This paper presents a theoretic and practical approach to confirm the dynamic errors induced by acceleration on CMMs during fast measuring. To simplify, a physical model, a mathematical model of the bridge component and the prismatic joint that move along X axis of a bridge CMM are presented.
Science & Technology Development, Vol 15, No.K1- 2012 INFLUENCE OF ACCELERATION ON THE CMMS ACCURACY IN FAST MEASURING Thai Thi Thu Ha, Pham Hong Thanh University of Technology, VNU-HCM (Manuscript Received on April 5th, 2012, Manuscript Revised November 20rd, 2012) ABSTRACT: This paper presents a theoretic and practical approach to confirm the dynamic errors induced by acceleration on CMMs during fast measuring To simplify, a physical model, a mathematical model of the bridge component and the prismatic joint that move along X axis of a bridge CMM are presented The research points out that there is sure to be the dynamic errors when there is acceleration This has been demonstrated by the simulation of the errors on the tip probe by Matlab sofware and the experimental results are obtained The dynamic errors will decrease the accuracy of CMMs so it is necessary to make out a compensation software to keep up the the accuracy of the measuring results Keywords: Fast measuring, Fast probing, CMMs, Accuracy, Velocity, Acceleration, Dynamic error INTRODUCTION Today, CMMs have been developed in the trend that they’re kept up their accuracy during fast measuring [2], [4], [12] When CMMs are measuring at high speed, there are some kinetic elements, which cause the dynamic errors and reduce the accuracy of CMMs They are the approach distance to the surface, the speed of the approach to the surface, t/he acceleration of Y X the approach to the surface and the direction of the approach to the surface [5] Therefore, it is Z necessary to research the influence of these kinetic elements on the accuracy of CMMs and suggest some methods or some directions of Figure The whole of structure of a moving solution to make sure the demand CMM’s bridge CMM accuracy in fast probing STRUCTURE BRIDGE CMM [14] Trang 60 OF Structure elements serve as the backbone THE MOVING of CMM The machine base, table to support the part to be measured, machine columns, TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 15, SỐ K1- 2012 slide ways and probe shaft are essential structural elements [1] Y X Y Z X Figure The prismatic joint that moves along X Z axis of a moving bridge CMM PHYSICAL MODEL OF THE BRIDGE COMPONENT AND THE PRISMATIC Figure Table, bridge and prismatic joint of a JOINT MOVING ALONG THE Y AXIS moving bridge CMM X eX m eY KX , KY δ1 L δ2 G O Y k’ k a) Dynamic model of δ X (t ) and δX(1) a b δX(2) k ε Z (t ) ; b) Simplified model of δ X (t ) and ε Z (t ) Figure Top view of the CMM physical model depicts two component errors δ X (t ) and ε Z (t ) To create the physical model of CMM as springs When the probe tip runs at hight speed, shown in figure 4, the bridge is regarded as a accelerates or decelerates, inertia force induced mass point m attached to an elastic beam L and it makes the bridge, the prismatic joint and The air bearings, which links the elastic beam the slideways deformed The translated error to the X slideway are supposed as three along the X axis ( δ X ) and the rotated error Trang 61 Science & Technology Development, Vol 15, No.K1- 2012 about the Z axis ( ε Z ) due to inertia force are Where: + M is the inertia matrix determined as [2]: + C is the damping matrix a × δ X (2) + b × δ X (1) δ X (t ) = (1) a+b δ ( ) − δ ( ) X X ε (t ) = Z a+b + K is the stiffness matrix + F is the external excitation matrix + x is the displacement matrix MATHEMATICAL MODEL OF THE THE Application of the Newton’s law for the PRISMATIC JOINT MOVING ALONG physical model in figure result in the THE X AXIS –THE DYNAMIC ERRORS following equations of motion: BRIDGE COMPONENT AND OCCURSED ON THE TIP OF THE In the Y direction: PROBE DURING FAST MEASURING me&&Y (t ) + K Y eY (t ) + K Y Lε Z (t ) = (3) In this case, the influence of acceleration By using the moment equilibrium on the accuracy of CMM will be investigated condition of the bridge about point G, the when probe tip run fast along Y axis To yield relationship of the KY, K, k and k’ stiffness is the wanted mathematical model from the shown: physical model of CMM (figure 4), some K Y= K following assumptions are used: - The stiffness in X and Y direction of the elastic beam are: KX, KY 2k k ' (a) a2 + b2 ; K= 2k + k ' L To produce the solutions of the differential equation (3), this equation is presented in other - The distant between the carriage takes Z axis and Y slideway is L and the weight of the carriage takes Z axis is m - The air bearings, which links the elastic beam to the Y slideway are regarded as three springs and their stiffness are k and k’ - The errrors of the probe tip in X and Y direction are ex and ey, respectively form: &e&Y (t ) + K a + b2 a2 + b2 ( ) e t = − K ε Z (t ) (b) Y mL mL2 The general solution of the homogeneous linear differential equation has to be found: &e&Y (t ) + K The The equation of motion of the multi-degree a + b2 eY (t ) = m.L2 characteristic equation (c) of the differential equation (c) has two root, they are: of freedom linear system has the following form [3]: λ1 = i K M&x& + Cx& + Kx = F (t ) Trang 62 (2) a2 + b2 a2 + b2 ; λ2 = −i K m.L m.L2 (d) TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 15, SỐ K1- 2012 By using Euler’s formula, the general solution of the homogeneous linear differential equation (c) is written : a + b a + b (e) eY =C 1cos K t + C sin K t m.L m.L2 To get the solution of the the differential equation (3) and it is presented by the following form: a + b2 eY = − Lε Z (t ) cos 2. K m.L2 The natural frequency in the X direction: non- homogeneous linear differential equation (b), we will use the variation of constants method with some assumptions: - C1, C2 are functions (4) .t ωY = ( K a2 + b2 m.L2 ) (5) The eY error on the probe tip in the Y direction in fast measuring is simulated by MatLab software: - C1, C2 are chosen: a + b a + b C1′ cos K t + C2′ sin K t = (f) m.L m.L2 Hence, the C1′;C 2′ are determined by the function d=ptvp(t,ey) F0=100;K=60000;m=66.44;a=230; b=230;L=500;t1=1.0;t2=5; set of following differential equations: w0=sqrt(K*(a^2+b^2)/(m*L^2)); a2 +b2 a2 +b2 C1′ cos K t +C2′ sin K t =0 m L m.L2 a2 +b2 a2 +b2 a2 +b2 a2 +b2 sin K cos K t +C2′ K t =0 −C1′ K 2 m.L m.L m.L2 m.L2 if t>=t1 & t=t1 & t