Vibration Modeling of Parallel Kinematic Mechanisms (PKMS) with

VIBRATION MODELING OF PARALLEL KINEMATIC MECHANISMS (PKMS) WITH FLEXIBLE LINKS: ADMISSIBLE SHAPE FUNCTIONS Masih Mahmoodi, James K. Mills and Beno Benhabib Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, Ontario, Canada E-mail: [email protected] Received April 2014, Accepted August 2014 No. 14-CSME-50, E.I.C. Accession 3712

ABSTRACT The accuracy of various admissible shape functions, for vibration modeling of flexible links of Parallel Kinematic Mechanisms (PKMs), is investigated as a function of the ratio of the mass of the moving platform to the mass of the link. Knowing that the commonly used shape functions based on “pinned”, “fixed”, or “free” boundary conditions do not incorporate the moving platform mass, "pinned-mass" and “fixed-mass” shape functions are presented herein, and are compared with finite-element based results for various mass ratios. The closest shape functions to the finite-element results are, then, utilized and compared with other shape functions in the subsequent vibration modeling to predict the tooltip response. Keywords: parallel kinematic mechanism, natural frequency, admissible shape function, flexible links, meso-milling, machine tool.

MODÈLISATION DE LA VIBRATION D’UN MÉCANISME CINÉMATIQUE PARALLÈLE À ARTICULATIONS FLEXIBLES: FORME DE FONCTIONS ADMISSIBLES RÉSUMÉ L’exactitude des différentes formes de fonctions admissibles pour la modélisation de la vibration des articulations flexibles d’un mécanisme cinématique parallèle est investiguée comme une opération du ratio de la masse de la plateforme en mouvement par rapport à la masse de l’articulation. En sachant que la forme de la fonction utilisée couramment est basée sur des conditions aux limites “rotulées”, “encastrées” ou “libres” n’incorporent pas la masse de la plateforme en mouvement, ces formes de fonction sont présentées dans cet article. Elles sont comparées avec les résultats basés sur des éléments finis pour différents ratios de masse. Les formes de fonction qui s’approchent le plus des résultats des éléments finis sont alors utilisées et comparées avec d’autres formes de fonctions dans la modélisation des vibrations suivante pour prédire la réponse en info-bulle. Mots-clés : mécanismes cinématique parallèle; fréquence naturelle; forme de fonction admissible; articulations flexibles; méso-cylindre de fraisage; machine-outil.

Transactions of the Canadian Society for Mechanical Engineering, Vol. 39, No. 1, 2015

97

1. INTRODUCTION Demand for high-accuracy, high-speed, and high-acceleration motion with efficient power consumption requires contemporary Parallel Kinematic Mechanisms (PKMs) to utilize lightweight moving links, yielding flexible structures [1]. The structural vibration of flexible elements of PKMs, caused by external forces and/or inertial forces, decreases the positional accuracy of the PKM. For example, for PKM-based machine tools, structural vibrations could have a significant undesirable effect when the cutting force frequency is close to the natural frequencies of the machine-tool structure [2, 3]. Therefore, accurate modeling of the structural vibration behavior of such PKMs is necessary. Numerous methods, such as lumped-parameter modeling [4, 5], Finite-Element Method [6–10], Component-Mode Synthesis [11], and Kane’s method [12] have been reported to model the structural vibration of PKMs. The lumped-parameter-modeling approach has limited accuracy, and the Finite-Element Method is usually computationally expensive. Analytical methods based on continuous-system formulation of the flexible-link PKMs can provide accurate and time efficient tools for prediction of the structural vibration response. Yet, analytical determination of the exact mode shapes and natural frequencies requires the solution of the frequency equation, which may be complex in the case of multi-link mechanisms [13]. This complexity arises from the existence of non-homogeneous natural (or dynamic) boundary conditions that must satisfy the shear force/bending moment of PKM links at the end joints. These shear force and bending moments are dependent on the mass/inertia properties of the adjacent structural components. Hence, the frequency equation, mode shapes, and natural frequencies, in general, are dependent on the relative mass/inertia properties of the flexible intermediate links of the PKM and their adjacent structural components [14]. In order to avoid the complexities of solution of the exact frequency equation for PKMs with flexible links, admissible shape functions based on “pinned”, “fixed”, or “free” boundary conditions have been widely used along with the Assumed Mode Methods in the literature to approximate the natural frequencies and mode shapes of flexible link mechanisms. Most of the works have been applied to serial single-link or two-link manipulators [15]. The use of pre-selected admissible shape functions does not take into account the effects of the inertia of adjacent structural components on the natural frequencies and mode shapes of the PKM links. Thus, a crucial issue is to determine how well a set of admissible functions can approximate the realistic behavior of the flexible links in the context of a full PKM structure considering the ratio of the mass of the links to the mass of the moving platform and spindle [16]. In particular, the accuracy of using admissible shape functions has been investigated for single-link and two-link manipulators in [17, 18] with respect to the exact (or unconstrained mode) solution for a range of beam-to-hub and beam-to-payload ratios. Yet, no work has been reported so far to examine the accuracy of the use of admissible shape functions for flexible intermediate links of PKMs for a given range of moving platform and spindle mass to link mass ratios. The contribution of this paper is a methodology to obtain a set of admissible shape functions that accurately represent the realistic vibration behavior of flexible intermediate links of PKMs for a range of ratios of the effective mass of the moving platform and spindle to the mass of the link which is defined, herein, as the mass ratio. Furthermore, to account for the effects of the mass of the moving platform and spindle on the natural frequencies and mode shapes of the PKM links, we consider the use of shape functions associated with the “pinned-mass” and “fixed-mass” boundary conditions. The natural frequencies obtained from these shape functions and the classical admissible shape functions are compared with the modal analysis results, which are calculated from the Finite Element Analysis (FEA) for a range of mass ratios. Depending on the mass ratio, the closest shape functions to the FEA results are, then, selected to be used as candidates in the structural dynamic-modeling methodology proposed in this paper. The structural dynamic response of the PKM at the tooltip, utilizing the selected shape functions, is compared to those of the PKM using pinnedmass and fixed-mass shape functions to examine the role of various shape functions in accurate prediction 98

Transactions of the Canadian Society for Mechanical Engineering, Vol. 39, No. 1, 2015

of the structural dynamic response at the tooltip. The results are applicable to any PKM architecture with intermediate links connected through revolute and/or spherical joints. The above methodology is applied to a 3-PPRS PKM-based meso-Milling Machine Tool (mMT) as an example. 2. DYNAMICS OF THE PKM WITH ELASTIC LINKS A general PKM consists of a fixed-base platform and a moving platform (see Fig. 1). A number of actuators are mounted on the base platform and are connected to the moving platform through intermediate links. An end-effector/tool or spindle is mounted on the moving platform. The intermediate links may exhibit unwanted vibrations and, hence, yield a “flexible” PKM. Herein, the extended Hamilton’s principle utilizing the Euler–Bernoulli beam assumption is used to systematically generate the dynamic equations of the flexible links and boundary conditions. 2.1. Modeling of Elastic Linkages The extended Hamilton’s principle for the elastic linkages of PKMs is given by Z t2 t1

(δ TP − δUP + δWPext )dt = 0,

(1)

where δ TP , δUP and δWPext denote the variations of the total kinetic energy, total potential energy, and the virtual external forces applied on the elastic linkages, respectively. 2.1.1. Kinetic Energy In order to derive the kinetic energy of the elastic links, we assume that they are detached from the moving platform. The resulting mechanism is a set of n serial sub-chains plus the moving platform and spindle. The dynamics of the n serial sub-chains is obtained and superimposed on the dynamics of the moving platform and spindle. The dynamics of the entire PKM structure is, then, obtained via the superimposed dynamics of the PKM structural components, and considering the PKM kinematic constraints. Let us define qia (t) and qip (t) as the joint-space position vectors of the actuated joints, and passive joints of the ith sub-chain of a general PKM, respectively (Fig. 1). Also, let us define wi (xi ,t) = [win−1 wout−i ]T as the local vector of the two elastic lateral displacements of the ith flexible links at an arbitrary point xi and time t, where win−i and wout−i are the in-plane and out-of-plane components of the lateral elastic displacements of the of the ith link, respectively. The absolute Cartesian position of an arbitrary point xi along the ith elastic link of a general PKM at time t is given by ri (qia , qip , wi ). The total kinetic energy of the elastic link is, then, given by Z 1 n L TP = ∑ ρ(˙ri , r˙ i )dx, (2) 2 i=1 0 where ρ and L are the mass per unit length and the total length of each flexible link, respectively. 2.1.2. Potential Energy The total potential energy of the elastic links is given by n

UP = ∑

i=1

Z L 0

 EIin

∂ 2 win−i (xi ,t) ∂ xi2

2

Z L

dxi +

0

 EIout

∂ 2 wout−i (xi ,t) ∂ xi2

2

Z L

dxi +

0

! ρgriz dxi ,

(3)

where Iin and Iout are the area moments of inertia of the links with respect to axes normal to in-plane and out-of-plane surfaces, and E is the Young’s modulus of the linkage. Also, riz is the vertical component of the position vector ri . Transactions of the Canadian Society for Mechanical Engineering, Vol. 39, No. 1, 2015

99

Fig. 1. Schematic of a general PKM with kinematic notations.

2.1.3. Virtual Work of External Forces The total virtual work done by external forces on the elastic links is given as n

δWPext = ∑

i=1

i (t)δ r x (0,t) + Pi (t)δ r y (0,t) + Pi (t)δ r z (0,t) P1x i i i 1y 1z i (t)δ r x (L,t) + Pi (t)δ r y (L,t) + Pi (t)δ r z (L,t) +P2x i i i 2z 2y

! ,

(4)

i i ]T and Pi = [Pi i i T where Pi1 = [P1x P1y P1z 1 2x P2y P2z ] are the two reaction forces acting on the two end joints of the ith elastic link (i.e., xi = 0 and xi = L), respectively, and δ rix , δ riy , and δ riz are the variations of the Cartesian components of vector, r, at the boundaries. Without loss of generality, we assume that the links are connected to revolute joints at xi =, and spherical joints at xi = L, respectively. Assume that win−i is measured in the same plane as the revolute joint angle is measured.

2.1.4. Boundary Conditions Evaluating the variations of Eqs. (2) and (3), and substituting this result, along with Eq. (4), into the extended Hamilton’s principle, yields a set of equations of motion that represents the motion of active joints, qia , passive joints, qip , and elastic vibration of the links, wi of the ith sub-chain. From the extended Hamilton’s principle, the boundary conditions for in-plane vibration of the links, win−i , at xi = 0 (i.e. revolute joint) are obtained as xi = 0 : win−i (xi ,t) = 0, (5a) xi = 0 :

Min−i (xi ,t) = EIin

∂ 2 win−i (xi ,t) = 0, ∂ xi2

(5b)

∂ 2 win−i (xi ,t) = 0, ∂ xi2

(6a)

and, at xi = L, (i.e., spherical joint) as follows: xi = L : 100

Min−i (xi ,t) = EIin

Transactions of the Canadian Society for Mechanical Engineering, Vol. 39, No. 1, 2015

xi = L :

Vin−i (xi ,t) = EIin

∂ 3 win−i (xi ,t) i i i = fin (P2x , P2y , P2z , qia , qip ). ∂ xi3

(6b)

Similarly, the boundary conditions for the out-of-plane vibration of the links, wout−i , at xi = 0, are obtained as xi = 0 : xi = 0 :

wout−i (xi ,t) = 0,

(7a)

∂ wout−i (xi ,t) = 0, ∂ xi

(7b)

and, at xi = L, as follows: xi = L :

xi = L :

∂ 2 wout−i (xi ,t) = 0, ∂ xi2

(8a)

∂ 3 wout−i (xi ,t) i i i = fout (P2x , P2y , P2z , qia , qip ), ∂ xi3

(8b)

Mout−i (xi ,t) = EIout

Vout−i (xi ,t) = EIout

where Min−i and Mout−i are the in-plane and out-of-plane components of the bending moment and Vin−i and Vout−i are the in-plane and out-of-plane components of the shear force, respectively. fin (·) and fout (·) are functions of the reaction forces at spherical joints of the ith chain for in-plane and out-of-plane, respectively. Since the Cartesian components of the reaction force vector, Pi2 , in fin (·) and fout (·) vary as a function of the mass of the moving platform and spindle, the realistic boundary conditions and the resulting mode shapes and natural frequencies of the PKM links are dependent on the mass of the moving platform and spindle. To complete the structural dynamic modeling methodology, we assume that there exist admissible shape functions φ (xi ) and ψ(xi ), that can approximate realistic in-plane and out-of-plane mode shapes of the ith PKM link, respectively. These admissible functions, although unknown at the moment, can be used in the Assumed Mode Method to express in-plane and out-of-plane elastic displacements of the ith link. Note that the accuracy of using various admissible shape functions in the context of the full PKM structure will be investigated after the procedure for structural dynamic modeling is complete. The admissible shape function, using the Assumed Mode Method can be expressed by the following: p

win−i (xi ,t) =

(m)

∑ φi

(m)

(xi )qi (t),

(9a)

m=1 p

wout−i (xi ,t) =

(m)

∑ ψi

(m)

(xi )qi (t),

(9b)

m=1 (m)

where qi (t) is the mth modal coordinate of the ith link. Assuming a p-mode truncation for the ith link, the vector of modal coordinates is as follows qif = [q1i q2i . . . q pi ]T ,

(10)

Considering the vector of modal coordinates q f = [q1f q2f . . . qnf ]T of the n sub-chains of the PKM, in conjunction with the rigid-body motion coordinates of the entire n sub-chains of the PKM, qr = [q1a . . . qna q1p . . . qnp ]T , the complete set of generalized coordinates of the PKM structure is given by q = [qr q f ]T . Details on substitution of Eqs. (9a) and (9b) into the variational dynamic model (Eq. 1) can be found in [19], which leads to the development of matrix expressions for the dynamics of the flexible links of the PKM. Transactions of the Canadian Society for Mechanical Engineering, Vol. 39, No. 1, 2015

101

2.2. Dynamics of PKM Actuators, Moving Platform, and Spindle In order to obtain the dynamics of the moving platform, spindle, and the actuators, the vector X(t) is defined to represent the Cartesian task-space position and orientation (pose) of the platform and spindle center of mass, with respect to an inertial frame {O} (Fig. 1). The kinetic and potential energies of the platform, spindle, and actuators are obtained with respect to task space coordinates and substituted into the Lagrange’s equations to derive the dynamic matrix expressions for the actuators, moving platform, and spindle [20]. 2.3. System Dynamic Modeling of the Overall PKM In order to derive the dynamics of the entire PKM, the matrix expressions of the dynamic equations for the flexible links is superimposed with the corresponding matrix expressions of dynamics of actuators, moving platform, and spindle. The kinematic constraint equations associated with the PKM closed-loop chains are given as Hi = 0, i = 1, . . . , l , (11) where l is the number of the closed kinematic chains. The dynamics of the PKM with n elastic links is given as   ∂H T M(qT )q¨ T + C(qT , q˙ T )q˙ T + KqT + q(qT ) = F + λ, (12) ∂ qT where qT = [qa q p X q f ]T , H = [H1 . . . Hl ]T , and λ = [λ1 . . . λ3l ]T is the vector of Lagrange multipliers. Eq. (12) can be partitioned with respect to the vector of active rigid/modal coordinates, [qa q f ]T , and the vector of passive/task space coordinates, [q p X]T . The dynamic equation for the active coordinates is given by Ma (qT )q¨T + Ka qT = Fic−a + Fext−a + Ja [λ1 . . . λ3l ]T , (13) and for the passive coordinates by M p (qT )q¨ T = Fic−p + Fext−p + J p [λ1 . . . λ3l ]T ,

(14)

Above, Fic−a = −Ca (qT , q˙ T )q˙ − ga (qT ), and Fic−p = −C p (qT , q˙ T )q˙ T − g p (qT ). Also, Ja , and J p represent the derivatives of the kinematic constraints with respect to active, and passive joints, respectively. The vector Fext−a = [Fqa Fq f ]T represents the external actuator forces, and the vector Fext−p = Fq p represents all external forces other than the actuator forces. Elimination of Lagrange multipliers from Eqs. (13) and (14) results in the following dynamic equation: ¨ T + Ka qT = Fext−a + Fic−a − Ja J−1 (Ma − Ja J−1 p M p )q p (Fext−p + Fic−p ).

(15)

An expression for the passive coordinates, in terms of active independent coordinates, can now be obtained via the kinematic analysis of the PKM [21] q˙ p = Jφ q˙ a ,

(16a)

˙ = Jq˙ a , X

(16b)

where Jφ and J are the Jacobian matrices. Evaluating the time derivative of Eqs. (16a) and (16b), the acceleration vector of dependent coordinates can be expressed in terms of independent coordinates as       q¨ a I1 0 0    q¨ p   Jφ 0  q¨ a  J˙ φ q˙ a  =   q¨ T =  + (17)  X  J˙ q˙ a  . ¨   J 0  vq ¨ f q¨ f 0 I2 0 102

Transactions of the Canadian Society for Mechanical Engineering, Vol. 39, No. 1, 2015

Substituting Eq. (17) into Eq. (15), the PKM equations of motion with independent coordinates, in closed form is given as     q¨ a qa Ms + Ks + Fs = Fext−a , (18) q¨ f qf where



I1   Jφ Ms = (Ma − Ja J−1 p Mp)  J 0

 0 0  , 0  I2

(19)

and Ks = Ka , and

(20) 

 0  J˙ φ q˙ a  −1  . Fs = −Fic−a + Ja J−1 p (Fic−p + Fext−p ) + (Ma − Ja J p M p )  ˙ ˙ Jqa  0

(21)

Equation (18) represents the explicit closed-form structural dynamics of a general PKM in terms of active joint and modal coordinates. 3. ADMISSIBLE SHAPE FUNCTIONS In order to avoid the complexities associated with solving the exact frequency equation for the entire PKM with flexible links, classical admissible shape functions that merely satisfy the geometrical boundary conditions (i.e., Eqs. 5a, 7a, and 7b) and not necessarily the dynamic boundary conditions (i.e., Eqs. 5b, 6a, 6b, 8a, and 8b), may be used. The classical admissible functions to be considered herein are “pinnedfree”, “pinned-pinned”, and “pinned-fixed” for in-plane and “fixed-free”, “fixed-pinned” and “fixed-fixed” for out-of-plane. The use of classical admissible shape functions as mentioned above leads to a frequency equation that is independent of the platform and spindle mass which might result in inaccurate mode shapes and natural frequencies. Thus, to incorporate the platform and spindle mass dependency on the natural frequencies and mode shapes, while avoiding the complexities of solving the exact frequency equations, we propose to consider pinned-mass and fixed-mass shape functions for in-plane and out-of-plane motions, respectively. Further, we also check the accuracy of these shape functions for various ratios of the moving platform and spindle mass to link mass. Also, we assume that the mass attached to each flexible link is equal to the total mass of the moving platform and spindle divided by the number of the PKM links, i.e. n, that is, we divide the platform/spindle into n equal mass segments. We assume that the shape functions for in-plane and out-of-plane motions can be expressed as φi (xi ) = Ain sin(βin x) + Bin cos(βin x) +Cin sinh(βin x) + Din cosh(βin x),

(22a)

ψi (xi ) = Aout sin(βout x) + Bout cos(βout x) +Cout sinh(βout x) + Dout cosh(βout x),

(22b)

and respectively, where βin and βout are the eigenvalue solutions associated with the in-plane and out-of-plane natural frequencies of the link, ωin , and ωout , as r EIin L 2 , (23a) ωin = βin m Transactions of the Canadian Society for Mechanical Engineering, Vol. 39, No. 1, 2015

103

r

EIout L , (23b) m where m is the mass of the link. Assuming harmonic motion and applying the boundary conditions on the platform end joint (i.e. where the platform and spindle masses are assumed to be attached to the link) for the pinned-mass shape function lead to the following frequency equation from which natural frequencies and mode shapes are calculated  0 1 ···  0 −1 ···  det   − cos(βin L) + rβin L sin(βin L) sin(βin L) + rβin L cos(βin L) ···  ωout =

− sin(βin L) −

Jo−in βin3 L m

cos(βin L) − cos(βin L) +

0

1

0

1

cosh(βin L) + rβin L sinh(βin L) sinh(βin L) −

2 βout

Jo−in βin3 L m

− sin(βout L) −

m

cosh(βin L) cosh(βin L) −

cos(βout L) +

3 L Jo−out βout

Jo−in βin3 L m

m

sinh(βin L)

1

···

0

···

···

sin(βout L) 1

1

0

3

(24a)

sin(βout L) + rβout L cos(βout L) · · ·

0 cosh(βout L) + rβout L sinh(βout L)

sin(βin L) · · · 

   = 0. sinh(βin L) + rβin L cosh(βin L)  

Similarly, for the “fixed-mass” shape function, we get  0  1  det   − cos(βout L) + rβout L sin(βout L)  3 L Jo−out βout

Jo−in βin3 L m



   = 0. sinh(βout L) + rβout L cosh(βout L)  

(24b)

3

sinh(βout L) − Jo−outmβout L cosh(βout L) cosh(βout L) − Jo−outmβout L sinh(βout L)

In the above, r = M/m is the mass ratio, where M = (Mpf + Msp )/n, with (Mpf + Msp ) being the total mass of the moving platform and spindle. Jo−in and Jo−out are the in-plane and out-of-plane components of the mass moment of inertia of the effective portion of the platform and spindle. The solution of Eqs. (24a) and (24b) is obtained numerically for different values of the mass ratio. 4. NUMERICAL SIMULATIONS Numerical simulations are performed to examine the accuracy of the proposed pinned-mass and fixed-mass admissible shape functions along with the classical shape functions for the flexible links of the PKM for a range of mass ratios. Once the most accurate set of shape functions have been obtained for a given mass ratio, they are used in the dynamic model of the PKM to predict the structural vibration response at the tooltip as shown in Fig. 1. Numerical simulations which model a 3-PPRS PKM-based meso-Milling Machine Tool (mMT), developed in our laboratory as an example architecture, are carried out (Fig. 2). 4.1. Architecture of the PKM-Based mMT As noted in Fig. 2, the PKM-based mMT consists of a circular base platform of radius, Rb , on which three circular prismatic joints, θi (i = 1, 2, 3), are mounted at points XAi . Three vertical columns are mounted to 104

Transactions of the Canadian Society for Mechanical Engineering, Vol. 39, No. 1, 2015

(a)

(b) Fig. 2. Example PKM-based mMT (a) mechanical structure, and (b) schematic.

(a)

(b)

Fig. 3. Elastic displacement component of the linkage for (a) in-plane, and (b) out-of-plane components.

the circular prismatic joints. The vertical prismatic joints, di (i = 1, 2, 3) are situated on these three columns, respectively, at points XCi . The moving platform is connected to the three columns through three flexible linkages of length L. The linkages are connected to the three columns through revolute joints ϕi . These linkages are connected to the moving platform through spherical joints at points XBi . The prismatic joints, θi , and di , are actuated joints and the revolute joints, ϕi , as well as the spherical joint at XBi , are passive joints. The moving platform is approximated with a cylindrical disk having a radius of Rpf , and the length between the center of the moving platform and the tooltip is denoted as Ltool . A stationary coordinate reference frame, {O}, is defined at the centre of the circular base platform of the system. A moving reference frame, {E}, is defined at the tooltip. The in-plane displacement component of the ith elastic linkage is defined as shown in Fig. 3(a), as the lateral displacement of the linkage in the plane formed by the linkage and the vertical column attached to it, denoted by win−i (xi ,t). The out-of-plane component is normal to the in-plane displacement and is given by wout−i (xi ,t) (Fig. 3b). Figure 4 shows the reaction forces at the spherical joints of the moving platform applied to one of the linkages of the mMT. Transactions of the Canadian Society for Mechanical Engineering, Vol. 39, No. 1, 2015

105

Fig. 4. Reaction forces at the spherical joints of the moving platform.

Table 1. Dimensions/physical parameters of structural components. Dimension/physical parameter Value Linkage inner diameter 0.016 m Linkage outer diameter 0.012 m Length of the linkage 0.230 m Radius of fixed-base 0.15 m Radius of moving platform 0.0225 m Thickness of moving platform 0.0225 m Tool length 0.015 m Elastic modulus of all components 205 GPa Density of all components 7850 Kg/m3 Circular actuators mass each 0.328 Kg Vertical actuators joint/housing mass each 0.545 Kg Vertical columns mass each 0.976 Kg Moving platform and spindle mass 0.158 Kg

The non-homogeneous boundary conditions of the PKM ith linkage for in-plane motion are obtained as xi = L :

EI

∂ 3 win−i (xi ,t) i i i = P2x cos θi sin ϕi − P2z cos ϕi + P2y sin ϕi cos θi , ∂ xi3

(25)

and for out-of-plane motion, the non-homogeneous boundary conditions are given as xi = L :

EI

∂ 3 wout−i (xi ,t) i i = P2y cos θi − P2x sin θi . ∂ xi3

(26)

Equations (25) and (26) contain the reaction forces that are dependent on the mass of the platform and spindle, as well as the joint space configuration of the PKM. The natural frequencies associated with each admissible function are obtained using the dimensions and physical parameters of the structural components given in Table 1. 106

Transactions of the Canadian Society for Mechanical Engineering, Vol. 39, No. 1, 2015

(a)

(b) Fig. 5. (a) Out-of-plane natural frequencies of the PKM links for the first mode. (b) Out-of-plane natural frequencies of the PKM links for the second mode.

4.2. The Accuracy of Admissible Shape Functions as a Function of Mass Ratio The focus of the simulations presented herein is to examine the accuracy of the proposed admissible shape functions to approximate the PKM link mode shapes for a wide range of platform and spindle mass to link mass ratios. For each mass ratio, the eigenvalue problem associated with in-plane and out-of-plane motion is solved for each shape function, and the natural frequencies for the first two vibration modes of the link along each direction are calculated. The natural frequencies associated with each shape function are then compared with the modal analysis results obtained from the FEA software package, ANSYS, with an aim to compare the accuracy of each admissible shape function for a given mass ratio. Figure 5(a) shows the values of the natural frequencies of the first out-of-plane mode obtained from the first mode of fixed-mass and the first mode of fixed-free shape functions compared with FEA versus mass ratio. It is noted that the natural frequencies obtained from fixed-mass shape function yields close results to those obtained from the fixed-free shape function when the mass ratio is very small (i.e., r ≤ 1/30). This result is expected, as the links behave dynamically close to the free boundary condition at the distal joint when the platform and spindle mass is small compared to that of the link. For r ≤ 1/30, both fixed-free and fixed-mass shape functions predict the realistic mode shape with an error of 15.6% compared with the result obtained with FEA. However, as the mass ratio increases above , the natural frequencies associated with fixed-mass shape function tends to give more accurate results than those of the fixed-free shape function. It is noted that the use of first mode fixed-pinned, and fixed-fixed shape functions yield the natural frequencies of 1170.3 and 1698.8 Hz, which are substantially far from first out-of-plane mode frequencies obtained from Transactions of the Canadian Society for Mechanical Engineering, Vol. 39, No. 1, 2015

107

(a)

(b) Fig. 6. (a) In-plane natural frequencies of the PKM links for the first mode. (b) In-plane natural frequencies of the PKM links for the second mode.

FEA and thus are not given in Fig. 5(a). Thus, the fixed-mass shape function is found to be best mode shape approximation for the first out-of-plane mode. The second out-of-plane mode frequencies versus mass ratio are given in Fig. 5(b). Here, in addition to the second fixed-mass and second fixed-free shape functions, the first mode fixed-pinned and first mode fixed-fixed shape functions are considered for analysis, since it is expected that the distal joint may act like a “pinned” or “fixed” connection for the second mode for large mass ratios. It is noted that for mass ratios of r < 10/3, the second fixed-mass shape function can better approximate the second out-of-plane mode shape than other shape functions with an error of about 15.1%. However, it is seen that as the mass ratio increases, the first mode fixed-pinned shape function gives a closer approximate of the second outof-plane mode than other shape functions leading to a maximum percentage error of about 5.7% for first fixed-pinned. Thus, the bottom end joint acts like a pinned connection for the second out-of-plane mode for r ≥ 10/3. Similar analysis was conducted for the first two in-plane modes of the links. Fig. 6(a) shows natural frequencies of the first in-plane mode. It is noted that the first mode pinned-pinned shape function can better approximate the first in-plane modes than other shape functions for the whole range of mass ratios. The second in-plane mode frequencies are given in Fig. 6(b). Similar to the case for the first in-plane mode, it is noted that the second pinned-pinned shape function gives a better approximation of the natural frequencies than other shape functions for the whole range of mass ratios. 108

Transactions of the Canadian Society for Mechanical Engineering, Vol. 39, No. 1, 2015

Table 2. Summary of the recommended shape functions for the PKM links with respect to the mass ratio. Type of motion Recommended shape function and Recommended shape function and maximum percentage error for maximum percentage error for r < 10/3 r ≥ 10/3 First out-of-plane First fixed-mass 14.9% First fixed-mass 2.6% Second out-of-plane Second fixed-mass 15.1% First fixed-pinned 5.2% First in-plane First pinned-pinned 11.6% First pinned-pinned 9.0% Second in-plane Second pinned-pinned 14.1% Second pinned-pinned 15.2%

Table 3. Shape functions used for comparison in the simulation Set 1. Mass ratio 1st out-ot-plane 2nd out-of-plane Set 1(A) 1/300 1st fixed-free 2nd fixed-mass Reference for set 1(a) 1/300 1st fixed-mass 2nd fixed-mass Set 1(b) 2/3 1st fixed-free 2nd fixed-mass Reference for set 1(b) 2/3 1st fixed-mass 2nd fixed-mass Set 1(c) 150/3 1st fixed-mass 1st fixed-pinned Reference for set 1(c) 150/3 1st fixed-mas 2nd fixed-mass

Table 2 summarizes the shape functions with the closest mode frequencies to the FEA results, as a function of the link to platform mass ratio. The recommended set of shape functions can predict the realistic structural vibration behavior of the PKM links within 15.2% error for the whole range of mass ratios. 4.3. Structural Vibration Response of the Entire PKM-Based mMT Simulations of the structural vibration of the entire PKM-based mMT were performed using the parameters given in Table 1. The purpose of the simulations was to examine the effect of using various shape functions on the time response of the tooltip for a given mass ratio. Assuming the moving platform to be a rigid body, the time response of the tooltip is a combination of contributions from the displacements due to the in-plane and out-of-plane modes at the distal end of the flexible links of the PKM. To examine these contributions, the simulations were carried out in two sets. In the first set of simulation results, the effects of using various out-of-plane shape functions on the tooltip response were examined with the in-plane shape functions unchanged. In the second set of simulation results, the effects of in-plane shape functions were considered, assuming that the out-of-plane shape functions were unchanged. Both sets of simulations were carried out for several mass ratios to examine the effects of the platform and spindle mass on the elastic response at the tooltip. The shape functions with the closest natural frequencies to the FEA results for a given mass ratio were selected for comparison with the presented fixed-mass and pinned-mass shape functions in each simulation set. Table 3 shows the shape functions used for comparison of the first simulation set for each mass ratio. As shown in Table 3, the first and second modes of fixed-mass shape functions are used as a reference for comparison of out-of-plane modes throughout the first simulation set. The MATLAB solver utilized was ode15s for stiff systems. The mechanism is initially positioned at the following configuration: q0a = [30◦ 140◦ 280◦ 0.164 0.163 0.163]T , and q0f = 012×1 . An impulse force of F = [1 1 10]T N was applied at the tooltip at t = 0.01, to excite the vibration modes of the linkages. Figure 7(a) corresponds to simulation Set 1(a), which shows the elastic response of the tooltip for the mass ratio of r = 1/300. The two responses are noted to have approximately the same frequency, as predicted by Fig. 5(a), for r = 1/300. However, the presence of the inertia force, due to the end-mass in the fixed-mass shape function, leads to a greater distal end displacement of the links than seen with the fixed-free shape Transactions of the Canadian Society for Mechanical Engineering, Vol. 39, No. 1, 2015

109

(a)

(b)

(c) Fig. 7. Tooltip time response for (a) “1st fixed-mass” and “1st fixed-free” shape functions for the 1st out-of-plane mode, r = 1/300, (b) “1st fixed-mass” and “1st fixed-free” shape functions for the 1st out-of-plane mode, r = 2/3 and (c) “2nd fixed-mass” and “1st fixed-pinned” shape functions for the 2nd out-of-plane mode, r = 150/3.

function. This leads to tooltip response amplitude of the fixed-mass shape function which is greater than that of the fixed-fee shape function. Thus, while the fixed-free shape function, accurately predicts the outof-plane natural frequency for low mass ratios, simulation with this mode shape tends to under-predict the response amplitude. Figure 7(b) is related to simulation Set 1(b), and shows the elastic response of the tooltip for mass ratio of r = 2/3. It is noted that the difference in response amplitudes and frequencies is more significant as the mass ratio increases from r = 1/300 (Fig. 7a), to r = 2/3 (Fig. 7b). Simulation Set 1(c) compares the effects of two shape functions as the 2nd out-of-plane mode in the tooltip response with the tooltip response, shown in Fig. 7(c). It is noted that unlike the previous cases, the use of the 1st fixed-pinned shape function for high mass ratios does not lead to a noticeable difference, compared with use of the 2nd fixed-mass shape function. The general trend from Figs. 7(a) to 7(c) demonstrates the expected trend of a decrease in natural frequency, with a corresponding increase in the response amplitude, as the mass ratio increases from r = 1/300 to r = 150/3. Such behaviour is due to the use of the proposed fixed-mass shape function which accounts for the dynamic effects of the adjacent structural components. The shape functions used for comparison in the second simulation set are given in Table 4. Since the FEA frequencies, as shown in Fig. 6 are close to the pinned-pinned shape functions for both in-plane modes, 110

Transactions of the Canadian Society for Mechanical Engineering, Vol. 39, No. 1, 2015

Table 4. Shape functions used for comparison in the simulation Set 2. Mass ratio 1st in-plane 2nd in-plane Set 1(a) 1/300 1st pinned-mass 2nd pinned-pinned Reference for set 1(a) 1/300 1st pinned-pinned 2nd pinned-mass Set 1(b) 2/3 1st pinned-mass 2nd pinned-pinned Reference for set 1(b) 2/3 1st pinned-pinned 2nd pinned-mass Set 1(c) 150/3 1st pinned-mass 2nd pinned-pinned Reference for set 1(c) 150/3 1st pinned-pinned 2nd pinned-mass

(a)

(b)

(c) Fig. 8. Tooltip time response for “1st and 2nd pinned-mass” and “1st and 2nd pinned-pinned” shape functions for the first and second in-plane modes at (a) r = 1/300, (b) r = 2/3, and (c) r = 150/3.

they are used as a reference for comparison with pinned-mass shape functions as given in Table 4. Figure 8 shows the time response at the tooltip for mass ratios of r = 1/300, r = 2/3 and r = 150/3. The use of pinned-pinned and pinned-mass shape functions leads to negligible difference in the tooltip response amplitude. In contrast, the use of these shape functions leads to significant differences in the natural frequencies of the response specially for very high and very low mass ratios (Fig. 6). The reason for such small difference is the negligible contribution of the in-plane modes due to the assumption of a pinned joint at the distal end of the flexible links. Thus, although the use of pinned-mass and pinned-pinned Transactions of the Canadian Society for Mechanical Engineering, Vol. 39, No. 1, 2015

111

shape functions leads to the same small contribution to the overall tooltip response, the pinned-pinned shape functions can more accurately predict the natural frequencies due to the in-plane modes. 5. CONCLUSIONS In this paper, a set of admissible shape functions have been proposed for accurate prediction of the structural vibration response of Parallel Kinematic Mechanisms (PKMs) with flexible intermediate links. The modes of each admissible shape function are calculated and compared to the modal analysis results of the PKM from Finite Element Analysis (FEA) with respect to the ratio of the effective mass of the moving platform and spindle to the mass of the flexible links. The shape functions with closest natural frequencies to the FEA results are selected for comparison with the proposed fixed-mass shape functions for out-of-plane modes, and pinned-pinned shape functions for in-plane modes in the vibration modeling methodology. As a result of the use of fixed-mass shape functions, the expected dependency of the natural frequencies and response amplitudes of the whole PKM structure to the mass ratio is taken into account. Comparison of the tooltip time responses shows that the use of fixed-mass and pinned-pinned shape functions can accurately predict the out-of-plane and in-plane vibration modes of the PKM with flexible links over a large range of mass ratios. Furthermore, the in-plane modes are seen to have negligible contribution to the overall response of the tooltip. Given the mass ratio, the results of this analysis can be used as a guide to the selection of the most accurate shape function, to represent the realistic behavior of the structural vibration of a generic PKM with revolute, and/or spherical joints. Unlike FEA-based modal analysis, the presented method provides a time-efficient solution for accurate prediction of the structural vibration response of the PKM. The approach to model boundary conditions for PKMs, leads to a better approximation to the actual dynamic behavior compared with other boundary conditions. The resultant dynamic model, with more accurate structural vibration modeling, can then be used for control system synthesis, as in our current and future work, to design controllers for both rigid body motion and suppression of the unwanted flexible linkage structural vibrations. ACKNOWLEDGEMENTS The authors thank the Natural Sciences and Engineering Research Council of Canada (NSERC) Strategic Network, Canadian Network for Research and Innovation in Machining Technology (CANRIMT), for supporting this project. REFERENCES 1.

2. 3. 4. 5.

6.

112

Zhang, X., Mills, J.K. and Cleghorn, W.L., “Dynamic modeling and experimental validation of a 3-PRR parallel manipulator with flexible intermediate links”, Journal of Intelligent Robotic Systems, Vol. 50, No. 4, pp. 323– 340, 2007. Altintas, Y., Manufacturing Automation: Metal Cutting Mechanics, Machine Tool Vibrations, and CNC Design, Cambridge University Press, 2000. Chung, B., Smith, S. and Tlusty, J., “Active damping of structural modes in high-speed machine tools”, Journal of Vibration and Control, Vol. 3, No. 3, pp. 279–295, 1997. Mukherjee, P., Dasgupta, B. and Malik, A., “Dynamic stability index and vibration analysis of a flexible Stewart platform”, Journal of Sound and Vibration, Vol. 307, No. 3–5, pp. 495–512, 2007. Mahboubkhah, M., Nategh, M.J. and Esmaeilzadeh Khadem, S., “A comprehensive study on the free vibration of machine tool’s hexapod table”, International Journal of Manufacturing Technology, Vol. 40, No. 11–12, pp. 1239–1251, 2009. Zhou, Z., Xi, J. and Mechefske, C.K., “Modeling of a fully flexible 3PRS manipulator for vibration analysis”, ASME Journal of Mechanical Design, Vol. 128, No. 2, pp. 403–412, 2006. Transactions of the Canadian Society for Mechanical Engineering, Vol. 39, No. 1, 2015

7.

8. 9.

10.

11. 12.

13. 14. 15. 16. 17. 18. 19.

20. 21.

Shanzeng, L., Zhencai, Z., Bin, Z., and Chusheng, L., “Dynamics of a 3-DOF spatial parallel manipulator with flexible links”, in Proceedings of IEEE International Conference on Mechanic Automation and Control Engineering (MACE), Wuhan, China, pp. 627–633, June 26–28, 2010. Katz, R. and Li, Z., “Kinematic and dynamic synthesis of a parallel kinematic high speed drilling machine”, International Journal of Machine Tools and Manufacture, Vol. 44, No. 12–13, pp. 1381–1389, 2004. Wang, X. and Mills, J.K, “A FEM model for active vibration control of flexible linkages”, Proceedings of the IEEE International Conference on Robotics and Automation (ICRA), Hong Kong, China, pp. 4308–4313, May 31-June 7, 2004. Gasparetto, A., “On the modeling of flexible-link planar mechanisms: Experimental validation of an accurate dynamic model”, ASME Journal of Dynamic Systems, Measurement, and Control, Vol. 126, No. 2, pp. 365–375, 2004. Wang, X. and Mills, J.K., “Dynamic modeling of a flexible-link planar parallel robot platform using a substructuring approach”, Mechanism and Machine Theory, Vol. 41, No. 6, pp. 671–687, 2006. Yun, Y. and Li, Y., “Modeling and control analysis of a 3-PUPU dual compliant parallel manipulator for micro positioning and active vibration isolation”, ASME Journal of Dynamic Systems, Measurement and Control, Vol. 134, No. 2, pp. 021001-1-9, 2012. Theodore, R.J. and Ghosal, A., “Comparison of the assumed modes and finite element models for flexible multilink manipulators”, The International Journal of Robotics Research, Vol. 14, No. 2, pp. 91–111, 1995. Milford R.I. and Asokanthan, S.F., “Configuration dependent eigenfrequencies for a two-link flexible manipulator: Experimental verification”, Journal of Sound and Vibration, Vol. 222, No. 2, pp. 191–207, 1999. Dwivedy, S.K. and Eberhard, P., “Dynamic analysis of flexible manipulators, a literature review”, Mechanism and Machine Theory, Vol. 41, No. 7, pp. 749–777, 2006. Meirovitch, L., Principles and Techniques of Vibrations, Prentice-Hall Inc., 1997. Morris K.A. and Taylor, K.J., “A variational calculus approach to the modelling of flexible manipulators”, Society for Industrial and Applied Mechanics (SIAM) , Vol. 38, No. 2, pp. 294–305, 1996. Barbieri E. and Ozguner, U., “Unconstrained and constrained mode expansion for a flexible slewing link”, in Proceedings American Control Conference, Atlanta, GA, pp. 83–88, June 15–17, 1988. Mahmoodi, M., Mills, J.K. and Benhabib, B., “Structural vibration modeling of a novel parallel mechanismbased reconfigurable meso-milling machine tool (RmMT)”, in Proceedings 1st International Conference on Virtual Machining Process Technology (VMPT), Montreal, Canada, May 2012. Le, A.Y., Mills, J.K. and Benhabib, B., “Dynamic modeling and control design for a parallel-mechanism-based meso-milling machine tool”, Robotica, Vol. 32, No. 4, pp. 515–532, 2013. Mahmoodi, M., Li, Y., Mills, J. K. and Benhabib, B., “An active dynamic model for a parallel-mechanism-based meso-milling machine tool”, in Proceedings of the 23rd Canadian Congress of Applied Mechanics (CANCAM), Vancouver, Canada, pp. 1032–1035, June 2011.

Transactions of the Canadian Society for Mechanical Engineering, Vol. 39, No. 1, 2015

113