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Research paper
Kinetostatic modeling of complex compliant mechanisms with serial-parallel substructures: A semi-analytical matrix displacement method Mingxiang Ling a,b, Junyi Cao a,∗, Larry L. Howell c, Minghua Zeng a a
State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an 710049, China Institute of Systems Engineering, China Academy of Engineering Physics, Mianyang 621999, China c Department of Mechanical Engineering, Brigham Young University, Provo, UT 84602, USA b
a r t i c l e
i n f o
Article history: Received 11 December 2017 Revised 25 March 2018 Accepted 26 March 2018 Available online xxx Keywords: Compliant mechanisms Flexure hinge Flexible manipulator Transfer matrix method
a b s t r a c t Kinetostatic analysis of compliant mechanisms are crucial at the early stage of design, and it can be difficult and laborsome for complex configurations with distributed compliance. In this paper, a kinetostatic modeling method for flexure-hinge-based compliant mechanisms with hybrid serial-parallel substructures is presented to provide accurate and concise solutions by combining the matrix displacement method with the transfer matrix method. The transition between the elemental stiffness matrix and the transfer matrix of flexure hinges/flexible beams is straightforward, enabling the condensation of a hybrid serial-parallel substructure into one equivalent two-node element simple. A general kinetostatic model of the whole compliant mechanisms is first established based on the equilibrium equation of the nodal force. Then, a condensed two-port mechanical network representing the input/output force-displacement relations of single-degree-of-freedom (DOF) compliant mechanisms and the Jacobian matrix for multi-DOF compliant mechanisms are respectively built. Comparison of the proposed method with the compliance matrix method in previous literature, finite element analysis and experiment for three exemplary mechanisms reveals good prediction accuracy, suggesting its feasibility for fast performance evaluation and parameter optimization at the initial stage of design. © 2018 Elsevier Ltd. All rights reserved.
1. Introduction Unlike traditional rigid-body mechanisms, compliant mechanisms transmit motion, force and energy through the elastic deformation of their components. Compliant mechanisms can provide highly accurate smooth motions without wear, friction, backlash and often need no assembly [1]. Therefore, they have attracted widespread attentions in variety of scientific and industrial applications, including precision positioning stage, micro griping manipulation, precision manufacturing [2–5], and so forth. However, one of the disadvantages in comparison to conventional rigid-body mechanisms is that their design and analysis require simultaneous consideration of kinematic and elasto-mechanical behaviors. Compliant mechanisms are often configured serially and/or in parallel with various kinds of flexure hinges. The kinetostatics of these flexure-hinge-based compliant mechanisms can be analyzed similar to the multi-rigid-body mechanisms by
∗
Corresponding author. E-mail address:
[email protected] (J. Cao).
https://doi.org/10.1016/j.mechmachtheory.2018.03.014 0094-114X/© 2018 Elsevier Ltd. All rights reserved.
Please cite this article as: M. Ling et al., Kinetostatic modeling of complex compliant mechanisms with serialparallel substructures: A semi-analytical matrix displacement method, Mechanism and Machine Theory (2018), https://doi.org/10.1016/j.mechmachtheory.2018.03.014
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modeling the flexure hinge as an equivalent joint with springs based on the pseudo-rigid-body model (PRBM) [1]. Serial compliant mechanisms can also be easily analyzed by employing the transfer matrix method or the chain algorithm [6]. However, for complex configurations with serial-parallel substructures and distributed/hybrid compliance, which are widely applied in engineering, the above mentioned modeling methods are incomplete and other methods are needed. Finite elemental analysis is one of the most popular methods for analyzing compliant mechanisms. It can provide excellent solutions for general structures with no constraint on geometry. However, the procedure is time consuming for frequent iterations at the early stage of design where many concepts should be evaluated in a short period of time. Meanwhile, a sufficiently large number of elements are inevitable for flexure hinges to obtain a reliable result. Thus, it requires too much time to be feasible for fast performance prediction and real-time controller design, though it is often used to verify the performance before fabrication. Considering the limitations of FEM, theoretical modeling for compliant mechanisms has been an interesting and popular topic since the pioneer works of analytical modeling for flexure hinges by Paros and Weisbord [7] and the subsequent works by Howell, Lobontiu and other scholars [1,8]. Many theoretical methods are now available such as the PRBM [5], the principle of virtual work [9–11], the Castigliano’s second theorem [12,13], the compliance matrix method [14,15] and the constraint-force-based method [16]. PRBM was proposed to solve the problem of large deformations [1,17–19] and now it is also used for theoretical modeling of compliant mechanisms by assuming the flexure hinge as a joint with springs [20]. Ma [21], Xu [22], Mottard [23] and Qi et al. [24] established kinetostatic models for rhombic and bridge-type amplifiers based on the elastic beam theory; and the accuracy of these models was enhanced recently by Ling et al. [9]. Meanwhile, the virtual work principle was also used to study the displacement attenuation in multistage compliant mechanisms [25]. Lobontiu and other scholars applied the Castigliano’s second theorem to analyze the static performance of typical compliant mechanisms [8,12,13]. Generally, the Castigliano’s second theorem and the elastic beam theory are popular for simple structures. For complex compliant mechanisms or their composed systems with all kinds of flexure hinges, kinetostatic modeling can resort to the compliance matrix method. Li and others carried out plenty of kinetostatic modeling for precision positioning stages using the compliance matrix method [14,15,26,27]. Pham et al. [28] and recently Lobontiu [29], Jiang et al. [30] separately proposed kinetostatic modeling methods for complex compliant mechanisms with serial-parallel substructures based on the compliance matrix method. However, the modeling procedures are still complicated. A modeling method similar to the rigid multi-body dynamics proposed by Ryu et al. [31] has been further developed and is now used for static and dynamic analyses of precision positioning stages [32,33]. It should also be noted that only the compliance of the flexure hinge was considered while that of the flexible beam was neglected in most of the aforementioned modeling methods, which may lead to low prediction accuracy. The contribution of this paper is to present a general kinetostatic modeling method for flexure-hinge-based compliant mechanisms with complex serial-parallel substructures having distributed/hybrid compliance by combining the matrix displacement method with the transfer matrix method, which is different from the compliance matrix method in [14,15,26–30]. The paper is limited to planar mechanisms due to their extensive applications. It can be extended to spatial cases with a similar procedure. Besides, this paper mainly deals with the problem of small deformations. For nonlinearly large deformation, the readers are recommended to refer to the previous works of Howell [1], Awta [34], Chen [35], Su [18] or others [36]. Interestingly, these previous pseudo-rigid-body or nonlinear models would be further modified and included as an element or a module into the presented general model for large deformations. The paper is organized as follows. A general kinetostatic modeling approach without condensation is presented in Section 2. The condensed modeling procedure is conducted in Section 3. The consideration of the rotational motion for multi-DOF compliant mechanisms is discussed in Section 4. Then, summary of the proposed modeling methodology as well as its numerical and experimental verifications are illustrated in Section 5 and Section 6, respectively. The conclusions are made in the final section. 2. Kinetostatic model based on the matrix displacement method In applications, most flexure-hinge-based compliant mechanisms are organized serially and/or in parallel. Fig. 1(a) provides the configuration of a piezo-actuated precision positioning stage. It is formed by serial and parallel branch chains with three parts: (i) flexure hinge; (ii) flexible beam (the guiding beam is also termed ‘flexible beam’ in the following); and (iii) lumped mass (e.g. output port). It can represent many cases in applications. Fig. 1(b) is a generic topology abstracted from Fig. 1(a), which will be used as the exemplary configuration for the proposed modeling procedure. Fig. 2 illustrates the serial and parallel branch chains cut from the topology in Fig. 1(b). A serial branch chain is composed of several flexible beams interconnected by flexure hinges with the characteristics of one junction node connecting only two elements, while one junction node has at least three elements interconnected in a parallel branch chain. The proposed modeling procedure consists of three steps: Step 1: Discretization and numbering A compliant mechanism is first discretized into the flexure hinge, the flexible beam and the lumped mass according to the configuration. The flexure hinge bears the largest deformation and the flexible beam bears different levels of deformation in different configurations. For example, the guiding beams in Fig. 1(a) bear large deformation while other flexible beams have small deformation. In the proposed modeling method, compliances of the flexure hinge and the flexible beam are both considered. The lumped mass bears little deformation compared to the other two elements. As shown in Fig. 3, the flexure Please cite this article as: M. Ling et al., Kinetostatic modeling of complex compliant mechanisms with serialparallel substructures: A semi-analytical matrix displacement method, Mechanism and Machine Theory (2018), https://doi.org/10.1016/j.mechmachtheory.2018.03.014
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Output port
Guiding beam
Enlarged view
Lumped mass
Clamped
Flexure hinge
Flexible beam Flexure hinge
3
Clamped Junction node
Output port
Flexible beam y
Clamped
Piezostacks
o
x
Clamped
fp Clamped
Fixed block (a)
Clamped Lumped mass
(b)
Input port
Fig. 1. A compliant mechanism with serial-parallel sub-chains and its topology. (a) Exemplary configuration. (b) General topology with flexure hinges shown in yellow and flexible beam in blue.
Serial
Flexible beam
Node
Flexure hinge
Parallel
Parallel
Fig. 2. Illustration of serial and parallel substructures.
(13)
(14) 10 0 (11) (16) 11 (15) Clamped 9 Clamped (10) (17) 8 12 (9) (18) (19) 14 (21) 15 (6) 6 (8) 5 7 (22) (5) (20) (7) y 13 4 0 0 16 (23) (4) Clamped Clamped x 3 17 o (24) (3) (2) 1 (34) (27) (25) 20 19 26 2 18 f p (28) (33) (1) (26) 21 0 25 0 Clamped 24 (32) (29) 22 Clamped (31) (30) 23 0
(12)
Fig. 3. Discretization and numbering of the mechanism.
wk ,Fky wj ,Fjy φj, Mj
φk, Mk uj , Fjx
wk ,Fky uk , Fkx
yi
Flexible beam
wj ,Fjy φj, Mj
y oi
θi
xi x
φk, Mk
uk , Fkx
uj , Fjx Flexure hinge
Fig. 4. Nodal displacement and nodal force of the ith beam-like element.
hinges and the flexible beams are denoted serially from (1) to (34) and are connected with nodes from 1 to 26; all the clamped nodes are numbered as 0. The actuating force is denoted as fp . It should be noted that the contact beam with the piezo-stacks is modeled as a rigid lumped mass, which is constant with practice where the contact beam is often designed as rigid as possible without deflection to enhance the output displacement. By discretizing, the compliant mechanism is transformed into a topology possessing a finite number of degrees of freedom. Step 2: Build the elemental stiffness matrix As shown in Fig. 4, each flexible beam is a uniform cross-section beam while the flexure hinge is regarded as a variable cross-section beam. Both of the resulting beam-like elements have two nodes, j and k, with three degrees of freedom per Please cite this article as: M. Ling et al., Kinetostatic modeling of complex compliant mechanisms with serialparallel substructures: A semi-analytical matrix displacement method, Mechanism and Machine Theory (2018), https://doi.org/10.1016/j.mechmachtheory.2018.03.014
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node: axial displacements uj and uk ; transverse deflections wj and wk and rotations ϕ j and ϕ k in the local coordinate system oi xi yi . The nodal displacement of the ith beam-like element in the local frame can be expressed as
T {q¯ i } = u j , w j , ϕ j , uk , wk , ϕk
(1)
where subscripts j and k denote the serial number of nodes connected to the ith beam-like element. Superscript T denotes the transpose of a matrix. The nodal force of the ith beam-like element, {F¯i } = [Fjx , Fjy , M j , Fkx , Fky , Mk ]T , in the local coordinate system can be expressed in the form of generalized Hooke’s law
F¯i = K¯ i · {q¯ i }
(2)
where K¯ i is the elemental stiffness matrix of the ith beam-like element. For the flexible beam with a uniform cross-section, its elemental stiffness matrix, K¯ i , in the local coordinate system can be founded based on the finite element theory
⎡ ⎢ ⎢
E Ai /li
0 12E Ii /(1 + β )li3
K¯ ib = ⎢ ⎢
⎣
0 6E Ii /(1 + β )li2 (4 + β )E Ii /(1 + β )li
Sym
−E Ai /li 0 0 E Ai /li
0 −12E Ii /(1 + β )li3 −6E Ii /(1 + β )li2 0 12E Ii /(1 + β )li3
⎤
0 6E Ii /(1 + β )li2 ⎥ (2 − β )E Ii /(1 + β )li⎥ ⎥ (3) ⎥ 0 ⎦ −6E Ii /(1 + β )li2 (4 + β )E Ii /(1 + β )li
where superscript b denotes the flexible beam. Ai and Ii are the area and moment of inertia about the neutral axis of the cross-section. E is the Young’s modulus. li is the length of the ith beam-like element. Here, a Timoshenko beam is considered and β = 12EIi /κ GAi li 2 , where G is the shear modulus, κ = 5/6; β = 0 means an Euler-Bernoulli beam. For the flexure hinge, since has a variable cross-section and large curvature, its elemental stiffness matrix can not be easily and accurately calculated. Actually, theoretical models [37,38] and empirical formulas [39,40] for the compliance matrix of flexure hinges featuring elliptical, corner-filleted, circular, and so on, were developed in the past investigations. Therefore, the elemental stiffness matrix of the flexure hinge needed here can be transformed from the existing analytical compliance matrix of flexure hinges. Assuming a planar flexure hinge as a 3-DOF elastic element, its compliance matrix, Ci , can be expressed as a standard form in its local coordinate system oi xi yi [8,29,41] as
Ci =
cx 0 0
0 cy cα
0 cα cθ
=
kx 0 0
0 ky kα
0 kα kθ
−1 (4)
where cx cy , cα , cθ and kx ky , kα , kθ are the coefficients of the compliance matrix Ci and its inverse. The detailed expressions can be found from previous papers [8,37–41]. The elemental stiffness matrix of the ith flexure hinge, K¯ ih (here superscript h denotes a flexure hinge), can be deduced as the function of the coefficients in Eq. (4). The detailed derivation is listed in Appendix A, and the ultimate expression is directly shown in Eq. (5).
⎡
kx
K¯ ih
⎢ ⎢ =⎢ ⎢ ⎣
0 ky
Sym
0 −kα kθ
−kx 0 0 kx
0 −ky kα 0 ky
⎤
0 −kα ⎥ −li · kα − kθ ⎥ ⎥ ⎥ 0 ⎦ kα kθ
(5)
Eq. (5) bridges the theoretical compliance matrix in previous literature with the elemental stiffness matrix needed here. It should be noted that since the ratio of length to thickness of the flexure hinge is small, shear deformation should be considered when choosing a compliance matrix for calculating Eq. (5). By performing rotation transformation, the nodal displacement and the nodal force in the local frame oi xi yi can be expressed in the reference frame oxyz, so Eq. (2) is expressed as
{Fi } = Ki · {xi }
(6)
where {Fi }= RTi {F¯i }, {xi }= RTi {q¯ i }, Ki = RiT K¯ i Ri . Rotation matrix Ri is determined by the orientation of the ith beam-like element. As illustrated in Fig. 4, θ i is the rotational angle of the ith local frame with respect to the reference frame, and Ri can be written as Please cite this article as: M. Ling et al., Kinetostatic modeling of complex compliant mechanisms with serialparallel substructures: A semi-analytical matrix displacement method, Mechanism and Machine Theory (2018), https://doi.org/10.1016/j.mechmachtheory.2018.03.014
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⎡
cos θi ⎢− sin θi ⎢ 0 Ri = ⎢ ⎢ 0 ⎣ 0 0
sin θi cos θi 0 0 0 0
0 0 1 0 0 0
0 0 0 cos θi − sin θi 0
0 0 0 sin θi cos θi 0
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⎤
0 0⎥ 0⎥ ⎥ 0⎥ ⎦ 0 1
(7)
in which the plus and minus signs of θ i is defined in Fig. 4. Step 3: Kinetostatic model without condensation Eq. (6), which describes the elastic relationships between the nodal force and nodal displacement of the ith beam-like element, can be further rewritten as
Fi, j Fi,k
ki,1 = ki,3
xj ki,2 · ki,4 xk
(8)
where Fi,j and Fi,k are, respectively, the nodal force vector of the j-end and the k-end in the ith beam-like element. {xj } = [uj , wj , ϕ j ]T and {xk } = [uk , wk , ϕ k ]T are the two nodal displacements of the ith beam-like element in the reference frame, as shown in Fig. 4. Taking each node as the study object, forces exerted on the node by its connected flexure elements are the sum of the inverse nodal force of the corresponding beam-like elements and external forces. Thus, the following force balance equation can be obtained for the nth node
F N i, j (k )
= Pn
(9)
where N is the total number of flexure elements connected to the nth node. Fi,j ( k ) is the nodal force of the j-end or k-end of the ith beam-like element. Because the nth node may be connected to the j-end or k-end of the ith beam-like element as shown in Fig. 3, so Fi,j ( k ) is set to Fi,j if the nth node is connected to the j-end; otherwise Fi,j ( k ) is set to Fi,k . Pn is the sum of external forces directly exerted on the nth node or equivalently on the flexure elements. For the mechanism in Fig. 3, only a lumped piezo-actuated force fp acts on the node 23 and no force acts on the intra-elements, so P23 = [0, −fp , 0]T and Pn = 0 for other nodes. By substituting Eq. (8) into Eq. (9), the following equation can be obtained
N
(ki,1 x j + ki,2 xk ) or (ki,3 x j + ki,4 xk ) = Pn
(10)
where the former is selected if the nth node is connected to the j-end of the ith beam-like element; the latter is valid when the elemental force exerted on the node is from the k-end. Eq. (10) is the force equilibrium equation of the nth node. Considering the force balance of all nodes in sequence, and noting that the nodal displacements of all clamped nodes are zero, then the kinetostatic model of the whole compliant mechanism taking the nodal displacement vector as the variables can be established. Here, only the equilibrium equations of representative nodes 1, 2 and 23 are given to illustrate the kinetostatic model; the balance equations for other nodes can be similarly obtained based on Eq. (10) but are not listed here
⎧ ⎪ ⎨node1 : node2 :
⎪ ⎩
(k1,4 + k2,1 + k34,4 )x1 + k2,2 x2 + k34,3 x26 = 0 k 2,3 x1 + ( k 2,4 + k 3,1 ) x2 + k 3,2 x3 = 0
(11)
node23 : k30,3 x22 + (k30,4 + k31,1 )x23 + k31,2 x24 = P23
where ki, 1 , ki, 2 , ki, 3 , ki, 4 are the coefficients of the elemental block stiffness matrix in Eq. (8). The force equilibrium equations of all nodes constitute a linear equation set, which is the kinetostatic model of the whole compliant mechanism and contains twenty-six vector equations for the compliant mechanism in Fig. 3. The equation set can be easily solved, and the static performance such as the displacement amplification ratio and the input/output stiffness can be then obtained. The y-component of x23 and x10 are respectively the input and the output displacement of the mechanism in Fig. 3. Therefore, when actuating force fp is applied, the displacement amplification ratio R and the input stiffness Kin of the compliant mechanism can be calculated as
R=
x10(y ) , x23(y )
Kin =
fp −x23(y )
(12)
where x10( y ) and x23( y ) are respectively the displacement of node 10 and node 23 in the y-axis. 3. Kinetostatic modeling with condensed configuration The kinetostatic model presented in Section 2 requires solving an equilibrium equation set with large degrees of freedom. In this section, the transfer matrix method is flexibly combined with the above matrix displacement method to condense a serial-parallel branch chain into an equivalent two-node element to further establish the input/output force-displacement Please cite this article as: M. Ling et al., Kinetostatic modeling of complex compliant mechanisms with serialparallel substructures: A semi-analytical matrix displacement method, Mechanism and Machine Theory (2018), https://doi.org/10.1016/j.mechmachtheory.2018.03.014
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The first condensation (13) Output port (14) Clamped 10 0 (12) (11)(16) (15) 11 Flexure Flexible 9 (10)(17) hinge beam 8 12 5 (6) 6 (8) (9) (18) (19) 14 (21) 15 7 y 13 (5) (20) (22) (7) 4 0 0 16 (4) Clamped Clamped17 (23) 3 o x (3) (2) 1 (34) (27)19 (25) (24) 26 fp 20 18 2 (1) (33) (28) (26) 21 0 25 0 Clamped 24 (32) (29) 22 Clamped (31) (30) 23
Clamped 0
(a)
Input port
The second condensation Output port
Clamped Condensed element 2
Output port Clamped
10
Clamped
Condensed Equivalent element 4 element 1
j
k y
Condensed element 1 (b)
o
Condensed element 3
Input port
Equivalent element 2
x fp
k
23
Clamped
k
fo y
j
k
o x j fp j
k
10
k
j
(c)
23
Input port
j
Fig. 5. Condensed configuration by the transfer matrix method.
relations of a compliant mechanism. As shown in Fig. 5, by condensing twice, the local parallel and serial substructures are simplified as two elements with equivalent stiffness matrix. As shown in Fig. 5(b), the locally parallel chain with three elements interconnected at one common node is first condensed as one equivalent two-node element. The equivalent elemental stiffness matrix is deduced for node 1 in the following and a similar procedure can be performed for the other three parallel sub-chains. Taking node 1 as the study object, forces on this node exerted by its connected flexure elements are the sum of the inverse nodal forces of these connected flexure elements, i. e.
F1,k + F2, j + F34,k = 0
(13)
By substituting the nodal force with the nodal displacement in Eq. (8), the force balance equation of node 1, i.e. Eq. (13), is transferred to
k1,3 · 0 + (k1,4 + k2,1 + k34,4 )x1 + k2,2 x2 + k34,3 x26 = 0
(14)
The nodal force of flexure beam (2) at the k-end, i.e. at the node 2, and the nodal force of flexure beam (34) at the j-end, i.e. at the node 26, can be directly obtained based on Eq. (8)
F2,k = k2,3 x1 + k2,4 x2 F34, j = k34,1 x26 + k34,2 x1
(15)
Substituting Eq. (14) into Eq. (15) and eliminating intermediate variable x1 , the equivalent elemental stiffness matrix of the condensed two-node element can be solved as
F34, j F2,k
=
−1
k34,1 − k34,2 · (k1,4 + k2,1 + k34,4 ) −1
−k2,3 · (k1,4 + k2,1 + k34,4 )
· k34,3
· k34,3
−1
−k34,2 · (k1,4 + k2,1 + k34,4 )
· k 2,2
−1
k2,4 − k2,3 · (k1,4 + k2,1 + k34,4 )
· k 2,2
x26 · x2
(16)
Through the first condensation, the hybrid serial-parallel chain becomes a sole serial chain, and the resulting serial chain can be further condensed as one equivalent two-node element with the transfer matrix method, as shown in Fig. 5(c). The transfer matrix of the ith beam-like element or the equivalent two-node element in the reference frame can be easily deduced from Eq. (8) as
xk −Fi,k
= Ti ·
xj Fi, j
=
ti,1 ti,3
xj ti,2 · ti,4 Fi, j
=
−ki,−1 · ki,1 2 ki,4 · ki,−1 · ki,1 − ki,3 2
xj ki,−1 2 · Fi, j −ki,4 · ki,−1 2
(17)
where the global transfer matrix Ti of the ith beam-like element or the equivalent two-node element relates the displacements and forces at both ends of the element. Through the condensation in Fig. 5(c), the transfer matrices of the new equivalent two-node elements transferred from the serial chains in Fig. 5(b), are
T1(II ) = T11 · T10 · T9 · T2(I ) · T5 · T4 · T3 · T1(I ) · T33 · T32 · T31 T2(II ) = T16 · T17 · T18 · T4(I ) · T22 · T23 · T24 · T3(I ) · T28 · T29 · T30
(18)
where superscript (II) denotes the second condensation. Ti ( I ) (i = 1, 2, 3, 4) are the four transfer matrices of the equivalent two-node element in Fig. 5(b). Please cite this article as: M. Ling et al., Kinetostatic modeling of complex compliant mechanisms with serialparallel substructures: A semi-analytical matrix displacement method, Mechanism and Machine Theory (2018), https://doi.org/10.1016/j.mechmachtheory.2018.03.014
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Clamped
Flexure hinge Lumped mass
xin1 Fp1
h1
7
Clamped
l2
l4
l1
l3
Clamped
Piezostacks
Plat form Flexible beam
l2
w
y
(a)
x
o
a b
Elliptic flexure hinge
he
(b)
l5 l6
Enlarged view
Guiding beam Fixed block
h2
Equivalent beam element
Fig. 6. Exemplary mechanism with rotational DOF. (a) Exemplary configuration. (b) Local structure and geometric parameters. (Noting: the configuration is not an optimal stage and is just used to illustrate the modeling procedure).
Then, the equivalent elemental stiffness matrix of the new condensed element in Fig. 5(c) can be again obtained from the corresponding transfer matrix T1 ( II ) and T2 ( II ) in Eq. (18)
Fi, j
Fi,k
(I I )
= Ki
·
xj
xk
=
ki,(I1I )
ki,(I2I )
(I I )
(I I )
ki,3
ki,4
·
xj
xk
=
−ti,−1 · ti,1 2
ti,4 · ti,−1 2
· ti,1 − ti,3
ti,−1 2 −ti,4 · ti,−1 2
·
xj
xk
(19)
As shown in Fig. 5(c), after performing condensation twice, the compliant mechanism is configured into a two-port mechanical network model. Considering node 10 at the output port and node 23 at the input port, and noting that the nodal displacements of all clamped nodes are zero, the equilibrium equation of the nodal force can be respectively established as
(k1(I,I1) + k2(I,I1) ) · x23 + (k1(I,I2) + k2(I,I2) ) · x10 = Fp (k1(I,I3) + k2(I,I3) ) · x23 + (k1(I,I4) + k2(I,I4) + k12,4 + k13,4 + k14,4 + k15,4 ) · x10 = Fo
(20)
where Fo is a virtual force acting on the output port to further calculate the output stiffness. Eq. (20) can be further rewritten as a more general form by defining input, output and coupled stiffness components [42]. Thus, the input/output force-displacement relations of the equivalent two-port mechanical network of the compliant mechanism have the standard form of the generalized Hooke’s law as
Fp Fo
=
Kin Kco
Kco · Kout
xin xout
(21)
where xin and xout are defined as the input and output displacements, and xin = x23 , xout = x10 . kin , kout and kco are the elements of the equivalent stiffness matrix of the two-port mechanical network of the compliant mechanism and are equal to
Kin
Kco
Kco
Kout
=
k1(I,I1) + k2(I,I1)
k1(I,I2) + k2(I,I2)
k1(I,I3) + k2(I,I3)
k1(I,I4) + k2(I,I4) + k12,1 + k13,1 + k14,1 + k15,1
(22)
In this way, the internal DOFs of each serial and parallel substructure do not appear in the kinetostatic model but only the input/output force-displacement relations. Based on Eq. (21), the displacement amplification ratio and the input stiffness of the compliant mechanism can be easily obtained by letting Fo = 0, which has the same results as Eq. (12). The significance of the two-port mechanical network model, i.e. Eq. (21), is that it comprehensively describes the kinetostatic characteristics of the compliant mechanism from the viewpoint of input-output ports without the need to care about the internal DOFs. With the two-port mechanical network model, the kinetostatic performance of a compliant mechanism can be conveniently calculated. 4. Consideration of rotational lumped mass The lumped mass in the preceding sections was simplified as a node without considering the rotary DOF, which is exactly accurate for compliant mechanisms with only translational motions, such as X, Y and XY flexible manipulators. However, the dimensions and the rotary DOF of the lumped mass should be included for compliant mechanisms with rotary motion. Fig. 6 Please cite this article as: M. Ling et al., Kinetostatic modeling of complex compliant mechanisms with serialparallel substructures: A semi-analytical matrix displacement method, Mechanism and Machine Theory (2018), https://doi.org/10.1016/j.mechmachtheory.2018.03.014
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The first condensation
0
0 (1)
(7) (3) (5) 5 xin1 (6) Fp1 (2) 1 2 (4) 36 4 (8) (11) (9) 9 0 7 (10) 8 (15) (14) xo (12) 12 (16) (17) 10 Limb 1 11 (13)
xin3
xin1 Fp1
Fp3
j
The second condensation xin3
k j k
xin1 Fp3 Fp1 j xo Equivalent k element 1
xo Condensed element
xin3 j Fp3
k
k Equivalent element 3
Limb 3 Limb 2 (a)
y
y
xin2 Fp2
x
o
xin2
(b)
Fp2
o
Equivalent element 2 x (c)
j F xin2 p2
y o
x
Fig. 7. Condensed configuration by the transfer matrix method.
shows a XYθ planar compliant mechanism. The representative configuration was widely investigated in previous applications [31,43] and is adopted here as an example to illustrate the modeling procedure for rotary lumped mass. As shown in Fig. 6(b), the triangular platform with lumped mass can be equivalent to six beam-like elements with four nodes. Each node has one rotational and two translational degrees of freedom, and the displacement vector of the central node is defined as the output displacement of the compliant mechanism. As shown in Fig. 7(a), only one limb and the platform are needed to number considering the symmetry of the configuration. Elements (1) to (3) constitute a local parallel chain interconnected at node 1. Considering node 1, forces exerted on the node by its connected flexure elements is the sum of the inverse nodal forces of the corresponding flexure elements and external forces. By substituting the nodal force with the nodal displacement in Eq. (8), also noting that the nodal displacements of all clamped nodes are zero, the force balance equation of node 1 is
Fp1 =(k1,4 + k2,4 + k3,1 )xin1 + k3,2 x2
(23)
The nodal force of element (3) at node 2 can be directly obtained based on Eq. (8)
F3,k = k3,3 xin1 + k3,4 x2
(24)
By combining Eq. (23) and Eq. (24), the equivalent elemental stiffness matrix of the new condensed two-node element is
Fp1
(I )
= K1 ·
F3,k
xin1
x2
=
k 1,4 + k 2,4 + k 3,1
k 3,2
k 3,3
k 3,4
·
xin1
(25)
x2
Similarly, Elements (5)–(8) constitute a local parallel chain interconnected at node 4. By using the similar procedure as Eq. (25), the equivalent elemental stiffness matrix of the new condensed two-node element can also be deduced as
F5, j
(I )
= K2 ·
F8,k
x3 x6
−1
k 5,1 − k 5,2 ( k x )
=
−1
−k8,3 (kx )
−1
−k5,2 (kx )
k 5,3
k 8,2 −1
k 8,4 − k 8,3 ( k x )
k 5,3
k 8,2
·
x3 x6
(26)
where intermediate variable kx = k5,4 + k8,1 + k6,1 − k6,2 (k6,4 + k7,1 )−1 k6,3 . As shown in Fig. 7(b), through the first condensation, the serial-parallel chain in limb 1 becomes a sole serial chain. The resulting serial chain is again condensed into one equivalent element with the transfer matrix method, as shown in Fig. 7(c). The transfer matrix of the new equivalent two-node element in Fig. 7(c) can be calculated as
Tin1 = T11 · T10 · T9 · T2(I ) · T4 · T1(I )
(27)
where the global transfer matrix T1 ( I ) and T2 ( I ) are, respectively, the transfer matrix of the equivalent stiffness matrix of K1 ( I ) and K2 ( I ) in Eqs. (25) and (26). Then, the equivalent elemental stiffness matrix Kin 1 of the new condensed two-node element of limb 1 in Fig. 7(c) can be again obtained from the corresponding transfer matrix Tin 1 in Eq. (27). The transition formula is the same as Eq. (19) in Section 3 and is omitted here. Based on the equivalent stiffness matrix, the input/output force-displacement relations of the condensed element of limb 1 is
Fp1
Fout1
= Kin1 ·
xin1 x9
=
kin1,1
kin1,2
kin1,3
kin1,4
·
xin1 x9
(28)
Please cite this article as: M. Ling et al., Kinetostatic modeling of complex compliant mechanisms with serialparallel substructures: A semi-analytical matrix displacement method, Mechanism and Machine Theory (2018), https://doi.org/10.1016/j.mechmachtheory.2018.03.014
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The equivalent stiffness matrices of other two limbs 2 and 3 can be obtained from Kin 1 by rotation transformation based on the symmetry of the compliant mechanism
Kin2 = R2T · Kin1 · R2
(29)
Kin3 = R3T · Kin1 · R3
where R2 and R3 are the rotation matrix in Eq. (7), the rotary angles in R2 and R3 are, respectively, −240° and −120°. Then, the input/output force-displacement relations of the condensed two-node element of these two limbs are, respectively
Fp2 Fout2 Fp3
xin2
= (R2T · Kin1 · R2 ) ·
Fout3
=(
R3T
x10
xin3
· Kin1 · R3 ) ·
=
=
x11
kin2,1
kin2,2
kin2,3
kin2,4
kin3,1
kin3,2
kin3,3
kin3,4
·
xin2
·
x10
xin3
(30)
(31)
x11
The following three equations can be directly obtained from Eqs. (28), (30) and (31)
⎧ F = kin1,1 xin1 + kin1,2 x9 ⎪ ⎨ p1
Fp2 = kin2,1 xin2 + kin2,2 x10
(32)
⎪ ⎩
Fp3 = kin3,1 xin3 + kin3,2 x11
As shown in Fig. 7(c), considering the nodes 9, 10, 11 and 12, the equilibrium equations of the nodal force can be established as
⎧ node9 : ⎪ ⎪ ⎪ ⎨
kin1,3 xin1 + (kin1,4 + k12,1 + k14,1 + k15,1 )x9 + k12,2 x10 + k14,2 x11 + k15,2 xout = 0 kin2,3 xin2 + (kin2,4 + k12,4 + k13,1 + k16,1 )x10 + k12,3 x9 + k13,2 x11 + k16,2 xout = 0
node10 :
⎪ node11 : kin3,3 xin3 + (kin3,4 + k13,4 + k14,4 + k17,1 )x11 + k14,3 x9 + k13,3 x10 + k17,2 xout = 0 ⎪ ⎪ ⎩ node12 : k15,3 x9 + k16,3 x10 + k17,3 x11 + (k15,4 + k16,4 + k17,4 )xout = 0
(33)
Based on Eqs. (32) and (33), the output displacement xout = [u, w, ϕ ]T of the central platform and the three input displacements xin 1 , xin 2 , xin 3 , can be analytically calculated as the explicit functions of the input actuating forces Fp 1 , Fp 2 , Fp 3
xin1 xin2 xin3
=
f11 f21 f31
f12 f22 f32
f13 f23 f33
{xout } = g1 g2 g3 ·
Fp1 Fp2 Fp3
·
Fp1 Fp2 Fp3
(34)
(35)
where coefficients fij and gi (i, j = 1, 2, 3) describe the stiffness properties between the input/output displacements and the input actuating forces. These coefficients are all explicit functions of the elemental stiffness matrix of the compliant mechanism. Also, the relationship between the output displacement and the input displacements can be further obtained by combining Eq. (34) with Eq. (35)
{xout } = J ·
xin1 xin2 xin3
= g1
g2
g3
f11
f21 f31
f12 f22 f32
f13 f23 f33
−1 ·
xin1 xin2 xin3
(36)
where J denotes the Jacobian matrix of the compliant mechanism, which represents the kinematic relations between the input and the output displacements. Here, the kinetostatic model of the triangular lumped mass is conducted and other shapes of the platform, such as the rectangular platform that is also frequently used in multi-DOF flexible manipulators with rotational motion, can be considered with a similar procedure. 5. Summary of the modeling method In Sections 2–4, three progressively kinetostatic modeling procedures are performed, respectively: (i) a general modeling method based on the matrix displacement method; (ii) further condensed two-port mechanical network model of compliant mechanisms with single-DOF; and (iii) consideration of rotary motion as well as the condensed Jacobian matrix for multiDOF compliant mechanisms. Please cite this article as: M. Ling et al., Kinetostatic modeling of complex compliant mechanisms with serialparallel substructures: A semi-analytical matrix displacement method, Mechanism and Machine Theory (2018), https://doi.org/10.1016/j.mechmachtheory.2018.03.014
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M. Ling et al. / Mechanism and Machine Theory 000 (2018) 1–16 Table 1 Results of the displacement amplification ratio with different methods. Thickness of the guiding beam (mm)
The proposed method
FEM
Experiment
Error with FEM
0.2 0.5 1.0 1.5
3.08 2.93 2.13 1.22
3.00 2.85 2.13 1.26
– – 1.95 –
0.08 0.08 0.00 0.04
(2.6%) (2.8%) (0.0%) (3.2%)
As a conclusion, the condensed two-port mechanical network model i.e. Eq. (21) is mainly used for compliant mechanisms with single input port-single output port (SISO); the Jacobian matrix i.e. Eq. (36) is mainly aimed at compliant mechanisms with multi input port-multi output port (MIMO) or multi input port-single output port (MISO). These models represent the kinetostatic performance of a compliant mechanism from the perspective of input-output ports and are explicit functions of the elemental stiffness matrices of the flexure hinge and the flexible beam. To obtain the expressions of these models, the matrix displacement method in conjunction with the transfer matrix method is developed. The proposed modeling method is essentially a finite element method except for the following two improvements: 1) An analytically elemental stiffness matrix with high accuracy and good flexibility is developed for all kinds of flexure hinges, which are typically variable-cross-section beams with large curvature. Therefore, the number of DOF is sharply reduced and the calculation efficiency is greatly improved compared to the usual FEM. So it can be regarded as a semi-analytical matrix displacement method. 2) The transfer matrix method is flexibly combined with the matrix displacement method to simplify the modeling complexity of serial-parallel compliant mechanisms. Condensing the serial branch chains with the transfer matrix method and establishing the equilibrium equation of nodal force for parallel branch chains are easy to operation because of the utilization of the elemental stiffness matrix but not the compliance matrix and procedures in previous compliance-matrix-based methods [14,15,26–33]. The presented method has the advantages of conciseness and generality and also exhibits a good flexibility for complex compliant mechanisms with serial-parallel substructures. Moreover, compliances of the flexure hinge and the flexible beam are both considered and included to enhance the prediction accuracy of compliant mechanisms with distributed/hybrid compliance. In Ref. [16], a constraint-force-based model was proposed for the kinetostatic modeling of compliant mechanisms based on the constraint force equilibrium equations and screw theory. The variable constraint force in this method is actually the nodal force in the proposed method. Meanwhile, nonlinear force-displacement relationship was included in the constraint-force-based model. For the proposed model, modified nonlinear or pseudo-rigid-body elemental stiffness matrix of large deformation can be further built and included as a module, and the preceding modeling procedure is still valid. 6. Verification and discussion Three case studies were conducted to verify the presented method: (i) the first compares the presented method with the results from the commercial finite element software ANSYS and experiment for the single-DOF compliant mechanism shown in Fig. 1; (ii) the second compares the presented method with the compliance matrix method in previous literature; (iii) the third application is an illustrative example of the multi-DOF compliant mechanism in Fig. 6 performed to highlight how to consider a lumped mass with rotary motion and provide an investigation on the influence of the geometric dimensions of the equivalent beam on the static performance. 6.1. Example one 6.1.1. Finite element calculation by ANSYS The compliant mechanism in Fig. 1 was geometrically modeled by Pro/E software and the static analysis was carried out with the commercial finite element software package ANSYS. The Solid45 element was chosen to build the model and the local mesh was refined for the flexure hinges and the guiding beams. The results were proven to be convergent and accurate enough. An actuating force of 100 N was exerted on the bottom flexible beam. The material is Aluminium with density ρ = 2770 kg/m3 , Young’s modulus E = 71 GPa and shear modulus G = 27 GPa. The flexure hinges are circular with identical parameters. Minimum thickness and radius of the circular flexure hinges are, respectively, 0.5 mm and 1.25 mm. Width of the whole stage is 10 mm. Thickness of all the flexible beams except for the bottom one are 3 mm. Other necessary geometric parameters are shown in Fig. 8(a). Four simulation sets were carried out by selecting different thickness of the guiding beams while keeping other parameters constant. Detailed results of the displacement amplification ratio and the input/output stiffness are comparatively listed in Tables 1 and 2. Fig. 8(b) provides one result of the static deformation by FEM. 6.1.2. Experiment A monolithic prototype was fabricated with the guiding beam’s thickness of 1 mm, while other material and geometric parameters are identical to the finite elemental model. The prototype and the experimental setup are shown in Fig. 9. A Please cite this article as: M. Ling et al., Kinetostatic modeling of complex compliant mechanisms with serialparallel substructures: A semi-analytical matrix displacement method, Mechanism and Machine Theory (2018), https://doi.org/10.1016/j.mechmachtheory.2018.03.014
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11
28mm
Constraint
26mm 8mm
20mm
13mm
20mm
fp
13mm
4mm 14mm
(a)
(b)
Fig. 8. Finite elemental calculation. (a) Finite elemental model. (b) Result of one simulation set.
Table 2 Results of the input and the output stiffness with different methods. Thickness of the guiding beam (mm)
Kin (N/μm)
0.2 0.5 1.0 1.5
Kou t (N/μm)
The proposed method
FEM
Error
The proposed method
FEM
Error
5.08 5.22 5.92 6.72
5.49 5.61 6.30 6.82
0.41 (7.5%) 0.39 (7.0%) 0.38 (6.0%) 0.10 (1.5%)
0.19 0.20 0.31 0.62
0.20 0.21 0.33 0.61
0.01 0.01 0.02 0.01
The prototype
Displacement measuring
Laser sensor
Data aquisition
Input to Piezo-stacks
Power amplifier
Signal generating
(5.0%) (4.8%) (6.1%) (1.6%)
Real-time simulator
Piezo-stacks Fig. 9. Prototype and experimental setup.
40 Input voltage Output displacement
50
0 1
20
2
3 Time (s)
4
Output displacement (μm)
Input voltage (V)
100
0 5
Fig. 10. Experimental results of the output displacement.
piezo-actuator and its power amplifier from the Physik Instrumente Inc. were used as the motion generator. The piezoactuator has a maximum output displacement of 15 μm at the driving voltage of 100 V. The fixed block of the compliant mechanism was mounted on an optical table to reduce the ground vibration. A real-time simulator with the data acquisition system from the Henrun Inc. of China was used to analyze the measurement data. A laser displacement sensor from KEYENCE with a 50 nm resolution was used to measure the output displacement. Up to 100 V non-negative sine voltages with 1 Hz were applied to the piezo-stacks. Fig. 10 shows the experimental results of the output displacement versus the Please cite this article as: M. Ling et al., Kinetostatic modeling of complex compliant mechanisms with serialparallel substructures: A semi-analytical matrix displacement method, Mechanism and Machine Theory (2018), https://doi.org/10.1016/j.mechmachtheory.2018.03.014
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7
30
6
25
Input stiffness (N/μm)
Displacement amplification ratio
12
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5 4 3 2
Thickness of the flexible beam: 3 mm 6 mm 9 mm 20 12 mm 15 15 mm 10
1
5
0 0 0.5 1 1.5 The thickness of the guiding beam (mm)
0 0 0.5 1 1.5 The thickness of the guiding beam (mm)
Fig. 11. Influence of the thickness of the flexible beam on the static performance.
input voltage. In order to calculate the displacement amplification ratio, the following formula was adopted.
R=
yout Kv · U · KP /(Kin + KP )
(37)
where yout is the experimental output displacement. Kp = 200 N/μm is the axial stiffness of the piezo-stacks. Kin is the input stiffness of the stage. Kv is the piezoelectric constant relating to the strain and is equal to 0.15 μm/V. U is the input voltage, V. 6.1.3. Discussion Tables 1 and 2 compare the results of the displacement amplification ratio, the input and output stiffness provided by the proposed method, four sets of FEM and experiment. Only one set of the experimental displacement amplification ratio is provided considering the high cost of the prototype. The displacement amplification ratio was computed by dividing the output displacement by input displacement under the actuating force of 100 N. The input and the output stiffness are the ratios of the input and output forces to their corresponding displacements. The analytical compliance matrix of the circular flexure hinge needed in the theoretical calculation was taken from [38]. The results show that the proposed method matches well with the results of FEM. The maximum deviation between the proposed method and FEM is less than 5% regarding the static displacement and less than 7.5% regarding the input and the output stiffness. It should be noted that the relative error of some simulation sets for the input and the output stiffness may appear large, but their absolute errors are small. In general, the prediction accuracy of the presented modeling method is satisfactory. On the other hand, there is an approximate 8.5% drop in the measured results of the displacement amplification ratio in comparing the results of the theoretical model and FEM. The main error source may be nonlinear hysteresis that was not considered in the theoretical model. The limited contact stiffness of Hertzian contact between the piezo-stacks and the compliant mechanism may also slightly attenuate the displacement of the piezo-stacks in the experiment, resulting in some reduced output displacement [5,20]. Compliances of the flexure hinge and the flexible beam are both considered in the proposed method, which is matched up with the actual deflection. To investigate the resulting difference of whether the compliance of the flexible beam is considered or not, the displacement amplification ratio and the input stiffness of the compliant mechanism shown in Fig. 1 are calculated under five different sets of thickness of the flexible beam while keeping other geometric parameters constant. The results are shown in Fig. 11. From the calculated results, two conclusions can be reached: •
•
Increasing the thickness of the flexible beam, which means that the flexible beam becomes stiffer, the displacement amplification ratio and the input stiffness monotonically increase and converge to a certain value when the thickness of the flexible beam further increases. This phenomenon is easy to understand that when the flexible beam becomes stiffer, less elastic energy will be stored in the mechanism and more actuating motions will be transformed and output by the compliant mechanisms. In the pervious investigations, the thickness of the flexible beam was often selected as thick as possible, thus the compliance of the flexible beam can be neglected when performing theoretical modeling without much prediction error in their theoretical models. However, if the flexible beam is designed too thick for applications, the mass moment of inertia will be large and the natural frequency will be reduced along with the frequency response of the whole compliant mechanisms limited. Therefore, suitable thickness of the flexible beam should be selected. In Fig. 11, the results when the thickness of the flexible beam is assigned to be 15 mm (curves with blue-filled dot) can be regarded as the case where the flexible beam is rigid and its compliance is not considered in the theoretical model. Therefore, prediction errors of more than 200% will be generated if the flexible beam is considered to be rigid in the theoretical model. And
Please cite this article as: M. Ling et al., Kinetostatic modeling of complex compliant mechanisms with serialparallel substructures: A semi-analytical matrix displacement method, Mechanism and Machine Theory (2018), https://doi.org/10.1016/j.mechmachtheory.2018.03.014
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13
Ref. [14] The proposed model
Items
FEM
Input stiffness (N/μm)
0.474
0.502
0.484
Displacement amplification ratio
3.935
4.171
4.088
(b) Comparison results
(a) The exemplary mechanism in Ref. [14]
Fig. 12. Comparison of the presented model with the compliance matrix method. Table 3 Key geometric and material parameters for FEM. Parameters
Values
Parameters
Values
h1 (mm) h2 (mm) he (mm) a (mm) b (mm) w (mm)
1.00 3.50 0.50 2.00 1.25 10.00
l1 l2 l3 l4 l5 l6
12.00 5.00 16.75 20.00 25.00 14.43
(mm) (mm) (mm) (mm) (mm) (mm)
the prediction error of non-consideration of the compliance of the flexible beam will be larger when the flexible beam is actually designed thinner. 6.2. Example two The next numerical example is to compare the proposed model with the compliance matrix method widely used in previous investigations [14,15,26–33]. The exemplary mechanism in Ref. [14], as shown in Fig. 12(a), is utilized as the calculating target. The geometric and material parameters are kept constant with those in Ref. [14]. It should be noted that the length of all flexible beams should be smaller 2.5 mm (the radius of the circular flexure hinge) than that was notated in Ref. [14] due to the different definition of geometric parameters and element discretization in the two methods. Compared to the results of displacement amplification ratio and the input stiffness provided by Ref. [14], the prediction accuracy of the proposed model matches well with the compliance matrix method and FEM. It can also be seen from Fig. 12(b) that the prediction accuracy of the proposed method is slightly enhanced due to the inclusion of the compliance of both flexure hinges and flexible beams. 6.3. Example three The next numerical example is to calculate the static displacement of the multi-DOF compliant mechanism shown in Fig. 6(a) with the proposed modeling method and the finite element software package ANSYS. The detailed values for the geometric parameters defined in Fig. 6(b) are listed in Table 3. The elliptical flexure hinge with the theoretical compliance matrix in [38] is adopted again in this example and its geometric parameters are also listed in Table 3. The material in this example is kept to be the same as that in example one. Since the rotational angle of the platform can not be directly read from the results of FEM, the following formula is adopted to calculate the rotational angle of the platform under the load conditions of fp 1 = fp 2 = fp 3 = 100 N:
θ = arcsin
u29 + w29 l6
(38)
where l6 = 14.43 mm is shown in Fig. 6(b). u9 , w9 are, respectively, the output displacement of the node 9 in the x- and ydirections, which can be directly read from the FEM results. Fig. 13 shows the total displacement distributions of the compliant mechanism by FEM when two sets of the load conditions are performed. Detailed results of the input and the output displacements obtained from the FEM are comparatively listed in Table 4 with those obtained from the proposed model. During the theoretical calculation, the thickness of the equivalent beams for simulating the platform with lumped mass was set as 5 mm. From the results in Table 4, one can see that the theoretical results are comparable with the FEM results and the maximum deviation is less than 10% in a large range except for the prediction error of the rotational angle is 13%. The reason of the larger deviation in the rotational angle’s result may be that an approximate model, i.e. Eq. (38), was adopted to indirectly calculate its value for FEM, which may Please cite this article as: M. Ling et al., Kinetostatic modeling of complex compliant mechanisms with serialparallel substructures: A semi-analytical matrix displacement method, Mechanism and Machine Theory (2018), https://doi.org/10.1016/j.mechmachtheory.2018.03.014
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Fig. 13. Finite element results. (a) fp 1 = 100 N, fp 2 = fp 3 = 0 N. (b) fp 1 = fp 2 = fp 3 = 100 N. Table 4 Results of the input and the output displacement with different methods. fp 1 = 100 N, fp 2 = fp 3 = 0 N
Method
FEM The proposed method Error
xin 1 (μm)
xo (μm)
u 29.4 29.6 0.7%
u 26.7 28.1 5.2%
fp 1 = fp 2 = fp 3 = 100 N
w −11.0 −11.4 3.6%
xin 1 (μm)
xin 2 (μm)
u 42.4 46.3 9.2%
u −21.3 −23.1 8.5%
xo (μm/rad)
xin 3 (μm) w 36.8 40.1 9.0%
u −21.3 −23.2 8.9%
w −36.7 −40.1 9.3%
u 0 0 0
w 0 0 0
θ 0.0115 0.0130 13%
65
x in1
60 55 50 45 0 2 4 6 8 10 Thickness of the equivalent beam (mm)
Input displacement (μm)
Input displacement (μm)
Note: u and w denote the displacement components in the x- and y- directions. θ is the rotary angle of the platform.
-22 -24 -26
xin2 xin3
-28 -30 -32 0 2 4 6 8 10 Thickness of the equivalent beam (mm)
Fig. 14. Influence of the thickness of the equivalent beams on the static performance.
brings about some errors. As a conclusion, the proposed modeling method can analytically predict the static performance of the multi-DOF compliant mechanism with an acceptable accuracy. The platform of compliant mechanisms with rotational motion is simplified as a series of equivalent beams in the proposed modeling method to deal with the rotary DOF. To illustrate the influence of the thickness of the equivalent beams on the static performance, the three input displacement components in the x direction were calculated with the change of the thickness of the equivalent beams while keeping other geometric parameters constant, as shown in Table 3. The calculated results are shown in Fig. 14. During the analysis, fp 1 = fp 2 = fp 3 = 100 N were exerted on the three input ports. One can see that the results begin to converge and become stationary after the threshold of about 4 mm under the given geometric parameters. The results mean that the thicker the equivalent beams, the higher accuracy of the presented modeling method.
7. Conclusions A general and accurate modeling methodology is proposed for analyzing the kinetostatics of complex serial-parallel compliant mechanisms with distributed/hybrid compliance. First, a semi-analytical matrix displacement method is developed based on the equilibrium equations of nodal force. Second, the transfer matrix method is flexibly combined with the matrix displacement method to condense the serial-parallel branch chains and then a two-port mechanical network model representing the input/output force-displacement relationships is established for single-DOF compliant mechanisms. Lastly, how to deal with the issue of rotational motion in multi-DOF compliant mechanisms is carried out and the Jacobian matrix are further established from the viewpoint of input and output ports. Besides, compliances of the flexure hinge and the flexible beam are both included in the presented model to enhance the prediction accuracy of distributed compliant mechanisms. Please cite this article as: M. Ling et al., Kinetostatic modeling of complex compliant mechanisms with serialparallel substructures: A semi-analytical matrix displacement method, Mechanism and Machine Theory (2018), https://doi.org/10.1016/j.mechmachtheory.2018.03.014
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15
Three examples, including a single-DOF compliant mechanism and multi-DOF precision positioning stages, are carried out to verify the proposed approach. The results provided by the proposed method are in agreement with those from the compliance matrix method, FEM and experiment. The results also indicate that there may be significant errors for compliant mechanisms with distributed/hybrid compliance if the compliance of the flexible beams is not included in the theoretical model. Acknowledgments This work was supported by the National Natural Science Foundation of China (grant numbers 51705487, 51575426), China; and the State Key Laboratory Program of Xi’an Jiaotong University (grant number SV2016-KF-19), China. Appendix A Eq. (4) is the compliance matrix of a planar flexure hinge. Its expression is as follows [37–40]
cx = cθ =
li
1 dx, EA(x )
li
1 , EI (x )
0
0
cy =
0 li
cα = −
li
0
x2 dx EI (x ) x dx EI (x )
(A.1)
As shown in Fig. 4, the ith flexure hinge has only three independent nodal forces, thus
Fjx
Fjy
Mj
Fkx
Fky
Mk
T
=
1 0 0
0 1 0
−1 0 0
0 0 1
0 −1 0
T
0 li 1
·
Fjx Fy j Mj
(A.2)
Based on the elastic beam theory, the translational and rotational displacements can be expressed as
⎧ li Fk ⎪ ⎪ u = dx + u j ⎪ k ⎪ 0 EA (x ) ⎪ ⎪ ⎨ li
M −F ·x
j yj θk = dx + θ j EI (x ) ⎪ 0 ⎪ ⎪ ⎪ li ⎪ M j − Fy j · x ⎪ ⎩wk = dxdx + θ j li + w j EI (x ) 0
(A.3)
By substituting Eq. (A.1) into Eq. (A.3), and then substitute the result into Eq. (A.2), the elemental stiffness matrix of the ith flexure hinge can be ultimately deduced by using relationship, ky = −2kα /li [41], as
⎧ ⎫ Fjx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Fjy ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨M ⎪ ⎬
⎡
kx
⎢ ⎢ ⎢ ⎢ j =⎢ ⎢ ⎪ Fkx ⎪ ⎪ ⎪ ⎢ ⎪ ⎪ ⎪ ⎪ ⎢ ⎪ ⎪ ⎪ ⎪ F ⎣ ky ⎪ ⎪ ⎪ ⎩ ⎪ ⎭ Mk
0
0
−kx
0
ky
−kα
0
−ky
0
kα
kx
0
kθ Sym
ky
⎤ ⎧ ⎫ uj ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎥ ⎪ ⎪ −kα wj⎪ ⎥ ⎪ ⎪ ⎪ ⎪ ⎪ ⎥ ⎪ ⎪ ⎨ ⎥ −li · kα − kθ θj ⎬ ⎥· ⎥ ⎪u ⎪ 0 k⎪ ⎥ ⎪ ⎥ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ kα w ⎦ ⎪ k⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎭ kθ θk 0
(A.4)
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Please cite this article as: M. Ling et al., Kinetostatic modeling of complex compliant mechanisms with serialparallel substructures: A semi-analytical matrix displacement method, Mechanism and Machine Theory (2018), https://doi.org/10.1016/j.mechmachtheory.2018.03.014
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Please cite this article as: M. Ling et al., Kinetostatic modeling of complex compliant mechanisms with serialparallel substructures: A semi-analytical matrix displacement method, Mechanism and Machine Theory (2018), https://doi.org/10.1016/j.mechmachtheory.2018.03.014
