- Email: [email protected]
A novel model of large deflection beams with combined end loads in compliant mechanisms
Accepted Manuscript Title: A Novel Model of Large Deflection Beams with Combined End Loads in Compliant Mechanisms Author: Yue-Qing Yu Shun-Kun Zhu Qi-Ping Xu Peng Zhou PII: DOI: Reference:
S0141-6359(15)00164-6 http://dx.doi.org/doi:10.1016/j.precisioneng.2015.09.003 PRE 6289
To appear in:
Precision Engineering
Received date: Revised date: Accepted date:
11-8-2014 11-4-2015 9-9-2015
Please cite this article as: Yu Y-Q, Zhu S-K, Xu Q-P, Zhou P, A Novel Model of Large Deflection Beams with Combined End Loads in Compliant Mechanisms, Precision Engineering (2015), http://dx.doi.org/10.1016/j.precisioneng.2015.09.003 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Highlights
PRR pseudo-rigid-body model is proposed for large deflection beams in compliant mechanisms. Lateral and axial deflections of beams are simulated by 3 kinematic pairs (PRR) with springs.
ip t
Characteristic parameters of PRR model are determined via optimization and fitting technique.
Ac ce p
te
d
M
an
us
cr
Superiority of PRR PRBM over 2R, PR and 3R model is shown in numerical examples.
1
Page 1 of 39
A Novel Model of Large Deflection Beams with Combined End Loads in Compliant Mechanisms
ip t
Yue-Qing Yu*, Shun-Kun Zhu, Qi-Ping Xu, Peng Zhou College of Mechanical Engineering and Applied Electronics Technology,
cr
Beijing University of Technology, Beijing 100124, P. R. China
us
*Corresponding author. E-mail: [email protected], Tel: 86-10-67391702
an
Abstract
M
Based on the Pseudo-Rigid-Body Model (PRBM), a new model with three degrees of freedom is proposed for the large deflection beams with combined end
d
loads in compliant mechanisms. The lateral and axial deflections of flexural beams
te
are modeled using four rigid links connected by one prismatic (P) pair with a
Ac ce p
compression spring and two revolute (R) joints with torsion springs. The flexural cantilever beam subject to end force and moment loads is simulated by PRR pseudo-rigid-body model. The characteristic parameters of the PRR PRBM are determined via the optimization and numerical fitting techniques. Compared with the 2R, PR and 3R models, the new PRR PRBM shows the superiority in simulating the large deflection beams of compliant mechanisms through numerical examples.
Keywords: Compliant mechanism; Pseudo-rigid-body model (PRBM); PRR model; Lateral and axial deflection; Combined end force and moment loads
2
Page 2 of 39
1 Introduction Traditional mechanisms are usually composed of rigid members connected with kinematic pairs. Some inevitable problems of such kind of mechanisms are exposed
ip t
for the requirements of high-speed, high-precision, high-performance, and
cr
high-efficiency mechanical systems. For rigid body mechanisms, the clearances and
us
frictions are the two main factors that affect the accuracy and especially the dynamic behaviors of mechanisms [1]. In addition, the assembly costs occupy a large
an
proportion in labor costs, and designers always try to reduce the assembly costs. These problems are very difficult to overcome for the conventional hinge mechanisms, Midha and Her [2] proposed a
M
but easy to solve for the compliant mechanisms.
mechanism that transfers motion and force mainly relying on the elastic deformation
te
d
of its links and named it as compliant mechanism for the first time. The compliant mechanism is different from general rigid mechanisms for it gains the mobility from
Ac ce p
the deflection of its flexible members, offers many advantages such as increased precision, reduced backlash and parts number, ability to store energy [3]. Therefore, compliant mechanisms have attracted more and more attentions of researchers in the field of mechanical engineering because of its particular functions and broad application prospects. However, a major difficulty of designing compliant mechanisms lies in modeling the large deflection of flexural beams. Many methods for modeling and designing compliant mechanisms have been proposed. In [4], a method was proposed to simulate the flexural beam using the linear torsion spring. This basic work laid the foundation of the pseudo-rigid-body model
3
Page 3 of 39
(PRBM) proposed later. In [5-7], the definition of elements in compliant mechanism was improved and important contributions were made to the study of compliant mechanism. The concept of fragment was developed in the classification of compliant
ip t
mechanisms that can be divided into several fragments in accordance with different cross sections [8]. In [9], an extensive investigation was made in the modeling,
cr
numerical analysis methods and software development of compliant mechanisms. A
us
finite element model was proposed [10] in which a flexural beam can be modeled as several segments of PRBM, however, the kinematic equations are very difficult to
an
solve in applications. In [11], a method to determine the degree of freedom of
M
compliant mechanisms was presented based on the segments and connection types among segments. Banerjee and et al [12] used a non-linear shooting method and
d
decomposition method to obtain large deflection of a cantilever beam subjected to
te
loading at intermediate locations besides end forces and moments. Chen and Zhang
Ac ce p
[13] used a comprehensive elliptic integral solution to solve large deflections of flexural beam with multiple inflection points. Based on the structural and kinematic analysis of rigid body mechanisms, Howell
and Midha proposed the Pseudo-Rigid-Body Model (PRBM) which played a very important role in simplifying the complicated modeling of compliant mechanism [14-15]. A PRBM of cantilever beam with end-force load was established to simulate the tip locus of beam end by 1R PRBM shown in Fig. 1(a) which consists of two rigid links joined with a revolute joint and a torsion spring [16]. The pseudo-rigid-body models subjected to three different end loads were established in [17]. A loop-closure
4
Page 4 of 39
theory for the analysis and synthesis of compliant mechanisms was presented later [18]. The parameters of 1R PRBM for compliant mechanisms are different with the various loads and the accuracy is not high enough. In [19], the PRBM was modified
ip t
by introducing two linear springs to restrain the change of characteristic radius factor for different load modes. Kimball and Tsai [20] used PRBM to model the large
cr
deflection of a cantilever beam with an inflection point. Furthermore, Su proposed a
us
3R pseudo-rigid-body model [21], as shown in Fig. 1(b), to simulate the large deflection of flexural beam subjected to tip loads. Although it has higher simulation
an
precision than that of 1R pseudo-rigid-body model, the inverse kinematic solution and
M
the iterative process are very difficult to obtain the characteristic parameters of the 3R PRBM. In [22], an improved particle swarm optimizer was used to determine the
d
optimal characteristic parameters of 3R PRBM. It showed better performance in
te
predicting large deflections of cantilever beams with pure end force load or moment
Ac ce p
load. However, deflections of cantilever beams with combined end force and moment load was not discussed. In [23], a 2R pseudo-rigid-body model was proposed to improve the simulation accuracy of 1R PRBM and simplify the iterative process of 3R PRBM. The 2R PRBM is composed of three rigid links joined by two pin joints and torsion springs, as shown in Fig. 1(c). The pseudo-rigid-body model was also applied to predict the motion of the human spine [24]. Based on the PRBM for large beam deflections, a method was introduced to synthesize the three-link and four-link compliant mechanisms that exhibit prescribed force-deflection responses [25]. Midha and et al [26] used the concept of PRBM to present a simple method of analyzing a
5
Page 5 of 39
fixed-guided compliant beam with an inflection point for several kinds of boundary conditions. From the point of view of degree of freedom of mechanisms, the simulating
ip t
precision of 1R PRBM is not very high because it has single degree of freedom. The 2R PRBM has two degrees of freedom and it can be used to simulate the end position
cr
and/or slope of a flexural link, but two out of three parameters are independent. The
us
3R pseudo-rigid-body model has three degrees of freedom and so it can describe both the end position and slop of a flexural beam. However, although the 1R, 2R and 3R
an
pseudo-rigid-models contain pin joints and can be used to simulate the bending
M
deformation of a flexural beam only, they are not useful to describe the axial deformation of the beam. If a flexible beam is subjected to combined end force and
d
moment loads, the axial deformation of the beam cannot be presented by the
te
pseudo-rigid-body models with revolute pairs only. In [27], a PR pseudo-rigid-body
Ac ce p
model with a prismatic pair was developed to simulate both of the axial and lateral deformation of a flexural beam. Compared with the 1R and 2R models, the PR PRBM has higher simulation precision for the flexural beam with force or moment load. However, the simulation accuracy of the PR PRBM is still needed to increase to deal with the flexible beam subjected to combined end force and moment loads. Therefore, a further study should be made to improve the simulating ability of the pseudo-rigid-body models in the analysis and design of compliant mechanisms.
6
Page 6 of 39
F0
a
b
2l
0l
EI , l
1l
K1
x
K2
3
2l
b
1l
2
EI , l
2
K2
b K1
1
0l
a
K3
.
K
F0
y
a 3l
1l
F0
y
O
x
0l
1
ip t
y
EI , l
x
us
Fig. 1 Three pseudo-rigid-body models.
cr
(a) 1R pseudo-rigid-body model. (b) 3R pseudo-rigid-body model. (c) 2R pseudo-rigid-body model.
an
A new PRR pseudo-rigid-body model is proposed in this study to simulate the
M
large deflection beam with combined end force and moment loads. The axial deformation of a flexural beam is described by one sliding link with a prismatic pair
d
and a compression spring. The lateral deflection of the beam is simulated by two
te
rotating links with pin joints and torsion springs. The characteristic parameters of the
Ac ce p
PRR PRBM are determined via the optimization and numerical fitting techniques. A numerical comparison among the 2R, PR, 3R and PRR pseudo-rigid-body models is presented. The result shows the superiority of the new PRR PRBM in simulating the large deflection beams with combined end force and moment loads in compliant mechanisms.
2 PRR pseudo-rigid-body model A large deflection cantilever beam with combined force and moment loads at end
7
Page 7 of 39
is shown in Fig. 2(a). 0 is the deflection angle of the beam, a and b are the horizontal and vertical coordinates of the tip point of the beam, respectively. l is the undeflected length of the flexible beam, and is the angle between the force
ip t
direction and horizontal direction. In order to simulate the large deflection of the flexural beam with end force and moment loads, a PRR Pseudo-Rigid-Body Model
cr
(PRR PRBM) is proposed here, as shown in Fig. 2(b). It consists of four rigid links
us
joined by a prismatic (P) pair with a compression spring and two revolute (R) pairs with torsion springs. In this way, the axial deformation of flexural beam can be
an
described by the sliding link with P pair and compression spring, and the main lateral
M
deflection of the beam can be simulated by the two rotating links joined by the two pin joints and torsion springs. Because the PRR PRBM has three degrees of freedom
d
and so it can be used to simulate accurately both the end position and slop of the
Ac ce p
te
flexural beam.
F0
y
a
y
0
nP 2l
M0
O
EI , l
F0
P
b
K2
1l
1
K
O
x
0 l 0l
M0 2
K1
EI , l
x
(a) Flexural beam with combined end force and moment loads. (b) PRR pseudo-rigid-body model. Fig. 2 Flexural beam with combined end force and moment loads and the PRR PRBM.
8
Page 8 of 39
The length of each rigid link in the PRR PRBM is presented as i l (i 0,1, 2) , and
i is called the characteristic radius factor [2], satisfying 0 1 2 1 .
0 is the variable characteristic factor and 0l indicates the axial displacement
ip t
of the sliding link. K1 , K 2 and K are the torsion spring stiffness constants and the compression spring stiffness constant, respectively. These factors and constants are
cr
the key characteristic parameters of the PRR PRBM to simulate the large deflection
us
beam with combined end force and moment loads in compliant mechanisms and they
an
will be determined in the following part.
M
3 Characteristic parameters
Because the process to determine the characteristic parameters of the PRR
d
PRBM is complicated, this section is divided into five subsections for better
te
understanding. It is organized as follows. An elliptic integration of deflection for a
Ac ce p
flexural beam is presented at first in Section 3.1. The kinematic and static equations of the PRR PRBM are derived in Section 3.2 and 3.3, respectively. The optimal characteristic parameters and spring stiffness coefficients of the PRR PRBM are determined in Section 3.4 and 3.5, respectively.
3.1 Elliptic Integration of Deflection
When a flexural beam is subjected to a combined force and moment at end, as shown in Fig. 2(a), the internal torque of the beam at any point can be presented as:
9
Page 9 of 39
M F0 a x sin b y cos M 0
(1)
Where, a is the horizontal distance from the fixed end to the free end, and b is the vertical distance of the free end from its undeflected position.
ip t
Substituting Eq. (1) into the Bernoulli-Euler equation, we have
M d d 2 y / dx 2 EI ds [1 ( dy / dx) 2 ]3/2
cr
(2)
us
From the theory of Elliptic Integrals [3], it can be derived from above differential equations that
an
d 2 F0 sin( ) ds 2 EI
d M 0 = when 0 , we have: ds EI
M
Integrating Eq. (3) with the boundary condition
(3)
(4)
d
M l d 1 2 F0l 2 [cos( 0 ) cos( )] ( 0 ) 2 ds l EI EI
te
Defining
F0 l 2 2 EI
Ac ce p
2
(5)
M 0l EI
(6)
Eq. (4) becomes:
d 1 4 [cos(0 ) cos( )] 2 ds l
(7)
It can be obtained by solving the differential equation that
2 2 Where,
d
0 0
cos 0 sin
(8)
n2 1
10
Page 10 of 39
Defining the load index =
2 , the dimensionless coordinates of the tip point 2 2
a b Q , of the flexural beam can be presented briefly as follows: l l
cos d
0
0
cos 0 cos sin d
0 0
cos 0 cos
(9)
ip t
b 1 l 2
(10)
cr
a 1 l 2
us
With these derivations above, the kinematic and static equations of the PRR pseudo-rigid-body model for the large deflection beam with end force and moment
M
an
loads in compliant mechanisms can now be derived as follows.
3.2 Kinematic Equations
d
The slope angle of the PRR PRBM can be presented as , and the deflection
te
angles of two torsion springs are described as 1 , 2 , as shown in Fig. 2(b). P
Ac ce p
indicates the vertical force and nP represents the horizontal force. So, the total force
F can be expressed as follows.
F0 P n 2 1
(11)
The angle between the force direction and the horizontal direction is
arctan
1 n
(12)
Because the slope angle of the PRR PRBM in Fig. 2(b) should be equal to the deflection angle 0 of the flexural beam in Fig. 2(a), the following equations can be written as:
11
Page 11 of 39
a l 0 0 1 cos 1 2 cos(1 2 ) b 1 sin 1 2 sin(1 2 ) l 0 1 2
ip t
(13)
us (14)
Ac ce p
3.3 Static Equations
te
d
M
an
b 2 sin 0 1 l 1 sin 1 b l 2 sin 0 1 2 0 sin 1 a 0 2 cos 0 1 cos 1 0 l
cr
So, the following relationship can be derived from the above equation:
For a given combined loads exerted at the end of the PPR PRBM, and the torque
and lateral force at the pivot are:
T1 Fxl (1 sin(1 ) 2 sin()) Fyl (1 cos(1 ) 2 cos()) M0 T2 Fxl ( 2 sin()) Fyl ( 2 cos()) M0 F K l 0 x
(15)
Where, the horizontal force and vertical force can be expressed as Fx F0 cos and
Fy F0 sin , respectively. It should be noted that K in Eq. (15) represents the stiffness constant of compression spring in the PPR PRBM. The torques T1 , T2 at the hinges are the product of the torsion spring stiffness constant K1 , K 2 and the angle 12
Page 12 of 39
1 , 2 , respectively: T1 K 1 1 T2 K 2 2
(16)
From Eq. (15), we have:
ip t
Fxl 0 K1 1 sin(1) 2 sin() 1 cos(1) 2 cos() 1 0 Fyl 1 0 0 0 K2 2 sin() 2 cos() 1 0 M0 2 0 0 1 Fx 0 0 0 K 0 l
us
cr
(17)
Let
1 sin(1) 2 sin() 1 cos(1) 2 cos() 1 0 J 2 sin() 1 0 2 cos() 0 0 0 1
an
T
M
and
(18)
(19)
Ac ce p
te
d
1 0 0 Fxl 1 F l K1 y K 0 1 T J 0 M0 2 2 K Fx 1 0 0 l 0
The resistance to deflection and flexibility of the flexural beam can be described by the dimensionless torsion spring stiffness coefficients K 1 , K 2 [3] and compression
spring stiffness coefficient K L , respectively, as follows:
EI K1 l K 0 2 K 0
0 EI l 0
0 K1 0 K2 EI KL l3
(20)
Therefore
13
Page 13 of 39
0 EI l 0
1 0 1 K 1 0 K 2 0 EI K L 0 l 3
0 1 2 0
0 Fxl F l T y 0 J M 0 Fx 1 l 0
or
l EI 0
1 0 1 0 0 3 l 0 EI
0 1 2 0
(22)
an
It can be obtained from the above equations that:
0 Fx l F l T y 0 J M 0 Fx 1 l 0
cr
0
us
l K 1 EI K 2 0 K L 0
(21)
ip t
EI l 0 0
te
d
M
l l l (1 sin(1) 2 sin()) (1 cos(1) 2 cos()) 0 Fxl EI EI EI 1 1 1 F l K1 y l l l K ( sin( )) ( cos( )) 0 M0 (23) 2 2 2 EI EI2 EI2 2 K Fx L l3 0 0 0 l 0EI
and
Ac ce p
F0 cos()l 2 F0 sin()l 2 Ml ( sin( ) sin( )) (1 cos(1) 2 cos()) 0 1 1 2 EI 1 EI 1 EI 1 K1 2 2 F0 cos()l F0 sin()l M0l (24) K2 EI ( 2 sin()) EI ( 2 cos()) EI 2 2 2 K L F cos()l 2 0 0 EI
In order to simplify the above equation, and the end force and the moment
F0 , M 0 can be express as the dimensionless force index index
2
F0l 2 and torque 2 EI
M 0l [3], respectively. Therefore, EI
14
Page 14 of 39
ip t
2 2 cos( ) 2 2 sin( ) K ( sin( ) sin( )) ( 1 cos(1 ) 2 cos()) 1 1 1 2 1 1 1 2 2 2 cos( ) 2 sin( ) ( 2 sin()) ( 2 cos()) (25) K 2 2 2 2 2 2 cos( ) KL 0
Now, it can be summarized that there are totally seven characteristic parameters
cr
in the PRR PRBM as follows: the characteristic radius factors 0 , 1 , 2 , the variable
us
characteristic factor 0 , the torsion spring stiffness coefficients K 1 , K 2 and the compression spring stiffness coefficient K L . The goal of this study is to find the
M
3.4 Optimal Characteristic Factors
an
optimal parameters, and it will be discussed in the following section.
It is well known that it is very difficult to obtain the characteristic parameters of
te
d
the PRR PRBM subjected to combined end force and moment loads. Therefore, a simple and effective way is proposed in this study to overcome this difficulty. At first,
Ac ce p
it is helpful to investigate the effects of load ratio n and load index on the
stiffness coefficients of springs in the PRR PRBM. Fig. 3 shows the plot of the spring stiffness when the load ratio n changes in a very large range from 1 to 10. It can be seen clearly from the figure that the spring stiffness is little correlated to the load ratio of horizontal force to vertical one. Hence, the effect of load ratio n is very small.
However, the effect of load index is quite different. It is known that the load condition is close to a pure force when 0 and it is a pure moment when . However, these two extreme load conditions are far from the case of PRR PRBM subjected to combined end force and moment loads and so they are not considered
15
Page 15 of 39
here. Fig. 4 shows the variation of spring stiffness coefficients with change of load index from 1 to 50. It can be seen in Fig. 4 that the stiffness coefficient changes monotonically while the load index increases greatly from 1 to 10 and almost
ip t
remains constant when changes from 10 to 50. This result shows that the change range from 1 to 50 is a sufficient scale to obtain a proper value of load index for
cr
combined end force and moment loads. So, the variation range of from 1 to 50 is
us
selected in this study. Therefore, it can be deduced that the optimal characteristic parameters i could be the one that has the minimum of the square of difference of
Ac ce p
te
d
M
an
the spring stiffness between the two cases ( =1 and =50 ).
Fig. 3 stiffness K i vs. load ratio n 1,10
16
Page 16 of 39
ip t cr us
M
an
Fig. 4 stiffness K i vs. load index 1,50
Based on the analysis above, two special cases of load index =1 and =50
d
are dealt with at first in this study. The optimal characteristic factors 0 , 1 , 2 and
can be determined finally. The optimal characteristic parameters
Ac ce p
K 1 , K 2 and K L
te
0 can be then obtained through optimization techniques. The stiffness coefficients
of the PRR PRBM subjected to combined end force and moment loads can be found. This is a three-dimensional search and the whole process to determine the characteristic parameters of the PRR PRBM can be presented as an optimization procedure as shown in the flow chart of Fig. 5. The optimal characteristic radius factors of the PRR PRBM subjected to combined end force and moment loads can be determined via the optimization as follows:
0 0.2, 1 0.52, 2 0.28
17
Page 17 of 39
Start
Input i (i =0,1,2) Satisfying 0 1 2 =1
ip t
i i +i (i 0,1,2)
No
i 1
cr
Yes
us
Set load condition
No
1 or 50
Yes
an
Substitute 0 into Eqs.(8)-(10) and compute a / l , b / l
and compute 1 , 2
1?
M
Substitute a / l , b / l , 0 into Eqs. (14)
No
d
Yes
te
Solve T1 and T2 from Eq. (15). Solve K a1 , K a 2
Ac ce p
through linear regression process
Solve T1 and T2 from Eq. (15). Solve K b1 , K b 2 through linear regression process
2
f ( i ) ( K aj K bj ) 2
Yes
j 1
f ( i i ) f ( i )
No
Output i
End
Fig. 5 Optimization procedure of characteristic parameters for PRR PRBM.
18
Page 18 of 39
3.5 Optimal Spring Stiffness Coefficients Now, the torsion spring stiffness coefficients of the PRR PRBM can be determined from the characteristic radius factors obtained above. When the PRR
ip t
PRBM is subjected to a combined end load ( =1) , a linear regression process [28] can be applied here to determine the approximate value of the torsion spring stiffness
cr
coefficient K a1 at the first pin joint from Eq. (25). The linear fitting curve is shown
us
in Fig. 6. Similarly, when the PRR PRBM is subjected to a combined end load
( =50) , the linear regression process can also be applied to determine the
an
approximate value of torsion spring stiffness coefficient K b1 at the first pin joint
Ac ce p
te
d
M
from Eq. (25). The linear fitting curve is shown in Fig. 7.
Fig. 6 Linear regression of stiffness coefficient
K a1 for ( =1) .
Fig. 7 Linear regression of stiffness coefficient
K b1 for ( =50) .
19
Page 19 of 39
In a similar way, the torsion spring stiffness coefficients K a2 ( =1) and
K b2 ( =50) , at the second pin joint of the PRR PRBM can be determined, too. The
an
us
cr
ip t
results are shown in Fig. 8 and Fig. 9, respectively.
K a2 for ( =1) .
Ac ce p
te
d
M
Fig. 8 Linear regression of stiffness coefficient
Fig. 9 Linear regression of stiffness coefficient
K b2 for ( =50) .
Therefore, the torsion spring stiffness coefficients of the PRR PRBM can be determined as follows. The corresponding values at the two cases are:
K a1 2.045, K b1 2.207 K a 2 1.828, K b 2 1.823
20
Page 20 of 39
As shown in Fig. 6-9, the spring stiffness K 1 increases from 2.045 ( 1 ) to 2.207 ( 50 ) while K 2 decreases from 1.828 ( 1 ) to 1.823 ( 50 ). It can be seen that the relative errors of the two cases ( 1 and 50 ) are small and the
ip t
spring stiffness changes slightly within 10,50 . So, the average value of the two cases can be selected as an acceptable solution for the PRR PRBM subjected to end
force
and
moment
loads.
Although
this
may
be
cr
combined
not
us
a mathematical justification, the numerical results obtained in Section 4 will show the effectiveness of this simple method. The torsion spring stiffness coefficients of the
an
PRR PRBM can be determined at =25.5 as follows:
M
K 1 1.930, K 2 2.032
The torsion spring stiffness constants K1 , K 2 depend on the torsion spring
d
stiffness coefficients K 1 , K 2 , geometrical relationship I / l , and material
te
properties E , respectively, and can be expressed as:
Ac ce p
K1 K 1
EI EI , K 2 K 2 l l
(26)
For the variable characteristic factor 0 in eqn. Eq. (14), it can be expressed
by the slope angle of PRR PRBM. Applying the Least Square Polynomial Fitting Method [29], a Quintic Polynomial of is used here to fit 0 . The value of 0 can be determined for each value of , and it can be fitted as shown in Fig. 10.
Therefore, the fitting curve of 0 can be presented in following equation: 0 0.033 5 +0.1528 4 0.25843 +0.1638 2 0.0494
(27)
21
Page 21 of 39
ip t cr
an
us
Fig. 10 Fitting curve of 0 .
When 0 has been obtained, the stiffness coefficient of compression spring K L
M
in the PRR PRBM can be determined with 2 solved from Eq. (8). Using the linear regression process [25], K L can be fitted as shown in Fig. 11.
Ac ce p
te
d
K L 0.871
Fig. 11 Fitting curve of stiffness coefficient
KL .
22
Page 22 of 39
Now, all the characteristic parameters of the PRR PRBM have been determined and, therefore, the new model has been developed to simulate the large deflection
ip t
beam with combined end force and moment loads in compliant mechanisms.
cr
4 Precision analyses
Fig. 12 shows the tip locus of the PRR, 2R, PR and 3R pseudo-rigid-body model,
us
as well as the flexural beam with combined end force and moment loads. It can be
an
seen from Fig. 12 that the four pseudo-rigid-body models can mainly follow the tip locus of the flexural beam in a small deformation condition. However, the curve of 2R
M
PRBM deviates off the tip locus of the flexural beam in the large deflection area. This is because the 2R model has two degree of freedom only and so its simulation
d
accuracy is low. Compared with the 2R PRBM, the PR PRBM has also two degrees of
te
freedom, but it can improve greatly the accuracy due to the sliding link with a
Ac ce p
prismatic pair and compression spring that describes the axial deformation of the flexural beam. Moreover, the PRR PRBM has three degree-of-freedom and so increases the simulation precision further. Although the 3R PRBM can well follow the tip locus of the flexural beam because it has three degrees of freedom too, the trajectory curve of the PRR PRBM is much closer to that of the actual flexural beam curve. So, the PRR PRBM is more accurate than the 3R PRBM. This result indicates further the important role of the prismatic pair in the pseudo-rigid-body model and the superiority of PRR PRBM in increasing the simulation accuracy for the large deflection beam with combined end force and moment loads in compliant
23
Page 23 of 39
mechanisms. At same time, it also shows the effectiveness of the simple method
M
an
us
cr
ip t
proposed in Section 3.
te
d
Fig. 12 Comparison of tip locus between the flexural beam and PRBMs.
Ac ce p
A discussion on the simulating errors of the four pseudo-rigid-body models can be made to illustrate further the advantage of the PRR PRBM. The relative deflection error between the tip locus of a pseudo-rigid-body model and that of the flexural beam can be defined here as:
e
(a a0 )2 (b b0 ) 2 l
(28)
Where, ( a0 , b0 ) is the tip locus of pseudo-rigid-body models. a and b are the horizontal and vertical distance of the flexural beam, respectively. The relative errors of the four pseudo-rigid-body models are shown in Fig. 13. The maximum errors at 0 max 2.3(rad )
for the 2R, PR, 3R and PRR
24
Page 24 of 39
pseudo-rigid-body model are 4.37%, 3.68%, 1.2%, and 0.98%, respectively, as listed in Tab. 1. It can be seen from the figures that the minimum relative error is obtained in the PRR PRBM while the maximum one is got in the 2R PRBM. The relative error of
ip t
the PR and that of 3R PRBM are in the middle. The average relative errors of the four PRBMs are also listed in Tab. 1. It can be seen that average errors of these PRBMs
cr
follow the same tendency as the maximum ones. The average error of the PRR, 3R,
us
PR and 2R PRBM increases from small to large. These results agree well to the
Ac ce p
te
d
M
an
conclusions obtained from Fig. 12.
Fig. 13 Relative errors of pseudo-rigid-body models.
Tab. 1 The relative deflection error of PRBMs
2R
PR
3R
PRR
max error
4.37%
3.68%
1.2%
0.98%
average error
2.37%
0.85%
0.63%
0.09%
25
Page 25 of 39
A further discussion about the effect of axial deflection in a flexural beam with combined force and moment loads is made here. The proportion of axial deflection over the total one can be useful to show clearly the advantage of the PRR
ip t
pseudo-rigid-body model. Let x l a , it is defined as the transverse deflection of the flexural beam, and 0l is defined as the axial displacement in the PRR PRBM.
cr
The axial deflection ratio pl can be used to represent the proportion of axial
us
deflection over the transversal one in a flexural beam as follows.
pl 0l / x 0l / (l a)
(29)
an
In Fig. 14, pl is 1% when 0 0.70( rad ) , and so it can be known that the axial deformation has a little effect in the small deflection condition. However, pl is
M
5% when 0 1.48( rad ) , and it is over 20% when 0 2.24(rad ) . This result
d
shows that the axial deflection has a great effect in the total deformation and cannot
te
be neglected for the large deflection beam with combined end force and moment loads
Ac ce p
in compliant mechanisms. Therefore, it can be concluded that the axial deflection of a flexural beam can be well simulated by the slider with a prismatic pair and compression spring in the PRR PRBM.
Fig. 14 Axial deflection ratio.
26
Page 26 of 39
It can be summarized now from the above numerical results that the PRR and 3R PRBM with three degrees of freedom have higher simulation precision than the PR
ip t
and 2R PRBM with two degrees of freedom in tracking the tip locus of a flexural beam. Obviously, the increase in degree of freedom can improve the simulation
cr
accuracy. In the same condition of degree of freedom, the PRR and PR PRBM with a
us
prismatic pair show better performance than the 3R and 2R PRBM with pin joints only, respectively. Therefore, the axial deflection of a flexural beam can be simulated
an
well by the sliding link with a prismatic pair and compression spring and the
M
simulation precision can be improved remarkably using the PRR PRBM.
d
5 Conclusions
te
The impacts of lateral and axial deformations of large deflection beams in
Ac ce p
compliant mechanisms have been investigated comprehensively and a new PRR pseudo-rigid-body model has been developed in this study. Although it is smaller than the lateral one, the axial deflection plays a significant role in the large deflection of flexural beam with combined end force and moment loads and can be well simulated using the sliding link with a prismatic pair and compression spring in the PRR PRBM. The difficulty in determining the characteristic parameters of the PRR PRBM has been overcome by a simple and effective approach. Compared with the 2R, PR and 3R pseudo-rigid-body models, the new PRR PRBM has superiority in representing the deformation behavior of the large deflection links in compliant mechanisms.
27
Page 27 of 39
Therefore, the new model developed in this paper has theoretical significance and application value in the analysis and design of compliant mechanisms.
ip t
Acknowledgements The financial support of this study was from the National Natural Science
us
cr
Foundation of China (Grant No.51175006 & No. 51575006).
References
an
[1] Jensen BD, Howell LL, Salmon LG. Design of two-link, in-plane, bistable compliant micro mechanisms. ASME Trans, J Mech Des 1999;121; 416-423
M
[2] Her AI, Midha A. Compliance number concept for compliant mechanisms and type synthesis. ASME Trans., J Mech Des 1987;109;348-355
d
[3] Howell LL. Compliant Mechanisms. New York: John Wiley & Sons; 2001.
te
[4] Burns RH,. Crossley FRE. Kinetostatic synthesis of flexural link mechanisms.
Ac ce p
ASME Paper 1968; 68-Mech-36 [5] Kota S, Ananthasuresh GK, Crary SB, Wise KD. Design and fabrication of micro-electromechanical
systems.
ASME
Trans,
J
Mech
Des
1994;116;1081-1088
[6] Saxena A, Ananthasuresh GK. On an optimal property of compliant topologies. Struct Multidisc Optim 2000;19;36-49
[7] Ananthasuresh GK, Kota S, Kikuchi N. Strategies for systematic synthesis of compliant mechanisms. ASME Trans, Dyn Syst Contr Divis 1994;116;677-686 [8] Midha A, Norton TW, Howell LL. On the nomenclature, classification, and abstractions of compliant mechanisms. ASME Trans, J Mech Des
28
Page 28 of 39
1994;116;270-279 [9] Frecker MI, Ananthasuresh GK, Nishiwaki S. Topological synthesis of compliant mechanisms using multi-criteria optimization. ASME Trans, J Mech Des 1997;119;38-45
compliant-segment
motion
generation.
ASME
Trans,
J
Mech
Des
cr
2001;123;535-541
ip t
[10] Saggere L, Kota S. Synthesis of planar, compliant four-bar mechanisms for
us
[11] Ananthasuresh GK, Howell LL. Case studies and a note on the degrees-of-Freedom in compliant mechanisms. ASME Trans, J Mech Des
an
1996;118;1-12
[12] Banerjee, A., B. Bhattacharya, and A. K. Mallik. "Large Deflection of
M
Cantilever Beams with Geometric Non-Linearity: Analytical and Numerical Approaches." International Journal of Non-Linear Mechanics 43, no. 5 2008:
d
366-76.
te
[13] Aimei Zhang, Gui min Chen. A Comprehensive Elliptic Integral Solution to the
Ac ce p
Large Deflection Problems of Thin Beams in Compliant Mechanisms. ASME Transactions， Journal of Mechanical Design, 2013, 5: 021006-1.
[14] Howell LL, Midha A. A method for the design of compliant mechanism with small-length
flexural
pivots.
ASME
Trans,
J
Mechanical
Des
1994;116;280-289
[15] Howell LL, Midha A. Evaluation of equivalent spring stiffness for use in a pseudo-rigid-body model of large deflection compliant mechanisms. ASME Trans, J Mech Des 1996;118;126-140 [16] Howell LL, Midha A. Parametric deflection approximations for end loaded large deflection beams in compliant mechanisms. ASME Trans, J Mech Des
29
Page 29 of 39
1995;117;156-165 [17] Lyon SM, Howell LL, Roach GM. Modeling flexible segments with force and moment end loads via the pseudo-rigid-body model. Proceedings of the ASME International Mechanical Engineering Congress & Exposition, 2000. 1-13
ip t
[18] Howell LL, Midha A. A loop-closure theory for the analysis and synthesis of compliant mechanisms. ASME Trans, J Mech Des 1996;118;121-125
cr
[19] Saxena A, Kramer SN. A simple and accurate method for determining large
ASME Trans, J Mech Des 1998;120;392-400
us
deflections in compliant mechanisms subjected to end forces and moments.
an
[20] Kimball C., Tsai L. W. Modeling of Flexural Beams Subjected to Arbitrary End Loads. ASME Transactions，Journal of Mechanical Design, 2002, 124(2): 223
M
–235.
[21] Su HJ. A pseudo-rigid-body 3R model for determining large deflection of
te
2009;1;021008-1~9
d
cantilever beams subject to tip loads. ASME Trans, J Mechanis Robot
Ac ce p
[22] Chen G, Xiong B, Huang X. Finding the optimal characteristic parameters for 3R pseudo-rigid-body model using an improved particle swarm optimizer. Prec Eng 2011;35;505–511
[23] Feng ZL, Yu YQ, Xu, QP. A pseudo-rigid-body 2R model of flexural beam in compliant mechanisms. Mech Mach Theory 2012;55;18-33
[24] Halverson AP,
Bowden AE,
Howell LL. A pseudo-rigid-body model of the
human spine to predict implant-induced changes on motion. ASME Trans, J Mechanis Robot 2011;3;041008-1~7 [25] Leishman LC,
Colton MB. A pseudo-rigid-body model approach for the
design of compliant mechanism springs for prescribed force-deflections.
30
Page 30 of 39
Proceedings of the 35th ASME Mechanisms and Robotics Conference, DETC2011, 2011. 47590 [26] Midha, A., S. G. Bapat, A. Mavanthoor, V. Chinta, and Asme. Analysis of a Fixed-Guided Compliant Beam with an Inflection Point Using the
ip t
Pseudo-Rigid-Body Model (Prbm) Concept. Proceedings of the Asme International Design Engineering Technical Conferences and Computers and
cr
Information in Engineering Conference 2012, Vol 4, Pts a and B. 2012.
us
[27] Zhou P, Yu YQ. A new PR pseudo-rigid-body model of compliant mechanisms subject to combined loads, Proceedings of the 2012 international Conference
an
on Mechanism and Machine Science (CCMMS2012), 2012. Paper GMD-1 [28] Weisberg S. Applied Linear Regression, 3rd Ed. New York: Wiley Interscience;
M
2005.
Ac ce p
te
d
[29] Xue G. Numerical Analysis, America: CRC Press; 2009.
31
Page 31 of 39
Figure Captions. Fig. 1 Three pseudo-rigid-body models. Fig. 2 Flexural beam with combined end force and moment loads and the PRR PRBM.
Fig. 4 stiffness
K i
vs. load index 1,50 .
K a1
for
( =1) .
Fig. 7 Linear regression of stiffness coefficient
K b1
for
( =50) .
Fig. 8 Linear regression of stiffness coefficient
K a2 for ( =1) .
Fig. 9 Linear regression of stiffness coefficient
K b2
an
us
Fig. 6 Linear regression of stiffness coefficient
cr
Fig. 5 Optimization procedure of characteristic parameters for PRR PRBM.
ip t
Fig. 3 stiffness K i vs. load ratio n 1,10 .
( =50) .
M
for
Fig. 10 Fitting curve of 0 .
KL .
d
Fig. 11 Fitting curve of stiffness coefficient
te
Fig. 12 Comparison of tip locus between the flexural beam and PRBMs.
Ac ce p
Fig. 13 Relative errors of pseudo-rigid-body models. Fig. 14 Axial deflection ratio.
Table Captions.
Tab. 1 The relative deflection error of PRBMs
32
Page 32 of 39
Figures.
a
b
2l
0l
1l
K1
x
EI , l
K2
3
2l
b
1l
2
1
K1
O
x
EI , l
2
K2
b 1
0l
a
K3
.
K
F0
y
a 3l
1l
F0
y
ip t
cr
F0
0l
EI , l
x
us
y
(a) 1R pseudo-rigid-body model. (b) 3R pseudo-rigid-body model. (c) 2R pseudo-rigid-body model.
F0
0
a
F0
Ac ce p
b
O
EI , l
O
1
K 0 l 0l
2 K2
1l
x
M0
2l
te
M0
y
d
y
M
an
Fig. 1. Three pseudo-rigid-body models.
K1
EI , l
x
(a) Flexural beam with combined end force and moment loads. (b) PRR pseudo-rigid-body model. Fig. 2. Flexural beam with combined end force and moment loads and the PRR PRBM.
33
Page 33 of 39
ip t cr
Ac ce p
te
d
M
an
us
Fig. 3 stiffness K i vs. load ratio n 1,10
Fig. 4 stiffness K i vs. load index 1,50
34
Page 34 of 39
Start
Input i (i =0,1,2) Satisfying 0 1 2 =1
ip t
i i +i (i 0,1,2)
No
i 1
cr
Yes
us
Set load condition
No
1 or 50
Yes
an
Substitute 0 into Eqs.(8)-(10) and compute a / l , b / l
Substitute a / l , b / l , 0 into Eqs. (14)
M
and compute 1 , 2
1?
No
d
Yes
te
Solve T1 and T2 from Eq. (15). Solve K a1 , K a 2
Ac ce p
through linear regression process
Solve T1 and T2 from Eq. (15). Solve K b1 , K b 2 through linear regression process
2
f ( i ) ( K aj K bj ) 2
Yes
j 1
f ( i i ) f ( i )
No Output i
End
Fig. 5 Optimization procedure of characteristic parameters for PRR PRBM.
35
Page 35 of 39
ip t cr
us
K a1 for ( =1) .
Ac ce p
te
d
M
an
Fig. 6 Linear regression of stiffness coefficient
Fig. 7 Linear regression of stiffness coefficient
K b1 for ( =50) .
Fig. 8 Linear regression of stiffness coefficient
K a2 for ( =1) .
36
Page 36 of 39
ip t cr
us
K b2 for ( =50) .
Ac ce p
te
d
M
an
Fig. 9 Linear regression of stiffness coefficient
Fig. 10 Fitting curve of 0 .
Fig. 11 Fitting curve of stiffness coefficient
KL .
37
Page 37 of 39
ip t cr us an M
Ac ce p
te
d
Fig. 12 Comparison of tip locus between the flexural beam and PRBMs.
Fig. 13 Relative errors of pseudo-rigid-body models.
38
Page 38 of 39
ip t cr
an
us
Fig. 14 Axial deflection ratio.
M
Tables
4.37%
average error
2.37%
Ac ce p
max error
PR
3R
PRR
3.68%
1.2%
0.98%
0.85%
0.63%
0.09%
te
2R
d
Tab. 1 The relative deflection error of PRBMs
39
Page 39 of 39
S0141-6359(15)00164-6 http://dx.doi.org/doi:10.1016/j.precisioneng.2015.09.003 PRE 6289
To appear in:
Precision Engineering
Received date: Revised date: Accepted date:
11-8-2014 11-4-2015 9-9-2015
Please cite this article as: Yu Y-Q, Zhu S-K, Xu Q-P, Zhou P, A Novel Model of Large Deflection Beams with Combined End Loads in Compliant Mechanisms, Precision Engineering (2015), http://dx.doi.org/10.1016/j.precisioneng.2015.09.003 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Highlights
PRR pseudo-rigid-body model is proposed for large deflection beams in compliant mechanisms. Lateral and axial deflections of beams are simulated by 3 kinematic pairs (PRR) with springs.
ip t
Characteristic parameters of PRR model are determined via optimization and fitting technique.
Ac ce p
te
d
M
an
us
cr
Superiority of PRR PRBM over 2R, PR and 3R model is shown in numerical examples.
1
Page 1 of 39
A Novel Model of Large Deflection Beams with Combined End Loads in Compliant Mechanisms
ip t
Yue-Qing Yu*, Shun-Kun Zhu, Qi-Ping Xu, Peng Zhou College of Mechanical Engineering and Applied Electronics Technology,
cr
Beijing University of Technology, Beijing 100124, P. R. China
us
*Corresponding author. E-mail: [email protected], Tel: 86-10-67391702
an
Abstract
M
Based on the Pseudo-Rigid-Body Model (PRBM), a new model with three degrees of freedom is proposed for the large deflection beams with combined end
d
loads in compliant mechanisms. The lateral and axial deflections of flexural beams
te
are modeled using four rigid links connected by one prismatic (P) pair with a
Ac ce p
compression spring and two revolute (R) joints with torsion springs. The flexural cantilever beam subject to end force and moment loads is simulated by PRR pseudo-rigid-body model. The characteristic parameters of the PRR PRBM are determined via the optimization and numerical fitting techniques. Compared with the 2R, PR and 3R models, the new PRR PRBM shows the superiority in simulating the large deflection beams of compliant mechanisms through numerical examples.
Keywords: Compliant mechanism; Pseudo-rigid-body model (PRBM); PRR model; Lateral and axial deflection; Combined end force and moment loads
2
Page 2 of 39
1 Introduction Traditional mechanisms are usually composed of rigid members connected with kinematic pairs. Some inevitable problems of such kind of mechanisms are exposed
ip t
for the requirements of high-speed, high-precision, high-performance, and
cr
high-efficiency mechanical systems. For rigid body mechanisms, the clearances and
us
frictions are the two main factors that affect the accuracy and especially the dynamic behaviors of mechanisms [1]. In addition, the assembly costs occupy a large
an
proportion in labor costs, and designers always try to reduce the assembly costs. These problems are very difficult to overcome for the conventional hinge mechanisms, Midha and Her [2] proposed a
M
but easy to solve for the compliant mechanisms.
mechanism that transfers motion and force mainly relying on the elastic deformation
te
d
of its links and named it as compliant mechanism for the first time. The compliant mechanism is different from general rigid mechanisms for it gains the mobility from
Ac ce p
the deflection of its flexible members, offers many advantages such as increased precision, reduced backlash and parts number, ability to store energy [3]. Therefore, compliant mechanisms have attracted more and more attentions of researchers in the field of mechanical engineering because of its particular functions and broad application prospects. However, a major difficulty of designing compliant mechanisms lies in modeling the large deflection of flexural beams. Many methods for modeling and designing compliant mechanisms have been proposed. In [4], a method was proposed to simulate the flexural beam using the linear torsion spring. This basic work laid the foundation of the pseudo-rigid-body model
3
Page 3 of 39
(PRBM) proposed later. In [5-7], the definition of elements in compliant mechanism was improved and important contributions were made to the study of compliant mechanism. The concept of fragment was developed in the classification of compliant
ip t
mechanisms that can be divided into several fragments in accordance with different cross sections [8]. In [9], an extensive investigation was made in the modeling,
cr
numerical analysis methods and software development of compliant mechanisms. A
us
finite element model was proposed [10] in which a flexural beam can be modeled as several segments of PRBM, however, the kinematic equations are very difficult to
an
solve in applications. In [11], a method to determine the degree of freedom of
M
compliant mechanisms was presented based on the segments and connection types among segments. Banerjee and et al [12] used a non-linear shooting method and
d
decomposition method to obtain large deflection of a cantilever beam subjected to
te
loading at intermediate locations besides end forces and moments. Chen and Zhang
Ac ce p
[13] used a comprehensive elliptic integral solution to solve large deflections of flexural beam with multiple inflection points. Based on the structural and kinematic analysis of rigid body mechanisms, Howell
and Midha proposed the Pseudo-Rigid-Body Model (PRBM) which played a very important role in simplifying the complicated modeling of compliant mechanism [14-15]. A PRBM of cantilever beam with end-force load was established to simulate the tip locus of beam end by 1R PRBM shown in Fig. 1(a) which consists of two rigid links joined with a revolute joint and a torsion spring [16]. The pseudo-rigid-body models subjected to three different end loads were established in [17]. A loop-closure
4
Page 4 of 39
theory for the analysis and synthesis of compliant mechanisms was presented later [18]. The parameters of 1R PRBM for compliant mechanisms are different with the various loads and the accuracy is not high enough. In [19], the PRBM was modified
ip t
by introducing two linear springs to restrain the change of characteristic radius factor for different load modes. Kimball and Tsai [20] used PRBM to model the large
cr
deflection of a cantilever beam with an inflection point. Furthermore, Su proposed a
us
3R pseudo-rigid-body model [21], as shown in Fig. 1(b), to simulate the large deflection of flexural beam subjected to tip loads. Although it has higher simulation
an
precision than that of 1R pseudo-rigid-body model, the inverse kinematic solution and
M
the iterative process are very difficult to obtain the characteristic parameters of the 3R PRBM. In [22], an improved particle swarm optimizer was used to determine the
d
optimal characteristic parameters of 3R PRBM. It showed better performance in
te
predicting large deflections of cantilever beams with pure end force load or moment
Ac ce p
load. However, deflections of cantilever beams with combined end force and moment load was not discussed. In [23], a 2R pseudo-rigid-body model was proposed to improve the simulation accuracy of 1R PRBM and simplify the iterative process of 3R PRBM. The 2R PRBM is composed of three rigid links joined by two pin joints and torsion springs, as shown in Fig. 1(c). The pseudo-rigid-body model was also applied to predict the motion of the human spine [24]. Based on the PRBM for large beam deflections, a method was introduced to synthesize the three-link and four-link compliant mechanisms that exhibit prescribed force-deflection responses [25]. Midha and et al [26] used the concept of PRBM to present a simple method of analyzing a
5
Page 5 of 39
fixed-guided compliant beam with an inflection point for several kinds of boundary conditions. From the point of view of degree of freedom of mechanisms, the simulating
ip t
precision of 1R PRBM is not very high because it has single degree of freedom. The 2R PRBM has two degrees of freedom and it can be used to simulate the end position
cr
and/or slope of a flexural link, but two out of three parameters are independent. The
us
3R pseudo-rigid-body model has three degrees of freedom and so it can describe both the end position and slop of a flexural beam. However, although the 1R, 2R and 3R
an
pseudo-rigid-models contain pin joints and can be used to simulate the bending
M
deformation of a flexural beam only, they are not useful to describe the axial deformation of the beam. If a flexible beam is subjected to combined end force and
d
moment loads, the axial deformation of the beam cannot be presented by the
te
pseudo-rigid-body models with revolute pairs only. In [27], a PR pseudo-rigid-body
Ac ce p
model with a prismatic pair was developed to simulate both of the axial and lateral deformation of a flexural beam. Compared with the 1R and 2R models, the PR PRBM has higher simulation precision for the flexural beam with force or moment load. However, the simulation accuracy of the PR PRBM is still needed to increase to deal with the flexible beam subjected to combined end force and moment loads. Therefore, a further study should be made to improve the simulating ability of the pseudo-rigid-body models in the analysis and design of compliant mechanisms.
6
Page 6 of 39
F0
a
b
2l
0l
EI , l
1l
K1
x
K2
3
2l
b
1l
2
EI , l
2
K2
b K1
1
0l
a
K3
.
K
F0
y
a 3l
1l
F0
y
O
x
0l
1
ip t
y
EI , l
x
us
Fig. 1 Three pseudo-rigid-body models.
cr
(a) 1R pseudo-rigid-body model. (b) 3R pseudo-rigid-body model. (c) 2R pseudo-rigid-body model.
an
A new PRR pseudo-rigid-body model is proposed in this study to simulate the
M
large deflection beam with combined end force and moment loads. The axial deformation of a flexural beam is described by one sliding link with a prismatic pair
d
and a compression spring. The lateral deflection of the beam is simulated by two
te
rotating links with pin joints and torsion springs. The characteristic parameters of the
Ac ce p
PRR PRBM are determined via the optimization and numerical fitting techniques. A numerical comparison among the 2R, PR, 3R and PRR pseudo-rigid-body models is presented. The result shows the superiority of the new PRR PRBM in simulating the large deflection beams with combined end force and moment loads in compliant mechanisms.
2 PRR pseudo-rigid-body model A large deflection cantilever beam with combined force and moment loads at end
7
Page 7 of 39
is shown in Fig. 2(a). 0 is the deflection angle of the beam, a and b are the horizontal and vertical coordinates of the tip point of the beam, respectively. l is the undeflected length of the flexible beam, and is the angle between the force
ip t
direction and horizontal direction. In order to simulate the large deflection of the flexural beam with end force and moment loads, a PRR Pseudo-Rigid-Body Model
cr
(PRR PRBM) is proposed here, as shown in Fig. 2(b). It consists of four rigid links
us
joined by a prismatic (P) pair with a compression spring and two revolute (R) pairs with torsion springs. In this way, the axial deformation of flexural beam can be
an
described by the sliding link with P pair and compression spring, and the main lateral
M
deflection of the beam can be simulated by the two rotating links joined by the two pin joints and torsion springs. Because the PRR PRBM has three degrees of freedom
d
and so it can be used to simulate accurately both the end position and slop of the
Ac ce p
te
flexural beam.
F0
y
a
y
0
nP 2l
M0
O
EI , l
F0
P
b
K2
1l
1
K
O
x
0 l 0l
M0 2
K1
EI , l
x
(a) Flexural beam with combined end force and moment loads. (b) PRR pseudo-rigid-body model. Fig. 2 Flexural beam with combined end force and moment loads and the PRR PRBM.
8
Page 8 of 39
The length of each rigid link in the PRR PRBM is presented as i l (i 0,1, 2) , and
i is called the characteristic radius factor [2], satisfying 0 1 2 1 .
0 is the variable characteristic factor and 0l indicates the axial displacement
ip t
of the sliding link. K1 , K 2 and K are the torsion spring stiffness constants and the compression spring stiffness constant, respectively. These factors and constants are
cr
the key characteristic parameters of the PRR PRBM to simulate the large deflection
us
beam with combined end force and moment loads in compliant mechanisms and they
an
will be determined in the following part.
M
3 Characteristic parameters
Because the process to determine the characteristic parameters of the PRR
d
PRBM is complicated, this section is divided into five subsections for better
te
understanding. It is organized as follows. An elliptic integration of deflection for a
Ac ce p
flexural beam is presented at first in Section 3.1. The kinematic and static equations of the PRR PRBM are derived in Section 3.2 and 3.3, respectively. The optimal characteristic parameters and spring stiffness coefficients of the PRR PRBM are determined in Section 3.4 and 3.5, respectively.
3.1 Elliptic Integration of Deflection
When a flexural beam is subjected to a combined force and moment at end, as shown in Fig. 2(a), the internal torque of the beam at any point can be presented as:
9
Page 9 of 39
M F0 a x sin b y cos M 0
(1)
Where, a is the horizontal distance from the fixed end to the free end, and b is the vertical distance of the free end from its undeflected position.
ip t
Substituting Eq. (1) into the Bernoulli-Euler equation, we have
M d d 2 y / dx 2 EI ds [1 ( dy / dx) 2 ]3/2
cr
(2)
us
From the theory of Elliptic Integrals [3], it can be derived from above differential equations that
an
d 2 F0 sin( ) ds 2 EI
d M 0 = when 0 , we have: ds EI
M
Integrating Eq. (3) with the boundary condition
(3)
(4)
d
M l d 1 2 F0l 2 [cos( 0 ) cos( )] ( 0 ) 2 ds l EI EI
te
Defining
F0 l 2 2 EI
Ac ce p
2
(5)
M 0l EI
(6)
Eq. (4) becomes:
d 1 4 [cos(0 ) cos( )] 2 ds l
(7)
It can be obtained by solving the differential equation that
2 2 Where,
d
0 0
cos 0 sin
(8)
n2 1
10
Page 10 of 39
Defining the load index =
2 , the dimensionless coordinates of the tip point 2 2
a b Q , of the flexural beam can be presented briefly as follows: l l
cos d
0
0
cos 0 cos sin d
0 0
cos 0 cos
(9)
ip t
b 1 l 2
(10)
cr
a 1 l 2
us
With these derivations above, the kinematic and static equations of the PRR pseudo-rigid-body model for the large deflection beam with end force and moment
M
an
loads in compliant mechanisms can now be derived as follows.
3.2 Kinematic Equations
d
The slope angle of the PRR PRBM can be presented as , and the deflection
te
angles of two torsion springs are described as 1 , 2 , as shown in Fig. 2(b). P
Ac ce p
indicates the vertical force and nP represents the horizontal force. So, the total force
F can be expressed as follows.
F0 P n 2 1
(11)
The angle between the force direction and the horizontal direction is
arctan
1 n
(12)
Because the slope angle of the PRR PRBM in Fig. 2(b) should be equal to the deflection angle 0 of the flexural beam in Fig. 2(a), the following equations can be written as:
11
Page 11 of 39
a l 0 0 1 cos 1 2 cos(1 2 ) b 1 sin 1 2 sin(1 2 ) l 0 1 2
ip t
(13)
us (14)
Ac ce p
3.3 Static Equations
te
d
M
an
b 2 sin 0 1 l 1 sin 1 b l 2 sin 0 1 2 0 sin 1 a 0 2 cos 0 1 cos 1 0 l
cr
So, the following relationship can be derived from the above equation:
For a given combined loads exerted at the end of the PPR PRBM, and the torque
and lateral force at the pivot are:
T1 Fxl (1 sin(1 ) 2 sin()) Fyl (1 cos(1 ) 2 cos()) M0 T2 Fxl ( 2 sin()) Fyl ( 2 cos()) M0 F K l 0 x
(15)
Where, the horizontal force and vertical force can be expressed as Fx F0 cos and
Fy F0 sin , respectively. It should be noted that K in Eq. (15) represents the stiffness constant of compression spring in the PPR PRBM. The torques T1 , T2 at the hinges are the product of the torsion spring stiffness constant K1 , K 2 and the angle 12
Page 12 of 39
1 , 2 , respectively: T1 K 1 1 T2 K 2 2
(16)
From Eq. (15), we have:
ip t
Fxl 0 K1 1 sin(1) 2 sin() 1 cos(1) 2 cos() 1 0 Fyl 1 0 0 0 K2 2 sin() 2 cos() 1 0 M0 2 0 0 1 Fx 0 0 0 K 0 l
us
cr
(17)
Let
1 sin(1) 2 sin() 1 cos(1) 2 cos() 1 0 J 2 sin() 1 0 2 cos() 0 0 0 1
an
T
M
and
(18)
(19)
Ac ce p
te
d
1 0 0 Fxl 1 F l K1 y K 0 1 T J 0 M0 2 2 K Fx 1 0 0 l 0
The resistance to deflection and flexibility of the flexural beam can be described by the dimensionless torsion spring stiffness coefficients K 1 , K 2 [3] and compression
spring stiffness coefficient K L , respectively, as follows:
EI K1 l K 0 2 K 0
0 EI l 0
0 K1 0 K2 EI KL l3
(20)
Therefore
13
Page 13 of 39
0 EI l 0
1 0 1 K 1 0 K 2 0 EI K L 0 l 3
0 1 2 0
0 Fxl F l T y 0 J M 0 Fx 1 l 0
or
l EI 0
1 0 1 0 0 3 l 0 EI
0 1 2 0
(22)
an
It can be obtained from the above equations that:
0 Fx l F l T y 0 J M 0 Fx 1 l 0
cr
0
us
l K 1 EI K 2 0 K L 0
(21)
ip t
EI l 0 0
te
d
M
l l l (1 sin(1) 2 sin()) (1 cos(1) 2 cos()) 0 Fxl EI EI EI 1 1 1 F l K1 y l l l K ( sin( )) ( cos( )) 0 M0 (23) 2 2 2 EI EI2 EI2 2 K Fx L l3 0 0 0 l 0EI
and
Ac ce p
F0 cos()l 2 F0 sin()l 2 Ml ( sin( ) sin( )) (1 cos(1) 2 cos()) 0 1 1 2 EI 1 EI 1 EI 1 K1 2 2 F0 cos()l F0 sin()l M0l (24) K2 EI ( 2 sin()) EI ( 2 cos()) EI 2 2 2 K L F cos()l 2 0 0 EI
In order to simplify the above equation, and the end force and the moment
F0 , M 0 can be express as the dimensionless force index index
2
F0l 2 and torque 2 EI
M 0l [3], respectively. Therefore, EI
14
Page 14 of 39
ip t
2 2 cos( ) 2 2 sin( ) K ( sin( ) sin( )) ( 1 cos(1 ) 2 cos()) 1 1 1 2 1 1 1 2 2 2 cos( ) 2 sin( ) ( 2 sin()) ( 2 cos()) (25) K 2 2 2 2 2 2 cos( ) KL 0
Now, it can be summarized that there are totally seven characteristic parameters
cr
in the PRR PRBM as follows: the characteristic radius factors 0 , 1 , 2 , the variable
us
characteristic factor 0 , the torsion spring stiffness coefficients K 1 , K 2 and the compression spring stiffness coefficient K L . The goal of this study is to find the
M
3.4 Optimal Characteristic Factors
an
optimal parameters, and it will be discussed in the following section.
It is well known that it is very difficult to obtain the characteristic parameters of
te
d
the PRR PRBM subjected to combined end force and moment loads. Therefore, a simple and effective way is proposed in this study to overcome this difficulty. At first,
Ac ce p
it is helpful to investigate the effects of load ratio n and load index on the
stiffness coefficients of springs in the PRR PRBM. Fig. 3 shows the plot of the spring stiffness when the load ratio n changes in a very large range from 1 to 10. It can be seen clearly from the figure that the spring stiffness is little correlated to the load ratio of horizontal force to vertical one. Hence, the effect of load ratio n is very small.
However, the effect of load index is quite different. It is known that the load condition is close to a pure force when 0 and it is a pure moment when . However, these two extreme load conditions are far from the case of PRR PRBM subjected to combined end force and moment loads and so they are not considered
15
Page 15 of 39
here. Fig. 4 shows the variation of spring stiffness coefficients with change of load index from 1 to 50. It can be seen in Fig. 4 that the stiffness coefficient changes monotonically while the load index increases greatly from 1 to 10 and almost
ip t
remains constant when changes from 10 to 50. This result shows that the change range from 1 to 50 is a sufficient scale to obtain a proper value of load index for
cr
combined end force and moment loads. So, the variation range of from 1 to 50 is
us
selected in this study. Therefore, it can be deduced that the optimal characteristic parameters i could be the one that has the minimum of the square of difference of
Ac ce p
te
d
M
an
the spring stiffness between the two cases ( =1 and =50 ).
Fig. 3 stiffness K i vs. load ratio n 1,10
16
Page 16 of 39
ip t cr us
M
an
Fig. 4 stiffness K i vs. load index 1,50
Based on the analysis above, two special cases of load index =1 and =50
d
are dealt with at first in this study. The optimal characteristic factors 0 , 1 , 2 and
can be determined finally. The optimal characteristic parameters
Ac ce p
K 1 , K 2 and K L
te
0 can be then obtained through optimization techniques. The stiffness coefficients
of the PRR PRBM subjected to combined end force and moment loads can be found. This is a three-dimensional search and the whole process to determine the characteristic parameters of the PRR PRBM can be presented as an optimization procedure as shown in the flow chart of Fig. 5. The optimal characteristic radius factors of the PRR PRBM subjected to combined end force and moment loads can be determined via the optimization as follows:
0 0.2, 1 0.52, 2 0.28
17
Page 17 of 39
Start
Input i (i =0,1,2) Satisfying 0 1 2 =1
ip t
i i +i (i 0,1,2)
No
i 1
cr
Yes
us
Set load condition
No
1 or 50
Yes
an
Substitute 0 into Eqs.(8)-(10) and compute a / l , b / l
and compute 1 , 2
1?
M
Substitute a / l , b / l , 0 into Eqs. (14)
No
d
Yes
te
Solve T1 and T2 from Eq. (15). Solve K a1 , K a 2
Ac ce p
through linear regression process
Solve T1 and T2 from Eq. (15). Solve K b1 , K b 2 through linear regression process
2
f ( i ) ( K aj K bj ) 2
Yes
j 1
f ( i i ) f ( i )
No
Output i
End
Fig. 5 Optimization procedure of characteristic parameters for PRR PRBM.
18
Page 18 of 39
3.5 Optimal Spring Stiffness Coefficients Now, the torsion spring stiffness coefficients of the PRR PRBM can be determined from the characteristic radius factors obtained above. When the PRR
ip t
PRBM is subjected to a combined end load ( =1) , a linear regression process [28] can be applied here to determine the approximate value of the torsion spring stiffness
cr
coefficient K a1 at the first pin joint from Eq. (25). The linear fitting curve is shown
us
in Fig. 6. Similarly, when the PRR PRBM is subjected to a combined end load
( =50) , the linear regression process can also be applied to determine the
an
approximate value of torsion spring stiffness coefficient K b1 at the first pin joint
Ac ce p
te
d
M
from Eq. (25). The linear fitting curve is shown in Fig. 7.
Fig. 6 Linear regression of stiffness coefficient
K a1 for ( =1) .
Fig. 7 Linear regression of stiffness coefficient
K b1 for ( =50) .
19
Page 19 of 39
In a similar way, the torsion spring stiffness coefficients K a2 ( =1) and
K b2 ( =50) , at the second pin joint of the PRR PRBM can be determined, too. The
an
us
cr
ip t
results are shown in Fig. 8 and Fig. 9, respectively.
K a2 for ( =1) .
Ac ce p
te
d
M
Fig. 8 Linear regression of stiffness coefficient
Fig. 9 Linear regression of stiffness coefficient
K b2 for ( =50) .
Therefore, the torsion spring stiffness coefficients of the PRR PRBM can be determined as follows. The corresponding values at the two cases are:
K a1 2.045, K b1 2.207 K a 2 1.828, K b 2 1.823
20
Page 20 of 39
As shown in Fig. 6-9, the spring stiffness K 1 increases from 2.045 ( 1 ) to 2.207 ( 50 ) while K 2 decreases from 1.828 ( 1 ) to 1.823 ( 50 ). It can be seen that the relative errors of the two cases ( 1 and 50 ) are small and the
ip t
spring stiffness changes slightly within 10,50 . So, the average value of the two cases can be selected as an acceptable solution for the PRR PRBM subjected to end
force
and
moment
loads.
Although
this
may
be
cr
combined
not
us
a mathematical justification, the numerical results obtained in Section 4 will show the effectiveness of this simple method. The torsion spring stiffness coefficients of the
an
PRR PRBM can be determined at =25.5 as follows:
M
K 1 1.930, K 2 2.032
The torsion spring stiffness constants K1 , K 2 depend on the torsion spring
d
stiffness coefficients K 1 , K 2 , geometrical relationship I / l , and material
te
properties E , respectively, and can be expressed as:
Ac ce p
K1 K 1
EI EI , K 2 K 2 l l
(26)
For the variable characteristic factor 0 in eqn. Eq. (14), it can be expressed
by the slope angle of PRR PRBM. Applying the Least Square Polynomial Fitting Method [29], a Quintic Polynomial of is used here to fit 0 . The value of 0 can be determined for each value of , and it can be fitted as shown in Fig. 10.
Therefore, the fitting curve of 0 can be presented in following equation: 0 0.033 5 +0.1528 4 0.25843 +0.1638 2 0.0494
(27)
21
Page 21 of 39
ip t cr
an
us
Fig. 10 Fitting curve of 0 .
When 0 has been obtained, the stiffness coefficient of compression spring K L
M
in the PRR PRBM can be determined with 2 solved from Eq. (8). Using the linear regression process [25], K L can be fitted as shown in Fig. 11.
Ac ce p
te
d
K L 0.871
Fig. 11 Fitting curve of stiffness coefficient
KL .
22
Page 22 of 39
Now, all the characteristic parameters of the PRR PRBM have been determined and, therefore, the new model has been developed to simulate the large deflection
ip t
beam with combined end force and moment loads in compliant mechanisms.
cr
4 Precision analyses
Fig. 12 shows the tip locus of the PRR, 2R, PR and 3R pseudo-rigid-body model,
us
as well as the flexural beam with combined end force and moment loads. It can be
an
seen from Fig. 12 that the four pseudo-rigid-body models can mainly follow the tip locus of the flexural beam in a small deformation condition. However, the curve of 2R
M
PRBM deviates off the tip locus of the flexural beam in the large deflection area. This is because the 2R model has two degree of freedom only and so its simulation
d
accuracy is low. Compared with the 2R PRBM, the PR PRBM has also two degrees of
te
freedom, but it can improve greatly the accuracy due to the sliding link with a
Ac ce p
prismatic pair and compression spring that describes the axial deformation of the flexural beam. Moreover, the PRR PRBM has three degree-of-freedom and so increases the simulation precision further. Although the 3R PRBM can well follow the tip locus of the flexural beam because it has three degrees of freedom too, the trajectory curve of the PRR PRBM is much closer to that of the actual flexural beam curve. So, the PRR PRBM is more accurate than the 3R PRBM. This result indicates further the important role of the prismatic pair in the pseudo-rigid-body model and the superiority of PRR PRBM in increasing the simulation accuracy for the large deflection beam with combined end force and moment loads in compliant
23
Page 23 of 39
mechanisms. At same time, it also shows the effectiveness of the simple method
M
an
us
cr
ip t
proposed in Section 3.
te
d
Fig. 12 Comparison of tip locus between the flexural beam and PRBMs.
Ac ce p
A discussion on the simulating errors of the four pseudo-rigid-body models can be made to illustrate further the advantage of the PRR PRBM. The relative deflection error between the tip locus of a pseudo-rigid-body model and that of the flexural beam can be defined here as:
e
(a a0 )2 (b b0 ) 2 l
(28)
Where, ( a0 , b0 ) is the tip locus of pseudo-rigid-body models. a and b are the horizontal and vertical distance of the flexural beam, respectively. The relative errors of the four pseudo-rigid-body models are shown in Fig. 13. The maximum errors at 0 max 2.3(rad )
for the 2R, PR, 3R and PRR
24
Page 24 of 39
pseudo-rigid-body model are 4.37%, 3.68%, 1.2%, and 0.98%, respectively, as listed in Tab. 1. It can be seen from the figures that the minimum relative error is obtained in the PRR PRBM while the maximum one is got in the 2R PRBM. The relative error of
ip t
the PR and that of 3R PRBM are in the middle. The average relative errors of the four PRBMs are also listed in Tab. 1. It can be seen that average errors of these PRBMs
cr
follow the same tendency as the maximum ones. The average error of the PRR, 3R,
us
PR and 2R PRBM increases from small to large. These results agree well to the
Ac ce p
te
d
M
an
conclusions obtained from Fig. 12.
Fig. 13 Relative errors of pseudo-rigid-body models.
Tab. 1 The relative deflection error of PRBMs
2R
PR
3R
PRR
max error
4.37%
3.68%
1.2%
0.98%
average error
2.37%
0.85%
0.63%
0.09%
25
Page 25 of 39
A further discussion about the effect of axial deflection in a flexural beam with combined force and moment loads is made here. The proportion of axial deflection over the total one can be useful to show clearly the advantage of the PRR
ip t
pseudo-rigid-body model. Let x l a , it is defined as the transverse deflection of the flexural beam, and 0l is defined as the axial displacement in the PRR PRBM.
cr
The axial deflection ratio pl can be used to represent the proportion of axial
us
deflection over the transversal one in a flexural beam as follows.
pl 0l / x 0l / (l a)
(29)
an
In Fig. 14, pl is 1% when 0 0.70( rad ) , and so it can be known that the axial deformation has a little effect in the small deflection condition. However, pl is
M
5% when 0 1.48( rad ) , and it is over 20% when 0 2.24(rad ) . This result
d
shows that the axial deflection has a great effect in the total deformation and cannot
te
be neglected for the large deflection beam with combined end force and moment loads
Ac ce p
in compliant mechanisms. Therefore, it can be concluded that the axial deflection of a flexural beam can be well simulated by the slider with a prismatic pair and compression spring in the PRR PRBM.
Fig. 14 Axial deflection ratio.
26
Page 26 of 39
It can be summarized now from the above numerical results that the PRR and 3R PRBM with three degrees of freedom have higher simulation precision than the PR
ip t
and 2R PRBM with two degrees of freedom in tracking the tip locus of a flexural beam. Obviously, the increase in degree of freedom can improve the simulation
cr
accuracy. In the same condition of degree of freedom, the PRR and PR PRBM with a
us
prismatic pair show better performance than the 3R and 2R PRBM with pin joints only, respectively. Therefore, the axial deflection of a flexural beam can be simulated
an
well by the sliding link with a prismatic pair and compression spring and the
M
simulation precision can be improved remarkably using the PRR PRBM.
d
5 Conclusions
te
The impacts of lateral and axial deformations of large deflection beams in
Ac ce p
compliant mechanisms have been investigated comprehensively and a new PRR pseudo-rigid-body model has been developed in this study. Although it is smaller than the lateral one, the axial deflection plays a significant role in the large deflection of flexural beam with combined end force and moment loads and can be well simulated using the sliding link with a prismatic pair and compression spring in the PRR PRBM. The difficulty in determining the characteristic parameters of the PRR PRBM has been overcome by a simple and effective approach. Compared with the 2R, PR and 3R pseudo-rigid-body models, the new PRR PRBM has superiority in representing the deformation behavior of the large deflection links in compliant mechanisms.
27
Page 27 of 39
Therefore, the new model developed in this paper has theoretical significance and application value in the analysis and design of compliant mechanisms.
ip t
Acknowledgements The financial support of this study was from the National Natural Science
us
cr
Foundation of China (Grant No.51175006 & No. 51575006).
References
an
[1] Jensen BD, Howell LL, Salmon LG. Design of two-link, in-plane, bistable compliant micro mechanisms. ASME Trans, J Mech Des 1999;121; 416-423
M
[2] Her AI, Midha A. Compliance number concept for compliant mechanisms and type synthesis. ASME Trans., J Mech Des 1987;109;348-355
d
[3] Howell LL. Compliant Mechanisms. New York: John Wiley & Sons; 2001.
te
[4] Burns RH,. Crossley FRE. Kinetostatic synthesis of flexural link mechanisms.
Ac ce p
ASME Paper 1968; 68-Mech-36 [5] Kota S, Ananthasuresh GK, Crary SB, Wise KD. Design and fabrication of micro-electromechanical
systems.
ASME
Trans,
J
Mech
Des
1994;116;1081-1088
[6] Saxena A, Ananthasuresh GK. On an optimal property of compliant topologies. Struct Multidisc Optim 2000;19;36-49
[7] Ananthasuresh GK, Kota S, Kikuchi N. Strategies for systematic synthesis of compliant mechanisms. ASME Trans, Dyn Syst Contr Divis 1994;116;677-686 [8] Midha A, Norton TW, Howell LL. On the nomenclature, classification, and abstractions of compliant mechanisms. ASME Trans, J Mech Des
28
Page 28 of 39
1994;116;270-279 [9] Frecker MI, Ananthasuresh GK, Nishiwaki S. Topological synthesis of compliant mechanisms using multi-criteria optimization. ASME Trans, J Mech Des 1997;119;38-45
compliant-segment
motion
generation.
ASME
Trans,
J
Mech
Des
cr
2001;123;535-541
ip t
[10] Saggere L, Kota S. Synthesis of planar, compliant four-bar mechanisms for
us
[11] Ananthasuresh GK, Howell LL. Case studies and a note on the degrees-of-Freedom in compliant mechanisms. ASME Trans, J Mech Des
an
1996;118;1-12
[12] Banerjee, A., B. Bhattacharya, and A. K. Mallik. "Large Deflection of
M
Cantilever Beams with Geometric Non-Linearity: Analytical and Numerical Approaches." International Journal of Non-Linear Mechanics 43, no. 5 2008:
d
366-76.
te
[13] Aimei Zhang, Gui min Chen. A Comprehensive Elliptic Integral Solution to the
Ac ce p
Large Deflection Problems of Thin Beams in Compliant Mechanisms. ASME Transactions， Journal of Mechanical Design, 2013, 5: 021006-1.
[14] Howell LL, Midha A. A method for the design of compliant mechanism with small-length
flexural
pivots.
ASME
Trans,
J
Mechanical
Des
1994;116;280-289
[15] Howell LL, Midha A. Evaluation of equivalent spring stiffness for use in a pseudo-rigid-body model of large deflection compliant mechanisms. ASME Trans, J Mech Des 1996;118;126-140 [16] Howell LL, Midha A. Parametric deflection approximations for end loaded large deflection beams in compliant mechanisms. ASME Trans, J Mech Des
29
Page 29 of 39
1995;117;156-165 [17] Lyon SM, Howell LL, Roach GM. Modeling flexible segments with force and moment end loads via the pseudo-rigid-body model. Proceedings of the ASME International Mechanical Engineering Congress & Exposition, 2000. 1-13
ip t
[18] Howell LL, Midha A. A loop-closure theory for the analysis and synthesis of compliant mechanisms. ASME Trans, J Mech Des 1996;118;121-125
cr
[19] Saxena A, Kramer SN. A simple and accurate method for determining large
ASME Trans, J Mech Des 1998;120;392-400
us
deflections in compliant mechanisms subjected to end forces and moments.
an
[20] Kimball C., Tsai L. W. Modeling of Flexural Beams Subjected to Arbitrary End Loads. ASME Transactions，Journal of Mechanical Design, 2002, 124(2): 223
M
–235.
[21] Su HJ. A pseudo-rigid-body 3R model for determining large deflection of
te
2009;1;021008-1~9
d
cantilever beams subject to tip loads. ASME Trans, J Mechanis Robot
Ac ce p
[22] Chen G, Xiong B, Huang X. Finding the optimal characteristic parameters for 3R pseudo-rigid-body model using an improved particle swarm optimizer. Prec Eng 2011;35;505–511
[23] Feng ZL, Yu YQ, Xu, QP. A pseudo-rigid-body 2R model of flexural beam in compliant mechanisms. Mech Mach Theory 2012;55;18-33
[24] Halverson AP,
Bowden AE,
Howell LL. A pseudo-rigid-body model of the
human spine to predict implant-induced changes on motion. ASME Trans, J Mechanis Robot 2011;3;041008-1~7 [25] Leishman LC,
Colton MB. A pseudo-rigid-body model approach for the
design of compliant mechanism springs for prescribed force-deflections.
30
Page 30 of 39
Proceedings of the 35th ASME Mechanisms and Robotics Conference, DETC2011, 2011. 47590 [26] Midha, A., S. G. Bapat, A. Mavanthoor, V. Chinta, and Asme. Analysis of a Fixed-Guided Compliant Beam with an Inflection Point Using the
ip t
Pseudo-Rigid-Body Model (Prbm) Concept. Proceedings of the Asme International Design Engineering Technical Conferences and Computers and
cr
Information in Engineering Conference 2012, Vol 4, Pts a and B. 2012.
us
[27] Zhou P, Yu YQ. A new PR pseudo-rigid-body model of compliant mechanisms subject to combined loads, Proceedings of the 2012 international Conference
an
on Mechanism and Machine Science (CCMMS2012), 2012. Paper GMD-1 [28] Weisberg S. Applied Linear Regression, 3rd Ed. New York: Wiley Interscience;
M
2005.
Ac ce p
te
d
[29] Xue G. Numerical Analysis, America: CRC Press; 2009.
31
Page 31 of 39
Figure Captions. Fig. 1 Three pseudo-rigid-body models. Fig. 2 Flexural beam with combined end force and moment loads and the PRR PRBM.
Fig. 4 stiffness
K i
vs. load index 1,50 .
K a1
for
( =1) .
Fig. 7 Linear regression of stiffness coefficient
K b1
for
( =50) .
Fig. 8 Linear regression of stiffness coefficient
K a2 for ( =1) .
Fig. 9 Linear regression of stiffness coefficient
K b2
an
us
Fig. 6 Linear regression of stiffness coefficient
cr
Fig. 5 Optimization procedure of characteristic parameters for PRR PRBM.
ip t
Fig. 3 stiffness K i vs. load ratio n 1,10 .
( =50) .
M
for
Fig. 10 Fitting curve of 0 .
KL .
d
Fig. 11 Fitting curve of stiffness coefficient
te
Fig. 12 Comparison of tip locus between the flexural beam and PRBMs.
Ac ce p
Fig. 13 Relative errors of pseudo-rigid-body models. Fig. 14 Axial deflection ratio.
Table Captions.
Tab. 1 The relative deflection error of PRBMs
32
Page 32 of 39
Figures.
a
b
2l
0l
1l
K1
x
EI , l
K2
3
2l
b
1l
2
1
K1
O
x
EI , l
2
K2
b 1
0l
a
K3
.
K
F0
y
a 3l
1l
F0
y
ip t
cr
F0
0l
EI , l
x
us
y
(a) 1R pseudo-rigid-body model. (b) 3R pseudo-rigid-body model. (c) 2R pseudo-rigid-body model.
F0
0
a
F0
Ac ce p
b
O
EI , l
O
1
K 0 l 0l
2 K2
1l
x
M0
2l
te
M0
y
d
y
M
an
Fig. 1. Three pseudo-rigid-body models.
K1
EI , l
x
(a) Flexural beam with combined end force and moment loads. (b) PRR pseudo-rigid-body model. Fig. 2. Flexural beam with combined end force and moment loads and the PRR PRBM.
33
Page 33 of 39
ip t cr
Ac ce p
te
d
M
an
us
Fig. 3 stiffness K i vs. load ratio n 1,10
Fig. 4 stiffness K i vs. load index 1,50
34
Page 34 of 39
Start
Input i (i =0,1,2) Satisfying 0 1 2 =1
ip t
i i +i (i 0,1,2)
No
i 1
cr
Yes
us
Set load condition
No
1 or 50
Yes
an
Substitute 0 into Eqs.(8)-(10) and compute a / l , b / l
Substitute a / l , b / l , 0 into Eqs. (14)
M
and compute 1 , 2
1?
No
d
Yes
te
Solve T1 and T2 from Eq. (15). Solve K a1 , K a 2
Ac ce p
through linear regression process
Solve T1 and T2 from Eq. (15). Solve K b1 , K b 2 through linear regression process
2
f ( i ) ( K aj K bj ) 2
Yes
j 1
f ( i i ) f ( i )
No Output i
End
Fig. 5 Optimization procedure of characteristic parameters for PRR PRBM.
35
Page 35 of 39
ip t cr
us
K a1 for ( =1) .
Ac ce p
te
d
M
an
Fig. 6 Linear regression of stiffness coefficient
Fig. 7 Linear regression of stiffness coefficient
K b1 for ( =50) .
Fig. 8 Linear regression of stiffness coefficient
K a2 for ( =1) .
36
Page 36 of 39
ip t cr
us
K b2 for ( =50) .
Ac ce p
te
d
M
an
Fig. 9 Linear regression of stiffness coefficient
Fig. 10 Fitting curve of 0 .
Fig. 11 Fitting curve of stiffness coefficient
KL .
37
Page 37 of 39
ip t cr us an M
Ac ce p
te
d
Fig. 12 Comparison of tip locus between the flexural beam and PRBMs.
Fig. 13 Relative errors of pseudo-rigid-body models.
38
Page 38 of 39
ip t cr
an
us
Fig. 14 Axial deflection ratio.
M
Tables
4.37%
average error
2.37%
Ac ce p
max error
PR
3R
PRR
3.68%
1.2%
0.98%
0.85%
0.63%
0.09%
te
2R
d
Tab. 1 The relative deflection error of PRBMs
39
Page 39 of 39