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A novel approach to fixture design on suppressing machining vibration of flexible workpiece
a r t i c l e i n f o
Article history: Received 29September2011 Received inrevisedform 20 February2012 Accepted 20February2012 Available online3March2012 .
a b s t r a c t
The machining vibration of the flexible workpiece is a major factor that greatly affects the machining accuracy of the final part. Improper fixture layout is apt to generate the machining vibration, which will seriously affect the machining quality of the surface, especially for the flexible workpiece. This paper is concerned with suppressing the machining vibration of the flexible workpiece by designing appropriate fixture layout scheme. A dynamic model on the workpiece–fixture–cutter system is built, where the cutting force is used as the disturbance input, and the fixture element is used as the control input. On the basis of this model, an approach to fixture design is proposed and introduced, for the first time, to suppress the machining vibration of the flexible workpiece originating from the cutting forces. It is worth noting that the location, the applied forces and the number of fixture elements can be simultaneously optimized. The effectiveness of the proposed method is verified by a machining example.
Keywords: Fixture design Machining vibrationsuppression Workpiece–fixture–cutter Flexible workpiece Machining accuracy .
1. Introduction
In the past few decades, the analysis and design of the fixtures for machining applications, especially for machining the flexible workpiece, has gained special attention due to its effects on the quality of the final part and the production cost. Milling of flexible workpiece is a common manufacturing process in the aerospace industry. As a result of its low rigidity, the vibration and surface errors of flexible workpiece are apparent during the machining process. Fixture design is especially important in machining of flexible workpiece, for its ability in suppressing the vibration of the workpiece. Numerous efforts have been made in modeling, analysis, and designing of fixtures for machining applications. The majorities of the prior works treat the workpiece–fixture system as quasi-static and ignore the system dynamics. Since machining is often characterized by the periodic cutting forces and vibration, taking the dynamic effects into consideration in the fixture design is very critical. For the flexible workpiece, the dynamic response of the workpiece–fixture system is a major factor that greatly affects the accuracy and efficiency of the machining process. This is because the workpiece-tool system change from the initial rigid-flexible system to the flexible-rigid system component which is prone to vibration under the influence of high speed cutting forces. To accurately design the fixture layout is especially essential for machining of flexible workpiece. with the effect of material removal, and the workpiece becomes a poor rigidity .
1.1. Review of researches related to vibration control on flexible workpiece
Vibration is a common phenomenon in the finishing machin- ing of the flexible workpiece due to its low rigidity, which has a significant influence in both the quality of the machined surface of workpiece and the machining process. Biermann et al. [1] presented a simulation system for computing regenerative work- piece vibrations during the five-axis milling of turbine blades and proposed a modeling method for visualizing the resulting surface. Bravo et al. [2] proposed a method for constructing the three- dimensional lobe diagrams by considering both the dynamic behaviors of the machine structure and the machined workpiece. The proposed method was validated by machining a series of thin walls. Campa et al. [3] presented a methodology based on the estimation of modal parameters of the part and the corresponding stability lobes for chatter avoidance in milling of flexible thin floors. A stability model for milling of compliant systems with a bull-nose end mill was constructed in the tool axis direction. Adetoro et al. [4] considered the nonlinearity of the flexible workpiece dynamics when predicting stable region in machining using a finite element and Fourier transform approach. Zhang and Sims [5] depicted an experimental investigation in order to assess the feasibility of piezoelectric active vibration control on milling a flexible workpiece using a positive position feedback control strategy. Kersting and Biermann [6] presented a simulation concept for predicting regenerative flexible workpiece vibrations during the five-axis milling process. This concept combined an accurate and fast simulation of the five-axis machining process including the material removal and the force calculation. A finite element model was used for computing the workpiece displace- ments. Arnaud et al. [7] proposed a model of flexible workpiece to study the stability of the cutting process and evaluate the machining vibrations of the workpiece. Man?e′ et al. [8] analyzed the dynamic interaction of a spindle-tool set and a flexible workpiece by a finite element approach. An accurate stability lobes diagram was also elaborated by coupling the dynamic behavior of the machine and the workpiece. Their experimental results indicate that the spindle speed regulation is a necessary constraint to guarantee the optimum stability during the machin- ing of flexible structures. All these studies focus on the vibration control of flexible workpiece by choosing the process parameters reasonably. However, the fixture is of importance in the machin- ing of the flexible workpiece, since it has the ability to suppress the excessive machining vibration of the workpiece and balance the cutting forces. By designing the fixtures appropriately, the dynamic rigidity of the flexible workpiece can be added to.
1.2. Review of researches related to fixture layout for machining applications
In general, machining fixtures are used to rigidly and accu- rately hold and support the workpieces with clamps and locators, so as to ensure the configuration of the workpiece relative to the cutting tool can be kept during the machining process. The 3-2-1 locating principle is commonly used to locate the workpiece properly. Many researchers focus on the fixture geometric pro- gramming design and optimization [9–14]. These research works are based on the conditions of the form-closure and force-closure under the assumption that the workpiece is completely rigid or there exists local contact deformation. However, for flexible workpiece, there is not only the local contact deformation but also the entire deformation occurs during the clamping and machining process. Moreover, the layout of the fixture elements and the number of the clamps and locators not only affect the form-closure and force-closure of the workpiece, but also affect the machining stability of the workpiece.
Refs. [15–17] use the finite element software to analyze the contact stiffness between the fixture elements and the workpiece as well as the contact deformation between the cutter and the work- piece. Both the cutting force and the displacement of the workpiece are predicted. However, the finite element software can predict some parameters in the case of given fixture layout and cutting process parameters. It cannot be directly applied to optimize the fixture layout. Liu et al. [18] proposed a method for dealing with the optimization of the number and positions of the locators on the secondary locating surface in the peripheral milling of a low-rigidity workpiece. This method only considered the static deformation of the flexible workpiece during the clamping process, but did not consider the effect of both the fixture layout and the number of fixture elements on the dynamics of the machining process. Qin et al. [19] proposed a new methodology considering the varying contact forces and friction force during clamping. Raghu and Melkote [20] considered the fixture geometric error and elastic deformation of the fixture and workpiece due to fixturing forces by modeling the process of part loading and clamping in a machining fixture. This method only considered the static clamping deforma- tion and entire rigid displacement of the workpiece, but did not consider the workpiece dynamic displacement and the effect of the fixture layout on workpiece dynamics. Chen et al. [21] established a multi-objectivemodel to reduce the degree of clamping deformation and increase the distributing uniformity of deformation by combin- ing the finite element method and genetic algorithm. Although this method is more effective in minimizing and uniforming the defor- mation than the traditional fixture design methods, it does not consider the effect of the fixture layout on workpiece dynamics. These methods are all based on the static analysis. The clamping deformation control mainly considers the layout of fixture elements and the clamping force. However, the effects of the cutting force on fixture layout optimization and clamping deformation control are not taken into consideration, which are essential factors when solving the deformation control problem in machining of flexible workpiece. Moreover, for the flexible workpiece, the objective of the fixture layout is not only on how to reduce the static clamping deformation and the residual stress induced by clamping. The most vital problem is how to improve the dynamic machinability of flexible workpiece.
1.3. Review of researches related to fixture layout on vibration control of flexible workpiece in machining
To the best of our knowledge, there are few studies in the area of the fixture layout on the vibration control of flexible workpiece in machining while taking into consideration the effect of the fixture layout on workpiece dynamics. Aoyama and Kakinuma [22] presented new fixture devices, which can support thin and compliant workpieces securely with little deformation, but did not study the effect of the fixture layout on deformation or vibration of thin and compliant workpieces.
Based on the above analysis, this paper developed a dynamic model of workpiece–fixture–cutter system, and then, a novel fixture design method is introduced to suppress the machining vibration of the flexible workpiece. The remainder of this paper is organized as follows. Section 2 constructs the dynamic model of the workpiece–fixture–cutter system. The methodology of fixture design for suppressing machining vibration has been presented in Section 3. Section 4 develops a machining example to verify the proposed fixture design method. Section 5 concludes the paper.
2.Workpiece–fixture–cutter system dynamic model building
The system studied here contains two subsystems, that is, the spindle-cutting tool subsystem and the workpiece–fixture sub- system. In this section, both the cutting force model and the flexible workpiece model are depicted. And then, a model of the coupled dynamic system, namely, the dynamic system on workpiece–fixture–cutter, is constructed.
2.1. Cutting force model
The cutting force model used in the present work is similar to the model proposed by Engin and Altintas [23]. The schematic illustration of the cutting force model with a flat end mill is shown in Fig. 1, considering the local cutting edge geometry and the three differential cutting forces along the tangential, radial, and axial directions at cutting points. The differential cutting forces are defined for the tangential, radial, and axial directions on an infinitesimal cutting edge segment as follows [24]:
where Kte, Kre, Kae and Ktc, Krc, Kac are theedgecuttingcoefficients and theshearforcecoefficients,respectively.Theycanbe obtained using the orthogonal cutting database and the oblique cutting model presented by Budak et al. [25,26]. ds is the cutting edge length. db is the projected length of an infinitesimal cutting flute in the direction along the cutting velocity. h(j,k) is the uncut chip thickness normal to the cutting edge, which changes with the position of the cutting point and cutter rotation. The chip thickness for the reference cutting tooth can be evaluated using the kinematics of milling [27]. The dynamic chip load can be evaluated according to Ref. [28]. Once the chip load and cutting coefficients are evaluated for the local edge geometry, the cutting forces in the Cartesian coordinate system along the x-, y- and z- directions can be given as [23]
The total cutting forces in the Cartesian coordinate system along the x-, y- and z-directions can be evaluated by integrating Eq. (2) with respect to the differential axial depth, dz. The expressions for the reference cutting tooth can be written as
where Nt is the number of flutes on the cutter, zl(t) and zu(t) are the lower and upper limits of integration for the current cutting region. For the flat endmill,according to its simple geometry, Eq. (3)can be represented as
Besides, for low speed milling process, building of the cutting forces model should consider the effect of the cutting process damping, which has been proved to be significant factor for accurate prediction of stability limits by Budak and Tunc [29].
2.2. Workpiece model
In order to have a workpiece being representative of the aerospace structural components in machining problems, the study concentrates on a typical flexible workpiece with a struc- ture to achieve a compromise between the weight reduction and the stiffness preservation (see Fig. 2(a)). The geometry embodies a part that can be easily found in the aerospace industry, for instance, the rib and the spar in the airframe. In general, the machining process in practice begins with the rough machining of a workpiece. In this stage, the tool can be considered as flexible compared to the relative rigid workpiece, whereas in the finishing machining operation stage of a flexible workpiece, the tool can be considered as rigid compared to the flexible workpiece. There- fore, in the stage of the finishing machining operation, the workpiece–fixture system dynamics is more dominant than the tool dynamics.
In order to determine a proper fixture layout for the vibration suppression of the flexible workpiece during the finishing machining operation stage, a model of the dynamic response of the workpiece–fixture system should be carried out. The work- piece can be discretized to the bottom side and four sidewalls. Assuming that the bottom side of the workpiece is fixed during the machining, namely, the bottom side is rigid compared to the four sidewalls, and only the dynamic responses of the four sidewalls need to be concerned. The boundary conditions for each sidewall should be modeled first to develop the transverse dynamic response model of each sidewall of the workpiece. The interaction between the adjacent sidewalls is modeled by the torsional springs and translational springs. The translational springs are in the transverse direction and the rotational springs act around the axis along the corresponding edge (see Fig. 2(b)). The critical issue is how to assign each torsional spring and translational spring a right stiffness value, so as to ensure the rationality of the constructed model. The optimum stiffness values of the torsional springs and translational springs can be achieved by minimizing the deviation of the dynamic response of the critical points between the finite element model of the matrix determined by the point of cutting force application.
3. Modeling methodology of fixture design for suppressing machining vibration
Based on the model built in Section 2, a novel fixture layout approach is presented for suppressing the machining vibration of the flexible workpiece. Eq. (5) can be written in the state space form as
where denotes the 2nxs tate 1 vector, the disturbance input matrix and v is the disturbance vector.The system output equation can be given as y=Ex, where y denotes the mx1 output vector and E denotes the mx2n output matrix. Assuming that an output feedback control is applied to the system, the input vector u can be expressed as u=Gy.