Modeling Large Spatial Deflections of Slender ...

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¯Uxi, ¯Uyi, ¯Uzi, ¯θyi, ¯θzi and ¯θxdi (an ”overbar” is placed over each parameter to indicate that it is measured with respect to its moving coordinate frame).
Journal of Mechanisms and Robotics. Received June 18, 2015; Accepted manuscript posted February 1, 2016. doi:10.1115/1.4032632 Copyright (c) 2016 by ASME

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Guimin Chen∗ and Ruiyu Bai

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Modeling Large Spatial Deflections of Slender Bisymmetric Beams in Compliant Mechanisms Using Chained Spatial-Beam-Constraint-Model (CSBCM)

School of Electro-Mechanical Engineering, Xidian University, Xi’an, Shaanxi 710071, China Abstract

Introduction

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Modeling large spatial deflections of flexible beams has been one of the most challenging problems in the research community of compliant mechanisms. This work presents a method called chained spatial-beam-constraint-model (CSBCM) for modeling large spatial deflections of flexible bisymmetric beams in compliant mechanisms. CSBCM is based on the spatial beam constraint model (SBCM), which was developed for the purpose of accurately predicting the nonlinear constraint characteristics of bisymmetric spatial beams in their intermediate deflection range. CSBCM deals with large spatial deflections by dividing a spatial beam into several elements, modeling each element with SBCM, and then assembling the deflected elements using the transformation defined by Tait-Bryan angles to form the whole deflection. It is demonstrated that CSBCM is capable of solving various large spatial deflection problems either the tip loads are known or the tip deflections are known. The examples show that CSBCM can accurately predict large spatial deflections of flexible beams, as compared to the available nonlinear FEA results obtained by ANSYS. The results also demonstrated the unique capabilities of CSBCM to solve large spatial deflection problems that are outside the range of ANSYS.

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This work will present a method called chained spatial-beam-constraint-model (CSBCM) for modeling large spatial deflections of flexible bisymmetric beams in compliant mechanisms. CSBCM is based on the spatial beam constraint model (SBCM) proposed by Sen and Awtar [1], which was developed for the purpose of accurately predicting the nonlinear constraint characteristics of a bisymmetric spatial beam over an intermediate translational and angular displacement range of 10% of the beam-length and 0.1 rad, respectively. CSBCM substantially extends SBCM’s prediction range by dividing a spatial beam into several elements and modeling each element with SBCM. A bisymmetric beam, as defined in Ref. [1], refers to a beam whose cross section has equal moments of area about the y- and z-axes (i.e., I y = Iz = I) and zero product of inertia (I yz = 0) (assuming that the longitudinal direction of the undeflected beam is along the x-axis, as shown in Figure 1). Commonly used bisymmetric beams in compliant mechanisms include beams with circular cross-sections and square cross-sections. Compliant mechanisms that achieve their mobility through large planar deflections of their flexible members have been extensively studied [2] (including planar linkages and spatial linkages [3, 4]). Pseudorigid-body models (PRBM) [5, 6, 7] are often used to approximately model these large deflections during early design stages of compliant mechanisms. There are also a number of methods available for accurate modeling of these large deflections, for example, the finite element method, the chain algorithm [8], the circular-arc method [9], the Adomian decomposition method [10, 11], the global coordinate model with an incremental linearization approach [12], the elliptical integral solutions [13, 14, 15, 16], the GaussianChebyshev quadrature [17] and the chained beam-constraint-model [18], to name a few. As we know, spatial deflections yield a great variety of possible motions and could be useful for accomplishing sophisticated tasks beyond what planar deflections are capable of providing (could be 1 Corresponding

author: [email protected]

JMR-15-1148, Chen 1

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Journal of Mechanisms and Robotics. Received June 18, 2015; Accepted manuscript posted February 1, 2016. doi:10.1115/1.4032632 Copyright (c) 2016 by ASME

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used for creating compliant mechanisms that are more compact than those utilizing planar deflections for accomplishing complex tasks). As early as 1971, Shoup and McLarnan [19] presented a few design concepts for compliant spatial mechanisms that utilize spatially deflected beams, but none of them has been modeled. A spherical bistable compliant micro-mechanism was reported in Ref. [20], in which two small-length flexural pivots undergoing spatial deflections were employed to achieve motion out of the fabrication plane. Tanık and Parlaktas¸ [22, 21] proposed a compliant spatial four-bar mechanism and a compliant spatial slider-crank mechanism which obtain their mobility through the multiple-axis deflections of the small-length flexural pivots. However, fewer work has been done on modeling spatial deflections of flexible beams in compliant mechanisms. In such a situation, compliant spatial mechanisms were often modeled using their simplified (or kinematically equivalent) PRBMs, for example, Smith and Lusk [23] proposed a PRBM for the spherical bistable compliant micro-mechanism [20] by considering the bending and torsion of the flexural pivots as independent parameters (that is to say, the bending and torsion interactions were neglected), and Tanık and Parlaktas¸ [22, 21] determined the deflections of the multiple-axis flexural pivots based on the kinematic models of the rigid-body counterparts of the mechanisms. A few 3D PRBMs have been specifically developed to approximate large spatial deflections of slender beams by two rigid beams connected by a spherical joint with torsional springs [24, 25, 26]. Although these PRBMs provide quick ways for design and analysis of spatial compliant mechanisms, they suffer from modeling inaccuracies. The 3D chain algorithm with PRBM elements proposed by Chase et al. [27] showed to be accurate for predicting the tip locus of spatially deflected beams but inaccurate for the tip angles due to the lack of an adequate parametric end angle of PRBM for superposition. The commercial nonlinear FEA (finite element analysis) software packages, e.g., ANSYS, can be easily applied to different large deflection problems, but it has been found that they become unstable when the couplings between the two bending directions and the torsional direction are significant, as will be illustrated in Sec. 4. Interestingly, there are three models available for determining the spatial deflections of slender beams in their intermediate deflection range, i.e., the nonlinear analytical model (NAM) [28] that combines two orthogonal planar beam constraint models (BCMs) [29] and a torsional deflection model, the simplified PRBM for spatial fixed-fixed beams (SPRBM) [30], and the spatial beam constraint model (SBCM) [1]. In comparison, SBCM captures all the critical geometric nonlinearities, thus is more accurate in modeling intermediate spatial deflections than NAM and SPRBM. Besides its accuracy, SBCM is also closed-form, parametric and easy-to-use, which motivates us to extend its use for large spatial deflection modeling. To bridge the gap between SBCM’s prediction range and large spatial deflection, we utilized a discretizationbased scheme with which the beam is divided into a number of elements such that the deflection of each element falls into SBCM’s prediction range. The rest of this paper is organized as follows: Sec. 2 describes the large spatial deflection problem of a flexible beam. Sec. 3 presents the basic formulation of CSBCM based on the constitutive equations of SBCM. Examples are given in Sec. 4 to illustrate the effectiveness of CSBCM for solving different large spatial deflections. The concluding remarks are made in the last section.

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2 Large Spatial Deflections

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Figure 1 illustrates the spatial deflection of a uniform cantilever beam subject to combined force and moment loads at its free end. The beam is assumed to be initially straight and its cross-section is square. A fixed coordinate frame xyz is chosen so that its origin is at the fixed end of the beam, its x-axis is along the longitudinal direction of the undeflected beam, and its y- and z-axes directed parallel to the two sides of the cross-section at the fixed end, respectively. The parameters of the beam include: the length L, the thicknesses along the y- and z-axes T y = Tz = T, and the Young’s modulus and the shear modulus of the material E and G, respectively. The area moments of inertia of the beam are calculated as I = I y = Iz =

JMR-15-1148, Chen

T4 12

(1)

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Journal of Mechanisms and Robotics. Received June 18, 2015; Accepted manuscript posted February 1, 2016. doi:10.1115/1.4032632 Copyright (c) 2016 by ASME

   

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Figure 1: Three dimensional deflection of a spatial cantilever beam subject to combined force and moment loads at its free end (the sign convention for forces and moments follows the right-hand rule).

1.167η5 + 29.49η4 + 30.9η3 + 100.9η2 + 30.38η + 29.41 η5 + 25.91η4 + 41.58η3 + 90.43η2 + 41.74η + 25.21

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and the torsional moment of inertia, which is different from the polar moment of inertia due to warping [1], can be determined as [31]

×

2T3y Tz3 7T2y + 7Tz2

≈ 0.1406T

(2)

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where η = T y /Tz . If the beam is of circular cross-section, the area moments of inertia are I = I y = Iz =

πD4 64

(3)

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and the torsional moment of inertia happens to equal the polar moment of inertia: J = I y + Iz =

πD4 32

(4)

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where D is the diameter. The loads applied at the beam’s free end (point A) include forces Fxo , F yo and Fzo and moments Mxo , M yo and Mzo . The 6 general displacements, including the end coordinates denoted by ao , bo and co and the rotations of the end cross-section by θ yo , θzo and θxdo (θxdo is the twisting angle along the deformed centroidal axis at the free end), are selected to represent the deflections of the beam end. At an arbitrary point P (x, y, z) on the beam, the intermediate loads are determined as (applying the static equilibrium

JMR-15-1148, Chen

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Journal of Mechanisms and Robotics. Received June 18, 2015; Accepted manuscript posted February 1, 2016. doi:10.1115/1.4032632 Copyright (c) 2016 by ASME

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zP

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= Fzo = Mxo + Fzo (bo − y) − F yo (co − z) = M yo − Fzo (ao − x) + Fxo (co − z)

(5)

= Mzo + F yo (ao − x) − Fxo (bo − y)

Chained Spatial-Beam-Constraint-Model (CSBCM)

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= Fxo = F yo

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  FxP      F yP        FzP    MxP      M  yP    M

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condition)

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Figure 2: Discretization of the spatial beam and the coordinates of the nodes with respect to the fixed coordinate frame.

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In CSBCM, the spatial beam is equally divided into n beam elements, each of which is termed as a spatial beam constraint element (SBCE) and is modeled by SBCM. The length of each SBCE equals L/n. As shown in Figure 2, the first node, i.e., the origin of the fixed coordinate frame xyz, corresponds to the fixed end of the beam, and the last node, Nn (i.e., point A), is the beam tip. The ith SBCE can be considered cantilevered at the free end of the (i − 1)th SBCE, i.e., node Ni−1 , and loaded at node Ni . The loads at node Ni with respect to the global coordinate frame are denoted as Fxi , F yi , Fzi , Mxi , M yi and Mzi , and the deflections of the free end of the ith SBCE are accordingly represented by ai , bi , ci , θ yi , θzi and θxdi , as illustrated in Figure 3.

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3.1 SBCE

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A moving coordinate frame xi yi zi is established for the ith (1 ≤ i ≤ n) SBCE so that its origin is located at node Ni−1 , the xi -axis directed perpendicular to the cross-section at node Ni−1 , and the yi - and zi -axes directed parallel to the y- and z-axes, respectively, when the beam is undeflected. The yi - and zi -axes are kept parallel to the two sides of the cross-section at node Ni−1 as the beam is deflected. This assignment of moving coordinate frames allows us to define the transformation that locates the orientation of the frame as a sequence of transformations (using Tait-Bryan angles) according to the deflection of the (i − 1)th SBCE, as will be discussed in the next subsection. For the ith SBCE, the load parameters with respect to its moving coordinate frame xi yi zi are denoted as ¯ xi , M ¯ yi and M ¯ zi , and the deflections can be represented by the following general displacements: F¯ xi , F¯ yi , F¯ zi , M ¯ ¯ ¯ ¯ ¯ ¯ Uxi , U yi , Uzi , θ yi , θzi and θxdi (an ”overbar” is placed over each parameter to indicate that it is measured with respect to its moving coordinate frame). The equivalent torsional moment expressed along the deformed JMR-15-1148, Chen

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Journal of Mechanisms and Robotics. Received June 18, 2015; Accepted manuscript posted February 1, 2016. doi:10.1115/1.4032632 Copyright (c) 2016 by ASME

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Figure 3: Tait-Bryan angles for ith SBCM element (the sign convention for the angles follows the right-hand rule).

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¯ xdi , is expressed as [1] centroidal axis at its free end, denoted as M

(6)

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¯ xdi = M ¯ xi + θ¯ zi M ¯ yi + θ¯ yi M ¯ zi M

These load and deflection parameters are normalized with respect to the element parameters as

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¯ yi L ¯ zi L ¯ xdi L M M M , m¯ yi = , m¯ xdi = , nEI nEI nEI F¯ yi L2 F¯ zi L2 F¯ xi L2 f¯zi = 2 , f¯yi = 2 , f¯xi = 2 , n EI n EI n EI ¯ yi ¯ zi ¯ xi nU nU nU , u¯ zi = , u¯ xi = u¯ yi = L L L

(7)

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¯ zi = m

SBCM [1] models the load-deflection relations of the ith SBCE using the following parametric and closed-form equations:

(8)

  u¯ yi   θ¯    + {u¯ yi θ¯ zi u¯ zi θ¯ yi }H2  zi   u¯ zi  ¯  θ yi   u¯ yi   θ¯  u¯ zi θ¯ yi }[ f¯xi H4 + 0.5m¯ xdi H5 ]  zi   u¯ zi  ¯  θ yi 5

(9)

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 ¯       f yi  u¯ yi  u¯ yi   m     θ¯   θ¯ zi   ¯ zi    ¯ xdi (2H3 + H7 )]  zi   ¯  =H1   − [2 f¯xi H2 + m  u¯ zi   fzi   u¯ zi  ¯  ¯ yi θ¯ yi θ yi m   u¯ yi   θ¯    2 2 ¯ ¯ ¯ xdi fxi H5 + m¯ xdi H6 ]  zi  − [ fxi H4 + m  u¯ zi  θ¯ yi

u¯ xi =

f¯xi ′ − k33

¯ 2xdi m ′ 2 k33 k44

+ {u¯ yi θ¯ zi JMR-15-1148, Chen

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Journal of Mechanisms and Robotics. Received June 18, 2015; Accepted manuscript posted February 1, 2016. doi:10.1115/1.4032632 Copyright (c) 2016 by ASME

¯ xdi m − ′ 2 θ¯ xdi = k44 k33 k44

+ {u¯ yi θ¯ zi u¯ zi

  u¯ yi   θ¯    + {u¯ yi θ¯ zi u¯ zi θ¯ yi }H3  zi   u¯ zi  ¯  θ yi   u¯ yi   θ¯    θ¯ yi }[m¯ xdi H6 + 0.5 f¯xi H5 ]  zi   u¯ zi  ¯  θ yi

where

(10)

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2m¯ xdi f¯xi

 0 0 0 0  12 6  6 4

(12)

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  12 −6 −6 4 H1 =  0  0 0 0

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GJ 12L2 (11) , k44 = 2 2 EI nT and the coefficient matrices H j (j = 1, . . . , 7) are non-dimensional beam characteristic coefficients given as [1] ′ k33 =

(13)

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  0 0  −3/5 1/20  1/20 −1/15 0 0   H2 =   0 −3/5 −1/20  0  0 0 −1/20 −1/15   −1/2 0 0  0  0 0 −1/2 −1/4  H3 =   −1/2 0 0   0  −1/2 −1/4 0 0   −1/1400 0 0   1/700 −1/1400 11/6300 0 0   H4 =   0 0 1/700 1/1400    0 0 1/1400 11/6300   0 0 1/60  0  0 0 1/60 0   H5 =   1/60 0 0   0  1/60 0 0 0   −1/10 0 0   1/5 −1/10 1/20 0 0   H6 =   0 1/5 1/10  0   0 0 1/10 1/20   0 0 0 0 0 0 0 1   H7 =   0 0 0 0   0 0 0 0

(14)

(15)

(16)

(17)

(18)

3.2 Tait-Bryan Angles As shown in Figure 3, the (i + 1)th SBCE (i = 1, ..., n − 1) can be considered being cantilevered at the free end of the ith element (Ni ). The rotation transformation from the ith element coordinate frame to the (i + 1)th element coordinate frame (i.e., the rotation of the cross-section at node Ni ), which is determined by the angular deflections of the ith SBCE (θ¯ yi , θ¯ zi and θ¯ xdi ), can be captured using a sequence of rotations defined JMR-15-1148, Chen

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Journal of Mechanisms and Robotics. Received June 18, 2015; Accepted manuscript posted February 1, 2016. doi:10.1115/1.4032632 Copyright (c) 2016 by ASME by three Tait-Bryan (or Euler) angles (θ¯ yi , θ¯ zdi and θ¯ xdi ): the first rotation about the yi -axis by an angle θ¯ yi , the second rotation about the new zi -axis (an intermediate axis) by an angle θ¯ zdi , and the last rotation about the new xi -axis (i.e., the xi+1 -axis) by an angle θ¯ xdi . θ¯ zi is the projection of θ¯ zdi in the xi yi plane. The transformation can be represented as sin θ¯ zdi cos θ¯ zdi 0

 0 cos θ¯ yi 0  0  1 sin θ¯ yi

 0 − sin θ¯ yi   1 0   0 cos θ¯ yi

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Ri = Rx (θ¯ xdi )Rz (θ¯ zdi )R y (θ¯ yi ) =   0 0   cos θ¯ zdi 1 0 cos θ¯ sin θ¯ xdi  − sin θ¯ zdi  xdi   0 − sin θ¯ xdi cos θ¯ xdi 0

(19)

θ¯ zdi = arctan(tan θ¯ zi cos θ¯ yi )

py ed

where θ¯ zdi can be obtained using θ¯ yi and θ¯ zi (note that all the angles are within interval (−π/2, π/2)): (20)

The whole transformation from the fixed end to the free end (from xyz to xn yn zn ) can be written as R=

n ∏

Rj

(21)

Co

j=1

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If R is given or obtained, the Tait-Bryan angles of the whole transformation (θ yo , θzdo and θxdo ) and the tip angles of the whole beam (θ yo , θzo and θxdo ), as illustrated in Figure 1, can be extracted from R by following the steps given below. Give a unit column vector [1 0 0]T with respect to xn yn zn and let     Sx  1 S y  = R−1 0 (22)     Sz 0 in which [Sx S y Sz ]T gives the representation of the vector with respect to xyz. We have −Sz Sx

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θ yo = arctan

Sy

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θzo = arctan

Sx

(θ yo = π/2 if Sx = 0)

(23)

(θzo = π/2 if Sx = 0)

(24)

Sy θzdo = arctan √ S2x + S2z (θ yo = π/2 and θzdo = π/2 if Sx = 0 and Sz = 0)

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Because

ce Ac

that is

pt

we have

(25)

R = Rx (θxdo )Rz (θzdo )R y (θ yo )

(26)

−1 Rx (θxdo ) = RR−1 y (θ yo )Rz (θzdo )

(27)

  0 0  1 0 cos θ sin θxdo  =  xdo   0 − sin θxdo cos θxdo    cos θ yo 0 sin θ yo  cos θzdo  0 1 0   sin θzdo R    − sin θ yo 0 cos θ yo 0

− sin θzdo cos θzdo 0

 0 0  1

(28)

Once Rx (θxdo ) is obtained, θxdo can be determined as θxdo = arctan JMR-15-1148, Chen

sin θxdo cos θxdo

(29)

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3.3

Loads and Deflections at Each Node

In the fixed coordinate frame xyz, the tip coordinates of the ith SBCE (node Ni ) can be calculated as (note that a1 = a¯1 , b1 = b¯ 1 and c1 = c¯1 )        j−1 i ∏ ai  a1  ∑  a¯ j  −1 bi  = b1  +  b¯ j  R (30)        k  ci c1 c¯ j j=2 k=1 where

ite d

  ¯  a¯ j  Uxj + L/n b¯   U ¯ yj   j  =      ¯ zj  U c¯ j

zi

zo

o

i

yo

o

(32)

Co

py ed

Then the loads acting on node Ni (including Fxi , F yi , Fzi , Mxi , M yi and Mzi ) can be expressed as   Fxi = Fxo = Fx1 = F¯ x1      F yi = F yo = F y1 = F¯ y1        Fzi = Fzo = Fz1 = F¯ z1    Mxi = Mxo + (bo − bi )Fzo − (co − ci )F yo      M yi = M yo − (ao − ai )Fzo + (co − ci )Fxo      M = M + (a − a )F − (b − b )F

(31)

i

xo

ip

tN

ot

where Fx1 , F y1 and Fz1 are the forces acting on tip of the first SBCE (node N1 ). These loads can transformed ¯ xi , M ¯ yi and M ¯ zi , into their moving coordinate frame using the following relations (denoted as F¯ xi , F¯ yi , F¯ zi , M respectively): ¯     i−1  Mxi  Mxi  ∏  M ¯  Ri−k  M yi  (33)  yi  =    ¯ zi M Mzi k=1 and

sc r

¯       i−1   i−1 Fxi  ∏  Fx1  ∏  Fxo  F¯   Ri−k  F y1  =  Ri−k  F yo   yi  =  ¯      Fzi Fz1 Fzo k=1 k=1

Ma nu

In addition, the forces at adjacent nodes have the following relations: ¯  ¯  ¯  Fx,i+1  Fxi  Fx,i−1  F¯       y,i+1  = Ri F¯ yi  = Ri Ri−1 F¯ y,i−1  ¯  ¯  ¯  Fz,i+1 Fzi Fz,i−1

(34)

(35)

Ac

ce

pt

ed

These equations for the loads and deflections, together with the SBCE equations for each element, constitute the CSBCM equations of the beam. Among the 6 load parameters Fxo , F yo , Fzo , Mxo , M yo and Mzo , and the 6 general displacements ao , bo , co , θ yo , θzo and θxdo , given any set of 6 parameters, the other 6 can be obtained by numerically solving the CSBCM equations. A considerable number of numerical tests suggest the following rule of thumb for choosing the number of SBCEs required in CSBCM: n = 20 should be used when |θ yo | and |θzo | are less than π/2 while |θxdo | is less than π/4; if |θxdo | exceeds π/4, n = 50 is suggested. Generally speaking, utilizing more SBCEs can always increase the modeling accuracy of CSBCM.

4 Results

This section presents a few examples to demonstrate the effectiveness of CSBCM in dealing with large spatial deflections of flexible bisymmetric beams. Cantilever beams subject to different tip-load cases were analyzed, which is followed by an example of a circular-guided spatial compliant mechanism containing a flexible bisymmetric beam. The flexible beams in all the examples were divided into 50 SBCE (n = 50). Some of the analytical results were verified by nonlinear FEA performed in ANSYSTM , with the flexible beam being meshed into 200 elements using BEAM188 and the geometric nonlinearity option turned on (using Command “NLGEOM, ON”). JMR-15-1148, Chen 8

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0.12 CSBCM ANSYS

0.1

ite d

z−axis (m)

0.08 0.06

py ed

0.04 0.02 0 0.12 0.1

0.2

0.06

Co

0.15

0.08

0.1

0.05

0.04

0

0.02 0

−0.05

y−axis (m)

ot

x−axis (m)

Cantilever Beam Subject to End Loads

ip

4.1

tN

Figure 4: The deflected configurations of the beam subject to pure end moments.

Ma nu

sc r

All the flexible beams analyzed in this subsection are assumed to be made of steel, with Young’s Modulus E = 2.1×1011 Pa and Poisson’s ratio ν = 0.26. The cross-section of the beam is square with T y = Tz = 0.001 m, and beam length L = 0.2 m. 4.1.1 Pure force load

ed

We assume the flexible beam is subjected to forces F yo and Fzo at its free end, both of which gradually increase from 0.2 N to 2 N in 10 load steps with equal increment. The deflected configurations of beam obtained by CSBCM at different load steps are shown in Figure 5. The nonlinear FEA results achieved by ANSYS are also plotted in the figure for the purpose of comparison. All the CSBCM and FEA results agree well. 4.1.2 Pure moment load

Ac

ce

pt

We assume the flexible beam is subjected to moments M yo and Mzo at its free end, among which M yo is gradually changed from −0.02 Nm to −0.2 Nm and Mzo from 0.02 Nm to 0.2 Nm in 10 load steps with equal increment/decrement. Figure 4 shows the deflected shapes of the flexible beam achieved by both CSBCM and the nonlinear FEA model. The CSBCM and FEA results are in good consistent for the first 6 load steps. However, the nonlinear FEA model failed in the last load steps. 4.1.3 Combined loads We assume the flexible beam is subjected to combined loads (forces and moments) at its free end, in which Fxo , F yo and Fzo are gradually increased from 0.05 N to 0.65 N, Mxo and Mzo from 0.01 Nm to 0.13 Nm, and M yo from −0.01 Nm to −0.13 Nm in 5 load steps with equal increment/decrement. The corresponding deflected configurations of the flexible beam achieved by both CSBCM and the nonlinear FEA model are plotted in Figure 6. From Figure 6 we see that the CSBCM results agree well with the FEA results for the first 3 load JMR-15-1148, Chen

9

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CSBCM ANSYS

0.1

ite d

0.09 0.08

0.06

py ed

z−axis (m)

0.07

0.05 0.04 0.03 0.02

0 0.1

Co

0.01

0.15

0.1

0.06

0.04

0

0

x−axis (m)

tN

y−axis (m)

0.05

0.02

ot

0.08

Ma nu

sc r

ip

Figure 5: The deflected configurations of the beam subject to pure end forces.

0.14 0.12

0.06

ed

0.08

pt

z−axis (m)

0.1

CSBCM ANSYS

ce

0.04 0.02

Ac

0 0.08

0.2

0.06

0.15

0.04

y−axis (m)

0.1

0.02

0.05 0

0

x−axis (m)

Figure 6: Deflected configurations for the beam subject to combined loads (a).

JMR-15-1148, Chen

10

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0.12 CSBCM ANSYS

ite d

0.08 0.06 0.04

py ed

z−axis (m)

0.1

0.02 0 0.12 0.1

0.2

0.15

0.08 0.06

0.1

0.04

Co

y−axis (m)

0.05

0.02 0

ot

0

x−axis (m)

tN

Figure 7: Deflected configurations for the beam subject to combined loads (b).

4.1.4

Ma nu

sc r

ip

steps. For the last 2 load steps, the nonlinear FEA model failed to yield reasonable results, although we did multiple trials, meshing the beam with 1000 and 10000 BEAM188 elements, respectively. Figure 7 plots the deflected configurations of the flexible beam subject to combined loads (forces and moments), in which Fxo and Fzo are gradually changed from 0.05 N to 0.65 N, F yo from 0.05 N to 1.05 N, Mxo from 0.01 Nm to 0.13 Nm, M yo from −0.01 Nm to −0.13 Nm, and Mzo from 0.01 Nm to 0.21 Nm in 5 load steps. Again, the CSBCM and FEA results agree well for the first 3 load steps, but the nonlinear FEA model failed to yield reasonable results for the last 2 load steps. Comparison

Ac

ce

pt

ed

Table 1 lists the tip coordinates for the last load step of each load cases predicted by ANSYSTM , together with those predicted by CSBCM (with n = 20 and n = 50). The results obtained by ANSYSTM and CSBCM are in good consistency, with the maximum error less than 0.1%. Table 2 compares the computation times for each load cases using ANSYSTM and CSBCM (with n = 20 and n = 50) implemented in MATLABTM . ANSYSTM was more efficient than CSBCM for pure force and pure moment load cases, but CSBCM with n = 20 outperformed ANSYSTM for the combined load cases. It should be noted that the efficiency of CSBCM can be further improved by switching to a more efficient programming language (e.g., FORTRAN or C) instead of MATLAB script, improving the programming styles and taking advantage the parallel processing capability of the multi-core processor, etc.

4.2 Circular-Guided Spatial Compliant Mechanism The circular-guided spatial compliant mechanism, as shown in Figure 8, is comprised of a flexible bisymmetric beam and a rigid crank. The flexible beam is cantilevered on the ground on the one end and rigidly attached to the free end of the crank on the other end. The angle between the flexible beam and the crank is denoted as α. The spatially oriented crank rotates in a plane (which will be referred to as the crank plane) that intersects the undeflected flexible beam at point A. The crank is driven by torque Tin applied on it, and the rotation of the crank deflects the flexible beam so that its tip follows a circle. This spatial compliant JMR-15-1148, Chen

11

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co (m) 0.106920 0.106396 0.106821 0.099201 0.098562 0.099088 0.092792 0.092663 0.092772 0.088576 0.088329 0.088558

py ed

bo (m) 0.106920 0.107313 0.106977 0.099203 0.099738 0.099287 0.062663 0.062826 0.062697 0.094691 0.094890 0.094703

Ma nu

sc r

ip

tN

ot

Co

ao (m) 0.109307 0.109447 0.109353 0.096200 0.096427 0.096230 0.151394 0.151396 0.151384 0.127688 0.127692 0.127679

PF (ANSYS) PF (CSBCM with n = 20) PF (CSBCM with n = 50) PM (ANSYS) PM (CSBCM with n = 20) PM (CSBCM with n = 50) C1 (ANSYS) C1 (CSBCM with n = 20) C1 (CSBCM with n = 50) hline C2 (ANSYS) C2 (CSBCM with n = 20) C2 (CSBCM with n = 50)

ite d

Table 1: Coordinates of the tip for different load cases. PF refers to the 10th load step of pure force load case shown in Figure 5, PM to the 6th load step of pure moment load case shown in Figure 4, C1 to the 3rd load step of combined load case shown in Figure 6, and C2 to the 3rd load step of combined load case shown in Figure 7.

ce

pt

ed

Table 2: Computation times for different load cases on a personal computer with a quad-core processor of 3.2 GHz. PF refers to all the 10 load steps of pure force load case shown in Figure 5, PM to the first 6 load steps of pure moment load case shown in Figure 4, C1 to the first 3 load steps of combined load case shown in Figure 6, and C2 to the first 3 load steps of combined load case shown in Figure 7.

Ac

PF PM C1 C2

JMR-15-1148, Chen

CSBCM (n = 20) 6.69 s 3.43 s 2.19 s 2.17 s

CSBCM (n = 50) 33.58 s 15.29 s 10.34 s 10.38 s

ANSYS (200 elements) 3.28 s 2.10 s 3.28 s 6.95 s

12

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ite d



py ed







Co





ip

tN

ot



sc r

Figure 8: Circular-guided spatial compliant mechanism.

pt

ed

Ma nu

mechanism is similar to the 1R mechanism presented in Ref. [19]. The lengths of the flexible beam and the crank are denoted as L and Lc , respectively. In this mechanism, the 6 general displacements ao , bo , co , θ yo , θzo and θxdo can be geometrically determined for a given crank angle ϕ (the detailed procedure is presented below), which are then used by CSBCM to calculate the 6 load parameters Fxo , F yo , Fzo , Mxo , M yo and Mzo . Locate the fixed coordinate frame xyz so the origin is at the fixed end of the flexible beam, the x-axis is along the undeflected position and the pivoted end of the crank is located in the xz-plane. The crank plane intersects the x-, y- and z-axes at points A (the free end of the flexible beam), B and C, respectively. LOA = L, LOC = L tan α and LOB can be used to determine the crank plane. The coordinates of point Q are given as xQ = L − Lc cos α, yQ = 0, zQ = Lc sin α

(36)

⃗ v1 = {cos α, 0, − sin α}

(37)

Ac

ce

The unit vector for the crank at its initial position is given as

The normal unit vector of the crank plane can be expressed as √ √ √ f f f ⃗ vn = { ,− , } LOA LOB LOC where f =

1 1

L2OA

JMR-15-1148, Chen

+

1 L2OB

+

1 L2OC

(38)

(39)

13

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and the angle between ⃗ vn and the xz plane β is √

f LOB

tN

β = arcsin

ot

Figure 9: Tip angles.

Co

py ed

 

ite d



(40)

ip

v1 lies in the crank plane thus is perpendicular to ⃗ vn . Let ⃗ v2 = ⃗ vn × ⃗ v1 . Obviously ⃗ v2 is a unit We know that ⃗ vector perpendicular to both ⃗ vn and ⃗ v1 and lies in the crank plane. By denoting ⃗ v1 and ⃗ v2 as (41)

⃗ v2 = {v2x , v2y , v2z }

(42)

sc r

⃗ v1 = {v1x , v1y , v1z }

Ma nu

the tip position of deflected flexible beam OA (point A’) for a given crank angle ϕ can be expressed as ao = L − Lc cos α + Lc (v1x cos ϕ + v2x sin ϕ)

bo = Lc (v1y cos ϕ + v2y sin ϕ) co = Lc sin α + Lc (v1z cos ϕ + v2z sin ϕ)

(43)

Ac

ce

pt

ed

Then we try to determine the deflected beam tip angles, θ yo , θzo , and θxdo . Suppose a moving coordinate frame xA yA zA is attached to the crank with the origin fixed at point A and the xA -, yA - and zA -axes parallel to the x-, y- and z-axes at the initial position, respectively. We assume point O rotates around ⃗ vn along with the rotation of the crank. The plane of the circular locus intersects the line of ⃗ vn at point P. For a given crank angle ϕ, the position of point O after rotation (denoted by point O’ in Figure 9) can be determined, as will be described in the following. Note that the crank lies in the xA zA plane (the x′A z′A plane at the deflected position) and plane O’QA’ determines the x′A z′A plane (in which plane x′A has an angle of α with QA’). LPQ equals the distance between point O and the crank plane LPQ = √

1 1

L2OA

Because

LOQ =

1

L2OB

+

1



f

LOP =



(44)

L2OC

√ (L − Lc cos α)2 + (Lc sin α)2

thus the radius of the circular locus is JMR-15-1148, Chen

+

=

L2OQ − L2PQ

(45) (46)

14

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Journal of Mechanisms and Robotics. Received June 18, 2015; Accepted manuscript posted February 1, 2016. doi:10.1115/1.4032632 Copyright (c) 2016 by ASME vn , we have Using the directional angles of ⃗ xQ − xP 1

=−

yQ − yP

=

1

LOA

zQ − zP 1 LOC

LOB

= f

(47)

thus the coordinates of point P are:

LOA

f LOB

zP = Lc sin α −

f LOC

py ed

yP =

f

ite d

xP = L − Lc cos α −

y

(48)

P P P , − LOP , − LzOP }. By denoting ⃗ We define ⃗ v3 as the unit vector along line PO, i.e., ⃗ v3 = {− LxOP v3 as

(49)

Co

⃗ v3 = {v3x , v3y , v3z }

the coordinates of point O’ for a given angle ϕ can be expressed as (note that the two planes of the circular paths are parallel to each other) f + LOP (v3x cos ϕ + v2x sin ϕ) LOA

ot

x′O = L − Lc cos α −

f + LOP (v3y cos ϕ + v2y sin ϕ) LOB f z′O = Lc sin α − + LOP (v3z cos ϕ + v2z sin ϕ) LOC

(50)

ip

tN

y′O =

sc r

We denote the unit vectors along the three axes of x′A y′A z′A coordinate frame as ⃗r′x , ⃗r′y and ⃗r′z , respectively. Considering that line O’A’ is collinear with the x′A -axis (

ao − x′O bo − y′O co − z′O = , , L L L

)

Ma nu

⃗r′x

(51)

and the y′A -axis is perpendicular to plane of O’QA’, we have ⃗ ′ × QO ⃗ ′ QA = RQ (ao − xQ , bo − yQ , co − zQ ) × (x′O − xQ , y′O − yQ , z′O − zQ )

(52)

RQ

pt

ed

⃗r′y =

ce

⃗ ′ × QO ⃗ ′ (i.e., ∥QA ⃗ ′ × QO ⃗ ′ ∥). Then the z′ -axis can be given as where RQ denotes the length of vector QA A ⃗r′z = ⃗r′x × ⃗r′y

(53)

Ac

Therefore, the transformation matrix from xyz (or xA yA zA ) to x′A y′A z′A is given as R = [⃗r′x ⃗r′y ⃗r′z ]T

and θ yo = arctan θzo = arctan JMR-15-1148, Chen

−co + z′O ao − x′O

bo − y′O ao − x′O

(54)

(55) (56)

15

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0.1 φ=84° Crank CSBCM (Forward) CSBCM (Reverse)

φ=72°

ite d

0.06

φ=60° 0.04

φ=48° φ=36°

py ed

z−axis (m)

0.08

0.02

φ=24°

0 0.1

φ=12°

0.08

Co

0.06 0.04 y−axis (m)

0.2

0.15

0.1

0.02

0.05 0

x−axis (m)

tN

ot

0

ip

Figure 10: Deflected configurations of the beam at different crank angles.

LOB (m) ∞

α (◦ ) 45

Ma nu

Lc (m) 0.1

sc r

Table 3: Parameters of the circular-guided spatial compliant mechanism

θzdo = arctan √

L (m) 0.2

T y = Tz (m) 0.001

bo − y′O (ao − x′O )2 + (co − z′O )2

(57)

pt

ed

Besides, θxdo can be calculated using Eq. (29). Once the load parameters are obtained by CSBCM, the driving torque Tin can be determined as Tin = −F∗xo Lc sin ϕ + F∗yo Lc cos ϕ + M∗zo

(58)

Ac

ce

in which F∗xo , F∗yo and M∗zo are the tip loads with respect to the moving coordinate frame of the crank:  ∗   0 Fxo  1 F∗  0 cos β  yo  =   ∗   0 − sin β Fzo

 0  cos α 0 sin β   0 1  cos β sin α 0

  − sin α Fxo  0  F yo    cos α Fzo

(59)

 ∗   0 Mxo  1 M∗  0 cos β  yo  =   ∗   0 − sin β Mzo

 0  cos α 0 sin β   0 1  cos β sin α 0

  − sin α Mxo    0  M yo    cos α Mzo

(60)

The parameters of a circular-guided spatial compliant mechanism example are listed in Table 3. The 6 deflection parameters were calculated using Eq. (43), Eq. (55), Eq. (56) and Eq. (29) at crank angle ϕ = 12◦ , JMR-15-1148, Chen 16

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0.25

0.2

in

T (Nm)

0.15

ite d

0.1

0 0

10

20

30

40

φ (°)

50

60

70

py ed

0.05

80

90

Co

Figure 11: Driving torque Tin vs. crank angle ϕ.

Conclusions

Ma nu

5

sc r

ip

tN

ot

24◦ , 36◦ , 48◦ , 60◦ , 72◦ and 84◦ , respectively. These deflection parameters were then employed by CSBCM to solve for the load parameters and the deflected configurations of the beam. The deflected configurations of the mechanism at different values of ϕ are plotted in Figure 10 (indicated as the forward solutions). Figure 11 plots the driving torque as a function of the crank angle. Because the nonlinear FEA model built in ANSYS failed to solve this problem, we made some preliminary verification of the CSBCM results. The load parameters obtained in each forward solution were then used as 6 known parameters to solve for the deflection of the beam by CSBCM. The corresponding deflected configurations are plotted in Figure 10 as the reverse solutions. The forward and reverse solutions are in good agreement. The differences between the forward and the reverse solutions were less than 10− 8, which can be attributed to numerical errors.

6

Ac

ce

pt

ed

In this work, CSBCM was proposed for modeling large spatial deflections of flexible bisymmetric beams by extending SBCM through a discretization-based scheme. CSBCM is capable of solving large spatial deflection problems whether the tip loads are known or the tip deflections are known. Generally speaking, the proposed method works well in modeling large spatial deflections of bisymmetric beams when the tip angles are in the range of (−π/2, π/2). If there are multiple solutions for a large spatial deflection problem, the proposed method can only locate one of them. The examples showed that CSBCM can accurately predict the large spatial deflections of flexible beams, as compared to the available nonlinear FEA results obtained by ANSYS. The results also demonstrated the unique capabilities of CSBCM to solve large spatial deflection problems that are outside the range of ANSYS. CSBCM may benefit the study of PRBMs for flexible segments undergoing spatial deflections and also the development of a variety of useful spatial compliant mechanisms.

Acknowledgment

The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China under Grant No. 51175396, the Specialized Research Fund for the Doctoral Program of Higher Education of China under Grant No. 20120203110015, the Shaanxi Coordinating Project for Science and Technology under No. 2014KTCQ01-27, the Young Rising Stars of Shaanxi Province in Science and Technology under Grant No. 2013KJXX-65 and the Fundamental Research Funds for the Central Universities under Grant No. K5051204021/SPSZ011407. JMR-15-1148, Chen

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Journal of Mechanisms and Robotics. Received June 18, 2015; Accepted manuscript posted February 1, 2016. doi:10.1115/1.4032632 Copyright (c) 2016 by ASME

References [1] Sen, S. and Awtar, S. “A closed-form nonlinear model for the constraint characteristics of symmetric spatial beams,” ASME J. Mech. Des., 2013, 135(3), 031003. [2] Howell, L. L., Magleby, S. P., and Olsen, B. M., 2013, “Handbook of Compliant Mechanisms,” Wiley, New York, NY.

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[3] Hoover, A. M. and Fearing, R. S., “Analysis of off-axis performance of compliant mechanisms with applications to mobile millirobot design,” Proceedings of IROS, Oct. 10-15, 2009, St. Louis, Mo: 27702776.

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[4] Chen, G., Zhang S. and Li, G., “Multistable behaviors of compliant Sarrus mechanisms,” ASME J. Mechan. Robot., 2013, 5(2): 021005. [5] Howell, L. L., 2001, “Compliant Mechanisms,” Wiley, New York, NY.

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[6] Su, H. J., 2009, “A pseudorigid-body 3r model for determining large deflection of cantilever beams subject to tip loads,” ASME J. Mechan. Robot., 1(2), May, p. 021008. [7] Yu, Y.-Q., Feng, Z.-L. and Xu, Q.-P., 2012, “A pseudo-rigid-body 2R model of flexural beam in compliant mechanisms,” Mechanism and Machine Theory, 55(9), May, pp. 18-23.

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[8] Midha, A., Her, I., and Salamon, B., 1992, “Methodology for compliant mechanisms design: Part i-Introduction and large-deflection analysis,” 18th Annual ASME Design Automation Conference, ASME, pp. 29-38.

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[9] Campanile, L. F., and Hasse, A., 2008, “A simple and effective solution of the elastica problem,” Proceedings of the Institution Of Mechanical Engineers Part C-Journal Of Mechanical Engineering Science, 222(12), pp. 2513-2516.

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[10] Banerjee, A., Bhattacharya, B., and Mallik, A. K., 2008, “Large deflection of cantilever beams with geometric non-linearity: analytical and numerical approaches,” International Journal of Non-Linear Mechanics, 43(5), pp. 366-376. [11] Tolou, N., and Herder, J. L., 2009, “A seminalytical approach to large deflections in compliant beams under point load,” Mathematical Problems in Engineering, 2009, p. 910896. [12] Lan, C. C., 2008, “Analysis of large-displacement compliant mechanisms using an incremental linearization approach,” Mechanism and Machine Theory, 43(5), May, pp. 641-658.

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[13] Kimball, C., and Tsai, W., 2002, “Modeling of flexural beams subjected to arbitrary ends loads,” ASME J. Mech. Des., 124(2), pp. 223-235.

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[14] Holst, G. L., Teichert, G. H., and Jensen, B. D., 2011, “Modeling and experiments of buckling modes and deflection of fixed-guided beams in compliant mechanisms,” ASME J. Mech. Des., 133(5), p. 051002.

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[15] Kim, C. and Ebenstein, D, 2011, “Curve decomposition for large deflection analysis of fixed-guided beams with application to statically balanced compliant mechanisms,” ASME J. Mechan. Robot., 4(4), p. 041009. [16] Zhang, A., and Chen, G., 2013, “A comprehensive elliptic integral solution to the large deflection problems of thin beams in compliant mechanisms,” ASME J. Mechan. Robot., 5(2), pp. 021006. [17] Saxena, A., and Kramer, S. N., 1998, “A Simple and accurate method for determining large deflections in compliant mechanisms subjected to end forces and moments,” ASME J. Mech. Des., 120(3), pp. 392-400.

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Journal of Mechanisms and Robotics. Received June 18, 2015; Accepted manuscript posted February 1, 2016. doi:10.1115/1.4032632 Copyright (c) 2016 by ASME

[18] Ma, F. and Chen, G., “Chained Beam-Constraint-Model (CBCM): A powerful tool for modeling large and complicated deflections of flexible beams in compliant mechanisms,” Proceedings of the ASME 2011 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, August 17-20, 2014, Buffalo NY, USA: DETC2014-34140. [19] Shoup, T. E. and McLarnan, C. W., 1971, “A survey of flexible link mechanisms having lower pairs,” Jnl. Mechanisms, vol. 6, pp. 97-105.

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[20] Choueifati, J. G., “Design and modeling of a bistable spherical compliant mechanism,” Ph.D Thesis, University of South Florida, 2007.

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[21] Parlaktas¸, V and Tanık, E., “Partially compliant spatial slider-crank (RSSP) mechanism,” Mechanism and Machine Theory, 2011, 46(5): 593-606. [22] Tanık, E. and Parlaktas¸, V., “A new type of compliant spatial four-bar (RSSR) mechanism,” Mechanism and Machine Theory, 2011, 46(11): 1707-1718.

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[23] Smith, C. L. and Lusk, C. P., “Modeling and parameter study of bistable spherical compliant mechanisms,” Proceedings of the ASME 2011 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference: DETC2011-47397.

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[24] Rasmussen, N. O., Wittwer, J. W., Todd, R. H., Howell, L. L. and Magleby, S. P., 2006, “A 3D pseudo-rigidbody model for large spatial deflections of rectangular cantilever beams,” ASME 2006 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, DETC2006-99465, pp. 191-198.

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[25] Ramirez, I. and Lusk, C. P., 2011, “Spatial beam large deflection equations and pseudo-rigid-body model for axisymmetric cantilever beams,” Proceedings of the ASME 2011 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference: DETC2011-47389.

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[26] Chimento, J., Lusk, C. P. and Alqasimi, A., 2014, “A 3-D pseudo-rigid body model for rectangular cantilever beams with an arbitrary force end-load,” Proceedings of the ASME 2011 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, August 17-20, 2014, Buffalo NY, USA: DETC2014-34292. [27] Chase, R. P. Jr., Todd, R. H., Howell, L. L. and Magleby, S. P., 2011, “A 3-D chain algorithm with pseudo-rigid-body model elements,” Mechanics Based Design of Structures and Machines, vol. 39, 1, pp. 142-156.

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[28] Hao, G., Kong, X. and Reuben, R. L., 2011, “A nonlinear analysis of spatial compliant parallel modules: multi-beam modules,” Mechanism and Machine Theory, vol. 46, pp. 680-706.

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[29] Awtar, S., Slocum, A. H., and Sevincer, E., 2007, “Characteristics of beam-based flexure modules,” ASME J. Mech. Des., 129(6), pp. 625-639.

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[30] Hao, G., 2013, “Simplified PRBMs of spatial compliant multi-beam modules for planar motion,” Mechanical Sciences, 4(2), pp. 311-318.

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[31] Chen, G. and Howell, L. L., 2009, “Two general solutions of torsional compliance for variable rectangular cross-section hinges in compliant mechanisms,” Precision Engineering, vol. 33, pp. 268-274. [32] Sen, S., “Beam constraint model: generalized nonlinear closed-form modeling of beam flexures for flexure mechanism design,” Ph.D Dissertation, the University of Michigan, 2013. [33] Chen, G. and Ma, F., “Kinetostatic modeling of fully compliant bistable mechanisms using Timoshenko beam constraint model,” ASME J. Mech. Des., 2015, 137(2), p. 022301.

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Journal of Mechanisms and Robotics. Received June 18, 2015; Accepted manuscript posted February 1, 2016. doi:10.1115/1.4032632 Copyright (c) 2016 by ASME

List of Figures

4 5 6 7 8 9 10 11

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Three dimensional deflection of a spatial cantilever beam subject to combined force and moment loads at its free end (the sign convention for forces and moments follows the righthand rule). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discretization of the spatial beam and the coordinates of the nodes with respect to the fixed coordinate frame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tait-Bryan angles for ith SBCM element (the sign convention for the angles follows the righthand rule). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The deflected configurations of the beam subject to pure end moments. . . . . . . . . . . . . . The deflected configurations of the beam subject to pure end forces. . . . . . . . . . . . . . . . Deflected configurations for the beam subject to combined loads (a). . . . . . . . . . . . . . . Deflected configurations for the beam subject to combined loads (b). . . . . . . . . . . . . . . Circular-guided spatial compliant mechanism. . . . . . . . . . . . . . . . . . . . . . . . . . . . Tip angles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deflected configurations of the beam at different crank angles. . . . . . . . . . . . . . . . . . . Driving torque Tin vs. crank angle ϕ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

List of Tables

4 5 9 10 10 11 13 14 16 17

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Coordinates of the tip for different load cases. PF refers to the 10th load step of pure force load case shown in Figure 5, PM to the 6th load step of pure moment load case shown in Figure 4, C1 to the 3rd load step of combined load case shown in Figure 6, and C2 to the 3rd load step of combined load case shown in Figure 7. . . . . . . . . . . . . . . . . . . . . . . . . Computation times for different load cases on a personal computer with a quad-core processor of 3.2 GHz. PF refers to all the 10 load steps of pure force load case shown in Figure 5, PM to the first 6 load steps of pure moment load case shown in Figure 4, C1 to the first 3 load steps of combined load case shown in Figure 6, and C2 to the first 3 load steps of combined load case shown in Figure 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Parameters of the circular-guided spatial compliant mechanism . . . . . . . . . . . . . . . . .

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