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5R pseudo-rigid-body model for inflection beams in compliant mechanisms
Mechanism and Machine Theory 116 (2017) 501–512
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Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmachtheory
Research paper
5R pseudo-rigid-body model for inflection beams in compliant mechanisms Yue-Qing Yu∗, Shun-Kun Zhu College of Mechanical Engineering and Applied Electronics Technology, Beijing University of Technology, Beijing 100124, PR China
a r t i c l e
i n f o
Article history: Received 18 October 2016 Revised 6 June 2017 Accepted 21 June 2017
Keywords: Inflection beam Compliant mechanism Pseudo-rigid-body model (PRBM) Free hinge Characteristic parameter
a b s t r a c t A 5R pseudo-rigid-body model (PRBM) is proposed for inflection beams in compliant mechanisms. This new model consists of six rigid links connected by five joints. Four joints with torsional spring are added at the joints to simulate the deflection and a free hinge without spring is used to present the inflection point of the flexural beam. An objective function is established according to relative angular displacement between the two rigid links jointed by the free hinge for finding the optimal characteristic parameters of the 5R PRBM. The spring stiffness coefficients of the 5R PRBM are obtained using a linear regression technique. Numerical examples are presented and the results are shown by comparing with elliptical integral solutions. This new pseudo-rigid-body model behaves as load independent and can be used to predict both the tip locus and inflection position of flexural beams. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction Compliant mechanisms, obtaining part or all their motion from the relative deflection of their flexibility, have been of interest in the field of mechanisms [1]. In designing and analyzing compliant mechanisms, however, it is difficult to investigate the nonlinearity of flexible beams with large deflection. A set of tools are available for working out this problem, such as finite element models, non-linear shooting method, Adomian decomposition method, pseudo-rigid-body model method, and elliptic integral solutions [2]. Compared with other methods, the pseudo-rigid-body model which requires less computation is essential to the design and synthesis of compliant mechanisms in early stages. Howell et al. [3,4] proposed a 1R PRBM with two rigid segments connected by a revolute joint. A torsional spring was placed at the joint to represent the ability of resistance to the deflection of flexible beams. However, the precision was guaranteed only in a limited range of the slope angle. Su [5] proposed a 3R pseudo-rigid-body model whose parameters were independent to the load. The accuracy of the model was relatively high when there was no inflection point on the flexible beam. In Ref. [6], a 2R pseudo-rigid-body model was proposed to improve the simulation accuracy of 1R PRBM and simplify the iterative process of the 3R PRBM. Moreover, a PR pseudo-rigid-body model was proposed to deal with the axial deflection of the flexible beam and the simulating precision was improved [7]. Based on the 3R and PR PRBM, a PRR pseudo-rigid-body model was presented further to simulate both lateral and axial deflections of flexural beams [8]. The PRBM method was one of the methods for the dynamic analysis approach of compliant mechanisms. Boyle et al. [9] developed a dynamic PRBM for the constant-force compression mechanisms. Based on the theory of dynamic equiv∗
Corresponding author. E-mail address: [email protected] (Y.-Q. Yu).
http://dx.doi.org/10.1016/j.mechmachtheory.2017.06.016 0094-114X/© 2017 Elsevier Ltd. All rights reserved.
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Fig. 1. Inflection beam and 5R PRBM.
alence, Yu et al. [10] derived a dynamic equation of general planar compliant mechanisms using the pseudo-rigid-body model. The PRBM method can also be used to design compliant mechanisms. Midha [11] developed a loop-closure method for the analysis and synthesis of compliant mechanisms using the PRBM method. Jensen et al. [12] used pseudo-rigid-body model to design bistable compliant mechanisms. Aten et al. [13] proposed a numerical method for finding the equilibrium position of a given PRBM of compliant mechanisms based on the principle of minimum potential energy. According to the same theory, Jin et al. [14] proposed a more efficient approach to analyze compliant mechanisms. A node coordinate of PRBM was selected as the variables to express the elastic deformation, which simplified the formulation for the analysis of compliant mechanisms. The study of pseudo-rigid-body model for a cantilever flexible beam with no inflection point is comparatively mature. Actually, the inflection normally exists on the flexural beams of compliant mechanisms, e.g., fixed-guided compliant mechanisms and compliant bistable mechanisms. There were only a few pseudo-rigid-body models can be applied to inflection beams. Kimball and Tsai [15] used the PRBM method to simulate the large deflection of a cantilever beam subjected to an arbitrary end load. Nevertheless, this model was oversimplified and the accuracy cannot be ensured. The 3R PRBM developed in Ref. [5] was also hard to improve the simulating precision of such flexible beam. Neither of the PRBM in Ref. [5] nor Ref. [15] were able to predict the inflection position of flexible beams. Midha et al. [16] used the concept of PRBM to generate the beam tip deflection domains where an inflection occurred in a flexible beam. The result obtained from this method was compared with those from FEM and elliptic integral solutions. However, this model was established by combining two 1R models into one model. In Ref. [17], two types of 4R PRBM were proposed to simulate the flexural beam with an inflection point. However, these two models were load dependent due to their asymmetric structures. In this paper, a new symmetric 5R PRBM is further developed to predict precisely both the location of end tip and inflection position of large deflection beams with inflection points in compliant mechanisms. The rest of the paper is organized as follows. In Section 2, a 5R pseudo-rigid-body model is proposed for inflection beams. In Section 3, the characteristic parameters of the 5R PRBM are determined through optimization and linear regression. In Section 4, a numerical example is presented to show the effectiveness and superiority of the new model. Finally, some conclusions are made in Section 5. 2. A 5R pseudo-rigid-body model The flexible beam is one of the major components in compliant mechanisms. When a force and a torque load are applied to the beam tip, an inflection may occur on the beam, as shown in Fig. 1(a). T is the free end of the flexible beam. l is the
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beam length. The coordinates and the slope of the beam tip are (a, b) and θ 0 , respectively. The force F0 and the torque M0 are applied to the beam tip, causing an inflection in the beam. F0 with the direction angle ϕ can be decomposed into a horizontal force nP and a vertical force P. The ratio between the horizontal and vertical forces is defined by n. We have
φ=
π
+ tan−1 (n )
2
(1)
From the theory of Elliptic integrals [1, 18], the curvature of an arbitrary point in a flexible beam can be written as
P dθ = ± 2 (sin θ0 − n cos θ0 − sin θ + n cos θ ) + EI ds
M 2 0
EI
(2)
where, E is the Young’s modulus of the material. I is the second moment of area of the flexible beam. When the arc of curved part for the flexible beam is concave, take the sign "+". When the arc is convex, take the sign "−". Eq. (2) can be rewritten as
dθ =± ds where
f ( θ )=
2P f (θ ) EI
(3)
λ− sin θ + n cos θ
and
λ = sin θ0 − n cos θ0 +
1 + n2 κ
(4)
(5)
κ is the load ratio, defined as
κ= √
M0 2
2 1 + n2 P EI
(6)
Setting Eq. (3) to be zero, the following equation can be obtained
λ = sin θ − n cos θ
(7)
Eq. (7) has infinite solutions for θ i representing the deflection angle at an inflection point. They are ∧
θ = φ ± cos−1 (cos (φ − θ0 ) + κ ) ± 2kπ k = 0, 1, 2 . . .
(8)
i
A 5R pseudo-rigid-body model shown in Fig. 1(b) is proposed here to simulate the large deflection of the flexural beam with an inflection point. The 5R model consists of 6 rigid links jointed by 5 revolute joints. 4 torsional springs are added at the joints with symbol K1 , K2 , K3 and K4 to present the ability to resist deflection of the beam, but the third hinge Qi . is a free hinge without torsional spring to indicate the inflection of flexible beam. The length of each rigid link in the 5R PRBM is presented as γi l (i = 0, 1, 2, 3, 4, 5 ), and γ i is called the characteristic radius factor [1], satisfyingγ0 + γ1 + γ2 + γ3 + γ4 + γ5 = 1. Ki (i = 1, 2, 3, 4 ) is the stiffness constant of the four torsion springs. These factors and constants are the characteristic parameters of the 5R PRBM to simulate the large deflection beam with an inflection point in compliant mechanisms. 3. Characteristic parameters The characteristic parameters of the 5R PRBM can be determined as follows. At first, the kinematics equations of the 5R PRBM are derived. The optimal characteristic radius factors are established according to the relative angular displacement between the two links jointed by the hinge without spring. This optimization problem is solved using the genetic algorithm [19]. Compared with the traditional optimization method (enumeration, heuristic, etc.), the genetic algorithm has a good convergence in terms of biological evolution. When the calculation accuracy is required for the complicated optimization problem, the computational time for the genetic algorithm is low and the robustness is high. Therefore, it is applied in this study. Secondly, the torque of each joint in the 5R PRBM is then determined through static analysis. Finally, the stiffness coefficients of the 5R PRBM are obtained using linear regression technique [20]. 3.1. Kinematic analysis The inflection beam can be divided into two segments from the inflection point I, as shown Segment I and Segment II in Fig. 2(a) and (b), respectively. Similarly, the corresponding 5R PRBM shown in Fig. 1(b) can also be divided into two parts from the hinge Qi , as shown PRBM-I and PRBM-II in Fig. 2(c) and (d), respectively. Therefore, Segment I and Segment II of the flexible beam can be simulated by PRBM-I and PRBM-II, respectively. The point Q can be considered as a fixed
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Fig. 2. Compliant beam considered as two segments and the corresponding PRBMs.
end, assuming that the PRBM is applied with end load of combined force and torque. The kinematic analysis of PRBM-I and PRBM-II can be presented as follows. In PRBM-I, the location of point Qi (Qix , Qiy ) is
Qix = [γ0 + γ1 cos 1 + γ2 cos ( 1 + 2 )]l
(9)
Qiy = [γ1 sin 1 + γ2 sin ( 1 + 2 )]l
(10)
The pseudo-rigid-body angles 1 , 2 can be obtained by inverse kinematic analysis
x = Qix − γ0 l
(11)
y = Qiy
(12)
2 = cos
−1
1 = tan−1
x2 +y2 − γ2 2 l 2 − γ3 2 l 2 2γ2 γ3 l 2
(13)
(γ1 l + γ2 l cos 2 )y − γ2 l sin 2 x (γ1 l + γ2 l cos 2 )x + γ2 l sin 2 y
(14)
The end slope of PRBM-I is
2R−I = 1 + 2
(15)
Similarly, the location of point Q(Qx , Qy ) in PRBM-II can be written as
Qx = Qix +l [γ3 cos( 0 + 3 + 4 )+γ4 cos( 0 + 4 )+γ5 cos 0 ]
(16)
Qy = Qiy +l [γ3 sin( 0 + 3 + 4 )+γ4 sin( 0 + 4 )+γ5 sin 0 ]
(17)
Through inverse kinematic analysis, we have
x = (Qx − Qix ) cos 0 + Qy − Qiy sin 0 − γ4 l
y = Qy − Qiy cos 0 − (Qx − Qix ) sin 0
3 = cos
−1
x2 + y2 − γ3 2 l 2 − γ4 2 l 2 2γ3 γ4 l 2
(18) (19)
(20)
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4 = tan
−1
(γ4 l + γ3 l cos 3 )y − γ3 l sin 3 x (γ4 l + γ3 l cos 3 )x + γ3 l sin 3 y
505
(21)
The slope of the free end of PRBM-II is
2R−II = 3 + 4 + 0
(22)
3.2. Optimal characteristic radius factors It is known that the curvature at the inflection point of a flexible beam is zero. Therefore, the angle between the two links jointed by the hinge without spring in the 5R PRBM can be used to represent the inflection point,ea = 2R−II − 2R−I , as shown in Fig. 2(c) and (d). This angle should be as small as possible to meet the requirement of zero-curvature at the inflection point. When the deflection locus of a flexible beam is determined, the value ea can be obtained through inverse kinematics of the PRBM-I or PRBM-II. For different value of γ i (i = 0, 1, 2, 3, 4), the value of ea may differ greatly. The objective of solution is to find a set of characteristic radius factors that minimize the value of ea . Therefore, the objective function can be established as the average value of ea . To determine the objective function, the load parameters, force ratio index n and load ratio κ , are set at first. Then, the maximum value of end slope for the flexible beam could be determined as θ 0max . If the beam tip slope exceeds θ 0max , no inflection point can be created. The expression of θ 0max can be determined from Eq. (8) as follows [21].
θ0 max = φ − cos−1 (1 − κ )
(23)
In order to calculate the deformation path of the flexible beam by elliptic integral solutions, the maximum value of end slope for the flexible beam, θ 0max can be divided into a number of steps. Enough positions should be taken into account so that the error of model with the optimized parameters remains small from the start to the end of the deflection. 50 positions were determined after several times of calculation and shown good enough in this study. As a result, the tip angle of the flexible rod increases gradually from zero, the value of θ 0 is determined as
θ0 j = θ0 j−1 + j θ0
(24)
Where, θ 0 =θ 0max /50, θ 01 = 0. After that, the deflection locus of the flexural beam can be determined by elliptic integral solutions. Because the 5R PRBM has five degree-of-freedom, it is possible that the end position, end slope and hinge position are set to be the same as that of inflection point. In this way, we have
(Qx , Qy ) = (a, b) Qix , Qiy = (ai , bi ) 0 =θ0
(25)
With a set of values of characteristic radius factors, the pseudo-rigid-body angles 1 , 2 , 3 , 4, 2R-I and 2R-II can be determined through the kinematic analysis using Eqs.(11)–(14) for the PRBM-I and Eqs.(18)–(21) for the PRBM-II. If one of the angles ( 1 , 2 , 3 , 4 ) is not a real one, the values of characteristic radius factors are not feasible. To avoid such infeasible values of characteristic radius factors, an objective function is added with a relatively big value M as penalty factor. We have
ea = | 2R−I − 1R−II | 1 ∈ R ∧ 2 ∈ R ∧ 3 ∧ 4 ∈ R . ea = M 1 ∈/ R ∨ 2 ∈/ R ∨ 3 ∨ 4 ∈/ R
(26)
Then, the objective function Ea can be determined as 50
Ea =
ea j
j=1
j
(27)
The optimization for the characteristic radius factors of the 5R PRBM can be described as: Minimize Fitness = Ea Subject to 4
γi =1 γi > 0 (i = 0, 1, 2, 3, 4, 5)
i=0
The process to determine Ea can be presented by the flowchart shown in Fig. 3. In this way, the optimal characteristic radius factors of the 5R PRBM can be obtained as follows when the load parameters are selected as n = 3 and κ = 1.5.
γ0 = 0.050, γ1 = 0.257, γ2 = 0.251, γ3 = 0.121, γ4 = 0.277, γ5 = 0.044
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Fig. 3. Flowchart for determining the objective function Ea of the 5R PRBM.
3.3. Stiffness coefficients The spring stiffness coefficients of the 5R PRBM describe the ability of flexible beam to resist deflection. After the characteristic radius factors have been obtained in above section, the torque of spring in the joints of the 5R PRBM can be calculated by static equations. Then, the stiffness coefficients can be determined through a linear regression technique. Define T1 , T2 , T3 and T4 as the torque of joints, t1 , t2 , t3 and t4 as the non-dimensional factors of torques, as shown in Fig. 4. Define K1 , K2 , K3 and K4 as the stiffness constants of torsion springs and k1 , k2 , k3 and k4 as the stiffness coefficients, satisfying
Ki =
EI l
· ki
Ti =
EI l
· ti
( i = 1, 2, 3, 4 )
(28)
For PRBM-I, the static equations are
T1 = P (n(γ2 l sin ( 1 + 2 ) + γ1 l sin 1 ) + γ2 l cos ( 1 + 2 ) + γ1 l cos 1 )
(29)
T2 = P γ2 l (n sin ( 1 + 2 ) + cos ( 1 + 2 ) )
(30)
The joint torques of PRBM-II can be presented as
T3 = P γ3 l (n sin ( 3 + 4 + 0 ) + cos ( 3 + 4 + 0 ) )
(31)
T4 = P γ3 l (n sin ( 3 + 4 + 0 ) + cos ( 3 + 4 + 0 ) )+ P γ4 l (n sin ( 4 + 0 ) + cos ( 4 + 0 ) )
(32)
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Fig. 4. Force analysis of the 5R PRBM.
From Eqs.(29)–(32), the non-dimensional equations can be derived as
t1 = α 2 (n(γ2 sin ( 1 + 2 ) + γ1 sin 1 ) + γ2 cos ( 1 + 2 ) + γ1 cos 1 )
(33)
t2 = α 2 γ2 (n sin ( 1 + 2 ) + cos ( 1 + 2 ) )
(34)
t3 = α 2 γ3 (n sin ( 3 + 4 + 0 ) + cos ( 3 + 4 + 0 ) )
(35)
t4 = α 2 γ3 (n sin ( 3 + 4 + 0 ) + cos ( 3 + 4 + 0 ) )+ α 2 γ4 (n sin ( 4 + 0 ) + cos ( 4 + 0 ))
(36)
Where, α is the normalized force load [1], defined as
α2 =
P l2 EI
(37)
Assuming that the model is in the status of static equilibrium, we have
⎧ ⎪ ⎨t1 = k1 1 t 2 = k 2 2 ⎪ ⎩t3 = k3 3 t 4 = k 4 4
(38)
Substituting Eq. (38) into Eqs.(33)–(36), we obtain
k1 1 = α 2 (n(γ2 sin ( 1 + 2 ) + γ1 sin 1 ) + γ2 cos ( 1 + 2 ) + γ1 cos 1 )
(39)
k2 2 = α 2 γ2 (n sin ( 1 + 2 ) + cos ( 1 + 2 ) )
(40)
k3 3 = α 2 γ3 (n sin ( 3 + 4 + 0 ) + cos ( 3 + 4 + 0 ) )
(41)
k4 4 = α 2 γ3 (n sin ( 3 + 4 + 0 ) + cos ( 3 + 4 + 0 ) )+ α 2 γ4 (n sin ( 4 + 0 ) + cos ( 4 + 0 ))
(42)
Then, the stiffness coefficients can be obtained in the following 4 steps. Firstly, set the boundary condition, and determine the tip point (a/l, b/l) of flexible beam by elliptic integral solutions for given n, κ , and θ 0. Secondly, set (Qx , Qy )=(a/l, b/l), (Qix , Qiy )=(ai /l, bi /l) , 0 =θ 0 . Determine the PRBM angles 1 , 2 , 3 and 4 from the inverse kinematics Eqs.(11)–(14) and (18)–(21). Thirdly, substitute the angles 1 , 2 , 3 and 4 into static equations Eqs.(33)–(36) to determine the joint torques of the 5R PRBM.
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Fig. 5. Fitting curve of stiffness coefficient of the 5R PRBM.
Finally, determine the stiffness coefficients by linear regression technique. In this study, ratio n and slope θ 0 are set to fixed values. The load ratio κ varies in the range of (0, κ max ). κ max determines the max value of κ satisfying the load condition where an inflection point is possible to occur.
κ = 1 − cos(φ − θ0 )
(43)
Set n = 1, θ 0 = 0.0126, the variation range of κ is determined as (0, 1.698) from Eqs. (8) and (43). From above procedure, the fitting curves of stiffness coefficients for the 5R PRBM can be obtained, as shown in Fig. 5. The stiffness coefficients of the 5R PRBM are obtained as follows.
k1 = 6.948, k2 = 3.160, k3 = 3.210, k4 = 6.418 4. Numerical examples In this section, the 5R PRBM developed above is applied to simulate the deflection of flexible beam with an inflection point, including the position of inflection point and beam tip. For comparison, the result obtained by elliptical integral solutions is also presented. First of all, defining the error of locus for the inflection point as
ei =
Qix −
ai l
2
+
Qiy −
bi l
2 (44)
Defining the error of locus for the beam tip as
et =
Qx −
a l
2
+
Qy −
b l
2 (45)
Hence, the average error of inflection point and beam tip can be defined as, respectively, 50
ei a =
ei j
j=1
50
et a =
(46)
j et j
j=1
j
(47)
Since the curvature at the inflection point is zero, it is reasonable to constrain the angle between the two links jointed by the free hinge Qi to be zero. We have
2R−I = 2R−II + 0 , The errors can be obtained in the following 4 steps.
(48)
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Table 1 Average errors of the 5R PRBM in different load conditions. Case
n
κ
Inflection
eta
eia
I II III IV V VI
0 1 3 0 1 3
0.4 0.6 1.5 0.4 0.6 1.5
yes yes yes no no no
0.16% 0.40% 1.14% 0.11% 0.19% 0.24%
1.58% 3.06% 3.30% – – –
First, obtain the locus of inflection point I(ai , bi ) and beam tip T(a, b), and the force load α by elliptic integral solutions. Second, substitute the force load α , force ratio n, and angle 0 =θ 0 into Eqs.(33)–(36) and (48) to obtain the pseudorigid-body angle 1 , 2 , 3 , 4 . Third, substitute the pseudo-rigid-body angle 0 , 1 , 2 , 3 and 4 into Eqs.(9), (10), (16) and (17) to obtain the locus of hinge Qi (Qix , Qiy ) and tip Q (Qx , Qy ) of the 5R PRBM. Fourth, obtain the errors of inflection point and beam tip from Eqs.(44)–(47). In order to show the effectiveness and superiority of the 5R PRBM, six cases are studied here for quite different load parameters n and κ ,. In addition to the case of n = 3 and κ = 1.5 discussed in above section for determining the characteristic radius factors of the 5R PRBM, the other cases of n = 0, 1 and κ = 0.4, 0.6 are also presented to illustrate the validation of the 5R PRBM for different load conditions. These 6 cases of load conditions are listed in Table 1 from case I to VI. In cases I–III, the flexible beam has one inflection point. For comparison, there is no inflection point in the beam for other three cases IV–VI. Fig. 6 shows the configurations of the 5R PRBM for the deflection of flexural beam in cases I–VI. The solid lines represent the configurations of the 5R PRBM and the dotted lines obtained by elliptical integral solutions describe those of the flexible beams. It can be seen from the figures that the tip of the 5R PRBM and that of the flexural beam almost coincide in most cases. In case III, the flexible beam with an inflection point is in a large-torque load condition of κ = 1.5 that causes two segments of the beam bend in two opposite direction (elbow up and elbow down). Moreover, it can be seen from the figures that the configurations of the 5R PRBM are closed to those of the flexible beam in all cases. The 5R PRBM can also be used to predict precisely the deformation of the flexural beam without inflection point as shown in cases IV–VI. This result indicates well the effectiveness of the 5R PRBM. Fig. 7 shows the errors on beam tip and/or inflection point of the 5R PRBM in cases I–VI. In case I, II and III, the errors of inflection point are relatively big while those of the slope of beam tip are very little. The tip locus errors fluctuate with the increase of tip slope. However, in case IV, V and VI without inflection point, the errors of tip locus increases monotonically with the increase of tip slope in a large scale. Moreover, it can also be seen in Fig. 7 that the errors of inflection point drop below 0.02 when the slope of the beam tip is over 0.2 rad. However, they are relatively big when the slope of beam tip is small. It is interesting to note that the inflection point locus is described by the PRBM-I that is part of the 5R PRBM while the tip locus of flexible beam is predicted by whole the 5R PRBM. Therefore, the errors of beam tips are obviously smaller than those of the inflection points. This result agrees with the fact that the more joints the PRBM has, the more accurate it becomes, such as the 1R, 2R to 3R PRBM developed in previous work. The average errors of the 5R PRBM in six cases are listed in Table 1. It can be seen that the average errors of beam tip locus are less than 1.14% and those of inflection point locus are less than 3.30%. This result shows well the accuracy of the 5R PRBM. It is true that the error increase as the slope of the beam tip increases. However, the average errors of the 5R PRBM in all cases of load are very small and kept in the acceptable range. This result shows that the 5R PRBM with the characteristic parameters determined in above section can also be applied to other different load conditions. That is to say, the 5R PRBM proposed in this paper behaves as load independent. This result indicates the superiority of the 5R PRBM over the 4R PRBM developed in Ref. [17] that is load dependent. This may be owing to that the 5R PRBM is a symmetric structure and it has more degrees-of-freedom than the 4R PRBM. The limit of the beam tip slope θ 0max can be used here to determine the condition where the inflection point exists. If the beam tip slope exceeds θ 0max , no inflection point can be created. In this section, the numerical results are obtained with non-dimensional factors only to show the accuracy of the 5R PRBM in general cases. If the model is used to analyze other compliant mechanisms, the parameters about the geometric and material properties of the flexural beams in the mechanisms are need to be determined while the characteristic parameters of the 5R PRBM obtained in this paper can be used directly. A brief flowchart is presented in Fig. 8 to show how to fully construct the 5R PRBM of a compliant mechanism. 5. Conclusions A 5R pseudo-rigid-body model was proposed in this study to simulate the inflection beams in compliant mechanisms. The characteristic parameters of the 5R PRBM were determined through optimization and linear regression. The numerical
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Fig. 6. Configurations of the 5R PRBM and the flexible beam in cases I–VI.
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Fig. 7. Errors of the 5R PRBM in cases I–VI.
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Fig. 8. Flowchart of using the 5R PRBM.
examples showed that this model behaved as load independent and it could be used to predict the deflection of flexible beams with or without inflection. This new model can also be used to the flexural beams in other load cases and it is, therefore, convenient in early stage of design or analysis of the flexural beams in complaint mechanisms. Acknowledgements The financial support of this study was from National Natural Science Foundation of China (Grant No. 51575006). References [1] L.L. Howell, Compliant Mechanisms, John Wiley & Sons, New York, NY, 2001. [2] A. Zhang, G. Chen, A comprehensive elliptic integral solution to the large deflection problems of thin beams in compliant mechanisms, ASME Trans J. Mech. Robot. 5 (2) (2012) 420–431. [3] L.L. Howell, A. Midha, T.W. Norton, Evaluation of equivalent spring stiffness for use in a pseudo-rigid-body model of large-deflection compliant mechanisms, ASME Trans J. Mech. Des. 118 (1) (1996) 126–131. [4] L. HowellL, A. Midha, Parametric deflection approximations for end-loaded, large-deflection beams in compliant mechanisms, ASME Trans J. Mech. Des. 117 (1) (1995) 156–165. [5] H.J. Su, Apseudorigid-body 3R model for determining large deflection of cantilever beams subject to tip loads, ASME Trans. J. Mech. Robot. 1 (2) (2009) 795–810. [6] Y.Q. Yu, Z.L. Feng, Q.P. Xu, A pseudo-rigid-body 2R model of flexural beam in compliant mechanisms, Mech. Mach. Theory 55 (9) (2012) 18–33. [7] Y.Q. Yu, P. Zhou, Q.P. Xu, A new pseudo-rigid-body model of compliant mechanisms considering axial deflection of flexural beams, Mech. Mach. Sci. 24 (2015) 851–858. [8] Y.Q. Yu, S.K. Zhu, Q.P. Xu, P. Zhou, A novel model of large deflection beams with combined end loads in compliant mechanisms, Precis. Eng. 43 (2016) 395–405. [9] C. Boyle, L.L. Howell, S.P. Magleby, M.S. Evans, Dynamic modeling of compliant constant-force compression mechanisms, Mech. Mach. Theory 38 (12) (2003) 1469–1487. [10] Y.Q. Yu, L.L. Howell, C. Lusk, Y. Yue, M.G. He, Dynamic modeling of compliant mechanisms based on the pseudo-rigid-body model, ASMR Trans. J. Mech. Des. 127 (4) (2005) 760–765. [11] L.L. Howell, A. Midha, A loop-closure theory for the analysis and synthesis of compliant mechanisms, ASME Trans. J. Mech. Des. 118 (1) (1996) 121–125. [12] B.D. Jensen, L.L. Howell, L.G. Salmon, Design of two-link, in-plane, bistable compliant micro-mechanisms, ASME Trans. J. Mech. Des. 121 (3) (1999) 416–423. [13] Q.T. Aten, S.A. Zirbel, B.D. Jensen, L.L. Howell, A numerical method for position analysis of compliant mechanisms with more degrees of freedom than inputs, ASME Trans. J. Mech. Des. 133 (6) (2011) 061009. [14] M. Jin, X. Zhang, B. Zhu, A numerical method for static analysis of pseudo-rigid-body model of compliant mechanisms, Proc. Inst. Mech. Eng. Part C 228 (17) (2014) 3170–3177. [15] C. Kimball, L.W. Tsai, Modeling of flexural beams subjected to arbitrary end loads, ASME Trans. J. Mech. Des. 124 (2) (2002) 223–235. [16] A. Midha, S.G. Bapat, A. Mavanthoor, V. Chinta, Analysis of a fixed-guided compliant beam with an inflection point using the pseudo-rigid-body model (PRBM) concept, ASME Trans J. Mech. Robot. 64 (4) (2012) 147–154. [17] S.K. Zhu, Y.Q. Yu, Pseudo-rigid-body model for the flexible beams with an inflection point in compliant mechanisms, ASME Trans J. Mech. Robot. 9 (3) (2017) 031005-031005-8. [18] A.G. Greenhill, The Applications of Elliptic Functions, Dover, New York, 1959. [19] V. Michael, The Simple Genetic Algorithm: Foundations and Theory, MIT Press, Cambridge, MA, 1999. [20] S. Weisberg, Applied Linear Regression, third ed., WileyInterscience, New York, NY, 2005. [21] A. Saxena, A New Pseudo-Rigid-Body Model for Flexible Members in Compliant Mechanisms MS thesis, University of Toledo, Toledo, OH, USA, 1997.
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Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmachtheory
Research paper
5R pseudo-rigid-body model for inflection beams in compliant mechanisms Yue-Qing Yu∗, Shun-Kun Zhu College of Mechanical Engineering and Applied Electronics Technology, Beijing University of Technology, Beijing 100124, PR China
a r t i c l e
i n f o
Article history: Received 18 October 2016 Revised 6 June 2017 Accepted 21 June 2017
Keywords: Inflection beam Compliant mechanism Pseudo-rigid-body model (PRBM) Free hinge Characteristic parameter
a b s t r a c t A 5R pseudo-rigid-body model (PRBM) is proposed for inflection beams in compliant mechanisms. This new model consists of six rigid links connected by five joints. Four joints with torsional spring are added at the joints to simulate the deflection and a free hinge without spring is used to present the inflection point of the flexural beam. An objective function is established according to relative angular displacement between the two rigid links jointed by the free hinge for finding the optimal characteristic parameters of the 5R PRBM. The spring stiffness coefficients of the 5R PRBM are obtained using a linear regression technique. Numerical examples are presented and the results are shown by comparing with elliptical integral solutions. This new pseudo-rigid-body model behaves as load independent and can be used to predict both the tip locus and inflection position of flexural beams. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction Compliant mechanisms, obtaining part or all their motion from the relative deflection of their flexibility, have been of interest in the field of mechanisms [1]. In designing and analyzing compliant mechanisms, however, it is difficult to investigate the nonlinearity of flexible beams with large deflection. A set of tools are available for working out this problem, such as finite element models, non-linear shooting method, Adomian decomposition method, pseudo-rigid-body model method, and elliptic integral solutions [2]. Compared with other methods, the pseudo-rigid-body model which requires less computation is essential to the design and synthesis of compliant mechanisms in early stages. Howell et al. [3,4] proposed a 1R PRBM with two rigid segments connected by a revolute joint. A torsional spring was placed at the joint to represent the ability of resistance to the deflection of flexible beams. However, the precision was guaranteed only in a limited range of the slope angle. Su [5] proposed a 3R pseudo-rigid-body model whose parameters were independent to the load. The accuracy of the model was relatively high when there was no inflection point on the flexible beam. In Ref. [6], a 2R pseudo-rigid-body model was proposed to improve the simulation accuracy of 1R PRBM and simplify the iterative process of the 3R PRBM. Moreover, a PR pseudo-rigid-body model was proposed to deal with the axial deflection of the flexible beam and the simulating precision was improved [7]. Based on the 3R and PR PRBM, a PRR pseudo-rigid-body model was presented further to simulate both lateral and axial deflections of flexural beams [8]. The PRBM method was one of the methods for the dynamic analysis approach of compliant mechanisms. Boyle et al. [9] developed a dynamic PRBM for the constant-force compression mechanisms. Based on the theory of dynamic equiv∗
Corresponding author. E-mail address: [email protected] (Y.-Q. Yu).
http://dx.doi.org/10.1016/j.mechmachtheory.2017.06.016 0094-114X/© 2017 Elsevier Ltd. All rights reserved.
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Fig. 1. Inflection beam and 5R PRBM.
alence, Yu et al. [10] derived a dynamic equation of general planar compliant mechanisms using the pseudo-rigid-body model. The PRBM method can also be used to design compliant mechanisms. Midha [11] developed a loop-closure method for the analysis and synthesis of compliant mechanisms using the PRBM method. Jensen et al. [12] used pseudo-rigid-body model to design bistable compliant mechanisms. Aten et al. [13] proposed a numerical method for finding the equilibrium position of a given PRBM of compliant mechanisms based on the principle of minimum potential energy. According to the same theory, Jin et al. [14] proposed a more efficient approach to analyze compliant mechanisms. A node coordinate of PRBM was selected as the variables to express the elastic deformation, which simplified the formulation for the analysis of compliant mechanisms. The study of pseudo-rigid-body model for a cantilever flexible beam with no inflection point is comparatively mature. Actually, the inflection normally exists on the flexural beams of compliant mechanisms, e.g., fixed-guided compliant mechanisms and compliant bistable mechanisms. There were only a few pseudo-rigid-body models can be applied to inflection beams. Kimball and Tsai [15] used the PRBM method to simulate the large deflection of a cantilever beam subjected to an arbitrary end load. Nevertheless, this model was oversimplified and the accuracy cannot be ensured. The 3R PRBM developed in Ref. [5] was also hard to improve the simulating precision of such flexible beam. Neither of the PRBM in Ref. [5] nor Ref. [15] were able to predict the inflection position of flexible beams. Midha et al. [16] used the concept of PRBM to generate the beam tip deflection domains where an inflection occurred in a flexible beam. The result obtained from this method was compared with those from FEM and elliptic integral solutions. However, this model was established by combining two 1R models into one model. In Ref. [17], two types of 4R PRBM were proposed to simulate the flexural beam with an inflection point. However, these two models were load dependent due to their asymmetric structures. In this paper, a new symmetric 5R PRBM is further developed to predict precisely both the location of end tip and inflection position of large deflection beams with inflection points in compliant mechanisms. The rest of the paper is organized as follows. In Section 2, a 5R pseudo-rigid-body model is proposed for inflection beams. In Section 3, the characteristic parameters of the 5R PRBM are determined through optimization and linear regression. In Section 4, a numerical example is presented to show the effectiveness and superiority of the new model. Finally, some conclusions are made in Section 5. 2. A 5R pseudo-rigid-body model The flexible beam is one of the major components in compliant mechanisms. When a force and a torque load are applied to the beam tip, an inflection may occur on the beam, as shown in Fig. 1(a). T is the free end of the flexible beam. l is the
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503
beam length. The coordinates and the slope of the beam tip are (a, b) and θ 0 , respectively. The force F0 and the torque M0 are applied to the beam tip, causing an inflection in the beam. F0 with the direction angle ϕ can be decomposed into a horizontal force nP and a vertical force P. The ratio between the horizontal and vertical forces is defined by n. We have
φ=
π
+ tan−1 (n )
2
(1)
From the theory of Elliptic integrals [1, 18], the curvature of an arbitrary point in a flexible beam can be written as
P dθ = ± 2 (sin θ0 − n cos θ0 − sin θ + n cos θ ) + EI ds
M 2 0
EI
(2)
where, E is the Young’s modulus of the material. I is the second moment of area of the flexible beam. When the arc of curved part for the flexible beam is concave, take the sign "+". When the arc is convex, take the sign "−". Eq. (2) can be rewritten as
dθ =± ds where
f ( θ )=
2P f (θ ) EI
(3)
λ− sin θ + n cos θ
and
λ = sin θ0 − n cos θ0 +
1 + n2 κ
(4)
(5)
κ is the load ratio, defined as
κ= √
M0 2
2 1 + n2 P EI
(6)
Setting Eq. (3) to be zero, the following equation can be obtained
λ = sin θ − n cos θ
(7)
Eq. (7) has infinite solutions for θ i representing the deflection angle at an inflection point. They are ∧
θ = φ ± cos−1 (cos (φ − θ0 ) + κ ) ± 2kπ k = 0, 1, 2 . . .
(8)
i
A 5R pseudo-rigid-body model shown in Fig. 1(b) is proposed here to simulate the large deflection of the flexural beam with an inflection point. The 5R model consists of 6 rigid links jointed by 5 revolute joints. 4 torsional springs are added at the joints with symbol K1 , K2 , K3 and K4 to present the ability to resist deflection of the beam, but the third hinge Qi . is a free hinge without torsional spring to indicate the inflection of flexible beam. The length of each rigid link in the 5R PRBM is presented as γi l (i = 0, 1, 2, 3, 4, 5 ), and γ i is called the characteristic radius factor [1], satisfyingγ0 + γ1 + γ2 + γ3 + γ4 + γ5 = 1. Ki (i = 1, 2, 3, 4 ) is the stiffness constant of the four torsion springs. These factors and constants are the characteristic parameters of the 5R PRBM to simulate the large deflection beam with an inflection point in compliant mechanisms. 3. Characteristic parameters The characteristic parameters of the 5R PRBM can be determined as follows. At first, the kinematics equations of the 5R PRBM are derived. The optimal characteristic radius factors are established according to the relative angular displacement between the two links jointed by the hinge without spring. This optimization problem is solved using the genetic algorithm [19]. Compared with the traditional optimization method (enumeration, heuristic, etc.), the genetic algorithm has a good convergence in terms of biological evolution. When the calculation accuracy is required for the complicated optimization problem, the computational time for the genetic algorithm is low and the robustness is high. Therefore, it is applied in this study. Secondly, the torque of each joint in the 5R PRBM is then determined through static analysis. Finally, the stiffness coefficients of the 5R PRBM are obtained using linear regression technique [20]. 3.1. Kinematic analysis The inflection beam can be divided into two segments from the inflection point I, as shown Segment I and Segment II in Fig. 2(a) and (b), respectively. Similarly, the corresponding 5R PRBM shown in Fig. 1(b) can also be divided into two parts from the hinge Qi , as shown PRBM-I and PRBM-II in Fig. 2(c) and (d), respectively. Therefore, Segment I and Segment II of the flexible beam can be simulated by PRBM-I and PRBM-II, respectively. The point Q can be considered as a fixed
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Fig. 2. Compliant beam considered as two segments and the corresponding PRBMs.
end, assuming that the PRBM is applied with end load of combined force and torque. The kinematic analysis of PRBM-I and PRBM-II can be presented as follows. In PRBM-I, the location of point Qi (Qix , Qiy ) is
Qix = [γ0 + γ1 cos 1 + γ2 cos ( 1 + 2 )]l
(9)
Qiy = [γ1 sin 1 + γ2 sin ( 1 + 2 )]l
(10)
The pseudo-rigid-body angles 1 , 2 can be obtained by inverse kinematic analysis
x = Qix − γ0 l
(11)
y = Qiy
(12)
2 = cos
−1
1 = tan−1
x2 +y2 − γ2 2 l 2 − γ3 2 l 2 2γ2 γ3 l 2
(13)
(γ1 l + γ2 l cos 2 )y − γ2 l sin 2 x (γ1 l + γ2 l cos 2 )x + γ2 l sin 2 y
(14)
The end slope of PRBM-I is
2R−I = 1 + 2
(15)
Similarly, the location of point Q(Qx , Qy ) in PRBM-II can be written as
Qx = Qix +l [γ3 cos( 0 + 3 + 4 )+γ4 cos( 0 + 4 )+γ5 cos 0 ]
(16)
Qy = Qiy +l [γ3 sin( 0 + 3 + 4 )+γ4 sin( 0 + 4 )+γ5 sin 0 ]
(17)
Through inverse kinematic analysis, we have
x = (Qx − Qix ) cos 0 + Qy − Qiy sin 0 − γ4 l
y = Qy − Qiy cos 0 − (Qx − Qix ) sin 0
3 = cos
−1
x2 + y2 − γ3 2 l 2 − γ4 2 l 2 2γ3 γ4 l 2
(18) (19)
(20)
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4 = tan
−1
(γ4 l + γ3 l cos 3 )y − γ3 l sin 3 x (γ4 l + γ3 l cos 3 )x + γ3 l sin 3 y
505
(21)
The slope of the free end of PRBM-II is
2R−II = 3 + 4 + 0
(22)
3.2. Optimal characteristic radius factors It is known that the curvature at the inflection point of a flexible beam is zero. Therefore, the angle between the two links jointed by the hinge without spring in the 5R PRBM can be used to represent the inflection point,ea = 2R−II − 2R−I , as shown in Fig. 2(c) and (d). This angle should be as small as possible to meet the requirement of zero-curvature at the inflection point. When the deflection locus of a flexible beam is determined, the value ea can be obtained through inverse kinematics of the PRBM-I or PRBM-II. For different value of γ i (i = 0, 1, 2, 3, 4), the value of ea may differ greatly. The objective of solution is to find a set of characteristic radius factors that minimize the value of ea . Therefore, the objective function can be established as the average value of ea . To determine the objective function, the load parameters, force ratio index n and load ratio κ , are set at first. Then, the maximum value of end slope for the flexible beam could be determined as θ 0max . If the beam tip slope exceeds θ 0max , no inflection point can be created. The expression of θ 0max can be determined from Eq. (8) as follows [21].
θ0 max = φ − cos−1 (1 − κ )
(23)
In order to calculate the deformation path of the flexible beam by elliptic integral solutions, the maximum value of end slope for the flexible beam, θ 0max can be divided into a number of steps. Enough positions should be taken into account so that the error of model with the optimized parameters remains small from the start to the end of the deflection. 50 positions were determined after several times of calculation and shown good enough in this study. As a result, the tip angle of the flexible rod increases gradually from zero, the value of θ 0 is determined as
θ0 j = θ0 j−1 + j θ0
(24)
Where, θ 0 =θ 0max /50, θ 01 = 0. After that, the deflection locus of the flexural beam can be determined by elliptic integral solutions. Because the 5R PRBM has five degree-of-freedom, it is possible that the end position, end slope and hinge position are set to be the same as that of inflection point. In this way, we have
(Qx , Qy ) = (a, b) Qix , Qiy = (ai , bi ) 0 =θ0
(25)
With a set of values of characteristic radius factors, the pseudo-rigid-body angles 1 , 2 , 3 , 4, 2R-I and 2R-II can be determined through the kinematic analysis using Eqs.(11)–(14) for the PRBM-I and Eqs.(18)–(21) for the PRBM-II. If one of the angles ( 1 , 2 , 3 , 4 ) is not a real one, the values of characteristic radius factors are not feasible. To avoid such infeasible values of characteristic radius factors, an objective function is added with a relatively big value M as penalty factor. We have
ea = | 2R−I − 1R−II | 1 ∈ R ∧ 2 ∈ R ∧ 3 ∧ 4 ∈ R . ea = M 1 ∈/ R ∨ 2 ∈/ R ∨ 3 ∨ 4 ∈/ R
(26)
Then, the objective function Ea can be determined as 50
Ea =
ea j
j=1
j
(27)
The optimization for the characteristic radius factors of the 5R PRBM can be described as: Minimize Fitness = Ea Subject to 4
γi =1 γi > 0 (i = 0, 1, 2, 3, 4, 5)
i=0
The process to determine Ea can be presented by the flowchart shown in Fig. 3. In this way, the optimal characteristic radius factors of the 5R PRBM can be obtained as follows when the load parameters are selected as n = 3 and κ = 1.5.
γ0 = 0.050, γ1 = 0.257, γ2 = 0.251, γ3 = 0.121, γ4 = 0.277, γ5 = 0.044
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Fig. 3. Flowchart for determining the objective function Ea of the 5R PRBM.
3.3. Stiffness coefficients The spring stiffness coefficients of the 5R PRBM describe the ability of flexible beam to resist deflection. After the characteristic radius factors have been obtained in above section, the torque of spring in the joints of the 5R PRBM can be calculated by static equations. Then, the stiffness coefficients can be determined through a linear regression technique. Define T1 , T2 , T3 and T4 as the torque of joints, t1 , t2 , t3 and t4 as the non-dimensional factors of torques, as shown in Fig. 4. Define K1 , K2 , K3 and K4 as the stiffness constants of torsion springs and k1 , k2 , k3 and k4 as the stiffness coefficients, satisfying
Ki =
EI l
· ki
Ti =
EI l
· ti
( i = 1, 2, 3, 4 )
(28)
For PRBM-I, the static equations are
T1 = P (n(γ2 l sin ( 1 + 2 ) + γ1 l sin 1 ) + γ2 l cos ( 1 + 2 ) + γ1 l cos 1 )
(29)
T2 = P γ2 l (n sin ( 1 + 2 ) + cos ( 1 + 2 ) )
(30)
The joint torques of PRBM-II can be presented as
T3 = P γ3 l (n sin ( 3 + 4 + 0 ) + cos ( 3 + 4 + 0 ) )
(31)
T4 = P γ3 l (n sin ( 3 + 4 + 0 ) + cos ( 3 + 4 + 0 ) )+ P γ4 l (n sin ( 4 + 0 ) + cos ( 4 + 0 ) )
(32)
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507
Fig. 4. Force analysis of the 5R PRBM.
From Eqs.(29)–(32), the non-dimensional equations can be derived as
t1 = α 2 (n(γ2 sin ( 1 + 2 ) + γ1 sin 1 ) + γ2 cos ( 1 + 2 ) + γ1 cos 1 )
(33)
t2 = α 2 γ2 (n sin ( 1 + 2 ) + cos ( 1 + 2 ) )
(34)
t3 = α 2 γ3 (n sin ( 3 + 4 + 0 ) + cos ( 3 + 4 + 0 ) )
(35)
t4 = α 2 γ3 (n sin ( 3 + 4 + 0 ) + cos ( 3 + 4 + 0 ) )+ α 2 γ4 (n sin ( 4 + 0 ) + cos ( 4 + 0 ))
(36)
Where, α is the normalized force load [1], defined as
α2 =
P l2 EI
(37)
Assuming that the model is in the status of static equilibrium, we have
⎧ ⎪ ⎨t1 = k1 1 t 2 = k 2 2 ⎪ ⎩t3 = k3 3 t 4 = k 4 4
(38)
Substituting Eq. (38) into Eqs.(33)–(36), we obtain
k1 1 = α 2 (n(γ2 sin ( 1 + 2 ) + γ1 sin 1 ) + γ2 cos ( 1 + 2 ) + γ1 cos 1 )
(39)
k2 2 = α 2 γ2 (n sin ( 1 + 2 ) + cos ( 1 + 2 ) )
(40)
k3 3 = α 2 γ3 (n sin ( 3 + 4 + 0 ) + cos ( 3 + 4 + 0 ) )
(41)
k4 4 = α 2 γ3 (n sin ( 3 + 4 + 0 ) + cos ( 3 + 4 + 0 ) )+ α 2 γ4 (n sin ( 4 + 0 ) + cos ( 4 + 0 ))
(42)
Then, the stiffness coefficients can be obtained in the following 4 steps. Firstly, set the boundary condition, and determine the tip point (a/l, b/l) of flexible beam by elliptic integral solutions for given n, κ , and θ 0. Secondly, set (Qx , Qy )=(a/l, b/l), (Qix , Qiy )=(ai /l, bi /l) , 0 =θ 0 . Determine the PRBM angles 1 , 2 , 3 and 4 from the inverse kinematics Eqs.(11)–(14) and (18)–(21). Thirdly, substitute the angles 1 , 2 , 3 and 4 into static equations Eqs.(33)–(36) to determine the joint torques of the 5R PRBM.
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Fig. 5. Fitting curve of stiffness coefficient of the 5R PRBM.
Finally, determine the stiffness coefficients by linear regression technique. In this study, ratio n and slope θ 0 are set to fixed values. The load ratio κ varies in the range of (0, κ max ). κ max determines the max value of κ satisfying the load condition where an inflection point is possible to occur.
κ = 1 − cos(φ − θ0 )
(43)
Set n = 1, θ 0 = 0.0126, the variation range of κ is determined as (0, 1.698) from Eqs. (8) and (43). From above procedure, the fitting curves of stiffness coefficients for the 5R PRBM can be obtained, as shown in Fig. 5. The stiffness coefficients of the 5R PRBM are obtained as follows.
k1 = 6.948, k2 = 3.160, k3 = 3.210, k4 = 6.418 4. Numerical examples In this section, the 5R PRBM developed above is applied to simulate the deflection of flexible beam with an inflection point, including the position of inflection point and beam tip. For comparison, the result obtained by elliptical integral solutions is also presented. First of all, defining the error of locus for the inflection point as
ei =
Qix −
ai l
2
+
Qiy −
bi l
2 (44)
Defining the error of locus for the beam tip as
et =
Qx −
a l
2
+
Qy −
b l
2 (45)
Hence, the average error of inflection point and beam tip can be defined as, respectively, 50
ei a =
ei j
j=1
50
et a =
(46)
j et j
j=1
j
(47)
Since the curvature at the inflection point is zero, it is reasonable to constrain the angle between the two links jointed by the free hinge Qi to be zero. We have
2R−I = 2R−II + 0 , The errors can be obtained in the following 4 steps.
(48)
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Table 1 Average errors of the 5R PRBM in different load conditions. Case
n
κ
Inflection
eta
eia
I II III IV V VI
0 1 3 0 1 3
0.4 0.6 1.5 0.4 0.6 1.5
yes yes yes no no no
0.16% 0.40% 1.14% 0.11% 0.19% 0.24%
1.58% 3.06% 3.30% – – –
First, obtain the locus of inflection point I(ai , bi ) and beam tip T(a, b), and the force load α by elliptic integral solutions. Second, substitute the force load α , force ratio n, and angle 0 =θ 0 into Eqs.(33)–(36) and (48) to obtain the pseudorigid-body angle 1 , 2 , 3 , 4 . Third, substitute the pseudo-rigid-body angle 0 , 1 , 2 , 3 and 4 into Eqs.(9), (10), (16) and (17) to obtain the locus of hinge Qi (Qix , Qiy ) and tip Q (Qx , Qy ) of the 5R PRBM. Fourth, obtain the errors of inflection point and beam tip from Eqs.(44)–(47). In order to show the effectiveness and superiority of the 5R PRBM, six cases are studied here for quite different load parameters n and κ ,. In addition to the case of n = 3 and κ = 1.5 discussed in above section for determining the characteristic radius factors of the 5R PRBM, the other cases of n = 0, 1 and κ = 0.4, 0.6 are also presented to illustrate the validation of the 5R PRBM for different load conditions. These 6 cases of load conditions are listed in Table 1 from case I to VI. In cases I–III, the flexible beam has one inflection point. For comparison, there is no inflection point in the beam for other three cases IV–VI. Fig. 6 shows the configurations of the 5R PRBM for the deflection of flexural beam in cases I–VI. The solid lines represent the configurations of the 5R PRBM and the dotted lines obtained by elliptical integral solutions describe those of the flexible beams. It can be seen from the figures that the tip of the 5R PRBM and that of the flexural beam almost coincide in most cases. In case III, the flexible beam with an inflection point is in a large-torque load condition of κ = 1.5 that causes two segments of the beam bend in two opposite direction (elbow up and elbow down). Moreover, it can be seen from the figures that the configurations of the 5R PRBM are closed to those of the flexible beam in all cases. The 5R PRBM can also be used to predict precisely the deformation of the flexural beam without inflection point as shown in cases IV–VI. This result indicates well the effectiveness of the 5R PRBM. Fig. 7 shows the errors on beam tip and/or inflection point of the 5R PRBM in cases I–VI. In case I, II and III, the errors of inflection point are relatively big while those of the slope of beam tip are very little. The tip locus errors fluctuate with the increase of tip slope. However, in case IV, V and VI without inflection point, the errors of tip locus increases monotonically with the increase of tip slope in a large scale. Moreover, it can also be seen in Fig. 7 that the errors of inflection point drop below 0.02 when the slope of the beam tip is over 0.2 rad. However, they are relatively big when the slope of beam tip is small. It is interesting to note that the inflection point locus is described by the PRBM-I that is part of the 5R PRBM while the tip locus of flexible beam is predicted by whole the 5R PRBM. Therefore, the errors of beam tips are obviously smaller than those of the inflection points. This result agrees with the fact that the more joints the PRBM has, the more accurate it becomes, such as the 1R, 2R to 3R PRBM developed in previous work. The average errors of the 5R PRBM in six cases are listed in Table 1. It can be seen that the average errors of beam tip locus are less than 1.14% and those of inflection point locus are less than 3.30%. This result shows well the accuracy of the 5R PRBM. It is true that the error increase as the slope of the beam tip increases. However, the average errors of the 5R PRBM in all cases of load are very small and kept in the acceptable range. This result shows that the 5R PRBM with the characteristic parameters determined in above section can also be applied to other different load conditions. That is to say, the 5R PRBM proposed in this paper behaves as load independent. This result indicates the superiority of the 5R PRBM over the 4R PRBM developed in Ref. [17] that is load dependent. This may be owing to that the 5R PRBM is a symmetric structure and it has more degrees-of-freedom than the 4R PRBM. The limit of the beam tip slope θ 0max can be used here to determine the condition where the inflection point exists. If the beam tip slope exceeds θ 0max , no inflection point can be created. In this section, the numerical results are obtained with non-dimensional factors only to show the accuracy of the 5R PRBM in general cases. If the model is used to analyze other compliant mechanisms, the parameters about the geometric and material properties of the flexural beams in the mechanisms are need to be determined while the characteristic parameters of the 5R PRBM obtained in this paper can be used directly. A brief flowchart is presented in Fig. 8 to show how to fully construct the 5R PRBM of a compliant mechanism. 5. Conclusions A 5R pseudo-rigid-body model was proposed in this study to simulate the inflection beams in compliant mechanisms. The characteristic parameters of the 5R PRBM were determined through optimization and linear regression. The numerical
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Fig. 6. Configurations of the 5R PRBM and the flexible beam in cases I–VI.
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Fig. 7. Errors of the 5R PRBM in cases I–VI.
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Fig. 8. Flowchart of using the 5R PRBM.
examples showed that this model behaved as load independent and it could be used to predict the deflection of flexible beams with or without inflection. This new model can also be used to the flexural beams in other load cases and it is, therefore, convenient in early stage of design or analysis of the flexural beams in complaint mechanisms. Acknowledgements The financial support of this study was from National Natural Science Foundation of China (Grant No. 51575006). References [1] L.L. Howell, Compliant Mechanisms, John Wiley & Sons, New York, NY, 2001. [2] A. Zhang, G. Chen, A comprehensive elliptic integral solution to the large deflection problems of thin beams in compliant mechanisms, ASME Trans J. Mech. Robot. 5 (2) (2012) 420–431. [3] L.L. Howell, A. Midha, T.W. Norton, Evaluation of equivalent spring stiffness for use in a pseudo-rigid-body model of large-deflection compliant mechanisms, ASME Trans J. Mech. Des. 118 (1) (1996) 126–131. [4] L. HowellL, A. Midha, Parametric deflection approximations for end-loaded, large-deflection beams in compliant mechanisms, ASME Trans J. Mech. Des. 117 (1) (1995) 156–165. [5] H.J. Su, Apseudorigid-body 3R model for determining large deflection of cantilever beams subject to tip loads, ASME Trans. J. Mech. Robot. 1 (2) (2009) 795–810. [6] Y.Q. Yu, Z.L. Feng, Q.P. Xu, A pseudo-rigid-body 2R model of flexural beam in compliant mechanisms, Mech. Mach. Theory 55 (9) (2012) 18–33. [7] Y.Q. Yu, P. Zhou, Q.P. Xu, A new pseudo-rigid-body model of compliant mechanisms considering axial deflection of flexural beams, Mech. Mach. Sci. 24 (2015) 851–858. [8] Y.Q. Yu, S.K. Zhu, Q.P. Xu, P. Zhou, A novel model of large deflection beams with combined end loads in compliant mechanisms, Precis. Eng. 43 (2016) 395–405. [9] C. Boyle, L.L. Howell, S.P. Magleby, M.S. Evans, Dynamic modeling of compliant constant-force compression mechanisms, Mech. Mach. Theory 38 (12) (2003) 1469–1487. [10] Y.Q. Yu, L.L. Howell, C. Lusk, Y. Yue, M.G. He, Dynamic modeling of compliant mechanisms based on the pseudo-rigid-body model, ASMR Trans. J. Mech. Des. 127 (4) (2005) 760–765. [11] L.L. Howell, A. Midha, A loop-closure theory for the analysis and synthesis of compliant mechanisms, ASME Trans. J. Mech. Des. 118 (1) (1996) 121–125. [12] B.D. Jensen, L.L. Howell, L.G. Salmon, Design of two-link, in-plane, bistable compliant micro-mechanisms, ASME Trans. J. Mech. Des. 121 (3) (1999) 416–423. [13] Q.T. Aten, S.A. Zirbel, B.D. Jensen, L.L. Howell, A numerical method for position analysis of compliant mechanisms with more degrees of freedom than inputs, ASME Trans. J. Mech. Des. 133 (6) (2011) 061009. [14] M. Jin, X. Zhang, B. Zhu, A numerical method for static analysis of pseudo-rigid-body model of compliant mechanisms, Proc. Inst. Mech. Eng. Part C 228 (17) (2014) 3170–3177. [15] C. Kimball, L.W. Tsai, Modeling of flexural beams subjected to arbitrary end loads, ASME Trans. J. Mech. Des. 124 (2) (2002) 223–235. [16] A. Midha, S.G. Bapat, A. Mavanthoor, V. Chinta, Analysis of a fixed-guided compliant beam with an inflection point using the pseudo-rigid-body model (PRBM) concept, ASME Trans J. Mech. Robot. 64 (4) (2012) 147–154. [17] S.K. Zhu, Y.Q. Yu, Pseudo-rigid-body model for the flexible beams with an inflection point in compliant mechanisms, ASME Trans J. Mech. Robot. 9 (3) (2017) 031005-031005-8. [18] A.G. Greenhill, The Applications of Elliptic Functions, Dover, New York, 1959. [19] V. Michael, The Simple Genetic Algorithm: Foundations and Theory, MIT Press, Cambridge, MA, 1999. [20] S. Weisberg, Applied Linear Regression, third ed., WileyInterscience, New York, NY, 2005. [21] A. Saxena, A New Pseudo-Rigid-Body Model for Flexible Members in Compliant Mechanisms MS thesis, University of Toledo, Toledo, OH, USA, 1997.