A novel method for constructing a dynamics model with the flexible characteristics

An accurate dynamics model is an essential part of ensuring the effectiveness of constructing flexible dynamical systems in the simulation. Finite element method is an effective method to build a flexible structural dynamics model, but this method will cause the model to have too many orders to increase the difficulty in the simulation. The reduction method based on the finite element principle is an effective method to build a streamlined model. This paper proposes a method that combines the critical modes of the structure and the reduction method based on the finite element model to construct an effective flexible dynamics model. Firstly, the determination of primary and secondary degrees of freedom is provided by the judgment of critical modes. Then, an adequate simulation model and an experimental setup are established respectively based on a flexible structure as an example. Finally, through the simulation comparison, the reduction model and the full-order model keep the same simulation results, but the simulation time consumed by the latter is 13 times of the former. In terms of experimental and simulation comparison, the simulation and the experimental results have the same transient oscillation frequency, and the steady-state angle deviation is 2 mrad.


Introduction
Virtual prototyping technology [1] is a digital design method based on computer simulation models, significantly shortening the product development cycle and reduce product development costs. This technology is widely used in the field of transportation [2], robotics [3], machine tools [4], and highprecision motion [5]. Whether the constructed virtual prototype is instructive for the system depends on how well the prototype model matches the actual model. The dynamics model is an essential part of the virtual prototype model. Typically, the dynamics model is usually constructed using the lumped parameter method [6]. This method transforms the control object into inertial, damping and elastic elements through a series of equivalences and simplifications [2], but this method ignores the flexible properties of the structure. In the precision motion system[7], the flexible vibration has affected the design of the motion system controller and the final system positioning accuracy. Therefore, it is no longer desirable to construct a simplified dynamics model.
Common dynamics modeling methods mainly include experimental modeling and theoretical modeling. The experimental modeling method mainly uses the system identification method in the time domain or frequency domain [9]to obtain the model. The prerequisite is that the control object must be available. In addition to the lumped parameter method shown above, the theoretical modeling also includes the finite element method that can reflect the flexibility of the structure. Its principle is to approximate the mechanical structure by a discrete method [10]. This method has been applied in the simulation of systems such as flexible robotic arms [11] and flexible components [12]. In this method, the small meshes can approximate the accurate model [13], but this implies the tremendous computational effort, complicated design of controllers and difficulty in ensuring real-time operations.
The model reduction method is an effective method to build a simplified model and has been widely used in precision machinery [14]. The basic idea of model reduction is to find a reduced-order basis, in a sense that the reduced-order system can optimally approximate the original large-scale system and maintain the main characteristics of the original-scale system. This method can be divided into three categories according to the reduction principle. The first type is based on the finite element model. This method obtains the reduced-order basis and reduced-order model on the basis of determining the primary and secondary degrees of freedom of the structure and removing the secondary degrees of freedom. Common methods include such as Guyan [15], dynamic condensation [16], improved static condensation system method [17], improved dynamic condensation system method [18], system equivalent expansion condensation method(SEREP) [19]; The second and third types are respectively based on the principles of projective and non-projective methods [20]to find a reduced-order basis and then obtain the reducedorder object. The former usually includes the moment matching method and the balanced truncation method based on Krylov subspace, and the latter includes methods such as singular perturbation and Hankel optimal reduction. The first type realizes the construction of the dynamics model based on the reduction of the finite element model. These two latter types are mainly from the perspective of control performance, and the ultimate goal of reduction is to obtain a low-dimensional controller to control the original high-order system [21]. Their mathematical meaning is clear, but the physical meaning is not. The first method is mainly used in the structural field to realize the reduction of the model from the primary and secondary degrees of freedom of the structure, and its physical meaning is evident [22]. This paper aims to construct an effective flexible dynamics model using the finite element method. In the condensation methods mentioned above, the SEREP method mainly uses modal information to construct the reduced-order basis. This method not only can capture the dynamic properties of the structure better with modal information and predict the system with the high-frequency dynamic properties with pre-set constraints better than the static and dynamic condensation methods but also can be suitable for condensing the large structures [23].
The prerequisite for realizing the FEM-based reduced order is to specify the primary and secondary degrees of freedom of the structure. The eigenvalues and eigenvectors of the structure are important factors for constructing a dynamics model. Therefore, the contribution of eigenvalues to the displacement of the structure can be used to determine the order of the important eigenvalues, and then use it to judge the primary and secondary degrees of freedom of the structure. Common eigenvalue ranking methods are these based on the DC gain or peak gain at the resonance peak of the model transfer function and these based on the observability and controllability of the system [24]. The former is generally limited to eigenvalue ranking for single-input, single-output systems, while the latter can be used for single-input, single-output and multiple-input, multiple-output systems.
In this paper, it proposes a method to establish a flexible dynamics model by combining the critical modes of the structure and the reduction method based on the finite element model, and then conduct a simulation and experimental comparison analysis of this modeling method by using a single-degree-offreedom electromechanical system with a flexible structure. The rest of the paper structure is as follows: Section 2 describes the SEREP method and the method of DC gain and peak gain for determining the critical modes, and on this basis, a comprehensive analysis method for constructing a flexible dynamics model is presented; In section 3, the method proposed in the previous section is used to construct a dynamics model of a flexible structure; Section 4 is a comparative analysis of motion system simulation and experimental results; Section 5 is a summary of the paper.

Construction of a reduced-order dynamics model
The motion of the equation for a linear flexible time-invariant structure with n DOFs is as follows: where M, C, K, F denote the mass matrix, damping matrix, stiffness matrix and excitation force, respectively, and , , x x x   the acceleration, velocity and displacement vectors, respectively. A coordinate transformation matrix is introduced to describe the transformation relationship between the existing coordinates and the reduced-order coordinates of the model.
where the superscript R is denoted as the reduced order; T R is the time-invariant reduced-order basis.
Combining Eqs (1) and (2), the obtained kinetic equations of the model after the reduced order are: ; the superscript T represents the transpose of the matrix.
The model-based order reduction method in this paper is mainly based on the principle of primary and secondary degrees of freedom. In order not to lose generality, the effect of damping is neglected, and the Eq. (1) Among them, m represents the number of primary degrees of freedom of the system, s represents the number of secondary degrees of freedom of the system, m∪s=n, m∩s=Ø.
Based on the reduced-order model, the dynamics model is constructed for the final simulation analysis. The mathematical model is as follows: where q(t) denotes the modal displacement vector, A mR , B mR , C mR respectively denote the system state matrix, input matrix and output matrix, being formed by the structure reduction in the modal coordinate system; y(t) denotes the output displacement, and u(t) denotes the system input force vector.
Since this paper mainly constructs the dynamics model from the modal information, this section mainly describes the SEREP method and the judgment method of critical modes. Then on this basis, a comprehensive analysis method combining these two methods is proposed.

SEREP Method
The reduced-order basis of the SEREP method is constructed based on the modal information of the structure. The variables are converted from being in the physical coordinate system to being in the modal coordinate system through transformation. Compared to the static and dynamic condensation, this method effectively predicts the system with high-frequency kinematic properties with predefined constraints [19].
First, the physical coordinate system is transformed into a modal coordinate system.

   
where Φ represents the eigenvector of the structure. Let  =0  where the superscript -1 represents the inverse of the matrix; T SEREP represents the reduced-order basis.
In general, the number of primary degrees of freedom is greater than the number of critical modes, that is, Φ mm is not a square matrix, then Φ mm -1 =Φ mm + .
  Among them, the superscript + represents the pseudo inverse of the matrix. The mass and stiffness matrix after the condensation can be obtained by Eq. (8):

Determination of the critical modes
The determination of the primary degrees of freedom is a prerequisite for obtaining the dynamics model. This paper takes the critical modes as the goal and then finds the primary degrees of freedom that can express the critical modes. The critical modes are determined by using the DC gain by judging the contribution of each mode to the system response. The premise of this application is to express the dynamic equation with the eigenvalues and eigenvectors as the uncoupled motion equation in the modal coordinate system. When the damping is included in the system, the gain is called peak gain, which has the same meaning as DC gain. ( 0 0 ) where φ nji φ nki is the product of the jth and kth row eigenvectors of the ith mode; ω i is the ith eigenvalue; ξ i is the ith damping ratio.

A comprehensive method for building the dynamics model
Based on the existing theory, this section proposes the following application procedures: 1) Determine the primary and secondary degrees of freedom and the reduction thresholds according to the critical modes and control indicators of the system; 2) Obtain the critical modal set according to Eq. (11).
3) Determine the primary and secondary degrees of freedom of the structure, and then organize the stiffness and mass matrices. 4) Determine the principal eigenvalues and the eigenvectors corresponding to the primary and secondary degrees of freedom based on the sorted stiffness and mass matrix; 5) Calculate the reduced-order basis of the model using Eq. (8); 6) Obtain the reduced-order stiffness and mass matrix, and their corresponding eigenvalues and eigenvectors； 7) The effect of the condensation is judged by whether the eigenvalues obtained after the condensation are equal to the original eigenvalues ({f R }={f orig }) and whether the modal confidence calculated from the eigenvectors after condensation and the original eigenvectors is equal to 1 (MAC=1).
where Φ orig represents the vector obtained under the original stiffness and mass matrix of the model; Φ R represents the eigenvector obtained under the reduced-order stiffness and mass matrix.
8)The dynamics model is obtained after the condensation.
The specific application flowchart of the method is shown in Fig. 1.

Dynamics model
In this section, a flexible beam is used as the analyzed object, and its dynamics model is constructed based on the method mentioned above.

Structural parameters of the flexible beam
The material of the flexible beam is aluminum-magnesium alloy, and its specific parameters are as follows.

Dynamics model of the flexible beam
As a continuum, the flexible beam is discretized to obtain 121 nodes, 20 elements, and each node has six degrees of freedom (x, y, z, Rx, Ry, Rz). Firstly, the eigenvalues, eigenvectors, stiffness matrix and mass matrix of the structure are obtained using the finite element software; secondly, the DC values corresponding to the eigenvalues of each order are calculated by Eq.10, and the accumulated DC values are then obtained. As can be seen from Fig. 2, there are 41 orders of eigenvalues corresponding to DC values above 10e-5, and the sum of the cumulative values is equal to 0.9999  , that is, these 41-order modes can accurately express the structural characteristics of the model. Since the servo cycle of the control system used is 50 us, the sampling frequency is 20 kHz. According to the Nyquist sampling theorem and considering the practical engineering application, the effective analysis signal is 1/2.56~1/4 times the sampling frequency, so 5 kHz is selected as the analysis frequency. Through the DC sorting, it can be seen that there is a total of 12 frequencies within 5 kHz, which account for 95% of the accumulated DC. The eigenvalue results are shown in Table 2.  .34 In the electromechanical system, the movement of the flexible beam is set for the rotational movement around Rz only. Based on the obtained eigenvalues corresponding to the vibration shape and the primary degrees of freedom that have a greater impact on the system motion, the rotational degree of freedom Rz of each node is extracted as the primary degree of freedom, and the rest are as the secondary degrees of freedom. Then the degrees of freedom of the flexible beam are reduced from 121*6 to 21. The extracted mass and stiffness matrices are shown in Fig. 3. According to the SEREP method mentioned above, the reduced-order basis is generated as shown in Fig. 4. Fig. 4 Reduced matrix based on the SEREP method The corresponding reduced-order stiffness and mass matrices are generated from the reduced-order basis, and the eigenvalues and eigenvectors are derived, as shown in Table 3 and Fig. 5. Combining with the original eigenvectors, the MAC calculated according to Eq. (12) is constantly equal to 1. Also, combining with the eigenvalues in Tables 2 and 3, it can be seen that the accuracy of the reduced model can be guaranteed.

Simulations and experiments
An electromechanical system with a flexible structure is used as an experimental setup for verifying the effectiveness of the mentioned technology based on the finite element model in the motion simulation system. This test device comprises a host computer, a control board, a driver and a motion system. Among them, the motion system mainly consists of a flexible beam and a drive motor. The flexible beam is driven by the motor to rotate. The detailed experimental setup is shown in Fig. 6.
To verify the accuracy of the above-mentioned flexible structure dynamics model, the prerequisite is to ensure that the parameter settings of the drive motor and its driver are correct. After parameter query and identification, the motor torque constant is 0.4157 N.m/Arms. The effective value current corresponding to the 1V voltage of the driver is 0.3119 Arms. In addition, an equivalent damping factor of 0.019 N*m/(rad/s) was identified at the motor spindle using the uniform speed section of the motor operation by using the equivalent method. The simulation and experiment are divided into two parts. The first part is to determine the correctness of the parameters of the system except for the flexible dynamics model. The second part is to verify the correctness of the flexible dynamics model based on the first part.
All simulation analyses in this chapter are carried out in the MATLAB-Simulink environment.