Vibration characteristics and modal analysis of a grinding machine

The machine industry has undergone several developments in the past years, and reducing the cost and time required for machine designing is important. In this study, the vibration characteristics of a precision grinding machine were obtained through experimental modal analysis and finite element analysis. The experimental modal analysis employed single point excitation, and the equipment used to determine the frequency response of the grinding machine comprised a hammer, an accelerometer, and a spectrum analyzer. In addition, the resonance frequency, damping factor, and modal shape of the grinding machine were determined. The natural frequency, modal shape, and interface stiffness were determined through finite element analysis. Finally, the theoretical model and the experimental modal analysis models were compared, and get closer to the actual situation of a model to conduct several times analysis. Thus, this paper presents a reliable and convenient method to study the characteristics of machine tools; this method can reduce unnecessary costs and find structural weaknesses in machine designs for improvement.


Introduction
At present, three types of dynamic structural analysis techniques are known: the distributed-mass beam method [1], the lumped-constant beam method [2], and the finite element method [3].In this paper, the dynamic characteristics of machine tools were studied through the finite element method.Moreover, the material properties, geometric properties, and boundary conditions were obtained.
Zhang [4] proposed computer-aided engineering to predict the dynamic behavior of machine tool structures.The dynamic flexible method was used for this purpose, and the experimental results presented the basic dynamic characteristics of a unit area.Sahoo [5] investigated the transient characteristics of the composite material layer plates, which included static properties, free deflection vibration, and natural frequency.These findings were confirmed by comparing them with the experimental results in the present study.Kono [6] used the contact stiffness model to establish the mathematical relationship between the load and rigidity of the fixed structure.The experiment results suggested that the proposed method can improve the rigidity of the machine support.
The structural system was subjected to an external force to produce a reciprocating movement in the reference direction.The structural system exhibited different frequency and amplitude under different operating conditions.In general, vibration is an undesirable characteristic of a structure; however, the vibration characteristics can be used for structural testing of a machine.Vibration can be classified into two types: First, free vibration is generated by the natural frequency of a structural system without an external force.Because the initial displacement and initial velocity of the system can periodically change its kinetic energy, the system performs repetitive motions.
Second, forced vibration is produced from the frequency of an external force.The system movement is then applied to a dynamic external load.
The vibration can be described as a set of modal parameters, which include natural frequency, mode shape, and damping.The natural frequency is an important component of the resonance effect.If the external excitation frequency is the same as that of the structure, a resonance effect can result.The modal analysis can be classified into experimental modal analysis and theoretical modal analysis.

Experimental modal analysis theory
For a damped mechanical member during resilient movement, the discrete component can be expressed as a linear differential equation as shown in Eq. (1) [7]: where [ ] is the mass matrix, [ ] is the damping matrix, [ ] is the stiffness matrix, { ( )} is the acceleration vector, { ( )} is the velocity vector, { ( )} is the displacement vector, and { ( )} is the force vector.By applying a Laplace transform, the linear differential equation is transformed from the real domain to Laplace domain.The Laplace domain equation is shown in Eq. ( 2): where { ( )} is the displacement of the Laplace transform, { ( )} is the force vector of the Laplace transform, { (0)} is the initial displacement vector, { (0)} is the initial velocity vector, and is the Laplace variable.
If the Laplace parameter is replaced by , the frequency information can be obtained as follows: (1) Homogeneous solution: If is the solution of the system matrix that makes the determinant of [ ( )] equal to zero, then is the eigenvalue.The corresponding eigenvector is called the modal shape.The eigenvalue and eigenvector are given as follows: where is the th modal damping and is the th modal frequency.(2) Particular solution: The force vector of the linear differential equation is nonzero, and eigenvectors describe the particular solution of the equation as follows: where [ ( )] is the transfer matrix.
The determinant of the system matrix [ ( )] is the quadratic function given in Eq. ( 3): where * is the complex conjugate of .The inverse matrix [ ( )] is shown in Eq. ( 4): The denominator of [ ( )] is a quadratic function; by using the remainder theorem, Eq. ( 4) can be rewritten as follows: where [ ] is the th complex matrix and [ * ] is the complex conjugate matrix of [ ].

Experimental modal analysis
Experimental modal analysis is illustrated in Fig. 1.The process can be divided into four stages: (1) Measuring technology: Appropriate measuring instruments were used to measure the analyte in order to obtain the input and output response.A hammer, an accelerometer, and a spectrum analyzer were used as shown in Fig. 2.
(2) Signal analysis: The resulting input and output signals of the time-domain response signal were used for signal processing, for example, through Fourier transformation.This study used a time-domain input signal and a frequency-domain output signal as shown in Fig. 3.
(3) Curve fitting: Curve fitting techniques were used to obtain the modal parameters of the system as shown in Fig. 4.
(4) Data analysis: Using curve fitting calculate the modal is necessarily all true [8], which through the damping ratio between 1 % and 5 % to obtain correct modal parameters.The natural frequency and modal damping ratio of the machine tools are shown in Table 1.

Finite element modeling and meshing
In this study, the machine tool model was prepared from casting drawings.The model included the internal structure and components, omitting the fillets and holes.The entire machine tool model is shown in Fig. 5.This study used the SOLID187 and COMBIN14 elements to simulate contact rigidity.The free mesh function and SOLID187 were used for the machine tool model, and COMBIN14 was used for simulation of contact rigidity.The finite element model is shown in Fig. 6.The convergence curve is shown in Fig. 7.The element mesh size(body size) is 0.05 m and in the circle model element size (body size) is 0.01 m.

Contact rigidity calculation
The contact rigidity between the linear guideway and linear slide was obtained from the design catalog.The contact rigidity has only two directions: vertical rigidity and lateral rigidity.The direction of the linear slide withstanding force is shown in Fig. 10.The contact rigidity is given in Table 3.
The linear ball screw nut and support bearing rigidity can be calculated using Eq. ( 7) [9]: where is the axial rigidity of the support bearing, is the withstanding pressure, and is the axial displacement.The equation for axial displacement is given below: where is the axial load, is the diameter of steel ball bearings, and is the bearing contact angle.The equation for axial load calculation is given in Eq. ( 9): where is the number of balls.The calculation parameters for the axial rigidity of support bearings can be obtained from the design catalog.In this study, the rigidity of the linear ball screw nut was calculated.Moreover, the linear ball screw prepressed value can be calculated as shown in Eq. ( 10) [10][11][12]: where is the reference torque (kgf•cm), is the lead, is the lead angle, and is the prepressed value (kgf).The unit length of the calculated prepressed was measured in centimeters.The axial rigidity can be calculated as shown in Eq. ( 11) [9,12,13]: where is the axial rigidity of the ball screw nut (N/μm), is the basic dynamic load rating (N), is the original rigidity of the ball screw (N/μm), is the basic dynamic load rating coefficient, which was assumed to be 10 % of the dynamic load rating for obtaining the nut axial rigidity.
The linear ball screw was subjected to axial rigidity, and the three-axis nut axial rigidity is 6295 shown in Table 4.The axial rigidity of the three-axis support bearing was 980 N/μm.

Construction of a cylindrical coordinate system and node coupling
In precision grinding, the finite element model of the three-axis linear ball screw was used to establish the respective cylindrical coordinate system to transfer the DOF nodes of COMBIN14.The cylindrical coordinates for each linear ball screw are shown in Fig. 11.The COMBIN14 elements were used to simulate the linear ball screw nut and bearing axial rigidity.Moreover, the COMBIN14 elements were used to choose the direction of the linear ball screw coupling two nodes DOF.To avoid self-rotation, they are bound by the rotation of the motor so that the elements nodes are coupled in the direction at the end of the COMBIN14 elements.The cylindrical coordinates coupling of two nodes DOF; the settings of all linear ball screws are shown in Fig. 12.
The ANSYS modal analysis model was used to solve the ideal model without damping; therefore, the results of ANSYS modal analysis and experimental modal analysis will have some error.In this study, the results of ANSYS modal analysis and ME'SCOPE modal analysis were obtained for only five frequency and mode shapes.The complete settings of the finite element model are shown in Fig. 13.

Natural frequency value comparison
The experimental process was assumed to be free from human error.The dynamic characteristics obtained through the experimental modal analysis were similar to the actual results.The experimental modal analysis results obtained in this study were the contact rigidity set reference.A comparison of the natural frequency of the five groups obtained through experimental modal analysis and finite element analysis is shown in Table 5 and Fig. 14.The error of 1.3 % for the second group was the optimal result obtained.The ANSYS finite element model ignores the fillets and the gap between the structures; the ANSYS modal analysis process has no damping, thereby resulting in the error.In the experimental modal analysis, the first group exhibited a low-frequency signal interference because of the size of the hammer.

Comparison of modal shapes
In the experimental modal analysis and finite element analysis, the natural frequency was divided into five groups; because of the size of the hammer, natural frequency error resulted in a signal interference of mode shape in the first group.The five groups corresponding to the natural frequency of mode shape are shown in Fig. 15.

Analysis of results for a precision grinding machine
In this study, the operating speed of the precision grinding machine was 15.000-26.000rpm; therefore, model shapes from 250-500 Hz were discussed.For the given operating speed, the spindle head causes larger vibration than do any other structures.The relative movement of the structure is shown in Table 6.The relative movement of the interface is the spindle head, the workplane and the vertical column.

Conclusions
In this paper, the vibration characteristics of a precision grinding machine were obtained through experiment modal analysis and finite element analysis.The following conclusions can be drawn from this study: (1) The axial rigidity of the three-axis support bearing was 980 N/μm.
(2) The natural frequencies of five groups were compared through experimental modal analysis and finite element analysis.The error of 1.3 % in the second group was the optimal result.(3) The operating speed of the precision grinding machine was 15.000-26.000rpm; therefore, the model shapes from 250 to 500 Hz were discussed.

Fig. 10 .
Direction of linear slide withstanding force

Fig. 14 .
Fig. 14.Natural frequency comparison a) Comparison of first group b) Comparison of second group c) Comparison of third group d) Comparison of fourth group 2743.VIBRATION CHARACTERISTICS AND MODAL ANALYSIS OF A GRINDING MACHINE.DYI CHENG CHEN, MING FEI CHEN, JING HAO KANG, CHIA CHUN LAI e) Comparison of fifth group Fig. 15.Five groups of corresponding mode shape 2743.VIBRATION CHARACTERISTICS AND MODAL ANALYSIS OF A GRINDING MACHINE.DYI CHENG CHEN, MING FEI CHEN, JING HAO KANG, CHIA CHUN LAI

Table 1 .
Natural frequency and modal damping ratio

Table 2 .
Material properties and support rigidity of machine tools

Table 3 .
Contact rigidity between linear guideway and linear slide

Table 4 .
Three-axis nuts axial rigidity of linear ball screw Linear ball screw nuts Ball diameter Dynamic load rating Cycle Axial rigidity

Table 5 .
Compare the natural frequency of the five groups ModelExperimental modal analysis Finite element analysis Error

Table 6 .
The relative movement of the structures