A novel model reduction technique for time-varying dynamic milling process of thin-walled components

Abstract

In this work, a novel model reduction technique is proposed, which is based on the hybrid coordinate space modal synthesis method and structural dynamics modification method. This technique can be used to construct the corresponding reduced-order model (ROM) to solve some problems, which focus on the updating of time-varying dynamic parameters for thin-walled components in the milling process. These characteristics of the pending workpiece are analyzed, and then the entire workpiece is divided into two substructure parts: constant workpiece and removed material workpiece. The substructure part of the constant workpiece is analyzed by the double-coordinated free interface method, and the substructure part of the removed material workpiece is analyzed by the finite element method (FEM) among them. Afterwards, the two substructure models in the hybrid coordinate space are coupled based on the interface coordination condition, and the ROM can be obtained about the degrees of freedom (DOFs) of thin-walled components. Moreover, the structural dynamics modification method is introduced, which is used to explore the modal characteristics and vibration response characteristics of the ROM, and is performed to update the dynamic parameters for thin-walled components systems in the milling process. Comparing the dynamical parameters obtained by the full-order model (FOM) method and the reference method under the same conditions, the results indicate that the proposed method can be applied to a variety of boundary conditions effectively, its normalized relative frequency difference (NRFD) values are all lower than 5%, and these values are continuously and stably close to 0. Its modal assurance criterion (MAC) values are all higher than 0.99, and its frequency response assurance criterion (FRAC) values are all 1. For the update speed, its maximum growth rate is 97.38%. Accordingly, the proposed method has nice universality and efficiency, which is embodied in the updating of time-varying dynamic parameters of thin-walled components.

Appendices

Appendix 1

The mode matrix [^αR] in reference [50] is as follows:

$$\left[{}^\alpha R\right]=\begin{bmatrix}{{}^\alpha\Phi}_{ik}&{{}^\alpha\psi}_{ib}\\0&{{}^\alpha I}_{bb}\end{bmatrix}$$

(A.1)

where [^αΦ_k] is related to the internal DOFs of substructure α and represents the modal matrix reserved by this system, and it is obtained by extracting the characteristic parameter matrix of the first k order. [^αΨ_ib] represents the attachment mode matrix of the internal DOFs. [^αI_bb] is the identity matrix.

Appendix 2

The mode matrix [^βR] in reference [51] is as follows:

$$\left[{}^\beta R\right]=\begin{bmatrix}{{}^\beta\Phi}_{ik}&{{}^\beta\psi}_{id}\\{{}^\beta\Phi}_{bk}&{{}^\beta\psi}_{bd}\end{bmatrix}$$

(A.2)

where [^βΦ_k] represents the reserved modal matrix of this system modal matrix [Φ], and it is obtained by extracting the characteristic parameter matrix of the first k order. [^βΨ_d] represents the remaining attached modal matrix of this system.

Appendix 3

It is necessary to reasonably select the number of reserved modes of the system to construct the mode transformation matrix, when the modal synthesis method is used to complete the order reduction process of the model. This current undamped transient analysis problem is discussed, and its linear governing equation can be described as follows.

$$M\ddot{X}(t)+KX(t)=F(t)$$

(A.3)

This modal transformation is substituted into the above equation, and then the new equation is obtained.

$$MR\ddot{P}(t)+KRP(t)=MR\Lambda {P}^{*}(t)$$

(A.4)

Left multiplication is used, and then the new relationship is obtained.

$${R}^{T}MR\ddot{P}(t)+{R}^{T}KRP(t)={R}^{T}MR\Lambda {P}^{*}(t)$$

(A.5)

It is assumed that the displacement of this system can be expressed as a superposition of wave functions, which is based on the Fourier transform conditions. Therefore, this single system wave function can be expressed as follows:

$$\left\{\begin{array}{l}X=\overline Xe^{i\omega t}\\P=\overline Pe^{i\omega t}\end{array}\right.$$

(A.6)

Then, a new transient equation is obtained.

$$-{\omega }^{2}{R}^{T}MR\ddot{P}(t)+{R}^{T}KRP(t)={R}^{T}MR\Lambda {\overline{P} }^{*}(t)$$

(A.7)

The above equation can be simplified as follows:

$$-{\omega }^{2}I\overline{P }(t)+\Lambda \overline{P }(t)=\Lambda {\overline{P} }^{*}(t)$$

(A.8)

The above equation is adjusted, and then this new relationship is obtained.

$$\overline{P }(t)=\frac{\Lambda }{\Lambda -{\omega }^{2}I}{\overline{P} }^{*}(t)$$

(A.9)

To sum up, this system mode interception criterion can be expressed as Λ ≈ ω², where ω represents the frequency range of concern.