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The effect of axis coupling on machine tool dynamics determined by tool deviation
International Journal of Machine Tools & Manufacture 88 (2015) 71–81
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International Journal of Machine Tools & Manufacture journal homepage: www.elsevier.com/locate/ijmactool
The effect of axis coupling on machine tool dynamics determined by tool deviation Lei Wang a, Haitao Liu b, Lei Yang a, Jun Zhang a, Wanhua Zhao a,n, Bingheng Lu a a b
State Key Laboratory for Manufacturing Systems Engineering, Xi'an Jiaotong University, Xi'an, Shaanxi 710054, China School of Mechanical Engineering, Xi'an Technological University, Xi'an, Shaanxi 710032, China
art ic l e i nf o
a b s t r a c t
Article history: Received 21 April 2014 Received in revised form 4 September 2014 Accepted 4 September 2014 Available online 16 September 2014
High acceleration forces of machine tool with kinetic coupling as the dominating coupling forces may deform the machine structure and result in the tool deviation. In this paper, a dynamic model of a threeaxis gantry milling machine tool considering axis coupling effects is proposed to model the varying dynamic behavior and evaluate the Tool Center Point (TCP) position deviations. The effect of axis coupling force on the stiffness changes of kinematic joints is analyzed. The variations of the frequencies and frequency response functions with respect to position parameters are calculated. And the TCP deviation affected by axial coupling in real-time motion state is discussed in detail. The results show that it is able to obtain an excellent match between the simulations and the measurements. The simulation and experimental results show that: (1) the natural frequencies and the receptance are greatly changed when the TCP is moving along the X-axis or the Z-axis, where the maximum changing of natural frequencies is up to 10% and response magnitude up to 2 times; (2) the elastic deformation and vibration of machine tool are caused by the coupling forces in acceleration and braking, which detrimentally affect dynamic response of the TCP. Thus, the model proposed in this paper represents the important effects for comprehension of machine dynamic behavior and for further compensation in future. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Axis coupling Dynamic model Tool Center Point Deviation
1. Introduction With the development of high-speed machine tool, the effect of kinetic coupling in typical multi-axis machine tool is dominated by acceleration-dependent disturbance forces, which may deform the machine structure and enhance the interaction between the motion axes. Due to coupling forces, the dynamic performance of machine tool is restricted. During the whole motion space of the Tool Center Point (TCP), because of the machine structure configuration changes and axis coupling force, the dynamic characteristics of system is complicated [1–3], which may result in elastic vibration and then affect the motion control accuracy of the TCP. In order to predict and avoid the unfavorable factors, the dynamic performance of machine tools must be obtained through simulations or measurements. Zaghbani et al. [4] presented a methodology for estimating of machine tool dynamic modal parameters during machining operation using operational modal analysis. Kono et al. [5] had developed the Axis Construction Kit (ACK) for dynamic simulations of machine tool. Ertürk et al. [6] proposed an analytical modeling of spindle-tool dynamics on machine tools using
n
Corresponding author. Tel.: þ 86 29 83399520. E-mail addresses: [email protected], [email protected] (W. Zhao).
http://dx.doi.org/10.1016/j.ijmachtools.2014.09.003 0890-6955/& 2014 Elsevier Ltd. All rights reserved.
Timoshenko beam model and receptance coupling for the prediction of frequency response function of tool point. Neugebauer et al. [7] integrated the control unit in the finite element modeling of machine tool for analysis of dynamic characteristics of machine tool. Mi et al. [8] and Hung et al. [9] used the finite element method to analyze the dynamic effect of preload of kinematic joints on the dynamic characteristics of machine tool. Altintas [10] described and summarized the application of the finite element method in the machine tool dynamics. Obviously, the static and dynamic characteristics of the machine tool will be position-dependent during motion. Therefore, the model based on single state in finite element method cannot predict these variations. Although the modified finite element analysis approach can be used, they are not “transparent” and do not allow an understanding the influence of state parameters. Law et al. [11,12] discussed the position dependent dynamics of serial– parallel kinematic machine and three-axis vertical milling machine tool using reduced order finite element models, without having modeled the stiffness at the joints. Symens et al. [13,14] studied the varying of eigenfrequencies of the machine with the variable configuration depending on the position of the tool for designing the high-performance motion controllers. Da Silva et al. [15] discussed the integrated design of mechatronic systems with varying dynamics. However, the coupling effects among multiple axes are neglected in most machine tool design and feed drive controllers [16].
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The movements of one axis can disturb other axes in a kinematic chain, while the disturbance forces and torques are called coupling forces. In machine tool terminology, this interaction is called “crosstalk” [17] or “coupling”.Uchiyama [16] presents a robust adaptive controller design for the X–Y axis feed drives systems to compensate the coupling effects. Thoma et al. [3], Zirn et al. [18] describe the coupling force of linear and rotary axes, and proposed the control strategy to reduce the coupling effect. These simplified dynamic model yields the coupling force model ignoring the effect of the stiffness of kinematic joints, although joint characteristics severely affect the response at tool point. Because inertial force from one axial motion affects the contact force between joints in other axes, the actual stiffness of kinematic joints may deviate from the initial conditions. Both Dhupia et al. [19] and Hung [20] found that the rolling guide joints exhibit a variation in stiffness owing to different preloads and external loads. This nonlinearity of joints may result in significant variation in magnitude and frequency of machine's FRF [21], which is not taken into account in analysis generally. It will limit the further design of the control system and increase the complexity of control algorithm for TCP position control accuracy. On the other hand, the virtual machine and virtual machining [10,22] need more accurate model to predict the dynamic performance of the system. Therefore, it is necessary to study the dynamic modeling of multi-axis coupling machine tool to analysis the effect of axis coupling on the dynamic performance of TCP. In this paper, a gantry milling machine tool is considered as a typical dynamic system to study the effect of axis coupling. The effect of axis coupling force on the stiffness changes of kinematic joints is analyzed. The variations of the frequencies and frequency response functions with respect to position parameters are also calculated. Finally, the TCP deviation affected by axial coupling in real-time motion state is discussed.
2. Dynamic modeling 2.1. Dynamic model of a milling machine tool The milling machine tool, which is shown in Fig. 1, is considered as the typical multi-axis coupling dynamic system. In
conventional machine tools, due to intensive FE optimization of the machine structure in design, the structural parts are much stiffer than the joints. The structural modal is at a relatively high frequency [23], and the predominant modes in the low-frequency and mid-frequency are entirely caused by limited joint stiffness [24,25]. Thus the dynamics of the system is mainly dependent on the stiffness of the joint rather than the stiffness of structures themselves. It is appropriate to model the machine tool by three lumped mass elements with 18 degrees of freedom, as shown in Fig. 1(b). The beam–ram–headstock-spindle system is modeled by three lumped mass elements and a set of translational and angular spring elements for each kinematic joint. The stiffness values may depend on the system configuration (coordinates X, Y and Z-axis) and the motion state in real time. In the Cartesian coordinates, it can position the tool block m4 in the three-dimensional space by means of three translational motions along the X–Y–Z axes, respectively, and we define that the nominal rigid body motion coordinates of the centers of gravity of m1 (beam), m2 (ram) and m3 (headstock-spindle) are (0, y1g), (x2g, y1g) and (x2g, y1g, z3g), which describe the system configuration respectively. Their actual motion coordinates are the synthesis of the nominal rigid body motion and the elastic terms caused by deformations of all springs. Since both translational and angular motions of the beam, ram and headstock-spindle have important effects, both masses and moments of inertia of these bodies must be considered. The mass m4 of the tool is also considered, but the inertia is neglected. So the expressions of kinetic energy Ek and potential energy Ep are described by Eqs. (1) and (2). Ek ¼
1 3 1 4 2 _ 2 þJ ϕ _2 _2 ∑ mi x_ 2i þ y_ 2i þ z_ i þ ∑ J ix ϕ iy iy þ J iz ϕiz ix 2i¼1 2i¼1
ð1Þ
Ep ¼
1 3 1 3 2 2 2 ∑ ki u2i þ v2i þ w2i þ ∑ kiϕx ϕix þ kiϕy ϕiy þ kiϕz ϕiz 2i¼1 2i¼1
ð2Þ
where k1x, k1y, k1z, k2x, k2y, k2z, k3x, k3y and k3z are the stiffness of joints for the X–Y–Z direction; u1, v1, w1, u2, v2, w2, u3, v3 and w3 are the elastic deformation of joints for the X–Y–Z direction;k1φx, k1φy, k1φz, k2φx, k2φy, k2φz, k3φx, k3φy and k3φz are the rotation angle
Fig. 1. A gantry milling machine tool and its dynamic model.
L. Wang et al. / International Journal of Machine Tools & Manufacture 88 (2015) 71–81
stiffness of joints for the A, B and C direction; ϕ1x, ϕ1y, ϕ1z, ϕ2x, ϕ2y, ϕ2z, ϕ3x, ϕ3y, and ϕ3z are the angle deformation of joints for the A, B and C direction; J1x, J1y, J1z, J2x, J2y, J2z, J3x, J3y and J3z are the inertia of beam, ram and headstock-spindle; m1, m2, m3, and m4 are the mass of beam, ram, headstock-spindle and tool. Compared with the magnitudes of the nominal velocity and acceleration of high speed machine, the increments of both deformation and velocities of system motions which are caused by structure compliances are small. So the terms containing products of more than two of them can be assumed to be neglected in the expression for kinetic energy Ek. The Lagrange equation is: d ∂Ek ∂E ∂Ep kþ ¼ Qi ð3Þ dt ∂q_ i ∂qi ∂qi Then, after the application of Lagrange equations for Eqs. (1) and (2), the elastic deformation motion equations are listed in Appendix section. The motion differential equations of the system can be written as the general form of the dynamics equation: MðgðtÞÞq€ ðtÞ þ CðgðtÞ;g_ ðtÞÞq_ ðtÞ þ K gðtÞ;g_ ðtÞ;g€ ðtÞ qðtÞ ¼ F gðtÞ;g_ ðtÞ;g€ ðtÞ þ Q
where δn is the normal deformation, and Kn represents the Hertz constant, which is determined by the contact geometry and material properties of the linear components. Qn denotes the normal contact force. The normal stiffness under a specific preload can be obtained: kgi ¼
dF dδ
ð4Þ
Q ¼ f 1 ; f 2 ; f 3 ; f 4 ; f 5 ; f 6 ; f 7 ; f 8 ; f 9 ; f 10 ; f 11 ; f 12 ; f 13 ; f 14 ; f 15 ; f 16 ; f 17 ; f 18
T
ð5Þ where f i ¼ ci q_ i , ci is the viscous damping coefficient.
In the example discussed here, the linear rolling guide joints connecting the beam and the bed are shown in Fig. 2. The contact force between a rolling ball and the raceway in the linear guide can be related to the local deformation at the contact point by the Hertz theory [27]: 2
ðm1 Z 1 þ m2 Z 2 þ m3 Z 3 þm4 Z 4 Þ y€ 1g 2b2
F 2_n ¼ F 3_n ¼
ð6Þ
ðm1 Z 1 þ m2 Z 2 þ m3 Z 3 þm4 Z 4 Þ y€ 1g 2b2
ð8Þ
In the X-axis motion process, the change of the external load of linear rolling guides is: ðm2 þ m3 þ m4 Þg x2g 2L ðm2 þ m3 þ m4 Þg L x2g F 3_n ¼ F 4_n ¼ 2L
F 1_n ¼ F 2_n ¼
F 1_τ ¼ F 2_τ ¼ ðm2 þ m3 þ m4 Þx€ 2g F 3_τ ¼ F 4_τ ¼ ðm2 þ m3 þ m4 Þx€ 2g
ð9Þ
In the Z-axis motion process, the change of the external load of linear rolling guides is: F 1_n ¼ F 4_n ¼
ðm3 Z 3 þ m4 Z 4 Þ z€ 3g þ ðm3 þ m4 Þ z€ 3g 2b2
F 2_n ¼ F 3_n ¼
2.2. Stiffness nonlinear change of kinematic joints under axis coupling
δn ¼ K n U Q 3n
ð7Þ
As shown in Eqs. (6) and (7), the contact stiffness depends nonlinearly on the contact force, which is essentially determined by the initial preload and the actual load. In the initial state, the external loads of the kinematic joints and the initial preload is in static equilibrium state, and the stiffness is a definite value. But in the motion process of TCP, the time-varying structure configuration and the axis coupling force of multi-axis machine will affect the contact force of the kinematic joints, and then the actual stiffness of kinematic joints. In the Y-axis motion process, the change of the external load of linear rolling guides is: F 1_n ¼ F 4_n ¼
where q(t)¼{u1, v1, w1, u2, v2, w2, u3, v3, w3, ϕ1x, ϕ1y, ϕ1z, ϕ2x, ϕ2y, ϕ2z, ϕ3x, ϕ3y, ϕ3z} is the generalized coordinates; g(t) is the rigid body motion coordinate; M, C and K are 18 18 inertia matrix, the damping matrix and the stiffness matrix, respectively. These coefficient matrices are the function of system motion state. The general dynamic model includes centrifugal, Coriolis and acceleration forces. For machine tools, the acceleration forces are by far the dominating coupling forces. The equation of motion is obviously nonlinear, which indicates the significant coupling between axes as well as the substantial influence of the machine tool configuration on the dynamic characteristics of the system. If it is necessary to consider the vibration damping [26], the damping effect can be equivalent to the viscous force, then substituted to Eq. (4):
73
ðm3 Z 3 þ m4 Z 4 Þ z€ 3g þ ðm3 þ m4 Þ z€ 3g 2b2
ð10Þ
Substituting Eqs. (8)–(10) into Eq. (7), we can obtain the nonlinear change of the stiffness of linear guides depending on the path planning. Here, the variation of stiffness with acceleration of the Y-axis of a linear guide is show in Fig. 3. It is also found that the difference of actual load reaches about 3 times among each slider, and varies with position shown in Fig. 4(a). Fig. 4(b) shows the position-dependent stiffness of the linear guide.
Fig. 2. The linear rolling guide joints of the beam and the bed: (a) side view, and (b) top-down view.
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3. Experimental verification of dynamic model Obviously, considering the effect of the instantaneous position of each axis motion, Eq. (4) is reduced to linear dynamic equations, and then the M, K, and C are dependent on the position of the tool. A gantry milling machine tool of DX1200 is considered as a typical dynamic system. In order to analyze the vibration characteristics of the machine tool, three typical spatial positions of TCP are chosen to solve the eigenvalue problem of Eq. (4). The values of the parameters in this paper are listed in Table 1. In order to validate the accuracy of the dynamic model, the experimental modal analysis is conducted by impact testing using an impact hammer(PCB, 086C20) and 4 three-directional accelerometers(PCB, 356A16), which are shown in Fig. 5. The number of measurement points is 27, and the bandwidth is 500 Hz in this experiment. The impact position is excited in the Y-direction, and then the sampling signal is analyzed by the LMS Test.Lab software. Since lower structural modes are focused on rigid body modes and the modes with the natural frequencies higher than 300 Hz are neglected. The first four modes are discussed here, as shown in Fig. 6. The mode shape of the first frequency is a hybrid of translatory TCP vibrations in the Y-direction and nodding TCP vibrations in the Zdirection. Mode 2 results in X-vibrations of the TCP. Mode 3 is a rotary mode of the beam about the Z-axis. Mode 4 is a deflection of the beam guide carriers in the Z-direction. The frequency responses of TCP in the Y-direction are picked up, respectively. The comparison of experimental and simulation results is shown in Fig. 7, where it can be seen that the simulation results of the modeled position-dependent dynamic characteristics are essentially coincident with the test,and the differences
between the calculated and the measured natural frequencies are listed in Table 2.
4. Varying dynamics analysis of machine tool The frequency responses of TCP in the Y-direction when the TCP is at 9 instantaneous positions are shown in Fig. 8(a)–(c). It can be seen that the frequency response functions change heavily in some frequency range, where the maximum changing of natural frequencies is up to 10%. Fig. 8(a) shows the variation of the frequency response function of the TCP depending on zg in the Zdirection. Fig. 8(b) shows the variation of the frequency response function of the TCP depending on xg in the X-direction, and the amplitude response of TCP changes up to 2 times greater than the minimum at the frequency of 230 Hz. Due to the very heavy gantry beam, variations of natural frequency and mode shape depend on the TCP position in the Y-direction are small as shown in Fig. 8(c). In the whole movement space range, the variation of the frequency response function of the TCP in the X-direction with the position parameter xg and zg changing are shown in Fig. 9, where the exciting force is also in the X-direction. It is shown that the natural frequencies and corresponding amplitude vary greatly when the machine tool is under different position states.
5. Effect of axis coupling on the dynamic response of the TCP The motion process is described by three order polynomial step function, which is plotted with dotted curves in Figs. 10–13. It is assuming that the time of axis motion is in the range of 0–2.5 s Table 1 Parameters for the gantry milling machine.
Fig. 3. The variation of stiffness with acceleration of the Y-axis of a linear guide.
Parameters
Value
m1 (kg) J1x,J1y,J1z (kg m2) L (mm) m2 (kg) J2x,J2y,J2z (kg m2) b2 (mm) c1 (mm) m3 (kg) m4 (kg) J3x,J3y,J3z (kg m2) c2 (mm) h3 (mm) h4 (mm)
1454 969,923,92 2320 240 19,8,12 420 172 638 8 210,14,206 283 420 630
Fig. 4. The variation of actual load and stiffness of the linear guide: (a) the variation of actual load, and (b) the variation of stiffness.
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Fig. 5. Impact modal testing.
Fig. 6. First four modes of the machine tool.
with the nominal feed speed v¼ 18 m/min, acceleration a ¼0.3 g. Thus the speed curve is divided into six phases: 0.5–0.6 s as the acceleration phase, 0.6–0.9 s as the uniform velocity phase, 0.9– 1.0 s as the deceleration phase; 1.5–1.6 s as the reverse acceleration phase, 1.6–1.9 s as the reverse uniform velocity phase and 1.9– 2.0 s as the reverse the deceleration phase. Using the speed curves as the input command in the servo motor, the real time position deviations curve of the TCP is obtained.
5.1. Coupling effects on the dynamic response of the TCP under single axis motion The effect of the coupling forces is shown in Fig. 10 for a fast positioning operation of the machine tool. As shown in Fig. 10, acceleration and braking caused by the Xaxis motion produce heavy coupling forces, which cause the elastic deformation and vibration of machine tool, then affect the
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Fig. 7. TCP frequency responses of modal test and simulation in the Y-direction: (a) the FRFs of TCP at the coordination of xg ¼0; yg ¼0; zg ¼ 0; (b) the FRFs of TCP at the coordination of xg ¼ 0; yg ¼ 0; zg ¼ 275 mm; and (c) the FRFs of TCP at the coordination of xg ¼0; yg ¼0; zg ¼ 550 mm. Table 2 The differences between the calculated and the measured natural frequencies. Order
1 2 3 4 5
xg ¼0; yg ¼ 0; zg ¼0
xg ¼0; yg ¼ 0; zg ¼275 mm
xg ¼0; yg ¼ 0; zg ¼550 mm
Simulations (Hz)
Test (Hz)
Errors (%)
Simulations (Hz)
Test (Hz)
Errors (%)
Simulations (Hz)
Test (Hz)
Errors (%)
51 140 220 – 295
48.89 134.54 224.10 – 288.01
4.32 4.06 1.43
52 155 230 245 285
50.17 151.25 228.92 239.82 280.92
3.74 2.79 0.48 2.16 1.42
51 142 225 – 298
49.99 135.08 223.84 – 293.51
2.07 5.14 0.52
2.43
1.57
Fig. 8. The frequency response of the TCP at the different position: (a) the frequency response of the TCP at the coordination of xg ¼ 0,yg ¼ 0,zg ¼ 0,275,550 (mm); (b) the frequency response of the TCP at the coordination of xg ¼ 425,425,0 (mm); yg ¼ 0; zg ¼ 0; and (c) the frequency response of the TCP at the coordination of xg ¼ 0; yg ¼ 900,0,900; zg ¼ 0 (mm).
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Fig. 9. The variation of the frequency response function of the TCP. (a) the FRF changing depends on xg in the X-direction (zg ¼ 500 mm); and (b) the FRF changing depend on zg in the Z-direction (xg ¼ 800 mm).
Fig. 10. The real time position deviations curve of the TCP: (a) Y-direction deviation caused by X-axis motion; (b) Z-direction deviation caused by X-axis motion; (c) X-direction deviation caused by Y-axis motion; (d) Z-direction deviation caused by Y-axis motion; (e) X-direction deviation caused by Z-axis motion; and (f) Y-direction deviation caused by Z-axis motion.
displacement deviation of the TCP. These coupling forces lead to significant position deviations in the range of 30 μm in the Ydirection, and 3 μm for the Z-axis. In the delayed phase of acceleration/deceleration, a period of vibration of elastic mechanism will continue for some seconds.
Y-axis accelerations are causing similar effects which cause the elastic deformation and vibration of machine tool, then affect the displacement deviation of the TCP. They lead to significant position deviations in the range of 38 μm in the Z-direction, and 2.8 μm for the X-axis. The influence of Y-axis movement to X-axis is small,
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Fig. 11. The position deviations of the TCP under two-axis motion: (a) the position deviation of Z-direction for X–Y axis motion; (b) the position deviation of X-direction for Y–Z axis motion; and (c) the position deviation of Y-direction for X–Z axis motion.
Fig. 12. The measurement system of TCP position deviations.
but a greater impact on the Z-axis, that because Y-axis acceleration and deceleration will cause the nodding vibration of spindle. The acceleration and braking of the Z-axis produce heavy coupling forces, which affect significant position deviations of the TCP in the range of 25 μm for the Y-direction, and 2.5 μm for the X-axis. The influence of Z-axis movement to Y-axis depends on the position of TCP in the Z-direction. As shown in Fig. 10(f), the position deviations of the tool have difference vibration amplitude
at different time, where the motion position changing with the corresponding time. 5.2. Coupling effects on the dynamic response of the TCP under twoaxis motion The effects of two axes movement for position deviations of the TCP is demonstrated in Fig. 11. The two-axis coupling has the
L. Wang et al. / International Journal of Machine Tools & Manufacture 88 (2015) 71–81
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Fig. 13. The measurement of the TCP position deviations: (a) Y-direction deviation caused by X-axis motion; (b) Z-direction deviation caused by X-axis motion; (c) X-direction deviation caused by Y-axis motion; (d) Z-direction deviation caused by Y-axis motion; (e) X-direction deviation caused by Z-axis motion; and (f) Y-direction deviation caused by Z-axis motion.
similar effects of position deviations, which detrimentally affect the dynamic response of the TCP. Because of the influence of nonlinear stiffness of kinematic joints caused by axis coupling, the effects are not the simple supervision of the effects of the single axis movement. Acceleration and braking produce heavy coupling forces, which cause the elastic deformation and vibration of machine tool, then affect the displacement deviation of the TCP. The effects of position deviations by X–Y axes and X–Z axes are serious in the range of 40 μm for the Z-direction and in the range of 52 μm for the Y-direction. But the effects by Y–Z axes are so small that can be ignored. In the delayed phase of acceleration/ deceleration, a period of vibration of elastic mechanism will continue for some seconds, and because of the gravity, it can be seen the static deformation error of the system.
5.3. Experimental measurements the axis coupling effects of the TCP A Laser TRACER measurement system is used to validate the results of the model, which is shown in Fig. 12. With the same
initial condition as modeling, the sampling signals of TCP are analyzed by the Trac-CAL software. In addition to the amplitude of position deviations, the twoaxis coupling has the similar axis coupling effects with the single axis motion. Thus the experiments are focused on the TCP position deviations under single axis motion, and the coupling effects of the TCP by two-axis coupling motion are not validated. The measurements of the TCP position deviations under single-axis motion are shown in Fig. 13. Since the straightness error is not taken into account in the simulation, but the static straightness error is observed in the measurement result. Comparing Figs. 10 and 13, it can be seen that the results are quite good in accordance with the amplitude during acceleration and braking of the machine tool between the simulation and the measurement. However, parameters like stiffness, especially damping, are difficult to estimate and are subject to changing due to the variation of lubrication and temperature, so far it is impossible to obtain more accurate results. Therefore, there are still moderate quantitative differences between the modeling and the actual motion, which is part of future research.
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€ þ m3 h3 þ m4 ðh3 þ h4 Þ ðm3 þ m4 Þz3g ϕ 2y
€ 2 ðm þ m Þz_ ϕ _ ðm þm Þz€ ϕ þ m4 h4 ϕ 3 4 3g 3 4 3g 3y 2y 2y þ ðm2 þ m3 þ m4 Þx€ 2g þ k2x u2 ¼ 0
However, the modeled position deviations of the TCP represent the important effects for comprehension of the TCP dynamic behavior and future compensation.
6. Conclusion (1) A dynamic model considering axis coupling effects is proposed to model the varying dynamic behavior and evaluate the position deviations of the Tool Center Point (TCP) of a threeaxis milling machine tool. It was demonstrated that the position-dependent response of the TCP as well as the axis coupling impact the position deviation of the TCP. The results showed that it is able to obtain an excellent match between the measurements and the simulations. Using explicit equations, the dynamic model has the ability to quickly set up and modify the parameters of a machine tool model, which is a very useful and efficient tool for future sensitivity and parametric analyses. (2) The axis coupling force on the nonlinear stiffness changes of kinematic joints are analyzed, and the variations of the frequencies and frequency response functions with respect to position parameters are also analyzed. It can be concluded that the natural frequencies and corresponding amplitude vary greatly when the machine tool is under different position states. One can see that the system parameters, such as xg, yg, zg, x_ g , y_ g , z_ g , x€ g , y€ g and z€ g are changed fast in working process, and then both natural frequencies and corresponding amplitude of dynamic response are also changed and the unstable condition would eventually appear, which can adversely affect the accuracy of the TCP of programmed motions. (3) The TCP deviation affected by axial coupling in real-time motion state is discussed. Because of the influence of nonlinear stiffness of kinematic joints caused by axis coupling, the effects are not the simple superposition of the effects of the single axis movement. It can also be concluded that acceleration and braking produce heavy coupling forces, which detrimentally affect the dynamic response of the TCP. The straightforward evaluation information on machine tool specific qualities (natural frequencies, frequency responses, tool deviations, etc.) during the conception phase will become practicable without time-consuming by using the dynamic model.
€ ðm3 þm4 Þu€ 1 þ ðm3 þ m4 Þu€ 2 þ ðm3 þ m4 Þu€ 3 þ ðm3 þ m4 Þc1 ϕ 1z
€ € þ ðm3 þ m4 Þc2 ϕ2z þ m3 h3 þ m4 ðh3 þh4 Þ ðm3 þ m4 Þz3g ϕ 2y
_ € _ þ m4 h4 ϕ3y 2 ðm3 þ m4 Þz3g ϕ2y
ðm3 þ m4 Þz€ 3g ϕ2y þ ðm3 þ m4 Þx€ 2g þ k3x u3 ¼ 0
This work is supported by the key project of the National Natural Science Foundation of China (No. 51235009). Authors are grateful to other participants of the projects for their cooperation.
ðm1 þm2 þ m3 þ m4 Þv€ 1 þ ðm2 þ m3 þ m4 Þv€ 2 þ ðm3 þ m4 Þv€ 3 þ ðm2 þ m3 € € þ m h þ m ðh þ h Þ ðm þ m Þz ϕ þ m4 Þx2g ϕ 3 3 4 3 4 3 4 3g 1z 2x
_ € þ m4 h4 ϕ3x þ2 ðm2 þ m3 þ m4 Þx_ 2g ϕ1z
_
€ þ ðm2 þ m3 þm4 Þx€ 2g ϕ1z 2 ðm3 þ m4 Þz_ 3g ϕ 2x ðm3 þ m4 Þz3g ϕ2x þ ðm1 þ m2 þ m3 þ m4 Þy€ 1g þk1y v1 ¼ 0
The elastic deformation motion equations are listed as: ðm1 þ m2 þ m3 þ m4 Þu€ 1 þ ðm2 þ m3 þ m4 Þu€ 2 þ ðm3 þm4 Þu€ 3 € þ ðm þ m Þc ϕ € þ ðm2 þ m3 þm4 Þc1 ϕ 3 4 2 2z 1z
€ € þ m3 h3 þ m4 ðh3 þ h4 Þ ðm3 þm4 Þz3g ϕ2y þ m4 h4 ϕ 3y
_
2 ðm3 þ m4 Þz_ 3g ϕ2y ðm3 þ m4 Þz€ 3g ϕ2y þ ðm2 þ m3 þm4 Þx€ 2g þk1x u1 ¼ 0 ðm2 þ m3 þ m4 Þu€ 1 þ ðm2 þ m3 þ m4 Þu€ 2 þ ðm3 þ m4 Þu€ 3 þ ðm2 þ m3 € þ ðm þ m Þc ϕ € þ m Þc ϕ 1z
3
ðA:4Þ
ðm2 þm3 þ m4 Þv€ 1 þ ðm2 þ m3 þ m4 Þv€ 2 þ ðm3 þ m4 Þv€ 3 € þ m h þ m ðh þ h Þ þ ðm2 þ m3 þ m4 Þx2g ϕ 3 3 4 3 4 1z € _ € þ 2 ðm þ m þm Þx_ ϕ ðm3 þ m4 Þz3g ϕ þm h ϕ 4 4 3x 2 3 4 2g 2x 1z
_
þ ðm2 þ m3 þm4 Þx€ 2g ϕ1z 2 ðm3 þ m4 Þz_ 3g ϕ2x ðm3 þ m4 Þz€ 3g ϕ2x þ ðm2 þ m3 þ m4 Þy€ 1g þ k2y v2 ¼ 0
ðA:5Þ
€ ðm3 þm4 Þv€ 1 þ ðm3 þ m4 Þv€ 2 þ ðm3 þ m4 Þv€ 3 þ ðm3 þ m4 Þx2g ϕ 1z
_
€ þ 2 ðm3 þm4 Þx_ 2g ϕ1z þ m3 h3 þ m4 ðh3 þ h4 Þ ðm3 þ m4 Þz3g ϕ 2x € þ ðm þ m Þx€ ϕ 2 ðm þm Þz_ ϕ _ þ m4 h4 ϕ 3 4 2g 3 4 3g 3x 1z 2x
ðm3 þ m4 Þz€ 3g ϕ2x þ ðm3 þ m4 Þy€ 1g þ k3y v3 ¼ 0 ðA:6Þ € 1 þ ðm2 þm3 þ m4 Þw € 2 þ ðm3 þ m4 Þw €3 ðm1 þm2 þ m3 þ m4 Þw € € ðm2 þ m3 þ m4 Þc1 ϕ1x ðm3 þ m4 Þc2 ϕ2x þ ðm3 þm4 Þz€ 3g þ k1z w1 ¼ 0 ðA:7Þ € 1 þ ðm2 þ m3 þm4 Þw € 2 þ ð m3 þ m4 Þ w €3 ðm2 þm3 þ m4 Þw € ðm þ m Þc ϕ € þ ðm þm Þz€ ðm2 þ m3 þ m4 Þc1 ϕ 3 4 2 3 4 3g 1x 2x þ k2z w2 ¼ 0
ðA:8Þ
€ € 1 þ ð m3 þ m4 Þ w € 2 þ ðm3 þ m4 Þw € 3 ðm3 þ m4 Þc1 ϕ ðm3 þm4 Þw 1x € € ðm þ m Þc ϕ þ ðm þ m Þz þ k w ¼ 0
ðA:9Þ
3
4
2
2x
3
4
3g
3z
3
ðm3 þ m4 Þc1 z€ 3g ¼ 0
h i h i € þ J þJ ϕ € þJ ϕ € J 1y þ J 2y þ J 3y ϕ 2y 3y 3y 3y þ k1ϕy ϕ1y ¼ 0 1y 2y h
Appendix
1
ðA:3Þ
€ J 1x þ J 2x þ J 3x þ ðm2 þ m3 þ m4 Þc21 ϕ 1x
€ € þ J 2x þ J 3x þ ðm3 þ m4 Þc1 c2 ϕ2x þJ 3x ϕ 3x € 2 ðm3 þ m4 Þc1 w € 1 þw € 3 þ k1ϕx ϕ1x ðm2 þ m3 þ m4 Þc1 w
Acknowledgments
4
ðA:2Þ
4
2
2z
ðA:1Þ
ðA:10Þ ðA:11Þ
i € J 1z þ J 2z þ J 3z þ ðm2 þ m3 þ m4 Þ c21 þ x22g ϕ 1z
€ € þ J 2z þ J 3z þ ðm3 þ m4 Þc1 c2 ϕ2z þ J 3z ϕ3z þ ðm2 þ m3 þ m4 Þc1 u€ 1 þ u€ 2 þ ðm3 þm4 Þc1 u€ 3 þ ðm2 þ m3 þ m4 Þx2g v€ 1 þ v€ 2 þ ðm3 þ m4 Þx2g v€ 3
€ þ m3 h3 z3g þ m4 h3 þ h4 z3g x2g ϕ 2x
€ þ m3 h3 z3g þ m4 h3 þ h4 z3g c1 ϕ2y h i _ þ ðm þm þ m Þx x€ þ k þ 2ðm2 þ m3 þ m4 Þx2g x_ 2g ϕ 2 3 4 2g 2g 1ϕz ϕ1z 1z _ 2ðm3 þ m4 Þz_ 3g c1 ϕ 2y
_ 2ðm3 þ m4 Þz€ 3g c1 ϕ2y 2 m3 x2g z_ 3g þ m4 x2g z_ 3g ϕ 2x m3 x2g z€ 3g þ m4 x2g z€ 3g þ m3 x_ 2g z_ 3g þ m4 x_ 2g z_ 3g
L. Wang et al. / International Journal of Machine Tools & Manufacture 88 (2015) 71–81
m3 x2g z_ 3g m4 x2g z_ 3g ϕ2x € þm h x ϕ € _ _ 2g ϕ þ m4 h 4 c 1 ϕ 4 4 2g 3x þ m4 h4 x 3y 3x þ ðm2 þ m3 þ m4 Þx€ 2g c1 þ ðm2 þ m3 þ m4 Þy€ 1g x2g ¼ 0
€ J 2x þ J 3x þ ðm3 þ m4 Þc1 c2 ϕ 1x h 2 2 i € þ J 2x þ J 3x þ ðm3 þ m4 Þc22 þ m3 h3 zg þ m4 h3 þh4 zg ϕ2x
€ þ J 3x ϕ3x þ m3 h3 zg þ m4 h3 þ h4 zg v€ 1 þ v€ 2 þ v€ 3 _ € 1 þw € 2 þw € 3 m4 h4 z_ 3g ϕ ðm3 þ m4 Þc2 w 3x
€ þ m3 h3 zg þ m4 h3 þ h4 zg x2g ϕ1z 2 m3 z_ 3g h3 z3g _ þ m4 z_ 3g h3 þ h4 z3g ϕ 2x h i þ k2ϕx m3 z€ 3g h3 zg þm4 z€ 3g h3 þ h4 zg ϕ2x
_ þ 2 m3 x_ 2g h3 zg þ m4 x_ 2g h3 þ h4 zg ϕ
1z þ m3 x€ 2g h3 zg þm4 x€ 2g h3 þ h4 zg ϕ1z € € þ m4 h4 h3 þ h4 z3g ϕ 3x þ m3 y1g h3 zg ðA:13Þ þ m4 y€ 1g h3 þ h4 zg ¼ 0
h
ðA:12Þ
i h i € þ J þ J þ m h z 2 þ m h þ h z 2 ϕ € J 2y þ J 3y ϕ g g 3 3 4 3 4 2y 3y 1y 2y
_ € þ J 3y ϕ3y 2 m3 z_ 3g h3 zg þ m4 z_ 3g h3 þ h4 zg ϕ2y h i þ k2ϕy m3 z€ 3g h3 zg þ m4 z€ 3g h3 þ h4 zg ϕ2y
€ þ m3 h3 zg þ m4 h3 þ h4 zg c1 ϕ 1z € _ _ z þ m4 h4 h3 þ h4 z3g ϕ m h ϕ 4 4 3g 3y 3y
€ þ m3 h3 zg þ m4 h3 þ h4 zg c2 ϕ
2z þ m3 h3 zg þ m4 h3 þ h4 zg u€ 1 þ u€ 2 þ u€ 3 ðA:14Þ þ m3 x€ 2g h3 zg þ m4 x€ 2g h3 þh4 zg ¼ 0
€
€ 2 € J 2z þJ 3z þ ðm3 þ m4 Þc1 c2 ϕ 1z þ J 2z þ J 3z þ ðm3 þ m4 Þc2 ϕ2z þ J 3z ϕ3z
€ € þ m3 h3 z3g þm4 h3 þh4 z3g c2 ϕ 2y þ m4 h4 ϕ3y _ ðm þm Þz€ c ϕ 2ðm3 þm4 Þz_ 3g c2 ϕ 3 4 3g 2 2y 2y þ ðm3 þ m4 Þc2 u€ 1 þ u€ 2 þ u€ 3 þ m3 x€ 2g c1 þ m4 x€ 2g c2 þ k2ϕz ϕ2z ¼ 0 ðA:15Þ
h
i
€ 2 € _ _ J 3x þ m4 h4 ϕ 3x þ m4 h4 h3 þh4 zg ϕ2x 2m4 z3g h4 ϕ2x € þ 2m x_ h ϕ _ m4 z€ 3g h4 ϕ2x þ m4 h4 x2g ϕ 4 2g 4 1z 1z € € € € þ m4 x2g h4 ϕ1z þ m4 h4 v1 þ v2 þ v3 þ m4 h4 y€ 1g þ k3ϕx ϕ3x ¼ 0 ðA:16Þ
h
J 3y þ m4 h4
2
i
ϕ€ 3y þ m4 h4 h3 þ h4 zg ϕ€ 2y 2m4 h4 z_ 3g ϕ_ 2y
€ m4 z€ 3g h4 ϕ2y þ m4 h4 u€ 1 þ u€ 2 þ u€ 3 þ m4 h4 c1 ϕ 1z € þ k ϕ þ m h x€ ¼ 0 þ m4 h 4 c 2 ϕ 4 4 2g 3ϕy 3y 2z
€ þk ϕ ¼ 0 J 3z ϕ 3ϕz 3z 3z
ðA:17Þ ðA:18Þ
Reference [1] A. Verl, S. Frey, Correlation between feed velocity and preloading in ball screw drives, CIRP Ann. – Manuf. Technol. 59 (1) (2010) 429–432. [2] R. Neugebauer, B. DenkenaK., Mechatronic Wegener, Systems for Machine Tools, CIRP Ann. – Manuf. Technol. 56 (2) (2007) 657–686.
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Contents lists available at ScienceDirect
International Journal of Machine Tools & Manufacture journal homepage: www.elsevier.com/locate/ijmactool
The effect of axis coupling on machine tool dynamics determined by tool deviation Lei Wang a, Haitao Liu b, Lei Yang a, Jun Zhang a, Wanhua Zhao a,n, Bingheng Lu a a b
State Key Laboratory for Manufacturing Systems Engineering, Xi'an Jiaotong University, Xi'an, Shaanxi 710054, China School of Mechanical Engineering, Xi'an Technological University, Xi'an, Shaanxi 710032, China
art ic l e i nf o
a b s t r a c t
Article history: Received 21 April 2014 Received in revised form 4 September 2014 Accepted 4 September 2014 Available online 16 September 2014
High acceleration forces of machine tool with kinetic coupling as the dominating coupling forces may deform the machine structure and result in the tool deviation. In this paper, a dynamic model of a threeaxis gantry milling machine tool considering axis coupling effects is proposed to model the varying dynamic behavior and evaluate the Tool Center Point (TCP) position deviations. The effect of axis coupling force on the stiffness changes of kinematic joints is analyzed. The variations of the frequencies and frequency response functions with respect to position parameters are calculated. And the TCP deviation affected by axial coupling in real-time motion state is discussed in detail. The results show that it is able to obtain an excellent match between the simulations and the measurements. The simulation and experimental results show that: (1) the natural frequencies and the receptance are greatly changed when the TCP is moving along the X-axis or the Z-axis, where the maximum changing of natural frequencies is up to 10% and response magnitude up to 2 times; (2) the elastic deformation and vibration of machine tool are caused by the coupling forces in acceleration and braking, which detrimentally affect dynamic response of the TCP. Thus, the model proposed in this paper represents the important effects for comprehension of machine dynamic behavior and for further compensation in future. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Axis coupling Dynamic model Tool Center Point Deviation
1. Introduction With the development of high-speed machine tool, the effect of kinetic coupling in typical multi-axis machine tool is dominated by acceleration-dependent disturbance forces, which may deform the machine structure and enhance the interaction between the motion axes. Due to coupling forces, the dynamic performance of machine tool is restricted. During the whole motion space of the Tool Center Point (TCP), because of the machine structure configuration changes and axis coupling force, the dynamic characteristics of system is complicated [1–3], which may result in elastic vibration and then affect the motion control accuracy of the TCP. In order to predict and avoid the unfavorable factors, the dynamic performance of machine tools must be obtained through simulations or measurements. Zaghbani et al. [4] presented a methodology for estimating of machine tool dynamic modal parameters during machining operation using operational modal analysis. Kono et al. [5] had developed the Axis Construction Kit (ACK) for dynamic simulations of machine tool. Ertürk et al. [6] proposed an analytical modeling of spindle-tool dynamics on machine tools using
n
Corresponding author. Tel.: þ 86 29 83399520. E-mail addresses: [email protected], [email protected] (W. Zhao).
http://dx.doi.org/10.1016/j.ijmachtools.2014.09.003 0890-6955/& 2014 Elsevier Ltd. All rights reserved.
Timoshenko beam model and receptance coupling for the prediction of frequency response function of tool point. Neugebauer et al. [7] integrated the control unit in the finite element modeling of machine tool for analysis of dynamic characteristics of machine tool. Mi et al. [8] and Hung et al. [9] used the finite element method to analyze the dynamic effect of preload of kinematic joints on the dynamic characteristics of machine tool. Altintas [10] described and summarized the application of the finite element method in the machine tool dynamics. Obviously, the static and dynamic characteristics of the machine tool will be position-dependent during motion. Therefore, the model based on single state in finite element method cannot predict these variations. Although the modified finite element analysis approach can be used, they are not “transparent” and do not allow an understanding the influence of state parameters. Law et al. [11,12] discussed the position dependent dynamics of serial– parallel kinematic machine and three-axis vertical milling machine tool using reduced order finite element models, without having modeled the stiffness at the joints. Symens et al. [13,14] studied the varying of eigenfrequencies of the machine with the variable configuration depending on the position of the tool for designing the high-performance motion controllers. Da Silva et al. [15] discussed the integrated design of mechatronic systems with varying dynamics. However, the coupling effects among multiple axes are neglected in most machine tool design and feed drive controllers [16].
72
L. Wang et al. / International Journal of Machine Tools & Manufacture 88 (2015) 71–81
The movements of one axis can disturb other axes in a kinematic chain, while the disturbance forces and torques are called coupling forces. In machine tool terminology, this interaction is called “crosstalk” [17] or “coupling”.Uchiyama [16] presents a robust adaptive controller design for the X–Y axis feed drives systems to compensate the coupling effects. Thoma et al. [3], Zirn et al. [18] describe the coupling force of linear and rotary axes, and proposed the control strategy to reduce the coupling effect. These simplified dynamic model yields the coupling force model ignoring the effect of the stiffness of kinematic joints, although joint characteristics severely affect the response at tool point. Because inertial force from one axial motion affects the contact force between joints in other axes, the actual stiffness of kinematic joints may deviate from the initial conditions. Both Dhupia et al. [19] and Hung [20] found that the rolling guide joints exhibit a variation in stiffness owing to different preloads and external loads. This nonlinearity of joints may result in significant variation in magnitude and frequency of machine's FRF [21], which is not taken into account in analysis generally. It will limit the further design of the control system and increase the complexity of control algorithm for TCP position control accuracy. On the other hand, the virtual machine and virtual machining [10,22] need more accurate model to predict the dynamic performance of the system. Therefore, it is necessary to study the dynamic modeling of multi-axis coupling machine tool to analysis the effect of axis coupling on the dynamic performance of TCP. In this paper, a gantry milling machine tool is considered as a typical dynamic system to study the effect of axis coupling. The effect of axis coupling force on the stiffness changes of kinematic joints is analyzed. The variations of the frequencies and frequency response functions with respect to position parameters are also calculated. Finally, the TCP deviation affected by axial coupling in real-time motion state is discussed.
2. Dynamic modeling 2.1. Dynamic model of a milling machine tool The milling machine tool, which is shown in Fig. 1, is considered as the typical multi-axis coupling dynamic system. In
conventional machine tools, due to intensive FE optimization of the machine structure in design, the structural parts are much stiffer than the joints. The structural modal is at a relatively high frequency [23], and the predominant modes in the low-frequency and mid-frequency are entirely caused by limited joint stiffness [24,25]. Thus the dynamics of the system is mainly dependent on the stiffness of the joint rather than the stiffness of structures themselves. It is appropriate to model the machine tool by three lumped mass elements with 18 degrees of freedom, as shown in Fig. 1(b). The beam–ram–headstock-spindle system is modeled by three lumped mass elements and a set of translational and angular spring elements for each kinematic joint. The stiffness values may depend on the system configuration (coordinates X, Y and Z-axis) and the motion state in real time. In the Cartesian coordinates, it can position the tool block m4 in the three-dimensional space by means of three translational motions along the X–Y–Z axes, respectively, and we define that the nominal rigid body motion coordinates of the centers of gravity of m1 (beam), m2 (ram) and m3 (headstock-spindle) are (0, y1g), (x2g, y1g) and (x2g, y1g, z3g), which describe the system configuration respectively. Their actual motion coordinates are the synthesis of the nominal rigid body motion and the elastic terms caused by deformations of all springs. Since both translational and angular motions of the beam, ram and headstock-spindle have important effects, both masses and moments of inertia of these bodies must be considered. The mass m4 of the tool is also considered, but the inertia is neglected. So the expressions of kinetic energy Ek and potential energy Ep are described by Eqs. (1) and (2). Ek ¼
1 3 1 4 2 _ 2 þJ ϕ _2 _2 ∑ mi x_ 2i þ y_ 2i þ z_ i þ ∑ J ix ϕ iy iy þ J iz ϕiz ix 2i¼1 2i¼1
ð1Þ
Ep ¼
1 3 1 3 2 2 2 ∑ ki u2i þ v2i þ w2i þ ∑ kiϕx ϕix þ kiϕy ϕiy þ kiϕz ϕiz 2i¼1 2i¼1
ð2Þ
where k1x, k1y, k1z, k2x, k2y, k2z, k3x, k3y and k3z are the stiffness of joints for the X–Y–Z direction; u1, v1, w1, u2, v2, w2, u3, v3 and w3 are the elastic deformation of joints for the X–Y–Z direction;k1φx, k1φy, k1φz, k2φx, k2φy, k2φz, k3φx, k3φy and k3φz are the rotation angle
Fig. 1. A gantry milling machine tool and its dynamic model.
L. Wang et al. / International Journal of Machine Tools & Manufacture 88 (2015) 71–81
stiffness of joints for the A, B and C direction; ϕ1x, ϕ1y, ϕ1z, ϕ2x, ϕ2y, ϕ2z, ϕ3x, ϕ3y, and ϕ3z are the angle deformation of joints for the A, B and C direction; J1x, J1y, J1z, J2x, J2y, J2z, J3x, J3y and J3z are the inertia of beam, ram and headstock-spindle; m1, m2, m3, and m4 are the mass of beam, ram, headstock-spindle and tool. Compared with the magnitudes of the nominal velocity and acceleration of high speed machine, the increments of both deformation and velocities of system motions which are caused by structure compliances are small. So the terms containing products of more than two of them can be assumed to be neglected in the expression for kinetic energy Ek. The Lagrange equation is: d ∂Ek ∂E ∂Ep kþ ¼ Qi ð3Þ dt ∂q_ i ∂qi ∂qi Then, after the application of Lagrange equations for Eqs. (1) and (2), the elastic deformation motion equations are listed in Appendix section. The motion differential equations of the system can be written as the general form of the dynamics equation: MðgðtÞÞq€ ðtÞ þ CðgðtÞ;g_ ðtÞÞq_ ðtÞ þ K gðtÞ;g_ ðtÞ;g€ ðtÞ qðtÞ ¼ F gðtÞ;g_ ðtÞ;g€ ðtÞ þ Q
where δn is the normal deformation, and Kn represents the Hertz constant, which is determined by the contact geometry and material properties of the linear components. Qn denotes the normal contact force. The normal stiffness under a specific preload can be obtained: kgi ¼
dF dδ
ð4Þ
Q ¼ f 1 ; f 2 ; f 3 ; f 4 ; f 5 ; f 6 ; f 7 ; f 8 ; f 9 ; f 10 ; f 11 ; f 12 ; f 13 ; f 14 ; f 15 ; f 16 ; f 17 ; f 18
T
ð5Þ where f i ¼ ci q_ i , ci is the viscous damping coefficient.
In the example discussed here, the linear rolling guide joints connecting the beam and the bed are shown in Fig. 2. The contact force between a rolling ball and the raceway in the linear guide can be related to the local deformation at the contact point by the Hertz theory [27]: 2
ðm1 Z 1 þ m2 Z 2 þ m3 Z 3 þm4 Z 4 Þ y€ 1g 2b2
F 2_n ¼ F 3_n ¼
ð6Þ
ðm1 Z 1 þ m2 Z 2 þ m3 Z 3 þm4 Z 4 Þ y€ 1g 2b2
ð8Þ
In the X-axis motion process, the change of the external load of linear rolling guides is: ðm2 þ m3 þ m4 Þg x2g 2L ðm2 þ m3 þ m4 Þg L x2g F 3_n ¼ F 4_n ¼ 2L
F 1_n ¼ F 2_n ¼
F 1_τ ¼ F 2_τ ¼ ðm2 þ m3 þ m4 Þx€ 2g F 3_τ ¼ F 4_τ ¼ ðm2 þ m3 þ m4 Þx€ 2g
ð9Þ
In the Z-axis motion process, the change of the external load of linear rolling guides is: F 1_n ¼ F 4_n ¼
ðm3 Z 3 þ m4 Z 4 Þ z€ 3g þ ðm3 þ m4 Þ z€ 3g 2b2
F 2_n ¼ F 3_n ¼
2.2. Stiffness nonlinear change of kinematic joints under axis coupling
δn ¼ K n U Q 3n
ð7Þ
As shown in Eqs. (6) and (7), the contact stiffness depends nonlinearly on the contact force, which is essentially determined by the initial preload and the actual load. In the initial state, the external loads of the kinematic joints and the initial preload is in static equilibrium state, and the stiffness is a definite value. But in the motion process of TCP, the time-varying structure configuration and the axis coupling force of multi-axis machine will affect the contact force of the kinematic joints, and then the actual stiffness of kinematic joints. In the Y-axis motion process, the change of the external load of linear rolling guides is: F 1_n ¼ F 4_n ¼
where q(t)¼{u1, v1, w1, u2, v2, w2, u3, v3, w3, ϕ1x, ϕ1y, ϕ1z, ϕ2x, ϕ2y, ϕ2z, ϕ3x, ϕ3y, ϕ3z} is the generalized coordinates; g(t) is the rigid body motion coordinate; M, C and K are 18 18 inertia matrix, the damping matrix and the stiffness matrix, respectively. These coefficient matrices are the function of system motion state. The general dynamic model includes centrifugal, Coriolis and acceleration forces. For machine tools, the acceleration forces are by far the dominating coupling forces. The equation of motion is obviously nonlinear, which indicates the significant coupling between axes as well as the substantial influence of the machine tool configuration on the dynamic characteristics of the system. If it is necessary to consider the vibration damping [26], the damping effect can be equivalent to the viscous force, then substituted to Eq. (4):
73
ðm3 Z 3 þ m4 Z 4 Þ z€ 3g þ ðm3 þ m4 Þ z€ 3g 2b2
ð10Þ
Substituting Eqs. (8)–(10) into Eq. (7), we can obtain the nonlinear change of the stiffness of linear guides depending on the path planning. Here, the variation of stiffness with acceleration of the Y-axis of a linear guide is show in Fig. 3. It is also found that the difference of actual load reaches about 3 times among each slider, and varies with position shown in Fig. 4(a). Fig. 4(b) shows the position-dependent stiffness of the linear guide.
Fig. 2. The linear rolling guide joints of the beam and the bed: (a) side view, and (b) top-down view.
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3. Experimental verification of dynamic model Obviously, considering the effect of the instantaneous position of each axis motion, Eq. (4) is reduced to linear dynamic equations, and then the M, K, and C are dependent on the position of the tool. A gantry milling machine tool of DX1200 is considered as a typical dynamic system. In order to analyze the vibration characteristics of the machine tool, three typical spatial positions of TCP are chosen to solve the eigenvalue problem of Eq. (4). The values of the parameters in this paper are listed in Table 1. In order to validate the accuracy of the dynamic model, the experimental modal analysis is conducted by impact testing using an impact hammer(PCB, 086C20) and 4 three-directional accelerometers(PCB, 356A16), which are shown in Fig. 5. The number of measurement points is 27, and the bandwidth is 500 Hz in this experiment. The impact position is excited in the Y-direction, and then the sampling signal is analyzed by the LMS Test.Lab software. Since lower structural modes are focused on rigid body modes and the modes with the natural frequencies higher than 300 Hz are neglected. The first four modes are discussed here, as shown in Fig. 6. The mode shape of the first frequency is a hybrid of translatory TCP vibrations in the Y-direction and nodding TCP vibrations in the Zdirection. Mode 2 results in X-vibrations of the TCP. Mode 3 is a rotary mode of the beam about the Z-axis. Mode 4 is a deflection of the beam guide carriers in the Z-direction. The frequency responses of TCP in the Y-direction are picked up, respectively. The comparison of experimental and simulation results is shown in Fig. 7, where it can be seen that the simulation results of the modeled position-dependent dynamic characteristics are essentially coincident with the test,and the differences
between the calculated and the measured natural frequencies are listed in Table 2.
4. Varying dynamics analysis of machine tool The frequency responses of TCP in the Y-direction when the TCP is at 9 instantaneous positions are shown in Fig. 8(a)–(c). It can be seen that the frequency response functions change heavily in some frequency range, where the maximum changing of natural frequencies is up to 10%. Fig. 8(a) shows the variation of the frequency response function of the TCP depending on zg in the Zdirection. Fig. 8(b) shows the variation of the frequency response function of the TCP depending on xg in the X-direction, and the amplitude response of TCP changes up to 2 times greater than the minimum at the frequency of 230 Hz. Due to the very heavy gantry beam, variations of natural frequency and mode shape depend on the TCP position in the Y-direction are small as shown in Fig. 8(c). In the whole movement space range, the variation of the frequency response function of the TCP in the X-direction with the position parameter xg and zg changing are shown in Fig. 9, where the exciting force is also in the X-direction. It is shown that the natural frequencies and corresponding amplitude vary greatly when the machine tool is under different position states.
5. Effect of axis coupling on the dynamic response of the TCP The motion process is described by three order polynomial step function, which is plotted with dotted curves in Figs. 10–13. It is assuming that the time of axis motion is in the range of 0–2.5 s Table 1 Parameters for the gantry milling machine.
Fig. 3. The variation of stiffness with acceleration of the Y-axis of a linear guide.
Parameters
Value
m1 (kg) J1x,J1y,J1z (kg m2) L (mm) m2 (kg) J2x,J2y,J2z (kg m2) b2 (mm) c1 (mm) m3 (kg) m4 (kg) J3x,J3y,J3z (kg m2) c2 (mm) h3 (mm) h4 (mm)
1454 969,923,92 2320 240 19,8,12 420 172 638 8 210,14,206 283 420 630
Fig. 4. The variation of actual load and stiffness of the linear guide: (a) the variation of actual load, and (b) the variation of stiffness.
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75
Fig. 5. Impact modal testing.
Fig. 6. First four modes of the machine tool.
with the nominal feed speed v¼ 18 m/min, acceleration a ¼0.3 g. Thus the speed curve is divided into six phases: 0.5–0.6 s as the acceleration phase, 0.6–0.9 s as the uniform velocity phase, 0.9– 1.0 s as the deceleration phase; 1.5–1.6 s as the reverse acceleration phase, 1.6–1.9 s as the reverse uniform velocity phase and 1.9– 2.0 s as the reverse the deceleration phase. Using the speed curves as the input command in the servo motor, the real time position deviations curve of the TCP is obtained.
5.1. Coupling effects on the dynamic response of the TCP under single axis motion The effect of the coupling forces is shown in Fig. 10 for a fast positioning operation of the machine tool. As shown in Fig. 10, acceleration and braking caused by the Xaxis motion produce heavy coupling forces, which cause the elastic deformation and vibration of machine tool, then affect the
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Fig. 7. TCP frequency responses of modal test and simulation in the Y-direction: (a) the FRFs of TCP at the coordination of xg ¼0; yg ¼0; zg ¼ 0; (b) the FRFs of TCP at the coordination of xg ¼ 0; yg ¼ 0; zg ¼ 275 mm; and (c) the FRFs of TCP at the coordination of xg ¼0; yg ¼0; zg ¼ 550 mm. Table 2 The differences between the calculated and the measured natural frequencies. Order
1 2 3 4 5
xg ¼0; yg ¼ 0; zg ¼0
xg ¼0; yg ¼ 0; zg ¼275 mm
xg ¼0; yg ¼ 0; zg ¼550 mm
Simulations (Hz)
Test (Hz)
Errors (%)
Simulations (Hz)
Test (Hz)
Errors (%)
Simulations (Hz)
Test (Hz)
Errors (%)
51 140 220 – 295
48.89 134.54 224.10 – 288.01
4.32 4.06 1.43
52 155 230 245 285
50.17 151.25 228.92 239.82 280.92
3.74 2.79 0.48 2.16 1.42
51 142 225 – 298
49.99 135.08 223.84 – 293.51
2.07 5.14 0.52
2.43
1.57
Fig. 8. The frequency response of the TCP at the different position: (a) the frequency response of the TCP at the coordination of xg ¼ 0,yg ¼ 0,zg ¼ 0,275,550 (mm); (b) the frequency response of the TCP at the coordination of xg ¼ 425,425,0 (mm); yg ¼ 0; zg ¼ 0; and (c) the frequency response of the TCP at the coordination of xg ¼ 0; yg ¼ 900,0,900; zg ¼ 0 (mm).
L. Wang et al. / International Journal of Machine Tools & Manufacture 88 (2015) 71–81
77
Fig. 9. The variation of the frequency response function of the TCP. (a) the FRF changing depends on xg in the X-direction (zg ¼ 500 mm); and (b) the FRF changing depend on zg in the Z-direction (xg ¼ 800 mm).
Fig. 10. The real time position deviations curve of the TCP: (a) Y-direction deviation caused by X-axis motion; (b) Z-direction deviation caused by X-axis motion; (c) X-direction deviation caused by Y-axis motion; (d) Z-direction deviation caused by Y-axis motion; (e) X-direction deviation caused by Z-axis motion; and (f) Y-direction deviation caused by Z-axis motion.
displacement deviation of the TCP. These coupling forces lead to significant position deviations in the range of 30 μm in the Ydirection, and 3 μm for the Z-axis. In the delayed phase of acceleration/deceleration, a period of vibration of elastic mechanism will continue for some seconds.
Y-axis accelerations are causing similar effects which cause the elastic deformation and vibration of machine tool, then affect the displacement deviation of the TCP. They lead to significant position deviations in the range of 38 μm in the Z-direction, and 2.8 μm for the X-axis. The influence of Y-axis movement to X-axis is small,
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Fig. 11. The position deviations of the TCP under two-axis motion: (a) the position deviation of Z-direction for X–Y axis motion; (b) the position deviation of X-direction for Y–Z axis motion; and (c) the position deviation of Y-direction for X–Z axis motion.
Fig. 12. The measurement system of TCP position deviations.
but a greater impact on the Z-axis, that because Y-axis acceleration and deceleration will cause the nodding vibration of spindle. The acceleration and braking of the Z-axis produce heavy coupling forces, which affect significant position deviations of the TCP in the range of 25 μm for the Y-direction, and 2.5 μm for the X-axis. The influence of Z-axis movement to Y-axis depends on the position of TCP in the Z-direction. As shown in Fig. 10(f), the position deviations of the tool have difference vibration amplitude
at different time, where the motion position changing with the corresponding time. 5.2. Coupling effects on the dynamic response of the TCP under twoaxis motion The effects of two axes movement for position deviations of the TCP is demonstrated in Fig. 11. The two-axis coupling has the
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79
Fig. 13. The measurement of the TCP position deviations: (a) Y-direction deviation caused by X-axis motion; (b) Z-direction deviation caused by X-axis motion; (c) X-direction deviation caused by Y-axis motion; (d) Z-direction deviation caused by Y-axis motion; (e) X-direction deviation caused by Z-axis motion; and (f) Y-direction deviation caused by Z-axis motion.
similar effects of position deviations, which detrimentally affect the dynamic response of the TCP. Because of the influence of nonlinear stiffness of kinematic joints caused by axis coupling, the effects are not the simple supervision of the effects of the single axis movement. Acceleration and braking produce heavy coupling forces, which cause the elastic deformation and vibration of machine tool, then affect the displacement deviation of the TCP. The effects of position deviations by X–Y axes and X–Z axes are serious in the range of 40 μm for the Z-direction and in the range of 52 μm for the Y-direction. But the effects by Y–Z axes are so small that can be ignored. In the delayed phase of acceleration/ deceleration, a period of vibration of elastic mechanism will continue for some seconds, and because of the gravity, it can be seen the static deformation error of the system.
5.3. Experimental measurements the axis coupling effects of the TCP A Laser TRACER measurement system is used to validate the results of the model, which is shown in Fig. 12. With the same
initial condition as modeling, the sampling signals of TCP are analyzed by the Trac-CAL software. In addition to the amplitude of position deviations, the twoaxis coupling has the similar axis coupling effects with the single axis motion. Thus the experiments are focused on the TCP position deviations under single axis motion, and the coupling effects of the TCP by two-axis coupling motion are not validated. The measurements of the TCP position deviations under single-axis motion are shown in Fig. 13. Since the straightness error is not taken into account in the simulation, but the static straightness error is observed in the measurement result. Comparing Figs. 10 and 13, it can be seen that the results are quite good in accordance with the amplitude during acceleration and braking of the machine tool between the simulation and the measurement. However, parameters like stiffness, especially damping, are difficult to estimate and are subject to changing due to the variation of lubrication and temperature, so far it is impossible to obtain more accurate results. Therefore, there are still moderate quantitative differences between the modeling and the actual motion, which is part of future research.
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€ þ m3 h3 þ m4 ðh3 þ h4 Þ ðm3 þ m4 Þz3g ϕ 2y
€ 2 ðm þ m Þz_ ϕ _ ðm þm Þz€ ϕ þ m4 h4 ϕ 3 4 3g 3 4 3g 3y 2y 2y þ ðm2 þ m3 þ m4 Þx€ 2g þ k2x u2 ¼ 0
However, the modeled position deviations of the TCP represent the important effects for comprehension of the TCP dynamic behavior and future compensation.
6. Conclusion (1) A dynamic model considering axis coupling effects is proposed to model the varying dynamic behavior and evaluate the position deviations of the Tool Center Point (TCP) of a threeaxis milling machine tool. It was demonstrated that the position-dependent response of the TCP as well as the axis coupling impact the position deviation of the TCP. The results showed that it is able to obtain an excellent match between the measurements and the simulations. Using explicit equations, the dynamic model has the ability to quickly set up and modify the parameters of a machine tool model, which is a very useful and efficient tool for future sensitivity and parametric analyses. (2) The axis coupling force on the nonlinear stiffness changes of kinematic joints are analyzed, and the variations of the frequencies and frequency response functions with respect to position parameters are also analyzed. It can be concluded that the natural frequencies and corresponding amplitude vary greatly when the machine tool is under different position states. One can see that the system parameters, such as xg, yg, zg, x_ g , y_ g , z_ g , x€ g , y€ g and z€ g are changed fast in working process, and then both natural frequencies and corresponding amplitude of dynamic response are also changed and the unstable condition would eventually appear, which can adversely affect the accuracy of the TCP of programmed motions. (3) The TCP deviation affected by axial coupling in real-time motion state is discussed. Because of the influence of nonlinear stiffness of kinematic joints caused by axis coupling, the effects are not the simple superposition of the effects of the single axis movement. It can also be concluded that acceleration and braking produce heavy coupling forces, which detrimentally affect the dynamic response of the TCP. The straightforward evaluation information on machine tool specific qualities (natural frequencies, frequency responses, tool deviations, etc.) during the conception phase will become practicable without time-consuming by using the dynamic model.
€ ðm3 þm4 Þu€ 1 þ ðm3 þ m4 Þu€ 2 þ ðm3 þ m4 Þu€ 3 þ ðm3 þ m4 Þc1 ϕ 1z
€ € þ ðm3 þ m4 Þc2 ϕ2z þ m3 h3 þ m4 ðh3 þh4 Þ ðm3 þ m4 Þz3g ϕ 2y
_ € _ þ m4 h4 ϕ3y 2 ðm3 þ m4 Þz3g ϕ2y
ðm3 þ m4 Þz€ 3g ϕ2y þ ðm3 þ m4 Þx€ 2g þ k3x u3 ¼ 0
This work is supported by the key project of the National Natural Science Foundation of China (No. 51235009). Authors are grateful to other participants of the projects for their cooperation.
ðm1 þm2 þ m3 þ m4 Þv€ 1 þ ðm2 þ m3 þ m4 Þv€ 2 þ ðm3 þ m4 Þv€ 3 þ ðm2 þ m3 € € þ m h þ m ðh þ h Þ ðm þ m Þz ϕ þ m4 Þx2g ϕ 3 3 4 3 4 3 4 3g 1z 2x
_ € þ m4 h4 ϕ3x þ2 ðm2 þ m3 þ m4 Þx_ 2g ϕ1z
_
€ þ ðm2 þ m3 þm4 Þx€ 2g ϕ1z 2 ðm3 þ m4 Þz_ 3g ϕ 2x ðm3 þ m4 Þz3g ϕ2x þ ðm1 þ m2 þ m3 þ m4 Þy€ 1g þk1y v1 ¼ 0
The elastic deformation motion equations are listed as: ðm1 þ m2 þ m3 þ m4 Þu€ 1 þ ðm2 þ m3 þ m4 Þu€ 2 þ ðm3 þm4 Þu€ 3 € þ ðm þ m Þc ϕ € þ ðm2 þ m3 þm4 Þc1 ϕ 3 4 2 2z 1z
€ € þ m3 h3 þ m4 ðh3 þ h4 Þ ðm3 þm4 Þz3g ϕ2y þ m4 h4 ϕ 3y
_
2 ðm3 þ m4 Þz_ 3g ϕ2y ðm3 þ m4 Þz€ 3g ϕ2y þ ðm2 þ m3 þm4 Þx€ 2g þk1x u1 ¼ 0 ðm2 þ m3 þ m4 Þu€ 1 þ ðm2 þ m3 þ m4 Þu€ 2 þ ðm3 þ m4 Þu€ 3 þ ðm2 þ m3 € þ ðm þ m Þc ϕ € þ m Þc ϕ 1z
3
ðA:4Þ
ðm2 þm3 þ m4 Þv€ 1 þ ðm2 þ m3 þ m4 Þv€ 2 þ ðm3 þ m4 Þv€ 3 € þ m h þ m ðh þ h Þ þ ðm2 þ m3 þ m4 Þx2g ϕ 3 3 4 3 4 1z € _ € þ 2 ðm þ m þm Þx_ ϕ ðm3 þ m4 Þz3g ϕ þm h ϕ 4 4 3x 2 3 4 2g 2x 1z
_
þ ðm2 þ m3 þm4 Þx€ 2g ϕ1z 2 ðm3 þ m4 Þz_ 3g ϕ2x ðm3 þ m4 Þz€ 3g ϕ2x þ ðm2 þ m3 þ m4 Þy€ 1g þ k2y v2 ¼ 0
ðA:5Þ
€ ðm3 þm4 Þv€ 1 þ ðm3 þ m4 Þv€ 2 þ ðm3 þ m4 Þv€ 3 þ ðm3 þ m4 Þx2g ϕ 1z
_
€ þ 2 ðm3 þm4 Þx_ 2g ϕ1z þ m3 h3 þ m4 ðh3 þ h4 Þ ðm3 þ m4 Þz3g ϕ 2x € þ ðm þ m Þx€ ϕ 2 ðm þm Þz_ ϕ _ þ m4 h4 ϕ 3 4 2g 3 4 3g 3x 1z 2x
ðm3 þ m4 Þz€ 3g ϕ2x þ ðm3 þ m4 Þy€ 1g þ k3y v3 ¼ 0 ðA:6Þ € 1 þ ðm2 þm3 þ m4 Þw € 2 þ ðm3 þ m4 Þw €3 ðm1 þm2 þ m3 þ m4 Þw € € ðm2 þ m3 þ m4 Þc1 ϕ1x ðm3 þ m4 Þc2 ϕ2x þ ðm3 þm4 Þz€ 3g þ k1z w1 ¼ 0 ðA:7Þ € 1 þ ðm2 þ m3 þm4 Þw € 2 þ ð m3 þ m4 Þ w €3 ðm2 þm3 þ m4 Þw € ðm þ m Þc ϕ € þ ðm þm Þz€ ðm2 þ m3 þ m4 Þc1 ϕ 3 4 2 3 4 3g 1x 2x þ k2z w2 ¼ 0
ðA:8Þ
€ € 1 þ ð m3 þ m4 Þ w € 2 þ ðm3 þ m4 Þw € 3 ðm3 þ m4 Þc1 ϕ ðm3 þm4 Þw 1x € € ðm þ m Þc ϕ þ ðm þ m Þz þ k w ¼ 0
ðA:9Þ
3
4
2
2x
3
4
3g
3z
3
ðm3 þ m4 Þc1 z€ 3g ¼ 0
h i h i € þ J þJ ϕ € þJ ϕ € J 1y þ J 2y þ J 3y ϕ 2y 3y 3y 3y þ k1ϕy ϕ1y ¼ 0 1y 2y h
Appendix
1
ðA:3Þ
€ J 1x þ J 2x þ J 3x þ ðm2 þ m3 þ m4 Þc21 ϕ 1x
€ € þ J 2x þ J 3x þ ðm3 þ m4 Þc1 c2 ϕ2x þJ 3x ϕ 3x € 2 ðm3 þ m4 Þc1 w € 1 þw € 3 þ k1ϕx ϕ1x ðm2 þ m3 þ m4 Þc1 w
Acknowledgments
4
ðA:2Þ
4
2
2z
ðA:1Þ
ðA:10Þ ðA:11Þ
i € J 1z þ J 2z þ J 3z þ ðm2 þ m3 þ m4 Þ c21 þ x22g ϕ 1z
€ € þ J 2z þ J 3z þ ðm3 þ m4 Þc1 c2 ϕ2z þ J 3z ϕ3z þ ðm2 þ m3 þ m4 Þc1 u€ 1 þ u€ 2 þ ðm3 þm4 Þc1 u€ 3 þ ðm2 þ m3 þ m4 Þx2g v€ 1 þ v€ 2 þ ðm3 þ m4 Þx2g v€ 3
€ þ m3 h3 z3g þ m4 h3 þ h4 z3g x2g ϕ 2x
€ þ m3 h3 z3g þ m4 h3 þ h4 z3g c1 ϕ2y h i _ þ ðm þm þ m Þx x€ þ k þ 2ðm2 þ m3 þ m4 Þx2g x_ 2g ϕ 2 3 4 2g 2g 1ϕz ϕ1z 1z _ 2ðm3 þ m4 Þz_ 3g c1 ϕ 2y
_ 2ðm3 þ m4 Þz€ 3g c1 ϕ2y 2 m3 x2g z_ 3g þ m4 x2g z_ 3g ϕ 2x m3 x2g z€ 3g þ m4 x2g z€ 3g þ m3 x_ 2g z_ 3g þ m4 x_ 2g z_ 3g
L. Wang et al. / International Journal of Machine Tools & Manufacture 88 (2015) 71–81
m3 x2g z_ 3g m4 x2g z_ 3g ϕ2x € þm h x ϕ € _ _ 2g ϕ þ m4 h 4 c 1 ϕ 4 4 2g 3x þ m4 h4 x 3y 3x þ ðm2 þ m3 þ m4 Þx€ 2g c1 þ ðm2 þ m3 þ m4 Þy€ 1g x2g ¼ 0
€ J 2x þ J 3x þ ðm3 þ m4 Þc1 c2 ϕ 1x h 2 2 i € þ J 2x þ J 3x þ ðm3 þ m4 Þc22 þ m3 h3 zg þ m4 h3 þh4 zg ϕ2x
€ þ J 3x ϕ3x þ m3 h3 zg þ m4 h3 þ h4 zg v€ 1 þ v€ 2 þ v€ 3 _ € 1 þw € 2 þw € 3 m4 h4 z_ 3g ϕ ðm3 þ m4 Þc2 w 3x
€ þ m3 h3 zg þ m4 h3 þ h4 zg x2g ϕ1z 2 m3 z_ 3g h3 z3g _ þ m4 z_ 3g h3 þ h4 z3g ϕ 2x h i þ k2ϕx m3 z€ 3g h3 zg þm4 z€ 3g h3 þ h4 zg ϕ2x
_ þ 2 m3 x_ 2g h3 zg þ m4 x_ 2g h3 þ h4 zg ϕ
1z þ m3 x€ 2g h3 zg þm4 x€ 2g h3 þ h4 zg ϕ1z € € þ m4 h4 h3 þ h4 z3g ϕ 3x þ m3 y1g h3 zg ðA:13Þ þ m4 y€ 1g h3 þ h4 zg ¼ 0
h
ðA:12Þ
i h i € þ J þ J þ m h z 2 þ m h þ h z 2 ϕ € J 2y þ J 3y ϕ g g 3 3 4 3 4 2y 3y 1y 2y
_ € þ J 3y ϕ3y 2 m3 z_ 3g h3 zg þ m4 z_ 3g h3 þ h4 zg ϕ2y h i þ k2ϕy m3 z€ 3g h3 zg þ m4 z€ 3g h3 þ h4 zg ϕ2y
€ þ m3 h3 zg þ m4 h3 þ h4 zg c1 ϕ 1z € _ _ z þ m4 h4 h3 þ h4 z3g ϕ m h ϕ 4 4 3g 3y 3y
€ þ m3 h3 zg þ m4 h3 þ h4 zg c2 ϕ
2z þ m3 h3 zg þ m4 h3 þ h4 zg u€ 1 þ u€ 2 þ u€ 3 ðA:14Þ þ m3 x€ 2g h3 zg þ m4 x€ 2g h3 þh4 zg ¼ 0
€
€ 2 € J 2z þJ 3z þ ðm3 þ m4 Þc1 c2 ϕ 1z þ J 2z þ J 3z þ ðm3 þ m4 Þc2 ϕ2z þ J 3z ϕ3z
€ € þ m3 h3 z3g þm4 h3 þh4 z3g c2 ϕ 2y þ m4 h4 ϕ3y _ ðm þm Þz€ c ϕ 2ðm3 þm4 Þz_ 3g c2 ϕ 3 4 3g 2 2y 2y þ ðm3 þ m4 Þc2 u€ 1 þ u€ 2 þ u€ 3 þ m3 x€ 2g c1 þ m4 x€ 2g c2 þ k2ϕz ϕ2z ¼ 0 ðA:15Þ
h
i
€ 2 € _ _ J 3x þ m4 h4 ϕ 3x þ m4 h4 h3 þh4 zg ϕ2x 2m4 z3g h4 ϕ2x € þ 2m x_ h ϕ _ m4 z€ 3g h4 ϕ2x þ m4 h4 x2g ϕ 4 2g 4 1z 1z € € € € þ m4 x2g h4 ϕ1z þ m4 h4 v1 þ v2 þ v3 þ m4 h4 y€ 1g þ k3ϕx ϕ3x ¼ 0 ðA:16Þ
h
J 3y þ m4 h4
2
i
ϕ€ 3y þ m4 h4 h3 þ h4 zg ϕ€ 2y 2m4 h4 z_ 3g ϕ_ 2y
€ m4 z€ 3g h4 ϕ2y þ m4 h4 u€ 1 þ u€ 2 þ u€ 3 þ m4 h4 c1 ϕ 1z € þ k ϕ þ m h x€ ¼ 0 þ m4 h 4 c 2 ϕ 4 4 2g 3ϕy 3y 2z
€ þk ϕ ¼ 0 J 3z ϕ 3ϕz 3z 3z
ðA:17Þ ðA:18Þ
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