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Advances in Mechanical Engineering
Volume 2013 (2013), Article ID 847194, 12 pages
Load Induced Error Identification and Camber Curve Design of a Large-Span Crossbeam
1College of Mechanical Engineering and Applied Electronics Technology, Beijing University of Technology, Beijing 100124, China
2Department of Mechatronics Engineering, Shantou University, Shantou, Guangdong 515063, China
3Beijing No. 1 Machine Tool Works, Beijing 101300, China
Received 11 March 2013; Revised 29 August 2013; Accepted 16 October 2013
Academic Editor: Adib Becker
Copyright © 2013 Qiang Cheng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Load induced errors aroused by internal masses of machine tool components and external cutting forces are a main contributor that influences machining accuracy. A mathematical model for load induced errors identification of large-span crossbeam in heavy-duty machine tool was proposed. According to the identified load induced errors, a method for optimisation of the guideway camber curve on crossbeam by particle swarm optimisation was proposed. Geometric errors of -axis along cambered crossbeam and noncambered crossbeam were calibrated, and multibody system theory was also introduced to predict theoretically the machining accuracy of machine tool with two kinds of crossbeam. The comparisons between two kinds of crossbeam show that after cambering, the geometric errors are decreased and can meet the requirements of ISO standard, and the machining accuracy is enhanced remarkably too. In addition, the indirect dynamic straightness error of -direction along cambered crossbeam was measured by displacement sensors, and the response results show that even though at the start-stop moment, the straightness error is also within the reasonable range of ISO standard.
The accuracy of machine tools is a critical factor that affects the quality of manufactured products and an important consideration for any manufacturer. Development of efficient techniques for performance verification and improvement of machine tools have been considered important for accuracy enhancement and quality assurance for both users and manufacturers of machine tools. The accuracy of a machine tool is primarily affected by those geometric errors caused by mechanical-geometric imperfections, misalignments, wear of the linkages, and elements of the machine tool structure, by the nonuniform thermal expansion of the machine structure and static/dynamic load induced errors . As a result, a volumetric error which is the relative error between the cutting tool and the work-piece is related . Samir and Tunde  and Ramesh et al.  have summarised the factors affecting the total volumetric accuracy of a machine tool and their interrelationships, and Samir and Tunde  have listed geometric errors, thermally induced errors, and load induced errors as the three main factors.
Geometric errors are caused by mechanical imperfections in the machine tool structure and the misalignment of machine tool elements: they arise under cold-start conditions and all change gradually due to component wear . Among these errors, geometric errors in machine tool components and structures are one of the biggest sources of inaccuracy . The systematic geometric error components of the machine tool directly affect the tool tip position because of dimensional and form errors of the kinematic linkage and linear/angular misalignment between them . Geometric errors are concerned with the quasistatic accuracy of the surfaces and their relative movement. The effect of the geometric inaccuracies is to produce errors between the machines’ moving elements. They manifest themselves as position and orientation errors of the tool with respect to the machined part .
Measurement, identification, and modelling of geometric errors inherent to a machine tool are closely related to its precision, which can be then used for error compensation. Some corresponding identification methods have also been developed such as the nine-line  and twelve-line methods  based on laser interferometer. In addition, some measuring instruments [8, 9] were proposed to measure these geometric errors quickly. Chen et al.  developed a method to calibrate the geometric errors in a multiaxis machine tool by an autoalignment laser interferometer system. Lei and Hsu  developed a 3D probe-ball measurement device to obtain the overall, 3D errors in a five-axis machine tool. Tsutsumi and Satio  proposed a method of identifying the eight deviations inherent in a five-axis machine centre by means of simultaneous four-axis control movements based on a ball-bar. Dassanayake et al.  investigated the effectiveness of the checking method specified in ISO10791.6 and proposed an additional method for identifying geometric deviations in a five-axis machine centre with a universal spindle head by a ball-bar. Bohez et al.  developed a way of identifying all the systematic angular errors in a five-axis machine tool separately and then used them further to identify the systematic translational errors. Zargarbashi and Mayer  developed a single set-up estimation of a five-axis machine tool based on Capball sensor measurement, which can estimate four tilt errors and centre line offsets of the rotary axes. However, these methods cannot identify all six angular geometric error parameters of each rotation axis separately and quickly. In error model development, as a general modelling method, kinematic chain and homogeneous transfer matrix (HTM) methods have been widely used [10, 15, 16]. Viewing a machine tool as a multibody system (MBS) composed of a few rigid bodies, Fan et al.  developed a universal kinematic model of a CNC machine tool based on MBS. Ekinci and Mayer  discussed the discrepancies between direct joint kinematic straightness and that obtained through the integration of joint kinematic angular error. The experiment in his paper verified the correctness of the method of the direct joint kinematic straightness while the latter is right on the condition that the wavelength () is considerably larger than the bearing spacing ().
External loads that cause errors in a machine include gravity (a function of the weight of the part to be machined), cutting load which arises from the cutting process, and axis acceleration load resulting from the displacement of the masses of the machine components . These errors are the load induced errors and badly affect the stiffness of a machine tool structure. They cause elastic strain in the machine tool structure with distributed and/or varying effects. This arises because of internal or external forces and produces unavoidable stresses and strains in machine tool components . Schellekens et al.  and Spaan  propose that these errors can be significant when compared with the kinematic errors in a machine tool. The main source of this error generation is the assumption of finite stiffness, as it is impossible to attain a fully rigid structural configuration. Moreover, the weight and configuration of the components in the structural loop also contribute to stiffness errors . The magnitude of load induced errors depends upon the object loading behaviour, object weight, machining forces, and unbalanced unforeseen forces in machine elements which affect the stiffness of the structural loop. Though it has been reported that in finish-machining the cutting force is small and the resultant deflection could be neglected , studies on load induced error compensation have been found to improve machine accuracy [22–25].
However, previous research overlooked nonrigid behaviour: the nature of the error components associated with each link will thus become more complex. Stiffness errors will depend not only on their own link variables but also on the remaining links’ variables within the machine tool and hence make the phenomenon more complex. However, as the stiffness in the machine’s components is not infinite, deformations arising from dynamic behaviour and load conditions may be significant and the errors thus produced cannot be neglected . Traditionally, researchers at design stage pay attention to make the structure more rigid by using heavy base frames and moving slides, but this affects the accuracy and power utilisation as the movement of heavier components is neither recommended nor appreciated. In addition, heavy parts may generate large inertial forces and increased dynamic load, especially for machines using high accelerations. Hence, error reduction to improve the accuracy of the machine tool is crucial. The errors/inaccuracies can be reduced with the structural improvement of the machine tool through better design, manufacturing, and assembly practices . Large machine tools are increasingly important because of the increasing demand for large parts, for example, in the production of wind turbines, which are growing in size with each new model . With the increasingly high demand for precision machining it is critical to establish an accurate processing model, especially for heavy-duty overloaded machines with the characteristics of large weight and high load strength. Therefore, constructing a machining accuracy model for heavy-duty machine tools with consideration of load induced errors can provide an important basis for design of machine tool structure and processing compensation technology.
In this paper, a load induced error identification method for heavy duty machine tools with large-span crossbeam was proposed, and also the influence of load induced error on the machining accuracy was analysed based on MBS theory. The paper is organised as follows. In Section 2, the identification of a mathematical model for load induced errors was established. Section 3 discusses the optimisation method of camber curve based on particle swarm optimisation algorithm. In Section 4, the effectiveness of the obtained camber curve was verified realistically by direct geometric errors calibration and theoretically by machining accuracy prediction with MBS theory. Section 5 shows the indirect experimental validation that the straightness error of -direction is eligible to ISO standard even at the start-stop moment of saddle. Conclusions are drawn in Section 6.
2. Identification of Load Induced Error
A heavy-duty CNC milling machine, as shown in Figure 1, is mainly made up of bed, left-hand column, right-hand column, crossbeam, milling head, saddle, ram, and working table. It is usually used to perform contour milling, or high accuracy surface machining. The crossbeam is a very important support component which holds the weight of the saddle, ram, and milling head and its self-weight when machining. The aforementioned weights plus the cutting force inevitably produce bending deformation of the crossbeam, and therefore load induced errors along the crossbeam arise and influence the machining accuracy. So, how to identify and eliminate load induced errors as far as possible is essential for machine tool companies and consumers in their quest for higher machining accuracy.
2.1. Establishment of the Deformation Curve
According to the location and function of the crossbeam in the machine, the crossbeam mechanical model is simplified as having two fixed ends, as shown in Figure 2(a).
In reality, the crossbeam is under a complex load regime. The weight of the beam is treated as a uniform load, which will cause static deformation of the crossbeam. The weights of the milling head and saddle are treated as point loads, and when they move along the guideway on the crossbeam, bending deformation of the crossbeam is induced. The stress state of the crossbeam contacts is shown in Figures 2(b) and 2(c). is the stress on the contact surface between saddle and upper guideway. and are the stresses on the contact surface stress between saddle and lower guideway. and lie along the -axis, while lies along the -axis.
To reflect the mechanical characteristics of the crossbeam during processing, software ADAMS was used to dynamically simulate the entire machine to obtain the contact forces and pressures. Because the bearing between the crossbeam and saddle is realized by a hydrostatic guideway with an oil cushion, the connection between the crossbeam and saddle can be simplified as a spring-damper system whose stiffness can be calculated according to Hydrostatic bearing principle. Here, according to the parameters of oil supplying systems, shape and size of the four oil cushions and hydrostatic load, the connection stiffness can be obtained as N/mm. Due to the complex geometry of the crossbeam, it is no longer suitable to apply classical elastic mechanics when analysing this nonlinear problem. Therefore, to obtain accurate displacements of the crossbeam, the finite element method (FEM) was used and the crossbeam divided into tetrahedral elements. With the above obtained stiffness value and the boundary conditions of fixed beam-end, the forces applied to the crossbeam can be gotten as , , and by ADAMS software. Correspondingly, the surface pressures of the contact area between saddle and beam are 2.785 MPa, 1.567 MPa, and 1.646 MPa.
Because the beam spans up to 14.350 m, 25 work points were marked on the crossbeam along the entire travel of the saddle. In reality, there are two contact surfaces (right and left, as shown in Figure 3.) on which the saddle can come into contact with the crossbeam at each work position. Thus, a total of six deformation data sets can be obtained for each work position shown in Figure 3. Because the machining accuracy is directly reflected by the milling head cutting point, the amount of deformation displacement of the centre point of the milling head is taken as the average of the two deformation data sets (right and left) on each contact surface. The deformation curve was obtained through data fitting along the -direction as shown in Figure 4.
2.2. Load Induced Errors Identification Model
The identification models are based on the assumption that the saddle is a rigid body and the deformation only comes from the contact interface and the structure of the crossbeam. Two body coordinate systems and are, respectively, fixed on the saddle and the crossbeam; meanwhile the deformation of the contact interface is also supposed to be along the direction of the force (e.g., , , and ) applied on this area. As shown in Figure 3, the body coordinates of the saddle are fixed on the centre of the lower edge, whose starting position coincides with the coordinate system of the crossbeam.
The linear errors between saddle and crossbeam are defined as the displacement of the origin in relation to . The angle errors are defined as the angle change of the line fixed on the saddle. A total of five load induced errors can be identified except the angle error along the -direction, because this error is almost ten times smaller than the others.
2.2.1. Errors in the xoz Plane
The xoz plane of the saddle is as shown in Figure 5(a). The contact area is at and points. Due to its own weight and cutting force, the origin of coordinate fixed on the saddle will shift to . and points move to , . The amount of deformation is and , respectively.
is defined as the angle between and , and is defined as the angle between and . is the vector from point a to point , and is the vector from point to point . According to the assumption, the saddle is deemed as rigid body. So the mathematic expression can be obtained as follows:
The vector of the linear error in xoz plane can be gotten as
So, the linear errors , can be obtained with the following mathematical functions:
The roll error can be obtained as
2.2.2. Errors in the yoz Plane
The yoz plane of the saddle is as shown in Figure 6(b). The contact area is at points , . The origin of coordinates fixed on the saddle will be shifted to . Points and move to , , and and are respectively the midpoint of lines and : the deformations are and , respectively.
is the vector from point to point , and is the vector from point to point . In the same way, the saddle is assumed rigid, so is perpendicular to . Then, the mathematical expression can be obtained as follows:
The vector of the linear error can be obtained from
The tilt error can be obtained from
2.2.3. Identification Results Analysis
The five load induced errors have been identified, and their mathematical models are as follows:
Based on the fitted deformation curve, with (10), the five load induced errors can be obtained as shown in Figure 6. It can be seen that the maximum of the straightness error in the yoz plane is 0.26 mm. According to the requirement of the accuracy in ISO standard  for gantry milling machines, the straightness requirement is 0.04 mm. So the straightness error cannot meet the accuracy requirement. Therefore, design of the crossbeam and elimination of load induced errors is essential.
3. Camber Curve Design
To reduce load induced errors, manufacturing companies traditionally make the guideway in the -direction into a cambered shape with artificial scraping undertaken as well as the experience of a usually skilled worker allows. In this section, the camber curve in the -direction of the crossbeam was optimised based on mathematical models of the load induced errors. As shown in Figure 7, define as the camber curve in the yoz plane, and , as the deformation curve of the left and right interfaces, respectively. So, the optimisation model, as shown in (11), can be established with the identified load induced errors , , and :
Because particle swarm optimisation (PSO)  is a stochastic global optimisation approach, and its main strength lies in its simplicity and rapid convergence, it was used here to carry out the optimization. Due to the fact that, in the plane, the relationship between identified load induced error and can be described as (because in (11), is much smaller than , so can be ignored in ) during optimisation, the weight coefficient is introduced to denote their relative importance and the multiobjective optimisation can be changed into a single-objective optimisation. Figures 8(a) and 8(b) are the optimised camber curve and the straightness error . It can be seen that after optimisation, the straightness error has been reduced greatly. In practice, the designer can adjust according to the actual situation’s requirements to achieve ideal accuracy limits.
4. Error Calibration and Volumetric Machining Accuracy Prediction
In this section, in order to validate the effectiveness of the optimised camber curve realistically and theoretically, on one hand autoptic calibration and comparison of geometric errors along -axis of cambered and noncambered crossbeams were performed; on the other hand, based on the calibrated geometric errors, the volumetric machining accuracy of machine tool with two kinds of crossbeam was predicted and compared by MBS theory.
4.1. Error Calibration
Figure 9 shows the 5-axis gantry milling machine analysed here whose 3D numerical model is shown in Figure 1. The guideway along the -axis was scraped according to the optimised camber curve in Section 3. 6D laser interferometer, dial indicator, and level were adopted to measure the geometric errors of the gantry milling machine. Figure 10 shows the calibration picture of the -axis geometric errors along the cambered crossbeam by means of 6D laser interferometer. The following may be drawn from the measurement and comparison: when the crossbeam is not cambered, the positioning error , straightness error , and yaw error in -axis along noncambered crossbeam are 0.54 mm, 0.4 mm, and 0.24/1000 rad, respectively, and those corresponding values in -axis along of the cambered crossbeam are 0.052 mm, 0.035 mm, and 0.03/1000 rad. And, in ISO standard , the positioning error , straightness error , and yaw error for fixed bridge-type (portal-type) milling machines are 0.04 mm, 0.0725 mm, and 0.04/1000 rad. So, it can be inferred that the geometric errors of cambered crossbeam are reduced down to the requirements of ISO standard.
4.2. Volumetric Machining Accuracy Prediction
Based on MBS theory [17, 29], the topological body structure of the machine tool in Figure 1 can be obtained as shown in Figure 11, which describes the connection and motion relationship of multibody.
To facilitate the precision of machine tool modelling, the coordinate system is required to be set specially. Consider the following settings. (1) In MBS theory, right-handed Cartesian coordinate systems are built on the inertial components and all moving parts. These coordinates are called generalised coordinates. In the generalised coordinates, the coordinate system on the inertial body is called the reference coordinate system, and that on the other moving body is the moving coordinate system. Three orthogonal matrices of each coordinate system are named -, -, and -axis, respectively, according to the right-hand rule. (2) The axes in different coordinate systems are correspondingly parallel. (3) The tool coordinate system is at the end of the milling head centre. Suppose that a tool forming point in the tool coordinate system lies at And the work-piece forming point in the work-piece coordinate system coordinate lies at
When the machine tool moves in its ideal, error-free mode, the tool forming point and work-piece forming point will overlap, and we get
The ideal forming function of the tool forming point in the work-piece coordinate system is
Machining accuracy is finally determined by the relative displacement error between the tool forming point of machine and work-piece. During the actual machining process, the actual position of the cutting tool forming point will inevitably deviate from its ideal location, thereby resulting in volumetric error. The comprehensive volumetric error caused by the gap between an actual forming point and the ideal forming point can be written as in which reflects the deviation of the tool forming point from its ideal location and its actual location in the work-piece coordinate system in -, -, and -axes. In the expression above, is the body ideal static or motion characteristic matrix, the ideal static characteristic matrix, and the ideal motion characteristic matrix. and denote the static and motion angle error characteristic matrices, respectively. Taking for example, , in which
Here, machining specimen in Figure 12 was selected to compare the machining accuracy in the -, -, and -axis, respectively. Based on the geometric errors calibrated in Section 4.1, the machining accuracy in the -, -, and -axis can be calculated by (16). The obtained ranges of machining accuracy in the whole machining region were listed in Table 1. It can be seen that the machining accuracy in the -, -, and -axis is improved, especially the machining accuracy in the -axis. The minimum machining accuracy in -axis is enhanced from 0.0082 to 0.0044, and the fluctuation range is reduced from 0.0038 to 0.0026. Therefore, the milling machine tool with cambered crossbeam has better machining accuracy than that with noncambered crossbeam, which illustrates the usefulness of the optimised camber curve theoretically.
5. Start-Stop Transient Error Measurement
In Section 4, the effectiveness of the optimised camber curve obtained in this paper was verified realistically and theoretically. However, the geometric errors calibrated at the end of ram in Section 4.1 are quasistatic errors, which reflect the geometric/kinematic errors of the machine due to mechanical imperfections and the static or dead weight of the machine’s components. In order to inspect whether the dynamical error caused by inertial forces of the machine tool with cambered crossbeam is satisfactory or not, the deformation at the end of ram was measured at start-stop transient with different accelerations of saddle. According to calibration in Section 4.1, the subsidence deformation of -direction has the maximum of 0.009 mm when saddle locates in the middle of cambered crossbeam. Therefore, here the transient vibration amplitude of the ram end point was measured by the displacement sensor setup in Figure 13 when the saddle locates in the middle of crossbeam. The relative value is shown in Figure 14. It can be seen that the amplitude is smaller than the quasistatic deformation value. Even though when the acceleration is , the relative subsidence deformation is 0.71 μm. So, the maximum subsidence deformation is 0.00971 mm, and the straightness error of -direction is 0.03571 mm and still within the required range of ISO standard . Therefore, although vibration is generally aroused at machine tool start-stop moment, the straightness error of -direction along cambered crossbeam remains meets the requirement of ISO standard.
To reduce load induced errors and their influence on machining accuracy, a method for identification load induced errors and design camber curve of the machine’s guideway was proposed. Based on bending deformation simulation analysis by the finite element method, the deformation curve was obtained by curve fitting. Then, a mathematical model was established to identify load induced errors on the basis of the deformation curve. Predeformation camber curve of the guideway on the crossbeam was optimised by particle swarm optimisation to reduce load induced errors. Geometric errors of -axis along the cambered crossbeam and noncambered crossbeam were calibrated. The calibration results show that the geometric errors are reduced after pre-deformation and can meet the requirements of ISO standard well. Furthermore, MBS theory was introduced to verify that the machining accuracy of machine tool with cambered crossbeam is improved more than that with noncambered one. Finally, the indirect dynamic straightness error of -direction along cambered crossbeam was measured by displacement sensor, and the response results show that even though at the start-stop moment, the straightness error is also within the reasonable range of ISO standard.
Due to the heavy-duty characteristics, most working tables in heavy-duty machine tools are used in static pressure supporting mode. The machining accuracy is usually influenced by the offset load of the work-piece on the working table. Therefore, future research will focus on identification load induced errors and design different work-piece installation techniques on static pressure work-supporting tables.
Conflict of Interests
The authors of this paper have purchased the ADAMS software from Mechanical Dynamics Inc. under an authorised license; there is no conflict of interest to declare.
The authors thank the National Natural Science Foundation of China (Grant no. 51005003), the National Science and Technology Great Special Programme (Grant no. 2010ZX04001-041), the Leading Talent Project led by Guangdong Province and Guangdong Provincial Natural Science Foundation of China (Grant no. 8351503101000001), the Rixin Talent Project led by Beijing University of Technology, and Beijing Education Committee Scientific Research Project all of whom supported this research.
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