2004; accepted 24 January 2005 Available online 14 March 2005
Abstract An iterative technique to perform the non-linear position analysis of planar compliant mechanisms is presented. The technique makes it possible to find the position and orientation (pose) of each link of a mechanism whose input link deflection is assigned. Unlike most papers, it does not rely on the finiteelement method. Its innovative part is in the fact that large deflections are considered, and the position analysis is solved without resorting to the linear approximation of small deflections. The technique is highly suitable for rigid-link mechanisms with compliant kinematic pairs: mathematical models for such pairs are used, and a modular approach for the study of compliant mechanisms is obtained. This modular approach reduces the time needed for the mechanism modelling, as different compliant pairs can be quickly embedded into any mechanism. Furthermore, using dedicated models for such pairs highly increases the computational efficiency. In the last part of the paper, the technique is applied to a four-bar mechanism. 2005 Elsevier Ltd. All rights reserved. Keywords: Compliant mechanisms; Position analysis; Flexures; Large deflections
*
Corresponding author. Tel.: +39 51 209 3459; fax: +39 51 209 3446. E-mail address: [email protected] (V. Parenti-Castelli).
0094-114X/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2005.01.009
S. Venanzi et al. / Mechanism and Machine Theory 40 (2005) 1224–1239
1225
1. Introduction The definition compliant mechanisms concerns those mechanisms which gain their mobility from the deflection of flexible elements rather than from kinematic pairs; if the flexible elements were considered as rigid bodies, such mechanisms would become structures and would thus have no mobility. Compliant mechanisms are becoming more and more common for a number of reasons. With respect to ordinary mechanisms, they allow for a cost reduction as no assembly is needed and the manufacturing process can often be simplified. Furthermore, they need reduced maintenance because no wear occurs and no lubrication has to be provided. This highly increases the mechanism reliability, and makes them suitable for working in difficult environments such as vacuum or high temperatures. On the other hand, compliant mechanisms present a reduced mobility, since the material strength poses a limit to deflection that cannot be overcome. Another disadvantage is the need for a driving force/torque to control the mechanism configuration and to deform its flexible parts; such a force/torque is needed even when no resistant loads act on the mechanism. Because of these peculiar features, compliant mechanisms are often intended as complementary to ordinary mechanisms, rather than alternative. In precision engineering—that is, in applications such as optics or metrology, where the positioning accuracy is by far the most relevant requirement—compliant mechanisms are the only acceptable solution. In most cases, they take the form of ordinary mechanisms; only, the kinematic pairs are replaced by flexible elements—flexures from here on. Flexures work without clearance or hysteresis, and their behavior is highly predictable. Unfortunately, a new problem arises: the lack of tools to analyze such mechanisms. Much theory has been developed for the analysis of ordinary mechanisms; on the contrary, a systematic approach dealing with compliant mechanisms has not been presented yet. In [1], a partial answer to the problem is given by the pseudo-rigid-body model. The model is used to reduce the study of compliant mechanisms to that of ordinary mechanisms, and can be very useful in mechanism design. However, it cannot cope with compliant mechanisms for precision engineering, as the assumptions and approximations behind the model are not acceptable when the requirements on accuracy become too strict. The alternative to [1] is the non-linear finite-element (FE) analysis of mechanisms, commonly used by designers. The FE analysis can be simplified by using the ‘‘chain calculation’’ [2–5]: the links are divided into small elements, and for each element loads and displacements are studied via the stiffness matrix. The main difference between the chain calculation and ordinary FE techniques lies in the fact that in the chain calculation each element is dealt with separately from the others, and the results are then assembled kinematically: no large stiffness matrices have to be assembled, and a higher computational efficiency is achieved. Both ordinary FE techniques and the chain calculation have to be performed iteratively to meet the imposed boundary conditions. The FE approach is time-consuming both in the modelling and in the computation, and not suitable to quickly compare different designs. As a result, most designers rely on a few standard designs whose accuracy has already been tested, like the leaf-spring mechanism or the doublecompound mechanism [6–9]. In this paper, an iterative technique to analyze compliant mechanisms is presented. The technique deals with planar mechanisms containing flexures. It can be used to perform the position analysis of a mechanism once the input variables are given (that is, when the input link deflection is prescribed), and provides both the exact pose of each link and the driving torques/forces needed
1226
S. Venanzi et al. / Mechanism and Machine Theory 40 (2005) 1224–1239
to achieve that configuration. Unlike previous methods, flexures are considered as whole parts undergoing large deflections, and are not divided into smaller elements. This allows for an efficient solution and, most of all, for a modular approach: flexures can be considered as effective hinges, and any mechanism with such hinges can be designed and analyzed. In the paper, some common flexure designs are listed and modelled. After its general description, the technique is applied to a planar four-bar linkage.
2. Technique outline The position analysis of ordinary mechanisms can be a challenging task because of its intrinsic non-linearity. The problem is even more complex for compliant mechanisms: because of the flexible elements, the kinematic relations used for ordinary mechanisms are not sufficient, and both equilibrium and elasticity equations have to be added. The solution of the problem has thus to satisfy kinematics, equilibrium and elasticity at the same time. Some techniques exist for structures where small displacements only are considered, and the linear beam theory can be used. However, the linear beam theory cannot express the large deflections typical of flexures. It is then necessary to use non-linear analysis. In this paper, non-linear analysis is used with the following assumptions: • Geometric non-linearity is considered. Geometric non-linearity refers to the fact that deflections are large, and have to be accounted for when considering the equilibrium configuration. • Material non-linearity is not considered. All materials are assumed to obey Hookes law, and effects like plasticity, non-linear elasticity, hyperelasticity and creep are neglected. • Strain is assumed to be small, so that it does not cause significant changes in geometrical features like section areas or beam thicknesses. It must be noted that small strain does not imply small deflections. In this paper, the three problems mentioned before—kinematics, equilibrium, and elasticity— are faced separately. The first problem is solved by a kinematic analysis. An ideal hinge is introduced between one of the extremities of each flexure and the adjacent rigid link (see Fig. 1). In this way, the relative rotation between the flexure and its adjacent rigid link becomes possible, and the congruence condition on rotations is broken. The compliant mechanism is thus replaced by an ordinary mechanism, whose mobility is given by the ideal hinges, and whose flexures do not deflect. If
Fig. 1. Ideal hinge.
S. Venanzi et al. / Mechanism and Machine Theory 40 (2005) 1224–1239
1227
the input variables for this new mechanism are given, the position analysis can be performed as for ordinary mechanisms; the position and orientation (pose) of each link are obtained. The second problem is solved by a static analysis. The static analysis is performed on the mechanism with ideal hinges, and provides the reaction force in each ideal hinge caused by the external loads acting on the mechanism. It depends on the link pose, and therefore on the kinematic analysis of the previous step. The third problem is solved by a deflection analysis, the most innovative point in the new technique. It considers each flexure in its whole, and uses a differential equation describing its large displacements. This differential equation relates the deflected shape of a flexure with the load acting on it. When working in a plane, the deflected shape of the ith flexure can be described by a three-component vector, [Xi, Yi, Hi], where Xi and Yi are linear displacements of one end section, and Hi is an angular displacement of the same section. Analogously, the load acting on the ith flexure can be expressed by a three-component vector, [Fxi, Fyi, Mi], where Fxi and Fyi are forces on one end section, while Mi is a moment on the same section. The elasticity theory relates the two vectors; in general, given three arbitrary components among the six Xi, Yi, Hi, Fxi, Fyi, Mi, it is possible to determine the remaining three. A numerical procedure has been developed, allowing the determination of the components Xi, Yi and Mi when the components Fxi, Fyi and Hi are given. Such a procedure is detailed in the following section. The choice of the components to be determined depends on the two previous steps: the two force components Fxi and Fyi have been determined by the static analysis, while the relative rotation Hi is the same as the rotation in the ideal hinge. Imposing the same value for the flexure rotation Hi and for the ideal hinge has the meaning of re-introducing the congruence condition on rotation, which had been broken when performing the kinematic analysis. The deflection analysis provides the components Xi and Yi, and thus a deflected shape for the ith flexure; it also provides Mi, bending moment acting on the same flexure. These values can be used to perform further kinematic and static analyses of the mechanism. The new kinematic analysis is performed on the mechanism with ideal hinges. But this time, flexures are deflected (and their deflected shape is expressed by the values Xi and Yi, previously obtained). The new pose of each link is evaluated. The new static analysis is performed on the mechanism with ideal hinges. With respect to the previous static analysis, however, the link pose has been updated; moreover, the bending moments Mi have to be considered. The results of the last kinematic and static analyses can be used for a new deflection analysis; in this way, a loop is built. The loop is iterated until all variables converge. The iterative procedure can be represented by the block diagram in Fig. 2. The next section will describe the mathematics used to perform the deflection analysis; the last section will illustrate how the iterative procedure can be applied to a well-known mechanism, the four-bar linkage.
3. Deflection analysis There are no standard procedures to analyze large displacements of flexures. In this paper, a mathematical procedure dealing with flexures has been developed and implemented. The
1228
S. Venanzi et al. / Mechanism and Machine Theory 40 (2005) 1224–1239
Fig. 2. Iterative procedure.
procedure is based on the Bernoulli–Euler equation, solved without the usual hypothesis of small deflections. A closed-form solution of the procedure would require elliptic integrals [3]; because of the complexity of such analytical approach, a numerical solution has been preferred. The numerical solution strongly depends on the boundary conditions and on the flexure design. In this paper, four common flexure designs are analyzed. 3.1. Leaf spring The most intuitive flexure is the simple leaf spring. According to the linear beam theory, it constrains one degree of freedom in the plane, the relative displacement along the axial direction, and allows a displacement in the shear direction and a rotation. However, axial displacement, shear displacement and rotation exist at the same time when large deflections are considered. Their value depends on the load acting on the leaf spring; an extensive study on the leaf spring, modelled by means of the pseudo-rigid-body model, can be found in [10]. In this paper the pseudo-rigid-body model is not used. On the contrary, the leaf spring is studied using Bernoulli–Euler equation, which states dhðsÞ MðsÞ ¼ ds k
ð1Þ
where s is a linear coordinate along the leaf spring, h(s) is the section rotation, M(s) is the bending moment acting on the section and k is the bending stiffness of the leaf spring. Narrow leaf springs can be considered like beams, so that their bending stiffness is given by E Æ I (E is the material elastic modulus, I is the area moment of inertia of the cross section). Wide leaf springs can be considered like plates, so that their bending stiffness is given by E Æ I/(1 m2), where m is the Poisson ratio of the material [11].
S. Venanzi et al. / Mechanism and Machine Theory 40 (2005) 1224–1239
1229
Fig. 3. Leaf spring.
The bending moment on a section can be expressed as MðsÞ ¼ M þ Fy ðX xðsÞÞ Fx ðY yðsÞÞ
ð2Þ
where M is the moment acting on the free end of the leaf spring, [Fx, Fy] is the force acting on the same end, [X, Y] are the center section coordinates of the free end, and [x(s), y(s)] are the cartesian coordinates of the section where the bending moment is evaluated (see Fig. 3). Furthermore, the cartesian coordinates of the section and its rotation are linked by the relation dxðsÞ=ds ¼ cos hðsÞ
ð3Þ
dyðsÞ=ds ¼ sin hðsÞ
Combining Eqs. (1)–(3), the large deflection of a leaf spring can be mathematically formulated by a first order differential equation 3 3 2 2 ½M þ Fy ðX xðsÞÞ Fx ðY yðsÞÞ=k hðsÞ d6 7 7 6 ð4Þ cosðhðsÞÞ 5 4 xðsÞ 5 ¼ 4 ds sinðhðsÞÞ yðsÞ subject to the boundary 3 2 3 2 0 hð0Þ 7 6 7 6 4 xð0Þ 5 ¼ 4 0 5; yð0Þ
0
conditions 3 2 3 2 H hðLÞ 7 6 7 6 4 xðLÞ 5 ¼ 4 X 5 yðLÞ
ð5Þ
Y
where H is the rotation of the leaf-spring end. In other words, the large deflection of the leaf spring can be expressed by a three-component function which has to satisfy the first order differential equation (4). Boundary conditions are applied to both bounds of the independent variable, the linear coordinate s, that varies between 0 and L, length of the leaf spring. In Eqs. (4) and (5) six parameters appear: M, moment on one end of the leaf spring; Fx and Fy, components of the force on the same end; H, rotation of the end section; and X and Y, expressing the center position of the end section. The problem can be solved numerically as an ordinary boundary-condition problem [12]. Three of the above listed parameters can be found, provided the other three are given.
1230
S. Venanzi et al. / Mechanism and Machine Theory 40 (2005) 1224–1239
The accuracy of Eq. (4) can be increased by adding the effects of the axial force acting in each section. The axial force is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Fy 2 2 ð6Þ N ðsÞ ¼ Fx þ Fy cos arctan hðsÞ Fx Eq. (4) then becomes 3 3 2 2 ½M þ FyðX xðsÞÞ FxðY yðsÞÞ=k hðsÞ d6 7 7 6 cosðhðsÞÞ 5ð1 þ N ðsÞ=EAÞ 4 xðsÞ 5 ¼ 4 ds sinðhðsÞÞ yðsÞ
ð7Þ
where A is the cross section area. A numerical procedure was developed in MATLAB. The procedure implements Eqs. (7) and (5), and solves the boundary value problem using standard numerical techniques. The procedure receives H, Fx and Fy as inputs, and provides M, X and Y as outputs. 3.2. Cross hinge Two leaf springs can be used at the same time to improve the flexure performance. The result of the assembly—which is shown in Fig. 4 and will be called cross hinge from here on—is another kind of flexure. Such a flexure has the advantages of being stiff to axial and shear forces, and to be compliant to rotation only. Its performances can be further increased by using three leaf springs, two narrow external ones and one wide internal one (see Fig. 5); in this way, the hinge has a higher torsional stiffness. To a first order approximation, the cross hinge behaves like a perfect hinge. Because of these interesting properties, it has been widely studied [13–16]. The large-deflection model developed for the leaf spring has been adapted to the cross hinge. A six-component function replaces the three-component function: three components for the first leaf spring, three for the second one. The differential equation to be satisfied is not reported, as it looks exactly like Eq. (7). However, 12 parameters instead of six appear: M1 and M2 are the moment at the end of the 1st and 2nd leaf spring respectively, [Fx1, Fy1] and [Fx2, Fy2] are the end forces, H1
Fig. 4. Cross hinge.
S. Venanzi et al. / Mechanism and Machine Theory 40 (2005) 1224–1239
1231
Fig. 5. Cross hinge.
Fig. 6. Cross hinge.
and H2 are the final section rotations, [X1, Y1] and [X2, Y2] identify the final section positions (see Fig. 6). The boundary conditions have to be slightly modified to express the link between the two leaf springs. The values of the function on the boundary are 3 3 2 3 2 3 2 2 h1 ðLÞ 0 H1 h1 ð0Þ 6 x ð0Þ 7 6 0 7 6 x ðLÞ 7 6 X 7 7 6 7 6 1 7 6 17 6 1 7 7 6 7 6 7 6 6 6 y 1 ð0Þ 7 6 0 7 6 y 1 ðLÞ 7 6 Y 1 7 7 7 6 7 6 7 6 6 ð8Þ 6 h ð0Þ 7 ¼ 6 0 7; 6 h ðLÞ 7 ¼ 6 H 7 7 6 27 6 2 7 6 7 6 2 7 7 6 7 6 7 6 6 4 x2 ð0Þ 5 4 0 5 4 x2 ðLÞ 5 4 X 2 5 y 2 ð0Þ
0
y 2 ðLÞ
Y2
Six more conditions have to be added. Three come from the fact that the deflected shape of flexures has to satisfy kinematic conditions:
1232
S. Venanzi et al. / Mechanism and Machine Theory 40 (2005) 1224–1239
H1 ¼ H2 ¼ H ðX 1 X 2 Þ sin a þ ðY 1 þ Y 2 Þ cos a ¼ L cos a sin H
ð9Þ
ðX 1 þ X 2 Þ cos a þ ðY 2 Y 1 Þ sin a ¼ L cos aðcos H þ 1Þ and three from the fact that it has to satisfy equilibrium: ðFx1 þ Fx2 Þ sin a þ ðFy 1 Fy 2 Þ cos a ¼ Fx ðFx2 Fx1 Þ cos a þ ðFy 1 þ Fy 2 Þ sin a ¼ Fy 1 ½ðX 2 2
þ
þ X 1 LÞ cos a þ ðY 2 Y 1 Þ sin a ½ðFx1 þ Fx2 Þ sin a ðFy 1 þ Fy 2 Þ cos a 1 ½ðX 2 2
ð10Þ
X 1 Þ sin a ðY 2 þ Y 1 Þ cos a ½ðFy 1 Fy 2 Þ sin a þ ðFx1 Fx2 Þ cos a
þ M1 þ M2 ¼ M In Eq. (10), M is the moment, [Fx, Fy] is the force acting on the hinge movable part, and a is a geometrical feature of the flexure—see Fig. 6. The equations were implemented in MATLAB and used to perform the deflection analysis. Similarly to the previous case, the numerical procedure receives H, Fx and Fy as inputs, and provides as output the moment M and the position [X, Y] of the point to which the force [Fx, Fy] is applied. 3.3. Notch hinge Another common design for flexures is that reported in Fig. 7, the so called notch hinge [17–20]. This design has the advantage of concentrating the strain in its central section, so that its kinematic behavior more closely resembles that of a theoretical hinge. This advantage is paid for in terms of maximum deflection, as the stress in the central section reaches high values even with limited rotations. Different shapes have been proposed for the notch hinge; a thorough comparison about this can be found in [21], where linear deflections only are considered. In this paper, the more traditional design—that with circular notches—is studied by means of Eqs. (7) and (5). To make these equation suitable, the hinge cross section moment of inertia I and area A cannot be considered constant, but have to be replaced by functions involving the linear coordinate s, the flexure width, its minimum thickness, and its curvature (see Fig. 7). Such an approach has already been adopted by other authors to evaluate the flexure stiffness [17,21], whereas a more sophisticated modelling requires FE techniques [19]. A numerical model similar to that for
Fig. 7. Notch hinge.
S. Venanzi et al. / Mechanism and Machine Theory 40 (2005) 1224–1239
1233
Fig. 8. Haberland hinge.
the leaf spring and the cross hinge has been developed and implemented. Eq. (7) has been used, with k replaced by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 , 12 ð11Þ k ¼ E b t þ 2R ð4R2 ð2s LÞ2 Þ with L hinge length, b hinge width, t hinge minimum thickness, R hinge curvature. A further factor 1/(1 m2) is to be added for plates. 3.4. Haberland hinge The last kind of flexure studied and implemented is the one reported in Fig. 8; the design is similar to the cross hinge, but the two leaf springs are connected in their central section. This design was first proposed by R. Haberland [22], and grants a higher accuracy than the cross hinge. The flexure is however stiffer to rotation than the cross hinge, and allows smaller rotations. This flexure has been studied as the union of four leaf springs, all connected to the central point; for the sake of brevity, the differential equation and the boundary conditions are not reported. A numerical model similar to the previous ones was developed and implemented in MATLAB.
4. Numerical example In this section, the technique is applied to a four-bar linkage. The technique is described in detail, and the iteration is shown step by step. The studied mechanism is shown in Fig. 9. The length of each link is reported in Table 1 (all lengths are in mm). The links are interconnected by leaf springs, whose dimensions are reported in Table 2. All leaf springs are assumed to behave like beams, with a bending stiffness k = EI. All links are assumed to be infinitely stiff, in the sense that their deformation can be ignored with respect to the deflection in the leaf springs. The whole mechanism is made of aluminium (elastic modulus E = 69 GPa). The reference configuration in which all flexures are not deflected is reported in Fig. 9, with the crank angle a0 = p/2. The orientation of the leaf springs in that configuration are those reported in Fig. 9. All angles in the
1234
S. Venanzi et al. / Mechanism and Machine Theory 40 (2005) 1224–1239
Fig. 9. Compliant four-bar linkage.
Table 1 Mechanism dimensions Link
Length (mm)
Crank, L1 Coupler, L2 Rocker, L3 Ground, L4
20 60 60 80
Table 2 Leaf spring dimensions Leaf spring
Length (mm)
Width (mm)
Thickness (mm)
1 2 3 4
l1 = 10 l2 = 10 l3 = 15 l4 = 15
b1 = 5 b2 = 5 b3 = 5 b4 = 5
t1 = 0.1 t2 = 0.1 t3 = 0.1 t4 = 0.1
Table 3 Angles—non-deflected configuration Angle
Name
Value (rad)
Crank angle Coupler angle Rocker angle
a0 b0 c0
1.570796 0.774981 1.996102
not-deflected configuration are reported in Table 3. The only external load acting on the mechanism is a horizontal force from right to left applied in the middle of the rocker, FR = 0.5 N. The crank angle a is assumed as input variable to control the mechanism; the orientation of coupler and rocker is expressed by the two angles b and c (see Fig. 9). Let a clockwise rotation
S. Venanzi et al. / Mechanism and Machine Theory 40 (2005) 1224–1239
1235
of 0.2 rad be given to the crank. The technique provides the pose of each link in the mechanism after such a rotation, and the moment that has to be applied to the crank to achieve it. It will now be applied step by step. 4.1. Step 1—guess values A guess value has to be assumed for the deflected shape of flexures and for the bending moment acting on them. The undeflected shape of flexures can be assumed, therefore X 1 ¼ l1
Y1 ¼ 0
M1 ¼ 0
X 2 ¼ l2
Y2 ¼ 0
M2 ¼ 0
X 3 ¼ l3
Y3 ¼ 0
M3 ¼ 0
X 4 ¼ l4
Y4 ¼ 0
M4 ¼ 0
ð12Þ
4.2. Step 2—kinematic analysis The ideal hinges are introduced to obtain an ideal rigid-link mechanism—see Fig. 10. Based on the guess values for flexures, the kinematic analysis of the ideal mechanism can be performed. The loop closure equations are 8 X 1 cos a0 Y 1 sin a0 þ L1 cos a þ X 2 cos a Y 2 sin a þ L2 cos b > > > > < ¼ L4 þ X 4 cos c Y 4 sin c þ L3 cos c þ X 3 cos c Y 3 sin c 0 0 ð13Þ > sin a þ Y cos a þ L sin a þ X sin a þ Y cos a þ L sin b X > 1 0 1 0 1 2 2 2 > > : ¼ þX 4 sin c0 þ Y 4 cos c0 þ L3 sin c þ X 3 sin c þ Y 3 cos c In the two equations (13), only two unknowns appear, b and c, which can be easily determined: b ¼ 0:826851 rad
ð14Þ
c ¼ 1:942198 rad
Fig. 10. Ideal rigid-link mechanism.
1236
S. Venanzi et al. / Mechanism and Machine Theory 40 (2005) 1224–1239
With this result, it is possible to evaluate the rotation in each ideal hinge: • the first hinge rotates from a0 to a, therefore H1 = a a0 = 0.200 rad; • the angle between crank and coupler changes from p a0 + b0 to p a + b, therefore H2 = b b0 a + a0 = 0.252 rad; • the angle between rocker and coupler becomes c b from c0 b0, therefore H3 = c c0 b + b0 = 0.106 rad; • the rocker angle passes from c0 to c, therefore H4 = c c0 = 0.054 rad. 4.3. Step 3—static analysis After the kinematic analysis, the static analysis has to be performed. Three equilibrium equations can be written for the crank, the coupler and the rocker: 8 Fx1 þ Fx2 ¼ 0 > > > Fy þ Fy ¼ 0 > > 1 2 > > > Fy ðL1 cos a þ X 2 cos a Y 2 sin aÞ Fx2 ðL1 sin a þ X 2 sin a þ Y 2 cos aÞþ > 2 > > > > M 1 þ M 2 þ M ¼ 0 > > > > > Fx2 Fx3 ¼ 0 < Fy 2 Fy 3 ¼ 0 ð15Þ > > > L cos b þ Fx L sin b M M ¼ 0 Fy 3 2 2 3 > 3 2 > > > þ Fx þ F ¼ 0 Fx > 4 3 R > > > > Fy 4 þ Fy 3 ¼ 0 > > > > > Fy ðL cos c þ X 3 cos c Y 3 sin cÞ Fx3 ðL3 sin c þ X 3 sin c þ Y 3 cos cÞþ > : 3 3 M 4 þ M 3 12 F R L3 sin c ¼ 0 where [Fxi, Fyi] is the force acting on the ith hinge, Mi is the moment acting on that hinge (null at this first step), M is the external moment to apply to the crank, FR is the resistant force on the rocker (see Fig. 11). The system above is a linear system of nine equations in the nine unknowns Fxi, Fyi, i = 1, . . ., 4, and M. The solution of the system is Fx1 ¼ 0:084 N
Fy 1 ¼ 0:092 N
Fx2 ¼ 0:084 N
Fy 2 ¼ 0:092 N
Fx3 ¼ 0:084 N
Fy 3 ¼ 0:092 N
Fx4 ¼ 0:216 N
Fy 4 ¼ 0:092 N
ð16Þ
M ¼ 1:933 N mm 4.4. Step 4—deflection analysis The values Hi, Fxi and Fyi, obtained in the previous analysis, can be used in Eqs. (7) and (5), to obtain new estimates for Xi, Yi and Mi. For each flexure, the force [Fx, Fy] has to be projected in the proper direction to obtain the axial and shear components.
S. Venanzi et al. / Mechanism and Machine Theory 40 (2005) 1224–1239
1237
Fig. 11. Static analysis.
H1 ¼ 0:200 Fx1 ¼ 0:092 Fy 1 ¼ 0:084 ) M 1 ¼ 0:943 X 1 ¼ 9:953 Y 1 ¼ 0:774 H2 ¼ 0:252 Fx2 ¼ 0:107 Fy 2 ¼ 0:064 ) M 2 ¼ 0:307 X 2 ¼ 9:862 Y 2 ¼ 1:482 H3 ¼ 0:106 Fx3 ¼ 0:055 Fy 3 ¼ 0:112 ) M 3 ¼ 1:040 X 3 ¼ 14:983 Y 3 ¼ 0:282
ð17Þ
H4 ¼ 0:054 Fx4 ¼ 0:172 Fy 4 ¼ 0:159 ) M 4 ¼ 1:215 X 4 ¼ 14:952 Y 4 ¼ 0:997 After this step, the procedure goes back to step 2—to be performed with the new values for Xi, Yi and Mi. 4.5. Final results After few iterations, the procedure converges towards a solution. The procedure has been tested with several planar mechanisms, different both in topology and in their geometrical features. The only convergence problems were found when buckling occurred in the flexures: since buckling is intrinsically connected with instability, it is possible to state that such convergence problems do not depend on the model. The numerical value of the two angles b and c after six iterations is b ¼ 0:817134 rad c ¼ 1:947244 rad
ð18Þ
whereas the correction from the 5th to the 6th step is 2.5e06 and 2.7e06 rad respectively. The counterclockwise moment to be applied to the crank to obtain the prescribed rotation is 4.023 N mm. The deflected shape of the mechanism is reported in Fig. 12. The same analysis was performed with non-linear FE software. The results obtained are b ¼ 0:817624 rad c ¼ 1:948497 rad
ð19Þ
There is a good match between the two models, as the difference is 0.5 and 1.3 mrad respectively.
1238
S. Venanzi et al. / Mechanism and Machine Theory 40 (2005) 1224–1239
Fig. 12. Deflected shape of the mechanism obtained with the proposed technique.
5. Conclusions In this paper, a new technique has been presented. The technique performs the position analysis of planar compliant mechanisms: it receives as input the mechanism deflection, and provides as output the actual pose of each link, the deflected shape of each flexible element, and the forces/moments needed to deflect the mechanism. Unlike other techniques, it does not rely on linear approximations and small deflections, but considers large elastic deflections for flexible elements. The technique is particularly suitable for mechanisms with compliant pairs (flexures), because it allows for a modular approach to their description. The paper describes some common flexures and mathematical models are provided for them. These hinge models can be used either as self-standing units, or as modules to study more complex mechanisms: they can be embedded as independent parts into any mechanism, so that the mechanism modelling becomes much quicker. Furthermore, using dedicated models to study flexures considerably reduces the computational time. In the last part of the paper, the technique is applied to a four-bar linkage. For this mechanism, the technique provides both the deflected shape of the entire mechanism and the moment to apply to deflect it to the desired shape. The technique shows a good match when compared with nonlinear finite-element analysis.
Acknowledgments The authors gratefully acknowledge the financial support of the Italian MIUR and CNR.
References [1] L.L. Howell, Compliant Mechanisms, John Wiley & Sons, 2001. [2] R.E. Miller, Numerical analysis of a generalized plane elastica, International Journal for Numerical Methods in Engineering 15 (1980) 325–332. [3] K. Mattiasson, Numerical results from large deflection beam and frame problems analysed by means of elliptic integrals, International Journal for Numerical Methods in Engineering 17 (1981) 145–153. [4] B.A. Coulter, R.E. Miller, Numerical analysis of a generalized plane elastica with non-linear material behavior, International Journal for Numerical Methods in Engineering 26 (1988) 617–630.
S. Venanzi et al. / Mechanism and Machine Theory 40 (2005) 1224–1239
1239
[5] T.C. Hill, A. Midha, A graphical, user-driven Newton Raphson technique for use in the analysis and design of compliant mechanisms, ASME Journal of Mechanical Design 112 (1990) 123–130. [6] G.H. Neugebauer, Designing springs for parallel motion, Machine Design 8 (1980) 119–120. [7] S. Smith, D. Chetwynd, Foundations of Ultraprecision Mechanism Design, Gordon & Breach, 1992. [8] J. van Eijk, On the Design of Plate-spring Mechanisms, Delft University of Technology, 1985. [9] R.V. Jones, Instruments and Experiences, Papers on Measurement and Design, John Wiley & Sons, 1988. [10] C. Kimball, L.W. Tsai, Modeling of flexural beams subjected to arbitrary end loads, ASME Journal of Mechanical Design 124 (2002) 223–235. [11] Timoshenko, Theory of Plates and Shells, McGraw-Hill, 1989. [12] G. Lindfield, J. Penny, Numerical Methods Using Matlab, Ellis Horwood Limited, 1995. [13] J.A. Haringx, The cross-spring pivot as a constructional element, Applied Scientific Research, A 1 (4) (1949) 313– 332. [14] W.H. Wittrick, The theory of symmetrical crossed flexure pivots, Australian Journal of Scientific Research, A 1 (2) (1948) 121–134. [15] S. Zelenika, F. De Bona, Analytical and experimental characterisation of high-precision flexural pivots subjected to lateral loads, Precision Engineering 26 (2002) 381–388. [16] B.D. Jensen, L.L. Howell, The modeling of cross-axis flexural pivots, Mechanism and Machine Theory 37 (5) (2002) 461–476. [17] J.M. Paros, L. Weisbord, How to design flexure hinges, Machine Design 11 (1965) 151–156. [18] S. Zhang, E.D. Fasse, A finite-element-based method to determine the spatial stiffness properties of a notch hinge, ASME Journal of Mechanical Design 123 (2001) 141–147. [19] G. Xu, L. Qu, Some analytical problems of high performances flexure hinge and micro-motion stage design, in: Proceedings of The IEEE International Conference on Industrial Technology, 1996, pp. 771–775. [20] J.W. Ryu, D.G. Gweon, Error analysis of a flexure hinge mechanism induced by machining imperfection, Precision Engineering 21 (2/3) (1997) 83–89. [21] N. Lobontiu, J.S.N. Paine, E. Garcia, M. Goldfarb, Design of symmetric conic-section flexure hinges based on closed-form compliance equations, Mechanism and Machine Theory 37 (2002) 477–498. [22] German Patents 2626800 and 2653427, and related US Patent 4261211.