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1 Scientia Iranica, Vol. 14, No. 3, pp 78{9 c Sharif University of Technology, June 7 Research Note Dynamic Error Analis of Gantry Type Coordinate Measuring Machines M.T. Ahmadian, G.R. Vossoughi 1 and S. Ramezani 1 Coordinate Measuring Machines CMMs are designed for precision inspection of complex industrial products. The mechanical accuracy of CMMs depends on both static and dynamic sources of error. In automated CMMs, one of the dynamic error sources is vibration of the probe, due to inertia forces resulting from parts acceleration and deceleration. Modeling of a gantry type CMM, based on the Timoshenko beam theory with moving mass eects, is developed and the dynamic errors of the probe resulting from the acceleration and deceleration of moving parts, are calculated. Findings from analytical solution and dynamic modeling software indicate high accuracy and good agreement between the results. INTRODUCTION Coordinate measuring machines are, nowada, widely used for a large range of measuring tasks. These tasks are expected to be carried out with ever increasing accuracy, speed, exibility and ability to operate under shop oor conditions. Research is necessary to meet these demands. CMMs are prone to many error sources. Based on functional components of a CMM, an overview has been given by Weekers [1] of the most important error sources aecting the accuracy of a CMM: Geometric Errors: Limited accuracy in manufacturing, assembling and adjustment of components, like guide wa and measurement stems; Drive Stem: For CNC operated CMMs, the axes are equipped with drives, transmission and a servocontrol unit causing errors, such as mechanical load and structural vibration; Measurement Stem: The actual coordinates of measuring points are derived from the values indicated by the linear scales of the CMM. The main errors introduced by scales are inaccuracy of scale pitch, the misalignment and adjustment of the reading device and interpolation errors; *. Corresponding Author, Department of Mechanical Engineering, Sharif University of Technology, Tehran, I.R. Iran. 1. Department of Mechanical Engineering, Sharif University of Technology, Tehran, I.R. Iran. Errors Due to Mechanical Loads: These are errors related to static or slowly varying forces on CMM components, in combination with the compliance of components, and are mainly caused by the weight of moving parts; Thermally Induced Errors: The dierence between the temperature of the measuring scales of CMM and the work piece and temperature gradient in machine components are sources of error; Dynamic Errors: These are errors mainly caused by the deceleration of moving parts before stopping. These errors depend on the CMMs structural properties, like mass distribution, component stiness and damping characteristics, as well as on the control stem and disturbing forces. This paper concentrates on the dynamic errors of CMMs caused by the deceleration of moving parts. Some studies on the error analis of CMMs have been done. Weekers and Schellekenes [] proposed a method for compensation of the dynamic errors of CMMs using inductive position sensors for online measurement of major dynamic errors. Barakat et al. [3] presented a kinematical and geometrical error compensation of CMMs, based on experiment. Nijs et al. [4] presented a very simple model of CMM for obtaining natural frequencies of a CMM. Several researchers have applied software compensation successfully on CMMs [5,6]. Most of these researchers have considered a very simple model for their analis, while most of the studies are based on experiment. In the present study, a full CMM modeling is analyzed. All columns

2 Gantry Type Coordinate Measuring Machines 79 and guide wa are modeled as a Timoshenko beam with moving mass eects [7,8], and exibility in all directions. All bearings are modeled as torsional springs and torsional deformation of the column and guide wa are considered, too. Dynamic equations of motion are derived using Hamilton's principle [9]. The derived equations are solved using the nite dierence method. The results are compared with those obtained from the dynamic modeling software ADAMS-FLEX. Determination of the natural frequencies of CMM at various positions of the probe and optimization of the stem using a Genetic Algorithm are presented by Ramezani [1]. MODELING OF CMM STRUCTURE Structural components of a gantry type CMM are shown in Figure 1. Because of the very small deformation of each component, the x-, y- and z-axes are assumed to remain in the same direction as in an undeformed state. The most important problem in this modeling is that the motions of the y-guide way, x- guide way and z-pinole are relative motions. They are the absolute deformation of each beam, but, they are not the absolute motion of each beam. So, the strain energy of each beam is only a function of its deformations, but, the kinetic energy of each component is inuenced by the motion of other components. The stem is assumed to be xed at each position of the probe. The modeling of each component is presented in the following sections. MODELING OF THE RIGHT COLUMN For the right column, bending, torsional and longitudinal vibration is considered. The kinetic and strain energy of the column can be written as: T c = 1 Zh [A 1 _A ; ti 1z _ ; t A 1 _B ; ti 1y _ ; ti 1z _ 1 ; t A 1 _C ; t]d; 1 U c = 1 where: Zh kg 1 A 1 ne 1 I 1x ; te 1 I 1y ; t A; t ; t B; t ; t! G 1 I 1z 1; te 1 A 1 C ; tod; A; t: bending of column in xz-plane, ; t: rotation of column around y-axis, B; t: bending of column in yz-plane, ; t: rotation of column around x-axis, 1 ; t: torsion of column around z-axis, C; t: longitudinal vibration in z-direction, A 1, I 1x, I 1y, I 1z, E 1, G 1, 1, h and k: cross sectional area, actual and equivalent moments of inertia, Young's modulus, shear modulus, density, length of column and shear factor, respectively. Note that the symbols _ and denote derivation, with respect to time and coordinates, respectively. MODELING OF THE RIGHT Y -GUIDE WAY For the right y-guide way, kinetic and strain energy have been considered to be as follows: T y = ZLy na [ _Ey; ty _h; t _Ch; t] _ A h; t[_dy; t _Bh; ty _ 1 h; t] I z _ 1y; ti y [ _ 3 y; t _ h; t] Figure 1. Schematic view of a gantry type CMM. I x _ y; tgdy; 3

3 8 M.T. Ahmadian, G.R. Vossoughi and S. Ramezani U y = 1 ZLy kg A 1 y; t ne I z 1y; te I x y; t y; t! where: t: torsion around y-axis, K bc3, M bc3, J bc3y and J bc3z : torsional stiness of bearing and drive stem, mass and mass moments of inertia of y-carriage and its bearing, respectively. MODELING OF THE X-GUIDE WAY where: G I y 3y; tody; 4 y s : location of y-carriage on the y-guide way, Dy; t: bending in xy-plane, 1 y; t: rotation around z-axis, Ey; t: bending in yz-plane, y; t: rotation around x-axis, 3 y; t: torsion around y-axis, A, I x, I z, I y, E, G, and L y are cross sectional area, actual and equivalent moments of inertia, Young's modulus, shear modulus and density and length of y-guide way, respectively. MODELING OF THE Y -CARRIAGE AND BEARING For the drive stem eect, only the rotation of the y- carriage bearing around the y-axis is considered. Note that this part acts as a moving mass on the y-guide way. The kinetic and strain energy in this member can be written as: T bc3 = 1 M bc3 [ _Ah; tv y ] _Ch; t y s _ 1 h; t _Bh; t y s _h; t 1 bc3y J 3 y; t h; _ t 1 J bc3z 1 y; t 1 y; t ; _ t 5 U bc3 = 1 K bc3 t ; 6 For the x-guide way, bending and torsion is considered. The kinetic and strain energy can be written as: Z T x = 1 Lx 3 A 3 y s 1 _ h; t _Bh; t x _ t 1y; t Ah; _ tv y 1 y; t _W x; ty s _h; tx 3 y; t _ h; t I 3z ZLx y; t U x = 1 Z Lx _Ch; t _' x; tdx I 3x dx dx I 3y Z Lx _V x; t ZLx _ x; tdx _x; t y; t ; 7 E 3 [I 3y x; ti 3z ' x; t] G 3 I 3x x; tkg 3 A 3 V x; t 'x; t! W x; t x; t dx; 8

4 Gantry Type Coordinate Measuring Machines 81 where: x s : location of y-carriage on the y-guide way, W x; t: bending in xz-plane, W 1 x; t: bending in xz-plane, x; t: rotation around y-axis, V y; t: bending in xy-plane, 'x; t: rotation around z-axis, x; t: torsion around x-axis, A 3, I 3x, I 3y, I 3z, E 3, G 3, 3 and L x are area, moments of inertia, Young's modulus, shear modulus, density and length of x-guide way, respectively. MODELING OF THE BEARINGS AT Z-PINOLE The kinetic and strain energy resulting from the motion of the z-pinole and torsion of bearings in the z-pinole assembly is, as follows: T bcz = 1 M bcz y s _ 1 h; t _Bh; tv x _h; t _Ch; t x s x s W x; t _h; t 3y; t _ t 1y; t V x; t v x 1 J bc4x y; t v x x; t W x; t v x 1 y; t _Ah; tv y V x; t x; t _ 4 t J bc6y J bc5x y; t _ 6t x; t _ 4 t _ 5 t x; t J 4y x; t J 4z 'x; t y; t v x x; t v x 'x; t y; t ; 9 U bcz = 1 [K bc4 4 t K bc5 5 t K bc6 6 t ]; 1 where: 4 t: torsion of x-carriage bearing around x-axis, 5 t: torsion of z-axis bearing around x-axis, 6 t: torsion of z-axis bearing around y-axis. The parameters K bc4, K bc5, K bc6, M bcz, J bc4x, J bc5x, J bc6y, J 4y and J 4z are torsional stiness of bearings and drive stem, mass and mass moments of inertia of z-pinole carriage about x-axis, z-bearing and housing about x-axis, z-bearing about y-axis, z-bearing and z- axis assembly about y- and z-axis, respectively. MODELING OF THE Z-PINOLE Here, the bending of z-pinole in the xz- and yz-planes is considered. Kinetic and strain energy can be written, as follows: T z = 1 4 ZLz A 4 _Sz; ty s _ 1 h; t _Bh; t v x zz _ 6 t v y 1 y; t _Rz; t _Ah; tv y x s z z x; t V x; t y; t v x V x; t _ t 1y; t _ 4 t _ 5 t y; t y s _h; t

5 8 M.T. Ahmadian, G.R. Vossoughi and S. Ramezani x s 3 y; t Zh ZLy E 5 A 5 C ; td E6 I6x y; t h; _ W x; t t _Ch; t l _ 4 t _ x; t 5 t y; t I 4y U x = 1 where: ZLz ZLz _ z; tdz dz I 4x ZLz W x; t v x y; t _ 1z; tdz ; 11 E4 I4x 1z; te 4 I 4x z; t Rz; t kg 4 A 4 1 z; t z Sz; t z; t z dz; 1 Rz; t: bending in yz-plane, 1 z; t: rotation around x-axis, Sz; t: bending in xz-plane, z; t: rotation around y-axis, A 4, I 4x, I 4y, I 4z, E 4, G 4, 4 and L z : cross sectional area, moments of inertia, Young's modulus, shear modulus, density and length of z-axis, respectively. MODELING OF THE LEFT COLUMN AND Y -GUIDE WAY For the left-side support column and y-guide way, the same motions C; t, Ey; t and y; t have been considered in a similar fashion to the right side column and y-guide way. The kinetic and strain energy can be written, as follows: T L = 1 Zh 5 A 5 _C ; td 6 A 6 Z Ly _Ey; t _Ch; t dy ; 13 U L = 1 kg 6 A 6 y; t dy ; 14 where A 5, A 6, I 6x, E 5, E 6, G 5, G 6, 5 and 6 are cross sectional area, moments of inertia, Young's modulus, shear modulus, density of left column and left y-guide way, respectively. Note that because of the eect of the drive stem, some motions, such as rotation of the x-carriage around the z-axis, longitudinal vibration of the z-axis and horizontal and vertical motion of the y-carriage, have been neglected. Furthermore, the gravitational strain energy is neglected. EQUATIONS OF MOTION Using the Hamilton's principle equations of motion can be found, as follows: Zt t 1 T U Wdt =; 15 where denotes the variation operator and: T = T c T x T y T z T bcz T L ; 16 U = U c U x U y U z U bcz U L : 17 Now, dening: M 1 = 1 A 1 h; M = A L y ; M 3 = 3 A 3 L x ; M 4 = 4 A 4 L z ; M = M 3 M 4 M bcz M bc3 ; 18 and neglecting higher order terms, virtual work done by the drive stem and gravitational eects can be written as: W = M a y y s Ch; tg x s h; tg x s 3 y; ty s g Ey; ty s g [M bcz M 4 ][a x x s W x; tx s g]: 19 Terms resulting from the moving mass eect are in the following form: N; t = N; t N; t v v N; t N; t a ;

6 Gantry Type Coordinate Measuring Machines 83 where the variable N can be E, D, 1,, V, W, ' and and the parameter can be y or x, respectively. Besides, other terms, resulting from the moving mass eect, will appear in the equations of the y- and x-guide wa. Now, the equations of each member can be written, as follows. EQUATIONS OF THE RIGHT COLUMN 1 A; tkg1 ; t 1 I 1x ; t E 1 I 1x ; t A; t A; t kg 1 A 1 ; t 1 B; tkg1 ; t 1 I 1y ; t E1 I 1y ; t B; t B; t kg 1 A 1 ; t = ; 1 = ; = ; 3 = ; 4 [ 1 A 1 5 A 5 ] C; t[e1 A 1 E 5 A 5 ]C ; t=; 5 1 I 1z 1 ; tg 1 I 1z 1 ; t =: 6 EQUATIONS OF RIGHT Y-GUIDE WAY M Dy; t Bh; ty 1 h; t] L y kg A G 6 A 6 M 1 y; t Dy; t Dy; tv y 1 _ h; t y s1 h; t Bh; t M bcz M 4 M 4 L z z z L z 6 t a x y s =; 7 I z 1 y; t E I z E 6 I 6z 1 y; t kg A G 6 A 6 1 y; t t 1 y; t J bc3z x sm bcz M 4 M 3L x 3 M bcz M 4 V x; t v x ay Ah; M 3 L x t x s M bcz M 4 x s v x v y Ah; _ t V x; t M 4 4 t a x V x; t y; t 5 t y s ;tx s ;tv y x; t L z z z a y y s =; 8 L z A [ Ey; t Ch; tyh; t] 6 A 6 [ Ey; t Ch; t] kg A G 6 A 6 y; t Ey; t M[Ey; ty s h; t L x v y _h; t][m 3 M 4x s ][ 3 y s ; t h; _ W t] [M bcz M 4 ] x; t W x; t v x a x W x; t 5 t y s ; tv y y; t x s ; t M 4 l 4 t x= a y x; t y s = MEy; ty s ; 9 A [ Ey; t Ch; tyh; t] 6 A 6 [ Ey; t Ch; t] kg A G 6 A 6 y; t

7 84 M.T. Ahmadian, G.R. Vossoughi and S. Ramezani Ey; t M[Ey; ty s h; t Ch; W x; t t W x; ta x v y _h; t] [M bcz M 4 ] W x; t a x x= y; t L x M3 M 4x s W x; t [ 3 y s ; t _ h; t] W x; t v x M 4 l[ 4 t 5 t y s ; t x; t a y x s ; t] y s = MEy; ty s ; 3 I x 6 I 6x y; t E I x E 6 I 6x y; t kg A G 6 A 6 y; t [ 3 I 3x J bc4x J bc5x M 4 ] x s ; tx= J bc4x J bc5x M 4 L z z 3 z 3 l M 4 L z z 3 z 3 l M 4 L z z z L z 1y; t V V x; t x; tax 1 y; t!! 4 t 5 t Ah; ta y v x y; t J bc5x _ t x s [ t 1 y; t] x s v x V x; t M 4 l v y _h; ty s h; tey; t v x [ _ 3 y; t h; _ t] x s 3 y; t h; t] v x W x; t x s y s =; 31 [ 3 y; t M [ 3 y; t h; t] L y G I y 3 y; t h; t] Ey; tay y s h; t J bc3y M 3 L x 3 M bcz M 4 x s L x M3 x s M bcz M 4 W x; t v x x= Ey; t M bcz M 4 x s W x; t W x; a x tx= 5 tx s ; tx= x; t a y Ch; t M 4 l _h; t x= 4 t y; tv y y; t y s =: 3 In the above equations, the term is the Kronecker delta function and is dened in the following form: y s = 1 if y = if y 6= y s 33 The coecients of y s are terms resulting from moving mass eects. EQUATION CORRESPONDING TO ROTATION J bc3z M 3 L x 3 M bczx s M 4 x s 1 y; tv y 1 y; t a y 1 x; t t! y s

8 Gantry Type Coordinate Measuring Machines 85 K bc3 t 4 t V x; t 5 tx s ; t y; t J 4z 'x; tx s =; 36 y; t a y x; t M 4 x s L z z z L z M 4 x s a x V x; t M bcz x s M 4 x s L z! Ah; tx s M bcz M 4 M3 L z M bczx s V x; t V x; t v x ]x s a y M3 L x = : 34 EQUATIONS OF X-GUIDE WAY M 3 [ V x; t Ah; t x ta y ] kg 3 A 3 L x 'x; t V x; t M 4 M bcz Ah; ta y v x 1y; t x s 1 t 1 y; t! _ t 1 y; t 1 x; t y; tvy a y L z z z M 4 L z x; tv x x; t!! 4 t 5 t y; tv y y; t x; t a y y s ] x s =; 35 3 I 3z 'x; t E 3 I 3z ' x; tkg 3 A 3 'x; t M 3 W x; t Ch; tx 3 y s ; t E ; t y s h; ta y L x kg 3 A 3 x; t Ey; t W x; t v y _ 1 h; ty s 1 h; t Bh; tax D ; ta y Dy; t _h; ty s h; t Ch; t E ; t a y Ey; t v x _ 3 y s ; t _ h; t h; t x s 3 y s ; tm bcz M 4 ] M 4 l W x; t 4 t 5 tx s ; t y s ; ta y y; t y; t x s = M bcz M 4 W x; tx s g; 37 3 I 3y x; t E3 I 3y x; tkg 3 A 3 W x; t 3 x; tg 3 x; t a y y; t x; t J bc6y J 4y x; tx s =; 38 4 tx s ; t y s ; t y; t J bc4x J bc5x

9 86 M.T. Ahmadian, G.R. Vossoughi and S. Ramezani l M 4 J bc5x l M 4 L z z z 5 t x s =: 39 K bc5 5 t x s M 4 t L z Again the coecients of x s are terms resulting from moving mass eects. EQUATIONS CORRESPONDING TO THE ROTATIONS 4, 5 AND 6 J bc4x J bc5x M 4 Lz K bc4 4 z 3 z 3 l 4 t J bc5x M 4 Lz z 3 z 3 l 5 t M 4 x s L z z z L z 1 y; t y; t M 4 Lz t 1 ; tay 1 y; t y s ; t ay y; t z 3 z 3 l V ; ta x V x; t v x _ V x; t M 4 l M 4 l a y J bc4x J bc5x L z z z M 4 x s L z W x s ; ta y W x; t Ah; ta y _W x; t v x Ch; taz y s h; t E ; t Ey; t M 4 lx s3 y s ;t=; 4 J bc5x M 4 Lz z 3 z 3 l 4 t 5 t] 1 y s ; ta y 1 y; t 1 y; t J bc5x M 4 l L z z 3 z 3 y s ; ta y y; t! y; t x s ; t L z z 3 z 3 M 4 Ah; ta y V ; t V x; t V x; t a x v x M 4 l W x s ; t W x; t a y W x; t x s 3 y s ; ty s h; t E ; t a y Ey; t L z z 3 z 3 J bc6y M 4 L z z z M 4 L z D ; ta y Ch; t a z = ; 6 tk bc6 6 t a x y s 1 h; t Bh; t Dy; t EQUATIONS IN Z-PINOLE 4 A 4 Rz; t Ah; t ta y 4 t 5 ta y y; t y s ; t zz ax V x; t x s y; t V v x 41 = : 4

10 Gantry Type Coordinate Measuring Machines 87 V ; t kg 4 A 4 1 z; t Rz; t solution is conditionally stable and should be used with z z =; a small time step size. A central dierence equation is 43 used for position dierentiation and forward dierence is used for time dierentiation. For every function, it 4 A 4 Sz; tz z 6 ta x Bh; t can be written that: Nnt Nnt Nnt Nnt = m1 m m 1 m ; 47 y s 1 h; ta y D ; t] kg 4 A 4 z; t z 4 I 4y z; t E 4 I 4y z; t Dy; t Sz; t z Sz; t kg 4 A 4 z; t z BOUNDARY CONDITIONS = 44 = : 45 Columns are assumed to be clamped at =. At = h, the boundary conditions are determined from Hamilton's principle. The y-guide way is clamped at y = and free at y = L y. The x-guide way is clamped-free in the xy-plane and simply supported in the xz-plane. The z-pinole is clamped free, too. Here, for brevity, representation of the equations of the boundary conditions are neglected, but, they can be easily obtained from Hamilton's principle. Furthermore, most of them are the well known \simple boundary conditions. As an example, boundary conditions corresponding to the right y-guide way are, as follows: 8>< >: 8>< >: D;t= 1 ;t= 1y;t Dy;t y=ly 1 L y ; t = E;t= ;t= y;t Ey;t y=ly 3 ;t= 3y; t L y ; t = y=ly METHOD OF SOLUTION y=ly = ; y=ly = = : 46 An explicit time integration method [11] is used for solving the equations of motion. This method of ; N Nn1t Nnt nt = m m m t ; 48 where is the step size in position, t is the time step, n is counter of time and m is the node number. The input of the stem is the prole of acceleration of moving parts versus time. Dening the time derivative of each variable as a new variable, the second order derivatives, with respect to the time, will be presented into the rst order derivative form and the descritized equations will be presented in two times nt and n 1t. The nal equation is in the following form: [A] n fv g n1 = [B] n fv g n fcg n ; 49 where the matrices [A] n and [B] n are known coef- cient matrices at time nt, the vector fv g n1 is a vector containing all variables in all nodes, except at the rst node, at time n 1t. The vector fcg n, is a known vector resulting from the acceleration of the moving parts. Note that because of the motion of x- and y-guide wa, each matrix should be constructed at each time step. Solving for velocities, one may obtain the corresponding displacements. In order to illustrate an example of modeling, a CMM with the following specications is modeled. i =785 kg/m 3 ; E i = Gpa; G i =7 Gpa; h=:8 m; L y =:6 m; L x =1: m; L z = :75 m; l =:16 m; z =:15 m; K bearings =5e7 N.m/rad; A 1 =7:6e 3 m ; A =4:6e 3 m ; A 3 =7:6e 3 m ; A 4 =1:e 3 m ; A 5 =4:6e 3 m ; A 6 =4:6e 3m ; I 1x =7:86e 5 m 4 ; I 1y =1:3e 5 m 4 ; I 1z =3:58e 5 m 4 ; I x =1:35e 5 m 4 ; I y =1:38e 6 m 4 ; I z =:4e 5 m 4 ; I 3x =5:94e 5 m 4 ; I 3y =:8e 5 m 4 ; I 3z =6:35e 5 m 4 ; I 4x =4:1e 7 m 4 ; I 4y =4:1e 7 m 4 ; I 5x =1:35e 5 m 4 ; I 5z =:4e 5 m 4 ; I 6x =1:35e 5 m 4 ; I 6y =:4e 5 m 4 ; J bc3z =:18 kgm 4 ; J bc3z =:3 kgm 4 ; J bc4x =:7 kgm 4 ; J bc5x =:5 kgm 4 ; J bc6y =:5 kgm 4 ; J 4y =3:4 kgm 4 ; J 4z =1:3 kgm 4 ; M bc3 =1 kg; M bcz =35 kg:

11 88 M.T. Ahmadian, G.R. Vossoughi and S. Ramezani In Figure, the prole of accelerations a x and a y is shown for two motions. In motion 1, the accelerations and decelerations are applied rapidly in the step function form and, in the second motion, the accelerations are applied in the semi-sinusoidal form. The maximum value of the accelerations is assumed to be 1.5 m/s and maximum velocities are.6 m/s. Note that, in both cases, the stems start from rest and the course of traveling is 1 cm and 6 cm in the x-and y-directions, respectively. The values of error for step time acceleration and deceleration in the x-, y- and z-directions are shown in Figures 3 to 5. The absolute value of error for this type of excitation is shown in Figure 6. It can be seen that the main part of the error is produced in the y- Figure. Acceleration and deceleration proles for two motions step and sinusoidal proles. Figure 4. Dynamic error of probe in y-direction for step type acceleration and deceleration. Figure 5. Dynamic error of probe in z-direction for step type acceleration and deceleration. Figure 3. Dynamic error of probe in x-direction for step type acceleration and deceleration. Figure 6. Magnitude of dynamic error of probe for step type acceleration and deceleration.

12 Gantry Type Coordinate Measuring Machines 89 direction, since the total stiness of the structure is small in this direction in comparison with the other two directions. Similarly, the values of error for the semi-sinusoidal acceleration and deceleration in the dierent directions are shown in Figures 7 to 9. The absolute value of error for this type of excitation is shown in Figure 1. Again, the main part of the error can be seen in the y-direction. The results indicate good agreement with those obtained from the dynamic modeling software ADAMS-FLEX. Furthermore, it is clear that, in order to decrease the oscillations of the probe, a smooth prole such as a semi-sinusoidal one of acceleration should be applied on the CMM. CONCLUSION The modeling of a gantry type CMM, based on the Timoshenko beam theory, subjected to moving mass was developed and the dynamic errors of the probe Figure 7. Dynamic error of probe in x-direction for semi-sinusoidal acceleration and deceleration. Figure 8. Dynamic error of probe in y-direction for semi-sinusoidal acceleration and deceleration. Figure 9. Dynamic error of probe in z-direction for semi-sinusoidal acceleration and deceleration. Figure 1. Magnitude of dynamic error of probe for semi-sinusoidal acceleration and deceleration. have been calculated. The equations of motion and boundary conditions have been obtained using Hamilton's principle. Results indicate that, in order to decrease the oscillatory motion of the probe, a smooth function of the acceleration or deceleration function should be applied on the CMM. The maximum value of the dynamic error in the step type excitation is larger than the value obtained from semi-sinusoidal excitation. So, a smooth function of excitation enforces a smaller value of absolute error to the machine. In both step and semi-sinusoidal types of excitation applied to the machine, the main part of the error is produced in the y-direction. The value of error in this direction is approximately three times the error induced in the x- and z-directions. This fact indicates that the total stiness of the stem in the y-direction is small, in comparison with the other two directions.

13 9 M.T. Ahmadian, G.R. Vossoughi and S. Ramezani The results of the authors' analytical modeling indicate good agreement and high accuracy in comparison with the results found by dynamic modeling software. REFERENCES 1. Weekers, W.G. \Compensation for dynamic errors of coordinate measuring machines, PhD Thesis, Eindhoven University of Technology Weekers, W.G. and Schellekens, P.H.J. \Compensation for dynamic errors of coordinate measuring machines, Measurement,, pp Barakat, N.A., Elbestawi, M.A. and Spence, A.D. \Kinematic and geometric error compensation of a coordinate measuring machine, Machine Tools & Manufacture, 4, pp de Nijs, J.F.C., Schellekens, M.G.M. and van der Wolf, A.C.H. \Modeling of a coordinate measuring machine for analis of its dynamic behavior, Annals of the CIRP, 37, pp Teewwsen, J.W.M., Soons, J.A. and Schelekens, P.H. \A general method for error description of CMMs using polynomial tting procedure, Annals of the CIRP, 38, pp Zhang, G. \Error compensation of coordinate measuring machines, Annals of the CIRP, 34, pp Lee, H.P. \Transverse vibration of a Timoshenko beam acted on by an accelerating mass, Applied Acoustics, 47, pp Esmailzadeh, E. and Ghorashi, M. \Vibration analis of a Timoshenko beam subjected to a traveling mass, Journal of Sound and Vibration, 199, pp Reddy, J.N., Energy and Variational Methods in Applied Mechanics, John Wiley & Sons Ramezani, S., \Vibration analis of gantry type CMMs and optimization using genetic algorithm, M.Sc. Thesis, Sharif University of Technology Belyteschko, T., Liu, W.K. and Moran, B., Nonlinear Finite Elements for Continua and Structures, John Wiley & Sons Inc..