Gantry Type Coordinate Measuring Machines 27 frequencies of a CMM. Vermeulen [5] generated highaccuracy 3-D coordinate machines using a new conguratio

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1 Scientia Iranica, Vol. 13, No. 2, pp 26{216 c Sharif University of Technology, April 26 Research Note Free Vibration Analysis 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. Even though CMMs have high accuracy in electrical systems, mechanical structures prevent very high accuracy in these machines. Mechanical accuracy of CMMs depends on both static and dynamic sources of error. In automated CMMs, one of the dynamic error sources is the vibration of probes, due to inertia forces resulting from parts deceleration. Modeling of a gantry type CMM, based on the Timoshenko beam theory, is developed and the natural frequencies of the CMM system, at dierent positions of the probe, are calculated. Findings from the analytical and nite elements method indicate high accuracy and good agreement between the results. INTRODUCTION Coordinate measuring machines are, nowadays, widely used for a large range of measuring tasks. These tasks are expected to be carried out with ever increasing accuracy, speed, exibility and the 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 ways and measurement systems; Drive system: 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 system: The actual coordinates of measuring points are derived from the values indicated by the linear scales of the CMM. The min. errors introduced by the scales are: Inaccuracy of *. 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. scale pitch, misalignment and adjustment of reading device and interpolation errors; 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 work piece and the temperature gradient in the machine components is a source of error; Dynamic errors: These are errors mainly caused by 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 system and disturbing forces. In this paper, the authors are interested in the dynamic errors of CMMs caused by the deceleration of moving parts. Some studies on the error analysis of CMMs have been done. Weekers and Schellekenes [2] proposed a method for compensation of the dynamic errors of CMMs using inductive position sensors for online measurement of the major dynamic errors. Barakat et al. [3] presented a kinematical and geometrical error compensation for CMMs based on experiment. Nijs et al. [4] presented a very simple model of CMM for obtaining natural

2 Gantry Type Coordinate Measuring Machines 27 frequencies of a CMM. Vermeulen [5] generated highaccuracy 3-D coordinate machines using a new conguration with fewer dynamic errors. Several researchers have applied software compensation successfully on CMMs [6,7]. Most of these researchers have considered a very simple model for their analysis while most of the studies are based on experiment. In the present study, a full CMM modeling is analyzed. All columns and guide ways are modeled as a Timoshenko beam [8,9] with exibility in all directions. All bearings are modeled as torsional springs and torsional deformation of the column and guide ways is considered. Dynamic equations of motion are derived using Hamilton's principle [1]. The derived equations are treated using a state space variables method. Natural frequencies of the CMM system are calculated at dierent positions of the probe in the system. Dynamic error analysis of CMM and optimization using a Genetic Algorithm, based on maximization of a tness function containing the eect of natural frequency, dynamic error and weight of the CMM structure, is presented by Ramezani [11]. 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 the undeformed state. The origin of the coordinate system is along the axis of the left column and at a height which is equal to the axis of the x-guideway. 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, however, the kinetic energy of each component is inuenced by the motion of other components. The system 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 2 ( h Z ha 1 ( A_ 2 (;t+ _B 2 (;t+ _C 2 (;t + I 1z _ 2 (;t+i 1y 2 _ (;t+i 1z _ 2 1(;tid U c = 1 2 ( h Z( ; (1 E 1 I 1x 2 (;t+e 1 I 1y 2 (;t+kg 1 A + G 1 I 1z 2 1(;t+E 1 A 1 C 2 (;t d ; (2 - 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 (for thin walled column moments of inertia, Young modulus, shear modulus, density, length of column and Timoshenko shear modication constant, respectively. Figure 1. Schematic view of a gantry type CMM. Note that the symbols ( : and ( denote derivation with respect to time and coordinates, respectively.

3 28 M.T. Ahmadian, G.R. Vossoughi and S. Ramezani 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 = 2 2 ( Zys na 2 ([ _E(y; t+y _(h; t _C(h; t] 2 + _A 2 (h; t+[_d(y; t+ _B(h; t+y _ 1 (h; t] 2 + I 2z _ 2 1(y; t+i 2y [ _ 3 (y; t+ _ (h; t] 2 + I 2x _ 2 2(y; tody + ZLy ys na 2 ([ _E 1 (y; t+y _(h; t _C(h; t] 2 + _A 2 (h; t+[ _D 1 (y; t+ _B(h; t +y _ 1 (h; t] 2 +I 2x _ 2 22(y; t+i 2y [ _ 3 (y; t+ _ (h; t] 2 + I 2z _ 2 11(y; t+i 2y [ _ 3 (y; t+ _ (h; t] 2 + I 2x _ 2 2(y; tody U y = 1 2 ( Zys ; (3 ne 2 I 2z 2 1(y; t+e 2 I 2x 2 2(y; t +kg 2 A 2h 2 (y; t ZLy ys ne 2 I 2z 2 + kg 2 A 2h t + h 1 (y; + G 2 I 2y 2 3(y; tody 11(y; t+h 22 (y; 1(y; 2 22(y; t+g 2 I 2y 2 33(y; tody 1(y; t + E 2 I 2x ; (4 - y s : Location of y-carriage on the y-guide way, - D(y; t: Bending in xy-plane for <y<y s, - D 1 (y; t: Bending in xy-plane for y s < y<l y, - 1 (y; t: Rotation around z-axis <y<y s, - 11 (y; t: Rotation around z-axis y s < y<l y, - E(y; t: Bending in yz-plane <y<y s, - E 1 (y; t: Bending in yz-plane y s < y<l y, - 2 (y; t: Rotation around x-axis <y<y s, - 22 (y; t: Rotation around x-axis y s < y<l y, - 3 (y; t: Torsion around y-axis <y<y s, - 33 (y; t: Torsion around y-axis y s < y<l y, - A 2, I 2x, I 2z, I 2y, E 2, G 2, 2 and L y : cross sectional area, actual and equivalent moments of inertia, Young modulus, shear modulus and density, length of y-guide way, respectively. MODELING OF THE Y -CARRIAGE AND BEARING For the drive system eect, only the rotation of a y- carriage bearing around the y-axis is considered. The kinetic and strain energy in this member can be written as: T bc3 = 1 2 M bc3n[ _ A(h; t] 2 + [y s _ 1 (h; t+ _D(y s ; t + _B(h; t] 2 + [y s _(h; t+ _E(y s ; t _C(h; t] 2o + J bc3z _ 2 (t 2 + J bc3y [ _ 3 (y s ; t+ _ (y s ; t] 2 ; (5 U bc3 = 1 2 K bc3 2 (t 2 ; (6-2 (t: Torsion around y-axis - K bc3, M bc3, J bc3y and J bc3z : Torsional stiness of bearing and drive system, mass and mass moments of inertia of y-carriage and its bearing, respectively. MODELING OF THE X-GUIDE WAY For the x-guide way, bending and torsion kinetic and strain energy can be written as: T x = 3 2 ( Zxs na 3 ([ V _ (x; t+ A(h; _ t x _ 2 (t] 2 + [y s _ 1 (h; t+ _D(y s ; t+ _B(h; t] 2 +[ _W (x; t _C(h; t + _E(y s ; t+y s _(h; t+x _ 3 (x; t] 2 + I 3y _ 2 (x; t + I 3z _' 2 (x; t+i 3x _ 2 (x; todx + ZLx xs na 3 ([ _D(y s ; t

4 Gantry Type Coordinate Measuring Machines 29 + _B(h; t+y s _ 1 (h; t] 2 + [ _W 1 (x; t+y s _(h; t + I 3z _' 2 1(x; t+i 3x _ x(x; to 2 dx + _E(y s ; t _C(h; t+x _ 3 (x; t] 2 + I 3y _ 2 1(x; t U x = 1 2 ( Zxs ne 3 [I 3y 2 (x; t+i 3z' 2 (x; t] +G 3 I 3x 2 (x; t+kg 3 A 3h (x; t +h'(x; (x; Z ; (x; Lx odx+ ne 3 I 3y[ 2 1(x; t +I 3z ' 2 1(x; t]+g 3 I 3x 2 x(x; t+kg 3 A 3h' 1 (x; 1(x; +h 1 (x; t xs (x; 1(x; t - x s : Location of y-carriage on the y-guide way, - W (x; t: Bending in xz-plane for <x<x s, - W 1 (x; t: Bending in xz-plane for x s < x<l x, - (x; t: Rotation around y-axis <x<x s, - 1 (x; t: Rotation around y-axis x s < x<l x, - V (y; t: Bending in xy-plane <x<x s, - V 1 (y; t: Bending in xy-plane x s < x<l x, - '(x; t: Rotation around z-axis <x<x s, - ' 1 (x; t: Rotation around z-axis x s < x<l x, - (x; t: Torsion around x-axis <x<x s, - x (x; t: Torsion around x-axis x s < x<l x, (8 - A 3, I 3x, I 3y, I 3z, E 3, G 3, 3 and L x are area, actual and equivalent moments of inertia, Young 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 the torsion of bearings is as follows: T bcz = 1 2nM bczn[y s _ 1 (h; t+ _D(y s ; t+ _B(h; t] 2 + [y s _(h; t+ _E(y s ; t _C(h; t] 2 + [ _ A(h; t + _ V (xs ; t x s _ 2 (t] 2o+J bc4x [ _ 4 (t+ _(x s ; t] 2 +J bc5x [ _ 4 (t+ _ 5 (t+ _(x s ; t] 2 +J bc6y _ 2 6(to; (9 U bcz = 1 2 [K bc4 4 (t 2 + K bc5 5 (t 2 + K bc6 6 (t 2 ]; (1-4 (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, - K bc4, K bc5, K bc6, M bcz, J bc4x, J bc5x and J bc6y : Torsional stiness of bearings and drive system, mass and mass moments of inertia of z-pinole assembly, respectively. MODELING OF THE Z-PINOLE Bending of the z-pinole in the xz- and yz-planes is considered. Kinetic and strain energy can be written as follows: T z = 4 2 A 4 ZLz ( [ _S(z;t+ _D(y s ; t+y s _ 1 (h; t + _B(h; t+(z + z _ 6 (t] 2 + [ _R(z;t+ _A(h; t + (z + z ( _ 4 (t+ _ 5 (t+ _(x s ; t + _ V (xs ; t x s _ 2 (t] 2 + [ _W (x s ; t+ _E(y s ; t+y s _(h; t _C(h; t+x s _ 3 (x; t+l( _ 4 (t+ _ 5 (t + _(x s ; t] 2 + I 4x _ 2 1(z;t+I 4y _ 2 2(z;t dz; (11

5 21 M.T. Ahmadian, G.R. Vossoughi and S. Ramezani U x = 1 2 ZLz ne 4 I 4x 2 1(z;t+E 4 I 4x 2 2(z;t + kg 4 A +h( 2 odz; EQUATIONS OF MOTION Using Hamilton's principle, equations of motion can be found. Zt2 t1 (T U + Wdt =; (15 where denotes the variation operator and: T = T c + T x + T y + T z + T bcz + T L ; (16 - R(z;t: Bending in yz-plane, - 1 (z;t: Rotation around x-axis, - S(z;t: Bending in xz-plane, - 2 (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 modulus, shear modulus, density and length of z-axis, respectively. MODELING OF LEFT COLUMN AND Y -GUIDE WAY For the left-side support column and y-guide way, the same motions C(;t, E(y; t and 2 (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 2 n 5 A 5 Z h _C 2 (;td + 6 A 6 ZLy [ _E(y; t _C(h; t] 2 dyo; (13 U L = 1 2 ne 5 A 5 Z h C 2 (;td + + kg 6 A 6h 2 (y; t Zys ne 6 I 6x 2 2(y; t odyo; 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 modulus, shear modulus and density of left column and left y- guide way, respectively. Note that because of the eect of the drive system, some motions, such as the rotation of the x-carriage around the z-axis and the longitudinal vibration of the z-axis, the horizontal and vertical motion of the y-carriage have been neglected. Furthermore, the gravitational strain energy is neglected. U = U c + U x + U y + U z + U bcz + U L : (17 In the case of a xed system, neglecting gravitational eects, it can be written that: W =: (18 It is interesting to nd natural frequencies using a state space variables approach. State variables are dened as follows: A = x 1 ; A = x 2 ; = x 3 ; = x 4 ; B = x 5 ; B = x 6 ; = x 7 ; = x 8 ; C = x 9 ; C = x 1 ; 1 = x 11 ; = x 1 12 ; D = x 13 ; D = x 14 ; 1 = x 15 ; = x 1 16 ; E = x 17 ; E = x 18 2 = x 19 ; = x 2 2 ; 3 = x 21 ; = x 3 22 ; D 1 = x 23 ; D = x 1 24 ; 11 = x 25 ; = x 1 26 ; E 1 = x 27 ; E = x 1 28 ; 22 = x 29 ; = x 22 3 ; 33 = x 31 ; = x ; W = x 33 ; W = x 34 ; = x 35 ; = x 36 ; V = x 37 ; V = x 38 ; ' = x 39 ; ' = x 4 ; = x 41 ; = x 42 ; W 1 = x 43 ; W = x 1 44 ; 1 = x 45 ; = x 1 46 ; V = x 47 ; V = x 48 ; ' 1 = x 49 ; ' = x 1 5 ; x = x 51 ; x = x 52 ; R = x 53 ; R = x 54 ; 1 = x 55 ; = x 1 56 ; S = x 57 ; S = x 58 ; 2 = x 59 ; = x 2 6 : (19 Note that for free vibration, each state variable can be separated into time and spatial coordinates, so: x i (coordinates, time = X i (coordinates:e i!t : (2 Now, the equations and boundary conditions of the system are written. Note that the attractiveness of the energy method is that it gives both equations of motion and corresponding boundary conditions. EQUATIONS OF THE RIGHT COLUMN X 1( =X 2 (; X 2( =X 4 ( ( 1! 2 =kg1x 1 (; (21a (21b

6 Gantry Type Coordinate Measuring Machines 211 X3( =X 4 (; X4( = ( 1! 2 =E1X 3 (+(kg 1 A 1 =E 1 I 1x [X 3 ( X 2 (]; X5( =X 6 (; X6( =X 8 ( ( 1! 2 =kg1x 5 (; X7( =X 8 (; X8( = ( 1! 2 =E1X 7 (+(kg 1 A 1 =E 1 I 1x (21c (21d (22a (22b (22c EQUATION CORRESPONDING TO ROTATION 2! 2 [J bc3z +M bcz x 2 s +A 3 L 3 x 3=3+x 2 s A 4L z 4 ]K bc3 2! 2 x s A 4 4 [(L z + z 2 z 2 ]=2( 4 + 5! 2 [M bcz x s +A 3 L 2 x 3 =2+x s A 4 L z 4 ]X 1 (h! 2 x s [M bcz + A 4 L z 4 ]X 27 (y s A 4 4 [(L z + z 2 z 2 ]=2X 31 (x s =: (28 [X 3 ( X 2 (]; (22d X9( =X 1 (; (23a X1( =! 2 ( 1 A A 5 [E 1 A 1 + E 5 A 5 ] 1 X 9 (; (23b X11( =X 12 (; (24a X12( =! 2 1 =G 1 X 9 (; (24b EQUATIONS OF RIGHT Y -GUIDE WAY X13(y =X 14 (y; (25a X14(y =X 16 (y! 2 2 A 2 [k(g 6 A 6 + G 2 A 2 ] 1 [X 13 (y+yx 11 (h+x 5 (h]; (25b X15(y =X 16 (y; (25c X16(y =[E 6 I 6z + E 2 I 2z ] 1 f! 2 2 I 2z X 15 (y + k(g 6 A 6 + G 2 A 2 [X 15 (y X 14 (yg; (25d EQUATIONS OF X-GUIDE WAY X 23(x =X 24 (x; X 24(x =X 26 (x! 2 3 =kg 3 fx 23 (x+x 17 (y s (29a + y s X 3 (h X 9 (h]g; (29b X 25(x =X 26 (x; X 26(x =1=E 3 I 3y f! 2 3 I 3y X 25 (x+kg 3 A 3 [X 25 (x X 24 (x]g; X 27(x =X 28 (x; (29c (29d (3a X 28(x=X 3 (x! 2 3 =kg 3 [X 27 (x x 2 +X 1 (h]; (3b X 29(x =X 3 (x; X 3(x =1=E 3 I 3z f! 2 3 I 3z X 29 (x (3c + kg 3 A 3 [X 29 (x X 28 (x]g; (3d X 17(y =X 18 (y; (26a X 31(x =X 32 (x; (31a X 18(y=X 2 (y! 2 [k(g 6 A 6 +G 2 A 2 ] 1 f( 2 A 2 X 32(x =! 2 3 =G 3 X 31 (x; (31b + 6 A 6 [X 17 (y X 9 (h] + 2 A 2 yx 3 (hg; (26b X 19(y =X 2 (y; X 2(y =[E 6 I 6x + E 2 I 2x ] 1 f! 2 [ 2 I 2x + 6 I 6x ] (26c X 19 (y+k(g 6 A 6 +G 2 A 2 [X 19 (y X 18 (y]g; (26d EQUATIONS CORRESPONDING TO THE ROTATIONS 4, 5 AND 6 fk bc4! 2 [J bc4x + J bc5x + A 4 4 (L z l 2 + [(L z + z 3 z 3 ]=3]g 4! 2 fj bc5x +A 4 4 (L z l 2 + [(L z +z 3 z 3 ]=3g 6 +x s (L z +z 2 z 2 ]=2! 2 2 X 21(y =X 22 (y; X 22(y =! 2 2 =G 2 [X 21 (y+x 7 (h]: (27a (27b =! 2 [J bc4x +J bc5x +A 4 4 (L z l 2 +[(L z + z 3 z 3 ]=3]X 31 (x s +! 2 A 4 4 f[(l z + z 2

7 212 M.T. Ahmadian, G.R. Vossoughi and S. Ramezani z 2 ]=2][X 1 (h+x 27 (x s ] +[L z l 2 X 27 (x s + L z ly s X 3 (h+lx s X 21 (y s + L z lx 17 (y s + L z lx 9 (hg; (32 A 4 4 x s [(L z +z 2 z 2 ]=2]! 2 2 fj bc5x +A 4 4 (L z l 2 + [(L z + z 3 z 3 ]=3]g! 2 ( K bc5 5 =! 2 fj bc5x +A 4 4 (L z l 2 +[(L z + z 3 z 3 ]=3]g +! 2 A 4 4 f[(l z +z 2 z 2 ]=2][X 1 (h+x 27 (x s ] +L z l 2 (y s X 3 (h+x s X 21 (h X 9 (h+x 23 (x s ]g; (33 f! 2 (J bc6y + A 4 4 [(L z + z 2 z 2 ]=2] K bc6y g 6 =! 2 A 4 4 [(L z + z 2 z 2 ]=2][y s X 11 (h + X 13 (y s + X 5 (h]: (34 EQUATIONS IN Z-PINOLE X 53(z =X 54 (z; X 54(z =X 56 (z! 2 4 =kg 4 fx 53 (z+(z + z [X 41 (x s ] x s 2 + X 1 (h (35a + X 37 (h]g; (35b X 55(z =X 56 (z; (35c BOUNDARY CONDITIONS FOR LEFT COLUMN The left column is considered to be clamped at =. At = h, the boundary conditions are determined from Hamilton's principle as follows: X 1 ( = X 3 ( = X 5 ( = X 7 ( = X 9 ( = X 11 ( = ; [2A 2 L y 2 +M]X 1 (h+a 3 L 2 x 3 =2 2 kg 1 A 1 [X 3 (h X 2 (h] A 4 4 [(L z + z 2 z 2 ]=2X 31 (x s = ; [2A 2 L y 2 +M]X 1 (h+a 3 L 2 x 3=2 2 kg 1 A 1 [X 3 (h X 2 (h] A 4 4 [(L z + z 2 z 2 ]=2X 31 (x s =; [ 2 A 2 L 3 y=3+y 2 sm]x 3 (h E 1 I 1x X 4 (h[ 2 A 2 L 2 y=2 + y s M]X 9 (h+ y s MX 17 (h+y s [x s M bcz +A 3 3 L 2 x=2 + A 4 L z 4 x s ]X 21 (y s + y s [M bcz + A 4 L z 4 ]X 23 (x s +y s [x s M bcz +A 3 3 L 2 x =2+A 4L z 4 x s ]X 31 (x s = ; [ 2 A 2 L 2 y=2+y 2 sm]! 2 X 5 (h+kg 1 A 1 [X 7 (h X 6 (h] [M M bc3 ]! 2 X 13 (y s +[ 2 A 2 L 2 y=2 + y s (M M bc3 ]! 2 X 11 (h X56(z =1=E 4 I 4x f! 2 4 I 4x X 55 (z+kg 4 A 4 [X 55 (z X 54 (z]g; X57(z =X 58 (z; X 58(z =X 6 (z! 2 4 =kg 4 [X 57 (z+(z + z 6 (35d (36a = ;! 2 [ 2 I 2y L y +J bc3y ]X 7 (h+e 1 I 1y X 7 (h+j bc3y X 21 (y s = ;! 2 [2 2 A 2 L y + 6 A 6 L y +M]X 9 (E 1 A 1 +E 5 A 5 X 1 (h + y s X 11 (h+x 5 (h+x 13 (y s ]; (36b! 2 M[y s X 3 (h+x 17 (y s ] X 59(z =X 6 (z; X 6(z =1=E 4 I 4y f! 2 4 I 4y X 59 (z+kg 4 A 4 [X 59 (z X 58 (z]g: (36c (36d! 2 (M bcz +A 4 L z 4 [X 23 (x s +x s X 21 (y s ]! 2 la 4 L z 4 lx 1 (x s =; [3 2 A 2 L 2 y=2+y s M]X 5 (h+g 1 I 1z X 12 (h Now, the corresponding boundary conditions for each member are written. [2 2 A 2 L 3 y =3+y2 s M]X 11(h y s MX 11 (h =; (37

8 Gantry Type Coordinate Measuring Machines 213 where M =(M bc3 + M bcz + A 3 L x 3 + A 4 L z 4. BOUNDARY CONDITIONS OF Y -GUIDE WAY Because the motions of the y-guide way are relative to the left column, it can be assumed that it is clamped at y = h and free at y = L y. Continuity conditions must be satised at y = y s. X 13 ( = X 15 ( = X 26 (L y = X 17 ( =; X 19 ( = X 3 (L y = X 21 ( = X 32 (L y =; X 13 (y s X 23 (y s = X 24 (L y X 25 (L y =; X 15 (y s X 25 (y s = X 16 (y s X 26 (y s =; X 17 (y s X 27 (y s = X 28 (L y X 29 (L y =; X 5 (h+y s X 11 (h+x 13 (y s =X 21 (y s X 31 (y s =; M[y s X 3 (h X 9 (h]+[m bcz +A 3 3 L 2 x=2+a 4 L z 4 ] X 21 (y s +(M bcz + A 4 L z 4 X 33 (x s + la 4 L z 4 X 31 (x s =; X 19 (y s X 29 (y s = X 2 (y s X 3 (y s =; A 4 L z 4 lx s X 31 (x s + x s [M bcz + A 4 L z 4 ] [X 23 (x s + X 9 (h+x s X 17 (y s + y s X 3 (h]=: (38 BOUNDARY CONDITIONS OF THE X-GUIDE WAY The x-guide way is assumed to be a simply supported beam in the xz-plane and to be clamped-free in the xy-plane. It's torsion along the x-axis is assumed to be clamped at both ends. Continuity conditions must be satised at x = x s. X 33 (=X 43 (L x =X 35 (=X 36 (L x =X 37 (=; X 39 ( = X 5 (L x = X 41 ( = X 51 (L x =; X 33 (x s X 43 (x s = X 35 (x s X 45 (x s =; X 36 (x s X 46 (x s = X 37 (x s X 47 (x s =; la 4 L z 4 X 41 (x s +[M bcz + A 4 L z 4 ][X 17 (y s + X 33 (x s + y s X 3 (h X 9 (h]=; X 49 (L x X 48 (L x = X 39 (x s X 49 (x s =; X 4 (x s X 6 (x s = X 41 (x s X 51 (x s =; [J bc4x +J bc5x +A 4 4 (L z l 2 +[(L z +z 3 z 3 ]=3] X 31 (x s +A 4 4 [(L z +z 2 z 2 ]=2][X 1 (h + X 37 (x s ] + A 4 4 L z l[ X 9 (h+x 33 (x s + y s X 3 (h]=: (39 BOUNDARY CONDITIONS OF THE Z-PINOLE The motion of the z-pinole in the zx- and zy-planes is assumed to be clamped-free. X 53 ( = X 55 ( = X 56 (L z = X 57 ( = X 59 ( = X 6 (L z =; X 54 (L z X 55 (L z = X 58 (L z X 59 (L z =: (4 METHOD OF SOLUTION In order to obtain the natural frequencies of the system, initially, the rotations 2, 4, 5 and 6 will be found from Equations 27 and 32 to 34 and substituted in the model equations. Note that the derived equation can be listed into two categories. The rst category contains equations corresponding to the motions of the left column and may be written in the following form: fx (g =[D]fX(g; (41 where fxg = fx i ; X j ; :::;X n g t represents the vector of corresponding state variables. The sign ( denotes derivation, with respect to. Furthermore, the matrix, [D] is in the form: [D] =[K]! 2 [M]; (42 where [K] and[m] can be found readily by arranging the equations in terms of! 2. The solution of Equation 41 can be written as: fx g =[U] t [][U]fCg; (43 where [U] is the matrix of the eigenvectors of [D] and fcg is a column vector of constants. The diagonal matrix, [], is in the form:

9 214 M.T. Ahmadian, G.R. Vossoughi and S. Ramezani [] = e i ; (44 where i 's are eigenvalues of the dynamic matrix [D] depending on! 2. Taking = h, the value of X i (h in terms of C i 's can be found and will be used in other formulas. The second category of equations contains equations corresponding to the motion of members in the y- guide way, x-guide way and z-pinole. These equations are in the form: fx g =[D]fXg +[F ]; (45 where, in the above equation, the vector, fxg, may be a function of x, y or z. In order to solve the equation, initially, the matrix of the eigenvectors of [D], i.e. [U], should be found. Using the following transformation: fy g =[U]fXg; (46 it can be obtained: or: fy g =[U] t [D][U]fY g +[U] t [F ]; (47 fy g = []fy g +[U] t [F ]: (48 Because the matrix, [], is diagonal, an ordinary dierential equation for each Y i is obtained and can be solved readily. Note that for each Y i, there is a constant of integration, C i. Next, using Equation 46, each X i can be found. Taking y = y s and x = x s, the values of X i (x s or X i (y s can be found in terms of C i 's and implemented in other formulas. So far, X i 's are obtained in terms of C i 's and i 's. Now, substituting X i 's into boundary conditions and rearranging in terms of C i 's, the following equation may be obtained: [B]fCg = fg: (49 In order to have a nontrivial solution, the determinant of coecient matrix [B] must be zero, i.e.: Det[B] =: (5 In order to illustrate an example of modeling, a CMM with the following specications is modeled. i = 785 kg/m 3 ; E i = 2 Gpa; G i = 7 Gpa; h =:825 m; L y = :6 m; L x = 1:2 m; L z = :75 m; l =:16 m; z = :125 m; K bearings = 5e7 N.m/rad; A 1 = 7:6e 3m 2 ; A 2 = 4:6e 3m 2 ; A 3 = 7:6e 3m 2 ; A 4 = 1:e 3m 2 ; A 5 = 4:6e 3m 2 ; A 6 = 4:6e 3m 2 ; I 1x = 7:86e 5m 4 ; I 1y = 1:3e 5m 4 ; I 1z = 3:58e 5m 4 ; I 2x = 1:35e 5m 4 ; I 2y = 1:38e 6m 4 ; I 2z = 2:4e 5m 4 ; I 3x = 5:94e 5m 4 ; I 3y = 2:82e 5m 4 ; I 3z = 6:35e 5m 4 ; I 4x = 4:21e 7m 4 ; I 4y = 4:21e 7m 4 ; I 5x = 1:35e 5m 4 ; I 6x =1:35e 5m 4 ; I 6y =2:4e 5m 4 ; I 6z = 6:95e 5m 4 ; J bc3z = :18 kgm 4 ; J bc3z = :3 kgm 4 ; J bc4x = 2:7 kgm 4 ; J bc5x = :5 kgm 4 ; J bc6y = :5 kgm 4 ; M bc3 = 1 kg; M bcz = 35 kg: The natural frequencies of the system depend on the location of the probe. The variation of the 1st and 2nd natural frequencies versus the fxy g position of the probe is illustrated in the 3-D plots of Figures 2 and 3, when the tip of the probe is at its maximum distance from the x-guideway (z = L z. It is clear that increasing the values of x s and y s reduces the natural frequencies of the system. Note that the driving systems of the x- and y-motions are located at the right side of the CMM, consequently, the x-guide way slides on the left y-guide way, and the minimum value of the natural frequency occurs when the probe is located at the end of the x-guide way. The variation of the 1st to 4th natural frequencies of the system at dierent fxy g positions of the probe (z = L z, obtained from the This yields an equation in terms of! 2. Because the presented model is based on continuous beam modeling, the obtained equation is transcendental and has an innite number of solutions. For each value of x s and y s, it can solve for! 2. Figure 2. First natural frequency (z = L z.

10 Gantry Type Coordinate Measuring Machines 215 Figure 3. Second natural frequency (z = L z. analytical and nite elements method, are presented in Figures 4 to 7. Findings from the analytical and nite elements method indicate high accuracy and good agreement between the results. Figure 5. Natural frequencies in Y = 2 for X = to X = 12 cm. CONCLUSION The modeling of a gantry type CMM, based on Timoshenko beam theory using Hamilton's principle, was developed and the natural frequencies of the system were calculated at various positions of the probe in the system. The parameters of motion are dened so that the kinetic energy of each element is inuenced by the motion of adjacent connected elements, while the strain energy is independent of the position and velocity of these elements. Results indicate that in the rst four natural frequencies, for any y-position of the probe, the system has the lowest natural frequency when the probe moves toward the end of the x-guide way. Moreover, the values of natural frequencies again decrease when the system moves toward the end of the y-guide way. However, when the system is at the end of both x- and y-guide ways, the minimum value of the Figure 6. Natural frequencies in Y = 4 for X = to X = 12 cm. Figure 4. Natural frequencies in Y = for X =to X = 12 cm. Figure 7. Natural frequencies in Y = 5 for X = to X = 12 cm (Z = L z.

11 216 M.T. Ahmadian, G.R. Vossoughi and S. Ramezani natural frequencies occurs. This is due to the minimum stiness of the system in that position. These frequencies help us to present any resonant when the system is under operation. Any interference between the systems natural frequencies and the frequencies of the driving system should be prevented. A good agreement and high accuracy in comparison with the results found by the nite element analysis can be observed. 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, 2, 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 (2. 4. De Nijs, J.F.C., Schellekens, M.G.M. and Van der Wolf, A.C.H. \Modeling of a coordinate measuring machine for analysis of its dynamic behavior", Annals of the CIRP, 37, pp ( Vermeulen, M. \High precision 3D-coordinate measuring machine, design and prototype development", PhD Thesis, Eindhoven University of Technology ( 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 analysis 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 analysis of gantry type CMMs and optimization using genetic algorithm", M.Sc. Thesis, Sharif University of Technology, Tehran, Iran (23.