Estimation of mass matrix in machine tool’s weak components research by using symbolic regression
Graphical abstract
Introduction
Weak component identification of machine tool structure is vital for machining stability. During the machining process, the weakest component of a machine tool has an effect on machine tool dynamic characteristics (Albertelli et al., 2012, Chen et al., 2004, Sekler et al., 2012). Law et al., 2013, Law et al., 2013 developed a position-dependent, multibody, dynamic FE model for a machine tool and used the developed model to identify the weak component of the machine tool structure. Moreover, the chatter of a machine tool structure can be predicted just from the dynamic properties of the spindle and tool system (Sulitka, Kolar, & Janota, 2009). Modal mass distribution matrix which is defined as the multiplication of mass matrix of a machine tool and its corresponding modal shape matrix reflects well the weakness distribution of machine tool’s components (He, Mao, & Liu, 2016).
Modal mass is the important dynamic parameter of the machine tool as a vibration system. For example, the modal mass and mode shape are used to calculate the equation coefficients of motion and dynamic load of the machine tool as a vibration system under certain working conditions, which provides an important basis for control design and load design. But to get it, the two items including mass matrix and scaling modal shape matrix must be acquired first. In generally, the scaling modal shape matrix of a machine is easily acquired by EMA. Kono, Lorenzer, Weikert, and Wegener (2010) obtained the natural frequencies and mode shapes of a machine using EMA. Kushnir (2004) obtained the mode shapes of a machine tool combining EMA with OMA. The mass matrix of a machine tool changes when the machine tool’s component position changes (Afazov et al., 2012, Beskos, 1987, Sadek and Knight, 1972). The modal mass acquisition methods of the vibration systems are currently divided into direct acquisition and indirect acquisition.
- (1)
The finite element method is mainly used, but the method currently has some problems in calculating the modal mass: it is necessary to take into account the mechanical properties of the material, such as the elastic modulus and Poisson's ratio, increasing the computational difficulty; for the classical formula of modal mass, the mode shape, can be obtained by modal analysis, and the system's overall mass matrix is very inconvenient to obtain directly. A: the overall mass matrix of complex systems cannot be given by experience; B: the mass matrix in finite element analysis is very large, and the data is so large that it can blast the computer hard disk.
- (2)
Experimental modal analysis methods is common in addition to the finite element method. At present, the experimental modal analysis method uses the UMM (He et al., 2016) method to obtain the mass matrix. The indirect acquisition method of the mass matrix depends on the modal mode, and the noise contained in the mode of the data processing is mixed into the mass matrix, resulting in the result is inaccurate and fails in severe cases.
In summary, there is an urgent need for a method of directly obtaining the mass matrix based on a general equation of multi-degree-of-freedom vibration system from a data source, and then used to calculate the modal mass.
The information is compressed in the form of correlations from which the mass matrix can be obtained. Assuming a functional relationship between the groups with a certain number of free variables and coefficients (constants), a regression analysis to minimize the error between estimation and experimental values is carried out to determine the appropriate values of the coefficients.
Evolutionary computation is a kind of calculation technology based on population operation. It can search for multiple solutions in solution space in implicit and parallel, and improve the efficiency of acquiring solution by utilizing the differences between different solutions. Therefore, evolutionary computation is more suitable for solving model search problems. Individual behavior is unpredictable but the whole is consistent, and the distance of two different individuals is the most appropriate. Similar to the behavior of biological groups, there exists a social information sharing mechanism in groups, which provides an advantage for the evolution of groups. Backtracking search algorithm (BSA) (Modiri-Delshad, Kaboli, Taslimi-Renani, & Rahim, 2016) is presented for solving economic dispatch (ED) problems and it is an evolutionary technique of optimization with simple structure and single control parameter to solve numerical optimization problems. Artificial cooperative search algorithm (Kaboli, Selvaraj, & Rahim, 2016) is a recently developed evolutionary algorithm with high probability of finding optimal solution in complex optimization problems. And artificial cooperative search algorithm using path coefficient analysis is implemented on linear, quadratic, exponential, and logarithmic models to determine the optimized weighting factors. The obtained adhesion strength and hardness of Ti67/Nb were modeled by particle swarm optimization (PSO) (Rafieerad et al., 2017) to predict the outputs performance. This paper proposes rain-fall optimization algorithm (RFO) (Kaboli, Selvaraj, & Rahim, 2017), a new nature-inspired algorithm based on behavior of raindrops, for solving of real-valued numerical optimization problems. And RFO has been developed from a motivation to find a simpler and more effective search algorithm to optimize multidimensional numerical test functions. A rotational d-q current control scheme is prepared with the Particle Swarm Optimization-PI (PSO-PI) controller (Sebtahmadi, Azad, Kaboli, Islam, & Mekhilef, 2018); which drives an induction motor (IM) through an Ultra Sparse Z-source Matrix Converter (USZSMC). And PSO performance is comparable to genetic algorithms but is faster and less complicated.
Above all, both regression analysis and most popular evolutionary computation methods are based on the assumption that the functional form used is fixed. So a disadvantage of these procedures is producing a lot of computing time spent. In addition, the fixed model functional form is selected empirically and the nature law might not be reflected well by it. It means that the complex model functional expression is selected for complex vibration system but the law implied in the experimental data is simple, or the simple model functional expression is selected for the complex law implied in the experimental data from the complex vibration system. It would thus be advantageous to have an algorithmic way to determine the best correlation that fits experimental data without assuming its functional form.
Symbolic regression is an important evolutionary computation method for data mining, and was first introduced by Michael Schmidt et al. in the science magazine in 2009 (Schmidt & Lipson, 2009). The symbol regression does not assume the form of a function at all, and automatically searches the data for the form of the mathematical formula and the parameters (Xu, Wang, & Zeng, 2012). The main methods of symbol regression are genetic programming (Koza, 1994a), gene expression programming (Ferreira, 2001), step-wise regression algorithm for randomly generating candidate factor sets (Liu, Feng, & Xie, 2000), and so on. The optimized GEP (Kaboli et al., 2017, Rafieerad et al., 2016) (gene expression programming) is applied to precisely formulate the relationships between historical data and estimation. In several ways, the research of genetic programming is the most profound. The process of genetic programming is an adaptive nonlinear search process under the guidance of fitness, and also a generalized hierarchical computer program to describe the problem (Huang & Li, 2001). Originally developed for the automatic generation of computer programs, it has been used in various applications, such as finance (Chen & Yeh, 1996), electronic design (Miller, Job, & Vassilev, 2000), signal processing (Uesaka & Kawamata, 2000) and system identification (Arkov et al., 2000). GP is discussed in detail in Koza's monograph (Koza, 1992).
The symbol regression method proposed in this paper solves the problem that the mass matrix is acquired directly based on a general equation of multi-degree-of-freedom vibration system from a data source, and then used to calculate the modal mass. It is searching the target model from the simple model structures to the complex ones and the efficiency can go up. In this paper, in comparison with the popular evolutionary computation method PSO, the accuracy is improved. Besides, compared to the regression analysis and the popular evolutionary computation method PSO, the model searched by GP from simple to complex can reflect the law in experimental data well for the lack of the redundant items in model structure. The method is: genetic programming is used to search out the equation expression structure and its coefficients among a group of variances including displacement, velocity, acceleration and external excitation force. And the mass matrix is contained in the equations’ coefficients. Moreover, the GP method is compared with linear regression analysis method (LRA) and the particle swarm optimization (PSO) (Rafieerad et al., 2017), and the superiority of the GP method is shown. In the present study, the main goals are as follows: (1) traditional finite element method cannot get the modal mass easily because the mass matrix is hard to get, especially for complex system, (2) UMM method as an experimental modal analysis method has the problem that the noise in the collected data of mode shape would be mixed into the mass matrix, and (3) Proposing a method of directly obtaining the mass matrix based on a general equation of multi-degree-of-freedom vibration system from a data source for acquiring the modal mass.
Organizations of the paper is as follow: Section 2 presents the theoretical analysis of the method to obtain the mass matrix of a machine tool by using genetic programming (GP). Section 3 presents a simulation verification of the estimation method of mass matrix. Section 4 presents the application of the method to a real machine tool and its verification using the result from LMS software. Section 5 gives a summary of the whole paper.
Section snippets
Method to obtain mass matrix
The modal mass distribution matrix can reflect the weakness distribution of machine tool structure. In this part, a theoretical derivation is introduced for mass matrix estimated by symbolic regression. As we know, modal mass of component can reflects the structural vibration energy. So the weaker structures of machine tool can be identified through modal mass distribution matrix (He et al., 2016). If the local amplitude in modal mass distribution matrix is higher, it means that the
Numerical simulation
We use a 4-DOF mass-spring-damper system to simply verify the effectiveness of proposed method. One set of parameters selected represents the model structure’s property at Table 1. And a third mass matrix is calculated in Eq. (1). In addition, the scaling modal shape matrix is calculated from the equation (10). Finally, the modal mass distribution matrix of all elements is calculated out. The estimated mass matrix, estimated damping matrix and estimated stiffness matrix are compared with the
Experiment verification
Before implementation, 0–2048 Hz was selected as the interest and analysis frequency band. In order to validate the proposed method, the following steps need to be implemented. Firstly, experimental modal analysis (EMA) was conducted through an impact testing to identify the natural frequencies and modal shapes of the spindle structures between 0 Hz and 2048 Hz, and the references of modal mass are acquired by using LMS software which are used to verify the correctness of the modal mass model.
Conclusions
Modal mass is the important dynamic parameter in weak component research of machine tool structure and also for its control design and load design. And the big problem is that the mass matrix of a machine tool is hard to get correctly. The symbol regression method proposed in this paper solves the problem that the mass matrix is acquired directly based on a general equation of multi-DOF vibration system from a data source, and then used to calculate the modal mass. It is searching the target
Acknowledgments
This work is supported by National Natural Science Foundation of China (NSFC) under Grant Nos. 51505084 and 51775212. The author also would like to express their appreciation to their supporter, especially Wei Long.
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