A novel identification technique of machine tool support stiffness under the variance of structural weight

The trial-and-error method was a technique commonly used in machine design; now, virtual prototyping technology is increasingly adopted to reduce the development cost and time of machine tool builders [1]. In particular, finite element analysis (FEA) has been widely applied in machine design [2‐5]. Garitaonandia et al. [6, 7] used FEA to analyze grinding machines, determine machining chatter, and predict the position-dependent dynamic behavior of a machine. Liang et al. [8] optimized the design and dynamic performance of an ultraprecision machine tool. Shen et al. [9] used FEA to identify structural weaknesses of a machine and optimized the machine’s dynamic characteristics through topology optimization. Accurate prediction of the vibration characteristics (including natural frequencies and mode shapes) of machines is critical at the design stage because it prevents workpiece defects caused by machining vibrations. In the simulation of dynamic characteristics for an assembled machine tool, joint interface characteristics are the most crucial information, apart from geometry and material properties. Various studies have investigated the stiffness of joint units, including linear guideways, ball screws, and bearings, in the machine tool. Lin et al. [10] developed a finite element (FE) model to discuss the effect of the preload of linear guideways on the dynamic stiffness of a spindle, Brecher et al. [11] conducted a comprehensive study on joint constraints, and Ealo et al. [12] examined the joint stiffness of linear guideways for the industrial milling machines. Support stiffness has a direct influence on the lowest frequency of a machine. It involves a much higher level of complexity than does the stiffness of joint units and is more difficult to determine its value compared with motion component stiffness; therefore, some studies have overlooked support stiffness in their FEA-based machine design [13‐20]. Disregarding support stiffness would lead to distorted modal simulations. To address this problem, scholars have researched support stiffness. Kono et al. [21] obtain the interface contact stiffness between two simple different material models through a static stiffness test. After that, the contact stiffness of support was calculated to estimate the modal analysis of the whole machine. The maximum difference between the experimental and simulation results was 55%. Kono et al. [22] had also established guidelines for determining the positions of a machine’s supports as well as obtained the maximum fundamental frequency of a machine. The load cell is required to acquire data in static stiffness test. It is difficult to set the load cell directly on the support of the whole machine, which will cause more uncertainty of the boundary conditions. Therefore, this study proposes a novel method to identify the stiffness of the support, which does not require a load cell and can effectively predict the natural frequency and vibration mode under the variance of structural loading. This method performs impact testing at each stage of a machine tool builder’s assembly process and the modal finite element modeling was validated to construct accurate equations between support stiffness and reaction force.

This study is organized as follows. Section 2 introduces the machine used in the experiment and the experimental methods. Section 3 verifies and simulates the material properties of castings, conducts a modal analysis for each assembly stage, compares the results with the experimental results for verification, and constructs equations for support contact stiffness for each assembly stage. Section 4 predicts natural frequency and mode shape on the basis of support stiffness. Section 5 discusses the results.

This study employed impact testing in conjunction with the experimental modal analysis (EMA) and excited a machine structure with an impact hammer to elicit the structure’s natural frequency. The experiment was conducted by moving an accelerometer to each measurement point and applying impact on a fixed point at different directions. Three impacts were applied on each measurement point to collect three signals, which were averaged to reduce experimental error. Table 1 provides specifications for the equipment used in the impact testing. The measurements of frequency ranges from 0 to 150 Hz. As shown in the left side of Fig. 1, a vertical machining center that weighed 5300 kg and had 11 supports was used. For accurate simulation of machine tools, which involve complex systems, the right side of Fig. 1 shows that the machine was disassembled into two parts: (1) casting structures, which contained structural components including the base, cross slide, table, column, and housing, and (2) joint units, which comprise joint interfaces such as the supports, linear guideway, ball screw, and bearing. Each casting structure was hung for impact testing to obtain the dynamic characteristics of the machine. Take the column for example. The column was hung as shown in Figs. 2a, b presents the measurement points for the impact testing. The number of measurement points for the casting structure is shown in Table 2. Then, the testing results were compared with those obtained in the FE modal analysis to determine the material properties of the structure. The details of the comparison are provided in Section 3.1.

Regarding the joint unit, various studies have researched the properties of linear guideways, ball screws, and bearings [23‐30]. Focusing on support stiffness, the present study revealed the dynamic characteristics of the machine structure by conducting impact testing at each stage of the machine’s assembly process. The support stiffness at each stage through FEA was analyzed, and the equations for support stiffness were established. The corresponding results were presented in Section 3.2. Figure 3 describes the assembly process and points of impact on the base. The assembly process comprised six stages. In the first stage, the base was placed on the floor, supported with 11 supports, and aligned to a horizontal position. The second stage involved adding the joint unit and cross slide to the part built in the first stage. Different components were added in subsequent stages until the machine was completely assembled at the sixth stage. The measurement points for each assembly stage are presented in Table 3. The experimental results are discussed and used to verify the simulated results in Section 3.2.

This study adopted the reverse finding numerical method as shown in Fig. 4 to determine Young’s modulus and support stiffness of the machining center according to the following process. Step 1: Establish the variables (e.g., Young’s modulus or support stiffness). Step 2: Conduct a modal analysis. Step 3: Compare the experimental and simulated natural frequency, with the experimental natural frequency as the objective value. The objective function (OF) is defined as follows:

where \({f}_{Exp}\) is the experimental natural frequency and \({f}_{FEM}\) is the FE-simulated natural frequency. The optimization was conducted using a Multi-Objective Genetic Algorithm. If the error was excessively large, the variables were modified. The reverse finding numerical method continued until the error was less than 10%.

The proposed research process facilitated effective establishment of equations between the reaction force and the stiffness of a machine’s supports. The study also constructed a modal analysis model as shown in Fig. 15 able to update support stiffness automatically from the static FE-model, produce results consistent with those of the EMA, and predict the natural frequencies and mode shapes of a different structural mechanism. In conventional machine design, the settings of support stiffness are unknown to machine builders. Therefore, the research results could provide machine builders with insights into the development of new machines and enable prediction of a machine’s modal changes when its weight changes due to changes in its structure, such as modifying the 3-axis machine tools to four or five axes machine.