Dynamic Modeling and Eigenvalue Evaluation of a 3-DOF ... - CiteSeerX

CHINESE JOURNAL OF MECHANICAL ENGINEERING Vol. 23,aNo. 2,a2010

·1·

DOI: 10.3901/CJME.2010.02.***, available online at www.cjmenet.com; www.cjmenet.com.cn

Dynamic Modeling and Eigenvalue Evaluation of a 3-DOF PKM Module ZHANG Jun1, LI Yonggang1,2, HUANG Tian1, * 1 School of Mechanical Engineering, Tianjin University, Tianjin 300072, China 2 Tianjin University of Technology and Education, Tianjin 300222, China Received May 26, 2009; revised February 27, 2010; accepted March 6, 2010; published electronically March 16, 2010

Abstract: Due to the structural complexity, the dynamic modeling and quick performance evaluation for the parallel kinematic machines (PKMs) are still to be remained as two challenges in the stage of conceptual design. By using the finite element method and substructure synthesis, this paper mainly deals with the dynamic modeling and eigenvalue evaluation of a novel 3-DOF spindle head named the A3 head. The topological architecture behind the proposed A3 head is a 3-RPS parallel mechanism, which possesses one translational and two rotational capabilities. The mechanical features of the A3 head are briefly addressed in the first place followed by inverse position analysis. In the dynamic modeling, the platform is treated as a rigid body, the RPS limbs as the continuous uniform beams and the joints as lumped virtual springs. With the combination of substructure synthesis and finite element method, an analytical approach is then proposed to formulate the governing equations of motion of system using the compatibility conditions at interface between the limbs and the platform. Consequently, by solving the eigenvalue problem of the governing equations of motion, the distribution of lower natural frequencies of the A3 head throughout the entire workspace can be predicted in a quick manner. Modal analysis for the A3 head reveals that the distributions of lower natural frequencies are strongly related to the mechanism configuration and are axially symmetric due to system kinematic and structural features. The sensitivity analysis of the system indicates that the dimensional parameters of the 3-RPS mechanism have a slight effect on system lower natural frequencies while the joint compliances affect the distributions of lower natural frequencies significantly. The proposed dynamic modeling method can also be applied to other PKMs and can effectively evaluate the PKM’s dynamic performance throughout the entire workspace.

Key words: dynamic modeling, parallel kinematic machine, natural frequency

1

∗

Introduction

Nowadays, high-speed machining of extra large components with complex geometries is one of the challenging issues in machine tool industry. For example, machining an aircraft wing over 30 meters long would require a huge gantry 5-axis machine tool with tons of weight and large footprint. One promising alternative solution is the use of parallel kinematic machine (PKM) technology. This proposition has been fully exemplified by the commercial success of Tricept robots[1–3] and Sprint Z3 head[4] in locomotive and aeronautical industries. Motivated by the success of Sprint Z3 head, HUANG, et [5] al proposed a novel PKM module named the A3 head, which possesses one translational and two rotational capabilities. Added by an x-y motion, the proposed module can be used as a multiple-axis spindle head to form a 5-axis high-speed machining unit. * Corresponding author. E-mail: [email protected] This project is supported by National Natural Science Foundation of China (Grant No. 50535010 and No. 50775158) and National Science and Technology Supporting Plan of China (Grant No. 2006BAF01B00)

Dynamic modeling and performance evaluation are two important issues in the design stage of a PKM if it is designed for machining where high rigidity and high dynamics are required. Finite element method (FEM) is the common use for the dynamic modeling of PKMs due to their complex geometries. With this method, FANG[6] proposed an FEM model for a 6-DOF parallel manipulator where six limbs were modeled using spatial beam elements. LEE[7] and FATTAH[8] dealt with dynamic modeling problem of 6-RPR spatial and 3-DOF planar parallel manipulators in the similar manner. The FEM was also applied to the dynamic modelings in Refs. [9–13]. It should be pointed out that in the above literature the joint compliances were not taken into account though they would bring significant bearings on the stiffness of the system[14–15]. With the aid of substructure synthesis in structural dynamics, this paper aims to present an analytical approach to formulate the governing equations of motion and to predict the distribution of lower natural frequencies of the A3 head throughout the entire workspace. The effects of component and joint compliances on the lower natural frequencies are also investigated using the sensitivity

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YZHANG Jun, et al: Dynamic Modeling and Eigenvalue Evaluation of a 3-DOF PKM Module

analysis in order to provide the designer with useful information in the stage of conceptual design.

2

Kinematic Modeling

2.1 Structural description Fig. 1 depicts the structure of the A3 head, which consists of a moving platform, a fixed base and three identical RPS limbs. Herein, R and S denote a revolute joint and a spherical joint, and the underlined P stands for an active prismatic joint. An electrical spindle is mounted on the moving platform to implement high-speed milling. Independently driven by 3 servomotors, one translation along z axis and two rotations about x and y axes can be achieved.

2.2 Coordinate systems Fig. 2 shows the schematic diagram of the A3 head, where Ai and Bi ( i = 1, 2,3 ) are the centers of spherical and revolute joint, respectively. ΔA1 A2 A3 and ΔB1 B2 B3 are assumed to be equilateral triangles. C3

Ball screw x

z

Block unit

Spindle Moving platform

Base Spherical joint Guideway Closed frame

Fig. 1.

Structure of the A3 head

As shown in Fig. 1, the mechanical features of the A3 head can be addressed in brief as follows. (1) The revolute joint is designed as a closed frame. Two opposing short half-shafts project from the frame, rigidly fixed to the inner rings of bearings that have outer rings mounted within block units attached to the base by bolts and pins. Internally, each side of the closed frame carries one element of a ball guideway; the other elements are mounted on each side of the limb body. The closed frame also carries the nut of the lead-screw assembly. (2) The limb body carries a servo-motor and lead-screw thrust-bearing at the rear end and a spherical bearing at the front end. The limb body is designed as a hollow rectangular structure with a dish on one side to accommodate the lead-screw. The cross-section dimensions of the limb body are set to keep the overall size and weight as small as possible while providing adequate bending rigidity against deflections caused by the constraint forces imposed at centre of the spherical joint along the direction of the axis of the revolute joint. (3) According to the equality criterion of rigidities, the geometry and component rigidities of the A3 head is optimized to achieve desirable rigidity and dynamic performance throughout the entire workspace. (4) The A3 head has only 60% overall weight of the Sprint Z3 head for the same spindle size and workspace.

x

z

p

v

w

y

B2

B

d3 s3

A2

a30

Limb body

rb

y

b3

A3

Servomotor

C2

B3

rp A

A1

Fig. 2.

y1

u

z1e

z1

y1e N1e

B1

C1

x1

x1e

Schematic diagram of the A3 head

To facilitate the formulation, the following Cartesian coordinate systems are placed: The reference coordinate system B − xyz that is attached at the center point B of uuuuuv the fixed base, with the x axis being parallel to B2 B3 and the z axis being normal to ΔB1 B2 B3 ; the body-fixed coordinate system A − uvw that is placed at the center point A of the moving platform, with the u axis being uuuuuv parallel to A2 A3 and w being normal to ΔA1 A2 A3 ; the limb reference frame Bi − xi yi zi that is established at the center point Bi of the ith revolute joint, with xi and zi being coincident with the axes of the revolute joints and the limb respectively; the element reference frame N ie − xie yie zie is set at the eth node and parallel to Bi − xi yi zi in the ith limb ( e = 1, 2,L , n + 1 ), where n is the total number of finite elements in each limb. For clarity, only one element reference frame in limb 1 is depicted in Fig. 2. The transformation matrix of the frame A − uvw with respect to the frame B − xyz can be formulated as ⎛ cψ cφ − sψ cθ sφ −cψ sφ − sψ cθ cφ sψ sθ ⎞ ⎜ ⎟ R = ⎜ sψ cφ + cψ cθ sφ −sψ sφ + cψ cθ cφ −cψ sθ ⎟ , ⎜ sθ sφ sθ cφ cθ ⎟⎠ ⎝

where “ s ”and “ c ” denote “sin()” and “cos()” functions, respectively; ψ , θ , φ are Euler angles in terms of precession, nutation and rotation. 2.3 Inverse position analysis As shown in Fig. 2, the position vector of point Ai measured in the B − xyz can be given as

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CHINESE JOURNAL OF MECHANICAL ENGINEERING Ai = ai + p = bi + di si ,

(1)

where p , bi represent the vectors of points A and Bi measured in B − xyz , respectively; di denotes the distance between Ai and Bi ; si is unit vector of the ith limb. And

plus the servomotor); and (c) the lumped spring of revolute joint (the closed frame plus the lead-screw assembly). Consequently, the entire limb can be modeled as a uniform spatial beam supported by two set of lumped springs as shown in Fig. 4.

Bi

Ai

di

fixed base respectively; βi = 2π(i − 1) / 3 − π/2 are the position angles of the joints; px , p y and pz are

Fig. 3.

ln

k r2zi

k r2xi

Bi

zi

Ai

xi

w2

k r2 yi

k r1yi

kszi

Ci

k r1zi

k r1xi

(2)

(3)

Assemblage of a RPS limb yi

coordinates of point A measured in B − xyz . Taking ψ , θ and pz as independent coordinates, and considering the constrains of revolute joint, we obtain the following:

⎧⎪di = ai + p − bi , ⎨ ⎪⎩ si = ( ai + p − bi ) / di .

Ci

L

where rp , rb are the radii of the moving platform and the

Eq. (2) gives the parasitic motions of the 3-RPS mechanism. Thus, the inverse position analysis can be conducted as follows:

Rear bearing Motor

Nut h

⎛ rp cos βi ⎞ ⎛ rb cos β i ⎞ ⎛ px ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ai = Rai 0 , ai 0 = ⎜ rp sin βi ⎟ , bi = ⎜ rb sin β i ⎟ , p = ⎜ p y ⎟ , ⎜ ⎜ ⎜p ⎟ 0 ⎟⎠ 0 ⎟⎠ ⎝ ⎝ ⎝ z⎠

⎧ px = rp (1 − c θ )s2ψ / 2, ⎪ ⎨ p y = rp (1 − c θ )c2ψ / 2, ⎪ φ = −ψ . ⎩

R joint

Lead-screw S joint Front bearing Limb body

h1

h2

ksxi

w1

ksyi

Fig. 4.

Spatial beam model of the limb body

Herewith, h1 , h2 , w1 and w2 are the dimensions of the cross section of the uniform beam; ksxi , ksyi and kszi are the stiffness coefficients of three virtual transverse linear springs of the spherical joint; and kr1xi , kr1yi , kr1zi , kr2xi , kr2 yi and kr2zi are stiffness coefficients of the

3

Dynamic Modeling

In the stage of conceptual design, the following hypotheses and approximations are made in the dynamic modeling of the A3 head. (1) The base and the moving platform are treated as rigid bodies. (2) The limb body is modelled as a hollowed spatial beam with constant rectangular cross-section. (3) The compliances of revolute and spherical joints are simplified as lumped springs with equivalent stiffness. (4) The coupling effect between rigid and elastic motions is negligible as the mechanism works at low or moderate speed. (5) Clearances, frictions and dampings in joint are neglected. 3.1 Dynamic equations of limbs Fig. 3 shows the assemblage of a RPS limb in the A3 head. For convenience, we classify the parts of the limb into 3 components: (a) the lumped spring of spherical joint; (b) the spatial beam of limb body assembly (the limb body

transverse and torsional springs of the revolute joints (plus the lead-screw assembly) in the ith limb, respectively. In the dynamic modeling using finite element method (FEM), the spatial beam can be meshed into elements with each node having three linear and three angular coordinates (along and about three axes)[16]. Fig. 5 shows the eth element

of e i

the e i

e e i i

ith

limb

frame N − x y z . Here,

e

in

element

reference

and e + 1 denote two

adjacent nodes of the element, ui ( i = 1 − 12 ) represents the nodal coordinates.

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YZHANG Jun, et al: Dynamic Modeling and Eigenvalue Evaluation of a 3-DOF PKM Module M iU&&i + K iU i = Fi ,

u3 u6 u5

e

wx

yie

u1 u9 u12

ux u8

u10

vx

θx

u11 u7

u4

where M i = Ti miTiT , K i = Ti kiTiT , U i = Ti ui , Fi = Ti fi , Ti = diag( Ri ,L , Ri ) . Here Ri is transformation matrix of Bi − xi yi zi with respect to B − xyz .

u2

x

3.2 Dynamic equations of moving platform The equations of motion of the moving platform can be formulated by using the free body diagram as shown in Fig. 6:

l

e +1

xie

N ie zie

To simplify the formulation, each limb body in the ith limb is discretized into n elements with Ai , Bi and Ci being one of the nodes of the elements. As a result, a set of equations of motion of the ith limb in the limb frame Bi − xi yi zi can be formulated with adequate boundary conditions: (4)

where mi and ki are mass and stiffness matrices, and ui and fi are the general coordinates vector and external load vector of the ith limb and can be expressed as follows: ⎛ ε Ai ⎜ ⎜ ξ Ai ⎜ ⎜ M ⎜εB i ui = ⎜ ⎜ ξ Bi ⎜ ⎜ M ⎜ε ⎜ Ci ⎜ ξC ⎝ i

3

3

i=1

i=1

m P ε&&P = −∑ FPLi + FP , I Pξ&&P = −∑ ri × FPLi + τ P , (7)

Fig. 5. Element of spatial beam

&&i + ki ui = fi , mi u

(6)

⎞ ⎛ f PLi ⎞ ⎟ ⎜ ⎟ ⎟ ⎜ 0 ⎟ ⎟ ⎜ M ⎟ ⎟ ⎜ ⎟ ⎟ ⎜ f BLi ⎟ ⎟ , fi = ⎜ ⎟, ⎟ ⎜ τ BLi ⎟ ⎟ ⎜ ⎟ ⎟ ⎜ M ⎟ ⎟ ⎜ 0 ⎟ ⎟ ⎜ ⎟ ⎟ ⎝ 0 ⎠ ⎠

where m P and I P are mass and inertial matrices of the moving platform measured in B − xyz ; ε P and ξ P are the linear and angular general coordinates of the moving platform; FPLi is the reaction force vector at the interface between the moving platform and the ith limb; ri is the vector pointing from the mass center of moving platform to the center of spherical joint. FP and τ P are external forces and moments acting on the moving platform, respectively. And there have I P = RI P 0 R T , FPLi = Ri f PLi ,

where I P 0 is the inertia of the moving platform measured in the body-fixed coordinate system A − uvw . τP FPL3

v

w A3

FP

FPL 2

A2

u

P

y A1

z

x

FPL1

where ε Ai , ξ Ai , ε Bi , ξ Bi , ε Ci and ξ Ci are linear and angular coordinates of nodes Ai , Bi and Ci in Bi − xi yi zi ; f PLi , f BLi and τ BLi are reaction forces and moments at Ai , Bi measured in Bi − xi yi zi , respectively. The nodal coordinates can be related to ui by

ε Ai = N cAi1 ui , ξ Ai = N cAi 2 ui , ε Bi = N cBi1 ui , ξ Bi = N cBi 2 ui (5),

Fig. 6.

Force diagram of moving platform

3.3 Deformation compatibility conditions As mentioned above, the moving platform is connected with the ith RPS limb by the spherical joint which can be simplified by three virtual transverse lump springs. Then, the displacement relationship between the platform and the limb can be generated as follows: ∇Ai

where N cA1 , N cB1 , N cA2 , N cB2 are connectivity matrices of i

i

i

i

nodes Ai , Bi with respect to ui in Bi − xi yi zi , respectively. Thus, the coordinate transformation can be made to express Eq. (4) in the reference coordinate system

Aim

ε Ai ksi

Moving platform

Fig. 7.

Αil Limb

Connection between moving platform and RPS limb

·5·

CHINESE JOURNAL OF MECHANICAL ENGINEERING As shown in Fig. 7, let Aim and Ail be the interface points associated with the moving platform and RPS limb, respectively. ∇Ai and ε Ai are displacements of Aim

equations of motion of the system % &&% + KU % % = F% , MU

(13)

and Ail measured in the limb reference frame Bi − xi yi zi . Thus, the deflections of the moving platform U P = (ε P , ξ P )T can be related to those of Ai of the RPS limbs by

% and K% are the global mass and stiffness where M matrices, U% and F% are the global general coordinates vector and external load vector, respectively. And we have

ΔAi = RiT D ri U P ,

% = diag( M , M , M , M ) , M = diag(m , I ) , M 1 2 3 4 4 P P

⎡1 0 0 0 ⎢ ri D = ⎢0 1 0 − z Ai ⎢ 1 y Ai ⎣⎢0

z Ai 0 − x Ai

(8) − y Ai ⎤ ⎥ x Ai ⎥ , ⎥ 0 ⎦⎥

⎛ U1 ⎞ ⎛ F1 ⎞ ⎜ ⎟ ⎜ ⎟ U F ⎛ε ⎞ ⎛F ⎞ U% = ⎜ 2 ⎟ U 4 = ⎜ P ⎟ , F% = ⎜ 2 ⎟ F4 = ⎜ P ⎟ , ⎜ U3 ⎟ ⎜ ⎟ ξ F ⎝ P⎠ ⎝τP ⎠ 3 ⎜ ⎟ ⎜ ⎟ ⎝U4 ⎠ ⎝ F4 ⎠

where x Ai , y Ai and z Ai are the coordinates of Ai measured in B − xyz . Therefore, the internal forces at the interface between the moving platform and the RPS limb can be expressed as

f PLi = −ksi (ε Ai − RiT D ri U P ) ,

⎛ K1,1 ⎜ % K =⎜ ⎜ ⎜⎜ ⎝ K 4,1

(9)

where ksi = diag(ksxi , ksyi , kszi ) is the stiffness matrix of

i

(10) K 4,4

Transforming ui measured in Bi − xi yi zi into B − xyz , leads to f PLi = −ksi ( N cA1 TiTU i − RiT D ri U P ) . i

i

( N cB2 )T Ri kr2 RiT N cB2 i i

Substituting Eq. (5) into Eq. (9), we have

i

K 4,2

K 3,3 K 4,3

K i ,i = ( N cA1 )T Ri ksi RiT N cA1 + ( N cB1 )T Ri kr1 RiT N cB1 +

spherical joint in corresponding directions measured in Bi − xi yi zi .

f PLi = − ksi ( N cA1 ui − RiT D ri U P ) .

K 2,2

K1,4 ⎞ ⎟ K 2,4 ⎟ , K 3,4 ⎟ ⎟ K 4,4 ⎟⎠

⎛ 3 T r ⎜ ∑ Ri ksi Ri D i 1 i = = ⎜⎜ ⎜ ⎜ ⎝

i

i

+ Ki ,

⎞ ⎟ ⎟, ⎟ 3 r ∑ ri × ( Ri ksi RiT D i ) ⎟⎟ i =1 ⎠

(11) ⎛ − Ri ksi RiT ⎞ K i ,4 = diag(− Ri ksi RiT D ri , 0) , K 4,i = ⎜ ⎟. ⎜ −r × R k RT ⎟ i si i ⎠ ⎝ i

Similarly, the internal forces and moments at the interface between the ith RPS limb and the base can be expressed as

where, M i and K i ( i = 1, 2,3 ) can be derived through the summation of standard finite element formulas.

f BLi = − kr1 N cB1 TiTU i , i

τ BLi = − kr2 N cBi 2 TiTU i ,

4 .

Eigenvalue Evaluation

(12)

kr1 = diag(kr1xi , kr1 yi , kr1zi ), kr2 = diag(kr2 xi , kr2 yi , kr2 zi )

Eqs. (11) and (12) are the deformation compatibility conditions of the system. 3.4 Governing equations of global system Substituting Eqs. (11) and (12) into Eq. (6) and then combining with Eq. (7), we finally obtain the governing

In this section, the eigenvalue evaluation for Eq. (13) is conducted to predict the distributions of lower natural frequencies of the A3 head throughout the entire workspace. Then, the sensitivity analysis is carried out to investigate the effect of the design parameters on the lower natural frequencies of the system. Let ω 2j , φj , η j ( j = 18n + 24 ) be, respectively, the system jth eigenvalue, eigenvector and modal mass associated with Eq. (13), we have

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YZHANG Jun, et al: Dynamic Modeling and Eigenvalue Evaluation of a 3-DOF PKM Module 1 2

4π η j

φjT K% φj .

(14)

Differentiating Eq. (14) with respect to a given design variable v leads to[17] ∂f j ∂v

=

1 8π2η j f j

(φjT

% ∂K% ∂M φj + φjT φj ) . ∂v ∂v

(15)

% M K% Because the matrices of and are configuration-related, numerical method is applied for solving the equations.

4.1 Example Eigenvalue analysis of a prototype of the A3 head shown in Fig. 1 is carried out to illustrate the effectiveness of the proposed method. The parameters for the computation are listed in Table. Herein, w1 , h1 , w2 and h2 can be referred to Fig. 4; I Pu 0 , I Pv 0 and I Pw0 can be calculated with Solidworks. Because the polar inertial moments of the platform are much smaller than the principal ones, they can be neglected here. And kSix , kSiy , kSiz ( i = 1, 2,3 ) are basic component

stiffness of spherical measured in the limb frame, krjx , krjy , krjz ( j = 1, 2 ) are constraint stiffness provided by

the revolute joints in related directions measured in the limb reference. These stiffness coefficients can be determined with FEA and linear interpolation method. Nomenclature of A3 head prototype

Nomenclature Radius of platform rp/mm Radius of base rb /mm Maximum distance between platform and base Hmax/mm Length of limb body L/mm Working stroke s/mm External width of limb body cross section w1/mm Internal width of limb body cross section w2/mm External height of limb body cross section h1/mm Internal height of limb body cross section h2/mm Maximum rotation angle of platform about x,y axes θ/(°) Mass of spherical (S) joint ms/kg Mass of revolute (R) joint mr/kg Mass of platform mp/kg Principal inertia of platform about u axis Ipu0/(kg•m2) Principal inertia of platform about v axis Ipv0 /(kg•m2) Principal inertia of platform about w axis Ipw0 /(kg•⋅ m2) Stiffness of short axis of S joint in x direction kS1x/(N•μm-1) Stiffness of short axis of S joint in y direction kS1y/(N•μm-1) Stiffness of short axis of S joint in z direction kS1z/(N•μm-1) Stiffness of long axis of S joint in x direction kS2x/(N•⋅ μm-1) Stiffness of long axis of S joint in y direction kS2y/(N•μm-1) Stiffness of long axis of S joint in z direction kS2z/(N•μm-1) Stiffness of cross axis of S joint in x direction kS3x/(N•μm-1) Stiffness of cross axis of S joint in y direction kS3y/(N•μm-1) Stiffness of cross axis of S joint in z direction kS3z/(N•μm-1) Stiffness of R joint in x direction kr1x/(N•μm-1) Stiffness of R joint in y direction kr1y/(N•μm-1)

Value 250 314 775 1150 200 220 200 120 100 40 14.61 92.91 157.4 3.56 3.76 3.74 31.4 33.3 436 1219 270 588 4 000 1 818 2 174 380 530

1 006 22.9 18.9 55

4.2 Simulation and discussions Fig. 8 shows the distribution of lower natural frequencies of the proposed prototype over the work plane of H max = 775 mm. From Fig. 8, it can be found that the first and the second orders of natural frequencies of the A3 head are axisymmetric over the given work plane. This is coincident with the axial symmetry of the structure of three RPS limbs in the parallel mechanism. The above figures also indicate that the distributions of lower natural frequencies of the A3 head are strongly configuration-related. For example, the first order natural frequency varies from the minimal value of 49.34 Hz to the maximal value of 52.01 Hz, and the second order natural frequency changes from 56.25 Hz to 62.90 Hz.

52.0 51.5 51.0 50.5 50.0 49.5 49.0 40

Ro

20

t a ti

on

40

ang

20

0

le θ

0

-20 y/ °

-20 -40

-40

R o ta

tio n a

ng

leθ x/°

(a) The first order natural frequency

Natural frequency f2 / Hz

Table

Stiffness of R joint in z direction kr1z/(N•μm-1) Stiffness of R joint about x direction kr2x/(N•μm-1) Stiffness of R joint about y direction kr2y/(N•μm-1) Stiffness of R joint about z direction kr2z/(N•μm-1)

Natural frequency f1/Hz

f j2 =

65

60

55 -40

Ro -20 t at io n

an 0 gl e 20 θy /° 40 -40

40 20

0 -20

R o ta

tio n

° e θ x/ ang l

(b) The second order natural frequency

Fig. 8.

Natural frequencies in ψ − θ − z work plane ( θ x = θ cosψ , θ y = θ sinψ )

Since the lower natural frequencies are related to the configuration of the parallel mechanism, designers may think about adjusting some design variables so that the A3

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CHINESE JOURNAL OF MECHANICAL ENGINEERING

Minimal freque nc y f1m in/H z

50.5

50

49.5

49

48.5

48

47.5 250 260

Fig. 9.

270

280 290

300 310 320

Radius of the base rb/mm

330 340

350

Variation of the minimum of the first order natural frequency versus the radius of the base

Obviously, f1min increases monotonically with the increment of rb . This can be interpreted as following: in the parallel mechanism of the A3 head, rP and H are related to rb due to kinematic and mechanical constrains[18]. Kinematic analysis reveals that increment of rb will result in a decrement in H at a given workspace. When H decreases, the rigidity of RPS limbs can be improved and then the 1st order natural frequency is increased. Nevertheless, the first order natural frequency is not very sensitive to the change of rb . The value of f1min only increases about 2 Hz when the radius of the base expands 1.4 times. Hence, we can conclude that increasing the dimension of the A3 head is not an effective way to improve system dynamic performance. Fig. 10 shows that the variation of the minimum of the first order natural frequency versus the joint stiffness of the A3 head.

Minimal freque nc y f1m in/H z

60 55

ks kR

50 45

50

45

k1x k1y k1z

40

35

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Relative stiffness coefficient λ

35

(a) Short axis

30

1

1.5

2

2.5

3

3.5

4

4.5

5

Relative stiffness coefficient λ

Fig. 10.

55

30 0.1 0.5

40

25 0.1 0.5

joints stiffness is denoted as a dimensionless factor λ , where λ = kSv / kS0 = kRv / kR0 . Here, kS0 , kR 0 are stiffness of spherical and revolute joints of the proposed prototype designed according to equality criterion of rigidities. Fig. 10 reveals that f1min is very sensitive to joints stiffness when λ < 1 . And when λ ≥ 1, f1min barely varies with joints stiffness. This means that the proposed prototype of the A3 head can achieve a compact, lightweight yet rigid structure with good dynamic performance. The variation of f1min also implies that the joints stiffness must be delicately controlled when making such a PKM module. For example, the stiffness of real joints is very likely to be much lower than that of ideal ones if clearances occur in joints. On such a circumstance, it can be predicted that the natural frequencies of the system may decrease dramatically from the theoretical values. Since joints stiffness has significant effects on the dynamic performance of the system, designers may feel obliged to concern the rigidity of each component of the joints. Fig. 11 shows the effects of compliances of three components of spherical joint on the minimal first order natural frequency. It can be observed that f1min varies with the stiffness of short axis when λ < 1 . Though the variations of f1min versus the stiffness of long and cross axes are also noticeable when λ < 1 , their variation amplitudes are much smaller than that of short axis. Thus, we can conclude that the compliance of short axis in spherical joint is a crucial concern at the stage of conceptual design of the A3 head. The effects of compliances of components of revolute joint on the lower natural frequencies can be investigated in a similar manner.

Minimal freque nc y f1m in/H z

head has desirable dynamic performance. To achieve this goal, a sensitivity analysis is needed. To simplify the analysis, only eigenvalue sensitivity of the minimum of the first order natural frequency f1min is conducted here. Fig. 9 shows the variation of f1min at the changes of rb .

Variation of the minimum of the first order natural frequency versus joint stiffness

In Fig. 10, the solid curve represents the variation of f1min versus the stiffness of spherical joint while the dashed one denotes that of revolute joint. The variation of

4.5

5.0

·8·

YZHANG Jun, et al: Dynamic Modeling and Eigenvalue Evaluation of a 3-DOF PKM Module Minimal freque nc y f1m in/H z

49.7 k2x

49.6

k2y

49.5

k2z

49.4 49.3 49.2 49.1 49.0

0.1 0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Relative stiffness coefficient λ

(b) Long axis

Minimal freque nc y f1m in/H z

49.45 49.40 49.35 49.30 k3x

49.25

k3y

49.20

k3z

49.15 49.10 49.05

49.00 48.95 0.1 0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Relative stiffness coefficient λ

4.5

5.0

(c) Cross axis

Fig. 11.

Variation of the minimum of the first order natural

frequency versus component stiffness of spherical joint

5

Conclusions

(1) This paper presents a semi-analytical approach to formulate the governing equations of motion for the A3 head by using substructure synthesis. With the proposed dynamic model, the distributions of lower natural frequencies of the A3 head throughout the entire workspace can be predicted in a quick manner. (2) The distribution of lower frequencies of the A3 head indicates that the dynamic characteristics of the system are strongly configuration-related and axially symmetric due to its kinematic and structural features. (3) The eigenvalue sensitivity analysis shows that the lower natural frequencies of the A3 head are slightly affected by the dimensional parameters but significantly influenced by the joints stiffness. Therefore, great effort should be made to ensure the joint rigidities. (4) Further investigations associated with experimental validations and modifications will be carried out and reported in separate articles. References [1] ZHANG Dan, GOSSELIN C M. Kinetostaic analysis and design optimization of the tricept machine tool family[J]. Journal of Manufacturing Science and Engineering, 2002, 124(8): 725–733. [2] NEUMANN K E. Tricept application[C]//Proceedings-3rd Chemnitz Parallel Kinematics Seminar, Verlag Wissenschaftliche Scripten, Zwickau, 2002: 547–551.

[3] CACCAVALE F, SICILIANO B, VILLANI L. The Tricept robot: dynamics and impedance control[J]. IEEE/ASME Transactions on Mechatronics, 2003, 8(2): 263–268. [4] HENNES N, STAIMER D. Application of PKM in aerospace manufacturing-high performance machining centers ECOSPEED, ECOSPEED-F and ECOLINER[C]//Proceedings-4th Chemnitz Parallel Kinematics Seminar, Verlag Wissenschaftliche Scripten, Zwickau, 2004: 557–577. [5] HUANG Tian, LIU Haitao. A parallel manipulator with two orientations and one translation[P]. WO/PCT/2007/124637, 2007. [6] FANG Yuefa, HUANG Zhen. Elastodynamic modeling for a 6-DOF parallel robot[J]. Mechanical Science and Technology, 1990, (1): 62–69. (in Chinese) [7] LEE J D, GENG Z. A dynamic model of a flexible steward platform[J]. Computer and Structures, 1993, 48(3): 367–374. [8] FATTAH A, ANGELES J, MISRA A K. Dynamics of a 3-DOF spatial parallel manipulator with flexible links[C]// Proceedings of the IEEE International Conference on Robotics and Automation, Nagoya, Japan, 1995: 627–632. [9] KANG B, MILLS J K. Dynamic modeling of structurally flexible planar parallel manipulator[J]. Robotics, 2002, 20(3): 329–339. [10] WANG X Y, MILLS J K. Dynamic modeling of a flexible-link planar parallel platform using a substructuring approach[J]. Mechanism and Machine Theory, 2006, 41(6): 671–687. [11] PIRAS G, CLEGHON W L, MILLS J K. Dynamic finite-element analysis of a planar high-speed, high-precision parallel manipulator with flexible links[J]. Mechanism and Machine Theory, 2005, 40(7): 849–862. [12] DU Zhaocai, YU Yueqing, ZHANG Xuping. Dynamic modeling of planar flexible parallel manipulators[J]. Chinese Journal of Mechanical Engineering, 2007, 43(9): 96–101. (in Chinese) [13] LI Bing, YANG Xiaojun, HU ying, et al. Dynamic modeling and design for the parallel mechanism of a hybrid type parallel kinematic machine[J]. Journal of Advanced Mechanical Design, System, and Manufacturing, 2007, 1(4): 481–492. [14] ZHOU Z L, JEFF X, CHRIS K M. Modeling of a Fully Flexible 3PRS Manipulator for Vibration Analysis[J]. ASME Journal of Mechanical Design, 2006, 128(2): 403–412. [15] SHIAU T N, TSAI Y J, TSAI M S. Nonlinear dynamic analysis of a parallel mechanism with consideration of joint effects[J]. Mechanism and Machine Theory, 2008, 43(4): 491–505. [16] DAVID H. Fundamentals of Finite Element Analysis[M]. New York: McGraw-Hill, 2004. [17] CHEN Suhuan. Matrix perturbation theory in structural dynamic design[M]. Beijing: Science Press, 2007. [18] LI Yonggang, HUANG Tian, LIU Haitao. Design of a 3-DOF PKM Module for Large Aircraft Component Machining[C]//Proceedings of the 7th World Congress on Intelligent Control and Automation, Chongqing, China, 2008: 370–375.

Biographical notes ZHANG Jun, born in 1981, is currently a lecturer in School of Mechanical Engineering, Tianjin University, China. He received his PhD degree from Tianjin University in 2007. His research interests include machinery dynamics and mechanical transmission. Tel: +86-22-27406263; E-mail: [email protected] LI Yonggang, born in 1975, is currently a post doctor in School of Mechanical Engineering, Tianjin University, China. He received his PhD degree from Tianjin University in 2007. His research interests include mechanisms and robotics. E-mail: [email protected] HUANG Tian, born in 1953, is currently a professor and a PhD candidate supervisor in School of Mechanical Engineering,

CHINESE JOURNAL OF MECHANICAL ENGINEERING Tianjin University, China. His main research interests include mechanisms and robotics. E-mail: [email protected]; [email protected]

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Tel: +86-571-87953096; E-mail: [email protected]