A new general method for assigning frequencies with low spillover through structural modifications

Abstract

By modifying the mass and stiffness parameters of a structure, a subset of its natural frequencies can be assigned. However, such an assignment is usually accompanied with changes of the remaining frequencies, known as ‘spillover’. Eigenvalue or frequency spillover often needs to be avoided in engineering.

In this paper, a new and general method is proposed for assigning a subset of the natural frequencies of a structure with a low level of spillover through structural modifications. The method combines a multi-population genetic algorithm and a radial basis function neural network and is implemented in a MATLAB-PYTHON-ABAQUS data interaction system. It is suitable for theoretical models (for example, finite element models) of general structures.

By means of three numerical examples, the ability of the method to assign subsets of frequencies to discrete and discretised continuous structures while maintaining low spillover is demonstrated.

Introduction

The natural frequencies (or eigenvalues) of a structure determine its dynamic characteristics. When a natural frequency of a structure is close to the frequency of excitation, resonance occurs, often resulting in noise, or malfunction or damage or even destruction. One effective way to avoid resonance is to redistribute the natural frequencies of a structure. A structure’s natural frequencies can be assigned different values using active control, ¹ passive control, ² semi-active control or a combination of active and passive control. ³ Active control uses feedback from sensors to produce the right control actions from actuators to change the ‘natural’ frequencies and/or modes of a structure. Passive control modifies the structural characteristics of a structure (such as its mass and stiffness matrices, and occasionally the damping matrix) to obtain desired frequencies and/or modes. There are two types of structural modifications: forward modification and inverse modification. A forward structural modification usually makes a series of modifications to the model, in a sort of trail-and-error manner or based on experience, until the desired frequencies and/or modes are found. These methods^4,5 are usually very time-consuming and cannot guarantee to find the desired frequencies. An inverse structural modification, on the other hand, takes the desired modal properties as the targets to determine the required modifications, so that the modified model certainly gains the desired dynamic characteristics.^6,7

In fact, usually, only some of the natural frequencies of a structure determine its dynamic behaviour, which means only a subset of its natural frequencies needs to be assigned. However, the theoretically determined structural modifications for the assignment of a natural frequency subset tend to cause the unassigned natural frequencies to shift as well, known as spillover of natural frequencies, which may lead to unpredictable or undesirable dynamics of the structure. Partial frequency assignment aims at keeping the unassigned natural frequencies unchanged, when reassigning a subset of natural frequencies of a structure. In recent years, there has been much research on partial eigenvalue/frequency assignment.

In early active control, a first-order system is used for partial eigenvalue assignment. Saad et al. ⁸ use a projection technique and a variant of Weilandt’s crunch technique for partial pole assignment of linear control systems. It is capable of making pole allocation for large systems. Datta et al. ⁹ use a parametric algorithm to solve some eigenvalue problems of linear systems. This method provides a parametric algorithm for making partial eigenvalue assignment. It offers an opportunity to devise a robust solution to the problem by exploiting the arbitrary nature of the parameters. Inspired by the research results of Brauer, ¹⁰ Araújo et al. ¹¹ propose an eigenvalue perturbation theorem and apply it to the partial eigenvalue assignment of second-order linear dynamic systems. This method employs multiplicative perturbation and introduces static output feedback of system acceleration to solve the partial eigenvalue assignment problem and is applicable to static acceleration output feedback in an undamped second-order system. Ram et al. ¹² propose a method based on measured receptances for the pole/zero assignment problem of active vibration control, and demonstrate the feasibility of the method with several theoretical examples. System matrices can be computed by using the finite element method, but there are always some modelling assumptions, and validated models are expensive to obtain. In contrast, receptances are readily available from modal tests. Mottershead et al. ¹³ propose an eigenvalue assignment method for active vibration suppression based on the output feedback and receptance method. The use of output feedback allows collocated actuator-sensor arrangements, but this necessitates the development of nonlinear characteristic equations in the control gains. Bai et al. ¹⁴ use the system matrix and receptance measurements to solve the partial quadratic eigenvalue assignment problem for a multi-input vibration system using a multi-step hybrid approach. Although the multi-step hybrid method is effective for solving the multi-input partial quadratic eigenvalue assignment problem, it is time-consuming. Zhang et al. ¹⁵ propose an algorithm to solve the partial eigenvalue assignment problem of multi-input state feedback based on the transfer eigenvalue matrix and the receptances. Ariyatanapol et al. ¹⁶ adopt the receptance method and propose a time-delay partial pole placement method for asymmetric systems based on single-input state feedback control, and the effectiveness of the method is proved by numerical examples of friction-induced vibration of a six-degree-of-freedom (DoF) structure and binary flutter of a rectangular wing.

Guzzardo et al. ¹⁷ present a method to find the optimal positions of the actuators and their gains in actively controlled vibration systems, thereby minimizing the control effort. Baddou et al. ¹⁸ propose a new partial eigenstructure assignment method, which can be used in cases where other methods are not applicable, especially those present by Benzaouia. ¹⁹ Liu et al. ²⁰ propose a method to solve the eigenvalue assignment problem of the output feedback part of the undamped structural system with the output matrix as the design parameter. Hu et al. ²¹ propose a constant output feedback control that handles actuator and sensor configurations and uses the output matrix as a design parameter, which can solve part of the eigenvalue and eigenstructure configuration problems of an undamped vibration system. Wei et al. ²² propose a new dual-input control theory including position, velocity and acceleration feedback, which is suitable for active vibration control with unreachable degrees of freedom, and obtain a new partial eigenvalue assignment method. Motta et al. ²³ propose a new partial eigenvalue assignment method, which is capable of reassigning a set of undesired eigenvalues by computing the feedback matrices for velocity and displacement and the influence matrix of actuators in a vibrating system. Belotti et al. ²⁴ propose a novel method for pole placement of linear systems through state feedback and rank-one control. Rather than assigning all the poles to the desired locations of the complex plane, the proposed method assigns only the dominant poles, while leaving others free to assume arbitrary positions within a pre-specified region in the complex plane. The methods described above are all improvements to the active control method, which increases its applicability and promotes the development of control theory. However, the disadvantages of expensive active control equipment and complicated repair and maintenance cannot be changed. Furthermore, the scope of applications of these improved active control methods remains limited.

In active control, spillover can be easily avoided, as manifested in the literature by the sufficient and necessary conditions ²⁵ to ensure that the unassigned eigenvalues remain unchanged. Although an active control method can avoid spillover of a structure, its use requires high cost and high technical requirements,^26,27,28 which becomes infeasible for structures with a large number of degrees-of-freedom and continuous structures. In contrast, passive control in the form of structural modifications is always stable, as long as some simple and easy-to-satisfy requirements are met (concerning the definiteness of the system matrices). A structural modification does not require sensors and actuators and is inexpensive and is applicable to large structures. ²⁹ In general, physical models^30,31,32,33 or modal models^34,35 of structures can be employed for inverse structural modification. However, for a complex structure, the theoretical model (usually a finite element model) is often simplified, often causing a modal truncation problem. It is difficult to obtain accurate structural data (material structure and geometry structure) and modal data, which impedes the use of passive control methods. In experiments, the modal data of complex structures are also difficult to measure accurately, which brings difficulties to active control methods. However, the emergence of the receptance^36,37,38 solves the problem that passive control cannot accurately obtain model structural data and accurate modal data. At the same time, the receptance method also solves the problem that it is difficult to obtain accurate measurement data in active control. The above-mentioned active control method for partial eigenvalue assignment of vibration systems has been widely used, but this methodology has a high technical requirement and is expensive. The passive control method adopted in this paper is easy to operate, inexpensive and easy to implement.

Mermertas et al. ³⁹ propose using springs to maintain the fundamental frequency of rectangular thin plates with an arbitrary number of point masses. Caracciolo et al. ⁴⁰ put forward a design method for a linear vibratory feeder based on the inverse structure modification method, improve the vibration characteristics of the linear vibratory feeder and produce a laboratory vibrating linear feeder to verify the design method. Zhu et al. ⁴¹ propose an optimal selection method and absolute value method to maintain a specific natural frequency based on the Sherman–Morrison formula. Both of these methods are achieved by installing springs on system and can eliminate the effect of the additional mass on specific frequencies. This method is not suitable for some structures that do not allow the installation of additional springs. Zhang et al. ⁴² propose a reception-based frequency assignment method for combined structures. Ren et al. ⁴³ use a genetic algorithm to add mass and stiffness to the original structure based on the frequency response function, and realize the assignment of the original structure’s natural frequency. Belotti et al. ⁴⁴ use the Taylor expansion approximation of the eigenfrequencies in the neighbourhood of a configuration choice, redefine the eigenfrequency assignment as a quadratic programming problem, and design flexible-link multibody systems with a desired eigenfrequency. The methods described above are all geared toward natural frequency assignment for simple structures and are incapable of or cumbersome in dealing with natural frequency assignment in large systems. Furthermore, when performing natural frequency assignment, the above methods do not take into account the problem of frequency spillover. The frequency spillover problem will seriously affect the dynamic characteristics of the structure, causing it to malfunction and even to fail.

Ouyang et al. ⁴⁵ seem the first researchers that succeed in making partial assignment of natural frequencies, albeit for simple-connected series mass-spring systems and multi-connected mass-spring systems. Belotti et al. ²⁵ propose a three-step procedure to assign a subset of natural frequencies with very low spillover, taking into account the prescribed eigenvalues and feasibility constraints, and demonstrate its application in examples of a simply connected series mass-spring system and an Euler–Bernoulli cantilever beam. Furthermore, Chen et al. ⁴⁶ extend Ouyang and Zhang’s work ⁴⁵ to simplified models of marine diesel engine propulsion systems and complex branch shaft systems of diesel generator sets, and partial eigenvalue assignment is accomplished through structural modification using the gradient flow method. Although some of the above methods consider frequency spillover, they are highly mathematical and are difficult to implement and apply to complicated structures. The inverse structural modification method proposed in this paper has low theoretical requirements, high versatility and ease of implementation.

At present, there are not many partial frequency assignment methods based on inverse structure modification. Partial frequency assignment method with inverse structure modification is used, most of which are only for systems with a specific structure. The majority of existing partial eigenvalue modification methods are valid for discrete structures, with very few studies on the assignment of partial eigenvalues for (discretised) continuous structures. Furthermore, profound mathematical theory for partial eigenvalue assignment must be used, and these methods are frequently restricted to some special structures, requiring strong mathematical ability of the users. Based on an intelligent algorithm, this paper proposes a general inverse structure modification method, which can make partial frequency assignment to not only discrete structures, but also discretised continuous structures, and the method is simple and efficient.

The paper is structured as follows. In Section 2, the theoretical approach is described in detail. In Section 3, the effectiveness of the method is demonstrated using an example of a discrete structure and two discretised continuous structures. Finally, in Section 4, conclusions are drawn.

Methodology

Method overview

The free vibration of an n-degree-of-freedom undamped discrete or discretised structure is governed by equation (1), where ${\mathbf{M}}_{\mathbf{0}}\in {\mathrm{R}}^{\mathrm{n}\times \mathrm{n}}$ is the mass matrix, ${\mathbf{K}}_{\mathbf{0}}\in {\mathrm{R}}^{\mathrm{n}\times \mathrm{n}}$ is the stiffness matrix, $\mathbf{q}\in {\mathrm{R}}^{\mathrm{n}\times 1}$ is the displacement vector and $\ddot{\mathbf{q}}$ is the acceleration vector. Equation (2) is obtained by substituting displacement vectors $\mathbf{q}\left(\mathrm{t}\right)=\mathbf{x}{\mathrm{e}}^{\mathrm{i}\mathrm{\omega }\mathrm{t}}$ and $\lambda ={\omega }^2$ into equation (1). The eigenvalue $\lambda$ and the corresponding natural frequency $f=\sqrt{\lambda }/2\pi$ of the undamped structure can be obtained from equation (2). Hereafter, structures under this study are represented by a pair of $\left({\mathbf{M}}_{\mathbf{0}},{\mathbf{K}}_{\mathbf{0}}\right)$ .

${\mathbf{M}}_{\mathbf{0}}\ddot{\mathbf{q}}+{\mathbf{K}}_{\mathbf{0}}\mathbf{q}=0$

(1)

$\left({\mathbf{K}}_{\mathbf{0}}-\mathrm{\lambda }{\mathbf{M}}_{\mathbf{0}}\right)\mathbf{x}=0$

(2)

In this paper, only undamped structures are studied. Due to the simple relationship between an eigenvalue λ and a natural frequency ω, the frequency assignment problem is equivalent to the eigenvalue assignment problem, and thus the two problems are not distinguished in this paper. It is intended to replace the subset ${\lambda }_i\left(i=j,\cdots ,j+p\right)$ of eigenvalues with ${\mu }_i\left(i=j,\cdots ,j+p\right)$ , where $j\in N$ , $p\in N$ and $j+p\le n$ , by making appropriate inverse structural modifications to the structure. Ideally, the remaining $n-p-1$ eigenvalues should remain unchanged after the structural modifications are made, which is a proper partial assignment and would be very challenging to achieve even in theory. Instead, if the remaining $n-p-1$ eigenvalues undergo an acceptably small variation, it would be equally useful to a proper partial assignment for all practical purposes. In fact, because of the avoidable uncertainty in theoretical models of structures and measurement data, a proper partial assignment in theory may not actually lead to a proper assignment in practice. Therefore, the aim of this paper is to assign a subset of eigenvalues while maintaining low spillover, which would less demanding than proper assignment in theory but virtually equally useful in practice. It is assumed that this subset of eigenvalues and their newly assigned values are different from the remaining eigenvalues, that is, $\left\{{\lambda }_j,\cdots ,{\lambda }_{j+p}\right\}\cap \left\{{\lambda }_1,\cdots ,{\lambda }_{j-p},{\lambda }_{j+p+1},\cdots ,{\lambda }_n\right\}=\varphi$ and $\left\{{\mu }_j,\cdots ,{\mu }_{j+p}\right\}\cap \left\{{\lambda }_1,\cdots ,{\lambda }_{j-1},{\lambda }_{j+p+1},\cdots ,{\lambda }_n\right\}=\varphi$ .

Figure 1 depicts the traditional information flow from GUI to ABAQUS/CAE in ABAQUS. The interactive program compiles the user-created structural finite element model in the GUI, which is then converted into an inp file by the ABAQUS/CAE pre-processor. The inp file contains all model data of the structural finite element model, such as nodes, elements and materials. The user submits the inp file to the ABAQUS solver for modal analysis, and the structure’s eigenvalue data is saved to the dat file.

Figure 1.

Traditional communication relationship between GUI and ABAQUS/CAE.

In this paper, MATLAB combines PYTHON to carry out secondary development of ABAQUS, builds a MATLAB-PYTHON-ABAQUS data interaction system and realizes structure modification, as shown in Figure 2. The precise steps are as follows: In the first step, MATLAB first determines the numerical values of the structure modification, and then calls PYTHON to modify the structure’s model data in the inp file to the values from MATLAB to realize the structure modification. In the second step, MATLAB submits the inp file to the ABAQUS solver to complete the structure’s modal analysis. In the third step, MATLAB calls PYTHON to read the structure’s eigenvalue data from the dat file. The optimization algorithm is introduced in MATLAB to continuously modify the structure iteratively by modifying the inp file, so that in the end the structure’s eigenvalues become the required target eigenvalues.

Figure 2.

MATLAB-PYTHON-ABAQUS data interaction system.

The eigenvalues read from the dat file by MATLAB-PYTHON-ABAQUS data interaction system are denoted by $\left\{E{V}_1,\cdots ,E{V}_n\right\}$ . In order to shift the eigenvalues $\left\{E{V}_1,\cdots ,E{V}_n\right\}$ in the finite element model to the target eigenvalues $\left\{{\lambda }_1,\cdots ,{\lambda }_{j-1},{\mu }_j,\cdots ,{\mu }_{j+p},{\lambda }_{j+p+1},\cdots ,{\lambda }_n\right\}$ and realize the partial eigenvalue assignment of the structure, the required structural parameter modification must be determined using an optimization algorithm so that $E{V}_1\to {\lambda }_1$ , $\cdots$ , $E{V}_{j-1}\to {\lambda }_{j-1}$ , $E{V}_j\to {\mu }_j$ , $\cdots$ , $E{V}_{j+p}\to {\mu }_{j+p}$ , $E{V}_{j+p+1}\to {\lambda }_{j+p+1}$ , $\cdots$ , $E{V}_n\to {\lambda }_n$ assigns eigenvalues ${\mu }_j,\cdots ,{\mu }_{j+p}$ and the spillover of eigenvalues ${\lambda }_1,\cdots ,{\lambda }_{j-1},{\lambda }_{j+p+1},\cdots ,{\lambda }_n$ is minimal. The chosen optimization algorithm is a multi-population genetic algorithm, which has good global optimization ability. It is incorporated into the eigenvalue assignment solution procedure of the MATLAB-PYTHON-ABAQUS data interaction system. As a finite element model may have many structural parameters, it is still a challenging to solve the problem at high accuracy only by a single algorithm (multi-population genetic algorithm). Hence, this paper employs a two-step procedure to achieve partial eigenvalue assignment of structure:

In the first step, the multi-population genetic algorithm is introduced into the ABAQUS-PYTHON-MATLAB data interaction system, which is a new approach for partial eigenvalue assignment of the structure, as shown in section 2.2.1. The multi-population genetic algorithm performs global optimization on the structural parameters within the given structural parameter constraint intervals, finds the optimal value of the structural parameters and realizes the partial eigenvalue assignment of the structure. For discrete or discretised structures, the multi-population genetic algorithm obtains the optimal values of the structural parameters mass $m_i^{*}\left(i=\mathrm{1,2},\cdots ,n\right)$ and stiffness $k_i^{*}\left(i=\mathrm{1,2},\cdots ,n\right)$ . The small intervals where the optimal values of the structural parameters are located are determined as ${\eta }_{m_1}=\left[m_1^{*}-\Delta m_1,m_1^{*}+\Delta m_1\right]$ , ${\eta }_{m_2}=\left[m_2^{*}-\Delta m_2,m_2^{*}+\Delta m_2\right]$ , $\cdots$ , ${\eta }_{m_n}=\left[m_n^{*}-\Delta m_n,m_n^{*}+\Delta m_n\right]$ , $\hspace{0.17em}{\eta }_{k_1}=\left[k_1^{*}-\Delta k_1,k_1^{*}+\Delta k_1\right]$ , ${\eta }_{k_2}=\left[k_2^{*}-\Delta k_2,k_2^{*}+\Delta k_2\right]$ , $\cdots$ , and ${\eta }_{k_n}=\left[k_n^{*}-\Delta k_n,k_n^{*}+\Delta k_n\right]$ . The $N^{*}$ groups of structural data for discrete structures are generated within the small intervals, and the eigenvalues corresponding to the $N^{*}$ groups of structural data are read by the MATLAB-PYTHON-ABAQUS data interaction system. The $N^{*}$ groups of structural data and corresponding eigenvalues will be used in the next step.

The same method can also be used for discretised continuous structures. For these structures, the multi-population genetic algorithm obtains the optimal value $\rho A_i^{*}\left(i=\mathrm{1,2},\cdots ,n\right)$ of the linear mass density, and the optimal value $E{J}_i^{*}\left(i=\mathrm{1,2},\cdots ,n\right)$ of the moment of inertia of the cross section to the neutral axis. The small intervals where the optimal values of the structural parameters are located are determined as ${\eta }_{\rho A_1}=\left[\rho A_1^{*}-\Delta \rho A_1,\rho A_1^{*}+\Delta \rho A_1\right]$ , ${\eta }_{\rho A_2}=\left[\rho A_2^{*}-\Delta \rho A_2,\rho A_2^{*}+\Delta \rho A_2\right]$ , $\cdots$ , ${\eta }_{\rho A_n}=\left[\rho A_n^{*}-\Delta \rho A_n,\rho A_n^{*}+\Delta \rho A_n\right]$ , ${\eta }_{E{J}_1}=\left[E{J}_1^{*}-\Delta E{J}_1,E{J}_1^{*}+\Delta E{J}_1\right]$ , ${\eta }_{E{J}_2}=\left[E{J}_2^{*}-\Delta E{J}_2,E{J}_2^{*}+\Delta E{J}_2\right]$ , $\cdots$ , and ${\eta }_{E{J}_n}=\left[E{J}_n^{*}-\Delta E{J}_n,E{J}_n^{*}+\Delta E{J}_n\right]$ . The $N^{*}$ groups of structural data for the structures are generated within the small intervals, and the eigenvalues corresponding to the $N^{*}$ groups of structural data are read by the MATLAB-PYTHON-ABAQUS data interaction system. The $N^{*}$ groups of structural data and corresponding eigenvalues will be used in the next step.

A pair of $\left({\mathbf{M}}_{\mathbf{G}},{\mathbf{K}}_{\mathbf{G}}\right)$ represents the structure corresponding to the optimal structure parameters obtained by global optimization of the structure $\left({\mathbf{M}}_{\mathbf{0}},{\mathbf{K}}_{\mathbf{0}}\right)$ using the multi-population genetic algorithm, where ${\mathbf{M}}_{\mathbf{G}}\in {\mathrm{R}}^{\mathrm{n}\times \mathrm{n}}$ and ${\mathbf{K}}_{\mathbf{G}}\in {\mathrm{R}}^{\mathrm{n}\times \mathrm{n}}$ are the mass matrix and stiffness matrix, respectively. Assume that the eigenvalue of structure $\left({\mathbf{M}}_{\mathbf{G}},{\mathbf{K}}_{\mathbf{G}}\right)$ is $\left\{{\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right\}$ , and the error between eigenvalues $\left\{{\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right\}$ and target eigenvalues $\left\{{\lambda }_1,{\lambda }_2,\cdots ,{\lambda }_{j-1},{\mu }_j,\cdots ,{\mu }_{j+p},{\lambda }_{j+p+1},\cdots ,{\lambda }_n\right\}$ is represented by equation (3).

${\delta }_i=\frac{\left|{\alpha }_i-{\lambda }_i\right|}{{\lambda }_i}\times 100\%\hspace{0.17em}i=1,\cdots ,n$

(3)

In the second step, for the discrete structure in this paper, the radial basis function neural network (RBF NN) is trained using the $N^{*}$ groups of structural data and the corresponding eigenvalue ${\lambda }_i\left(i=\mathrm{1,2},\cdots ,n\right)$ , and the relationship between the structural data of the discrete structure and the corresponding eigenvalue is simulated to complete the training of RBF NN model. According to the target eigenvalue, the eigenvalue $\left\{{\lambda }_1^{*},{\lambda }_2^{*},\cdots ,{\lambda }_n^{*}\right\}$ is input into RBF NN, and the exact optimal solution of a set of structural data is obtained. At this time, the discrete structure is represented by a pair of $\left(\mathbf{M},\mathbf{K}\right)$ . The error between the eigenvalues of discrete structure $\left(\mathbf{M},\mathbf{K}\right)$ and the target eigenvalues is smaller than that of ${\delta }_i\left(i=\mathrm{1,2},\cdots ,n\right)$ . In this paper, the same method as for the discrete structure is used to obtain the discretised continuous structure $\left(\mathbf{M},\mathbf{K}\right)$ with a smaller error between the eigenvalue and the target eigenvalue than ${\delta }_i\left(i=\mathrm{1,2},\cdots ,n\right)$ .

The specific process of these two steps will be described separately in the following subsections.

Algorithm implementation

Multi-population genetic algorithm

Multi-population genetic algorithm (MPGA) is a parallel implementation of a conventional genetic algorithm ⁴⁷ (SGA), which can overcome the premature shortcoming of conventional genetic algorithm^48,49,50 and is more suitable for our research. Three populations are used to form a multi-population genetic algorithm. In order to maintain the population diversity of MPGA, the population size N, crossover rate p_c and mutation rate p_m of SGA in MPGA are shown in Table 1. The optimal individuals of the elite population are kept for at least 10 generations.

Table 1.

Parameter values of SGA in MPGA.

Parameter	SGA1	SGA2	SGA3
N	500	500	500
p c	0.70	0.75	0.80
p m	0.02	0.01	0.03

There will be many degrees of freedom because the structure is discretized using the finite element method. If the theoretical method is used, the resulting mathematical model can be complex and massive. The multi-population genetic algorithm is combined with the MATLAB-PYTHON-ABAQUS data interaction system to form a new approach for making partial eigenvalue assignment of the structure, which can avoid the complex formulations required to make partial eigenvalue assignment in other researcher’s works, as shown in Figure 3.

Figure 3.

A new approach to the structure’s partial eigenvalue assignment.

Since structures usually have a large number of degrees of freedom, Zhang et al. ⁵¹ proposed a method of changing a subset of eigenvalues and keeping another subset of eigenvalues, without considering the remaining eigenvalues (without considering those of remaining eigenvalues that are sufficiently far away from the target eigenvalues), to realize the partial eigenvalue assignment of the structure. Because the example used in this paper has few degrees of freedom, all degrees of freedom of the structure are considered, though this will slow down the algorithm’s solution speed. In order to realize the partial eigenvalue assignment of the structure, the objective function of SGA in MPGA is given in equation (4). MPGA performs a global optimization of the structural parameters within the structural parameter constraint interval of the finite element structure in a new approach to the structure’s partial eigenvalue assignment, so that the objective function error obtains the minimum value. When the error approaches zeros, satisfying $E{V}_1\to {\lambda }_1$ , $\cdots$ , $E{V}_{j-1}\to {\lambda }_{j-1}$ , $E{V}_j\to {\mu }_j$ , $\cdots$ , $E{V}_{j+p}\to {\mu }_{j+p}$ , $E{V}_{j+p+1}\to {\lambda }_{j+p+1}$ , $\cdots$ , $E{V}_n\to {\lambda }_n$ results in the assignment of eigenvalues ${\mu }_j,\cdots ,{\mu }_{j+p}$ and the spillover of eigenvalues ${\lambda }_1,\cdots ,{\lambda }_{j-1},{\lambda }_{j+p+1},\cdots ,{\lambda }_n$ is small.

$\text{error}={\displaystyle \sum }_{i=1}^{j-1}{\left(E{V}_i-{\lambda }_i\right)}^2+{\displaystyle \sum }_{i=j}^{j+p}{\left(E{V}_i-{\mu }_i\right)}^2+{\displaystyle \sum }_{i=j+p+1}^n{\left(E{V}_i-{\lambda }_i\right)}^2$

(4)

For discrete structures, such as the mass-spring system shown in section 3.1, MPGA optimizes $m_i\left(i=\mathrm{1,2},\cdots ,5\right)$ and $k_i\left(i=\mathrm{1,2},\cdots ,5\right)$ within the constraint intervals of the given mass parameters $m_i\left(i=\mathrm{1,2},\cdots ,5\right)$ and stiffness parameters $k_i\left(i=\mathrm{1,2},\cdots ,5\right)$ , and realizes partial eigenvalue assignment for the mass-spring system. The optimal values of the obtained structural parameters are shown in the second column of Table 2. According to the optimal value of the structural parameters in the second column of Table 2, the small intervals where the optimal value of the structural parameters is located is determined as shown in the third column of Table 2. In the intervals shown in the third column of Table 2, $N^{*}$ groups of $m_i\left(i=\mathrm{1,2},\cdots ,5\right)$ and $k_i\left(i=\mathrm{1,2},\cdots ,5\right)$ data are generated, and the $N^{*}$ groups of eigenvalues ${\lambda }_i\left(i=\mathrm{1,2},\cdots ,5\right)$ corresponding to the $N^{*}$ groups of data are read by the MATLAB-PYTHON-ABAQUS data interaction system, as shown in Table 3.

Table 2.

Example of the mass-spring system structural parameters.

Parameter	Values obtained by multi-population genetic algorithm	The interval of the value obtained by the multi-population genetic algorithm
${\mathit{m}}_{\mathit{i}}\left(\mathit{i}=\mathbf{1,2},\cdots ,\mathbf{5}\right)\left[\text{kg}\right]$	$m_1^{*}$ , $m_2^{*}$ , $m_3^{*}$ , $m_4^{*}$ and $m_5^{*}$	${\eta }_{m_1}$ , ${\eta }_{m_2}$ , ${\eta }_{m_3}$ , ${\eta }_{m_4}$ and ${\eta }_{m_5}$
${\mathit{k}}_{\mathit{i}}\left(\mathit{i}=\mathbf{1,2},\cdots ,\mathbf{5}\right)\left[\mathbf{N}/\mathbf{m}\right]$	$k_1^{*}$ , $k_2^{*}$ , $k_3^{*}$ , $k_4^{*}$ and $k_5^{*}$	${\eta }_{k_1}$ , ${\eta }_{k_2}$ , ${\eta }_{k_3}$ , ${\eta }_{k_4}$ and ${\eta }_{k_5}$

Table 3.

Example of $N^{*}$ groups structural parameters and eigenvalues of the mass-spring system.

Parameter	${\mathit{m}}_{\mathbf{1}}$	${\mathit{m}}_{\mathbf{2}}$	${\mathit{m}}_{\mathbf{3}}$	${\mathit{k}}_{\mathbf{1}}$	${\mathit{k}}_{\mathbf{2}}$	${\mathit{k}}_{\mathbf{3}}$	${\mathit{\lambda }}_{\mathbf{1}}$	$\cdots$	${\mathit{\lambda }}_{\mathbf{5}}$
Value	$a_{11}$	$a_{12}$	$a_{13}$	$b_{11}$	$b_{12}$	$b_{13}$	$c_{11}$	$\cdots$	$c_{15}$
	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$\ddots$	$⋮$
	$a_{N^{*}1}$	$a_{N^{*}2}$	$a_{N^{*}3}$	$b_{N^{*}1}$	$b_{N^{*}2}$	$b_{N^{*}3}$	$c_{N^{*}1}$	$\cdots$	$c_{N^{*}5}$

For discretised continuous structures, such as the cantilever beam shown in section 3.2, MPGA optimizes $\rho A_i\left(i=\mathrm{1,2,3}\right)$ and $E{J}_i\left(i=\mathrm{1,2,3}\right)$ within the constraint intervals of the given linear mass density parameters $\rho A_i\left(i=\mathrm{1,2,3}\right)$ and the product parameters $E{J}_i\left(i=\mathrm{1,2,3}\right)$ of Young’s modulus and the moment of inertia of the cross section about the neutral axis, and realizes partial eigenvalue assignment for the cantilever beam. The optimal values of the obtained structural parameters are shown in the second column of Table 4. According to the optimal value of the structural parameters in the second column of Table 4, the small intervals where the optimal value of the structural parameters is located is determined as shown in the third column of Table 4. In the intervals shown in the third column of Table 4, $N^{*}$ groups of $\rho A_i\left(i=\mathrm{1,2,3}\right)$ and $E{J}_i\left(i=\mathrm{1,2,3}\right)$ data are generated, and the $N^{*}$ groups of eigenvalues ${\lambda }_i\left(i=\mathrm{1,2},\cdots ,6\right)$ corresponding to the $N^{*}$ groups of data are read by the MATLAB-PYTHON-ABAQUS data interaction system, as shown in Table 5. Similarly, $N^{*}$ groups of structure and eigenvalue data for the door frame can be obtained.

Table 4.

Example of the cantilever beam structural parameters.

Parameter	Values obtained by multi-population genetic algorithm	The interval of the value obtained by the multi-population genetic algorithm
$\mathit{\rho }{\mathit{A}}_{\mathit{i}}\left(\mathit{i}=\mathbf{1,2,3}\right)\left[\text{kg}/\mathbf{m}\right]$	$\rho A_1^{*}$ , $\rho A_2^{*}$ and $\rho A_3^{*}$	${\eta }_{\rho A_1}$ , ${\eta }_{\rho A_2}$ and ${\eta }_{\rho A_3}$
$\mathit{E}{\mathit{J}}_{\mathit{i}}\left(\mathit{i}=\mathbf{1,2,3}\right)\left[\mathbf{N}{\mathbf{m}}^{\mathbf{2}}\right]$	$E{J}_1^{*}$ , $E{J}_2^{*}$ and $E{J}_3^{*}$	${\eta }_{E{J}_1}$ , ${\eta }_{E{J}_2}$ and ${\eta }_{E{J}_3}$

Table 5.

Example of $N^{*}$ groups structural parameters and eigenvalues of the cantilever beam.

Parameter	$\mathit{\rho }{\mathit{A}}_{\mathbf{1}}$	$\mathit{\rho }{\mathit{A}}_{\mathbf{2}}$	$\mathit{\rho }{\mathit{A}}_{\mathbf{3}}$	$\mathit{E}{\mathit{J}}_{\mathbf{1}}$	$\mathit{E}{\mathit{J}}_{\mathbf{2}}$	$\mathit{E}{\mathit{J}}_{\mathbf{3}}$	${\mathit{\lambda }}_{\mathbf{1}}$	$\cdots$	${\mathit{\lambda }}_{\mathbf{6}}$
Value	$u_{11}$	$u_{12}$	$u_{13}$	$v_{11}$	$v_{12}$	$v_{13}$	$z_{11}$	$\cdots$	$z_{16}$
	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$\ddots$	$⋮$
	$u_{N^{*}1}$	$u_{N^{*}2}$	$u_{N^{*}3}$	$v_{N^{*}1}$	$v_{N^{*}2}$	$v_{N^{*}3}$	$z_{N^{*}1}$	$\cdots$	$z_{N^{*}6}$

Radial basis function neural network

MPGA realizes partial eigenvalue assignment for the finite element structure in section 2.2.1, and obtains the optimal values of the structure parameters. $N^{*}$ groups of structural data and their corresponding eigenvalues are obtained based on the optimal value of structural parameters. Next, the RBF NN ⁵² is used to simulate the relationship between $N^{*}$ groups of structural data and the corresponding eigenvalues. There are many topologies of an RBF NN. The RBF NN with high fitting accuracy is used to accurately design in this paper, as shown in Figure 4.

Figure 4.

The topology of the radial basis function neural network model.

The radial basis function of the radial base layer is a Gaussian radial basis function, as shown in equation (5).

$\text{radbas}\left(n\right)=e^{-n^2}$

(5)

The number of neurons in the radial base layer is equal to the number of input vectors, and each neuron has a centre point, which is the weight $\mathrm{I}{\mathrm{W}}_{\mathrm{i}\mathrm{j}}$ from the input layer to the radial base layer. The net input to the Gaussian radial basis function is the element-wise product of the Euclidean distance $\Vert \mathbf{x}-\mathrm{I}{\mathrm{W}}_{\mathrm{i}\mathrm{j}}\Vert$ between the input vector $\mathbf{x}$ and its weight $\mathrm{I}{\mathrm{W}}_{\mathrm{i}\mathrm{j}}$ and its bias $b_j$ , as shown in equation (6).

$n=\Vert \mathbf{x}-\mathrm{I}{\mathrm{W}}_{\mathrm{i}\mathrm{j}}\Vert .*b_j,i=\mathrm{1,2},\cdots ,\mathrm{M},j=\mathrm{1,2},\cdots ,\mathrm{M}$

(6)

Therefore, the output of the radial base layer neuron is shown in equation (7).

${\mathrm{R}}_{\mathrm{j}}=e^{-{\left(\Vert \mathbf{x}-\mathrm{I}{\mathrm{W}}_{\mathrm{i}\mathrm{j}}\Vert .*b_j\right)}^2},i=\mathrm{1,2},\cdots ,\mathrm{M},j=\mathrm{1,2},\cdots ,\mathrm{M}$

(7)

The radial base layer to the output layer is a linear layer, and the transfer function of the neurons in the output layer is the purelin function. Knowing the output value ${\mathrm{R}}_{\mathrm{j}}$ of the radial base layer neuron and the output value $T_l$ in the training sample, the weight $\mathrm{L}{\mathrm{W}}_{\mathrm{j}\mathrm{l}}$ from the radial base layer to the output layer and the bias $b_l$ of the output neuron can be obtained by equation (8).

$\mathrm{L}{\mathrm{W}}_{\mathrm{j}\mathrm{l}}*{\mathrm{R}}_{\mathrm{j}}+b_l=T_l,\mathrm{j}=\mathrm{1,2},\cdots ,\mathrm{M},l=\mathrm{1,2},\cdots ,\mathrm{K}$

(8)

The RBF NN completes the simulation of the relationship between the structural parameters and eigenvalues of the finite element structure by training on the $N^{*}$ groups of structural data and the corresponding eigenvalue data in section 2.2.1. The eigenvalues $\left\{{\lambda }_1^{*},{\lambda }_2^{*},\cdots ,{\lambda }_n^{*}\right\}$ are input into the trained RBF NN, and the precise optimal value of the finite element structure’s structural parameters is obtained. At this time, the error between the structure’s eigenvalues and the target eigenvalues is smaller than ${\delta }_i\left(i=\mathrm{1,2},\cdots ,n\right)$ .

In this section, MPGA is described to assign eigenvalues of the structure in order to obtain the optimal values of the structure parameters and determine the structure $\left({\mathbf{M}}_{\mathbf{G}},{\mathbf{K}}_{\mathbf{G}}\right)$ . $N^{*}$ groups of structural data and corresponding eigenvalues are obtained based on the optimal values of the structural parameters. At this time, the error between the eigenvalues of structure $\left({\mathbf{M}}_{\mathbf{G}},{\mathbf{K}}_{\mathbf{G}}\right)$ and the target eigenvalues is ${\delta }_i\left(i=\mathrm{1,2},\cdots ,n\right)$ , but the error ${\delta }_i\left(i=\mathrm{1,2},\cdots ,n\right)$ is relatively large. Then, $N^{*}$ groups of structural data and corresponding eigenvalues are used to complete the training of the RBF NN. The eigenvalues $\left\{{\lambda }_1^{*},{\lambda }_2^{*},\cdots ,{\lambda }_n^{*}\right\}$ are input to the trained RBF NN to obtain the exact optimal values of the finite element structure’s structural parameters and determine the structure $\left(\mathbf{M},\mathbf{K}\right)$ . At this time, the error between the eigenvalues of structure $\left(\mathbf{M},\mathbf{K}\right)$ and the target eigenvalues is smaller than ${\delta }_i\left(i=\mathrm{1,2},\cdots ,n\right)$ .

Numerical examples

A new approach to the structure’s partial eigenvalue assignment and the RBF NN are used to realize partial eigenvalue assignment for structures, and this method has been implemented in MATLAB. This section provides three examples of partial eigenvalue assignment for structures to demonstrate the method’s capability.

Mass-spring system

Spillover-free partial eigenvalue assignment for a single-connected mass-spring system was addressed in Ouyang. ⁴⁵ The example is used in this paper to demonstrate the proposed method’s ability to perform partial eigenvalue assignment to discrete structures. This is a five-DOF ‘fixed-free’ mass-spring system, as shown in Figure 5.

Figure 5.

Five degrees of freedom ‘fixed-free’ mass-spring system.

In the second column of Table 6, the initial values of the mass and stiffness of the mass-spring system are shown, and the corresponding eigenvalues are shown in the second column of Table 7. It is intended to determine the mass and stiffness values of the mass-spring system so that the first two lowest eigenvalues are equal to ${\mu }_1=0.15$ and ${\mu }_2=0.95$ , and the remaining eigenvalues remain unchanged. Three scenarios (a), (b) and (c) are described as follows and their results are listed in Tables 6–8.

(a) The constraint interval of mass $m_i\left(i=\mathrm{1,2},\cdots ,5\right)$ and stiffness $k_i\left(i=\mathrm{1,2},\cdots ,5\right)$ of the single-connected mass-spring system is set to $\left[\mathrm{0,5}\right]$ in the MPGA for partial eigenvalue assignment. MPGA is run to obtain the optimal values of $m_i\left(i=\mathrm{1,2},\cdots ,5\right)$ and $k_i\left(i=\mathrm{1,2},\cdots ,5\right)$ , as shown in the third column of Table 6 and to determine the structure $\left({\mathbf{M}}_{\mathbf{G}},{\mathbf{K}}_{\mathbf{G}}\right)$ . The eigenvalues of structure $\left({\mathbf{M}}_{\mathbf{G}},{\mathbf{K}}_{\mathbf{G}}\right)$ are shown in the third column of Table 7. The structural parameter intervals of the optimal values of $m_i\left(i=\mathrm{1,2},\cdots ,5\right)$ and $k_i\left(i=\mathrm{1,2},\cdots ,5\right)$ determined are shown in column (a) of Table 8. In the optimization, 3000 sets of structural data were generated, and the eigenvalues corresponding to these data sets are read into the MATLAB-PYTHON-ABAQUS data interaction system to produce 3000 sets of training data for the RBF NN. Eigenvalues ${\lambda }_1^{*}=0.15$ , ${\lambda }_2^{*}=0.95$ , ${\lambda }_3^{*}=1.715$ , ${\lambda }_4^{*}=2.831$ and ${\lambda }_5^{*}=3.683$ are taken by the RBF NN to obtain the exact optimal values of the structure parameters, shown in the sixth column of Table 6, and form the structure $\left(\mathbf{M},\mathbf{K}\right)$ . Finally, the eigenvalues of modified structure $\left(\mathbf{M},\mathbf{K}\right)$ are shown in the sixth column of Table 7.

(b) The constraint interval of $m_i\left(i=\mathrm{1,2},\cdots ,5\right)$ and $k_i\left(i=\mathrm{1,2},\cdots ,5\right)$ is determined as $\left[\mathrm{1,10}\right]$ . The same solution procedure as in (a) is used. The optimal values of the obtained structural parameters are shown in the fourth column of Table 6. The eigenvalues corresponding to structure $\left({\mathbf{M}}_{\mathbf{G}},{\mathbf{K}}_{\mathbf{G}}\right)$ are shown in the fourth column of Table 7. Column (b) of Table 8 shows the parameter intervals of the optimal structural parameters. The eigenvalues of modified structure $\left(\mathbf{M},\mathbf{K}\right)$ are shown in the seventh column of Table 7.

(c) The constraint interval of $m_i\left(i=\mathrm{1,2},\cdots ,5\right)$ and $k_i\left(i=\mathrm{1,2},\cdots ,5\right)$ is determined as $\left[\mathrm{5,40}\right]$ . The same solution procedure as in (a) is used. The optimal values of the obtained structural parameters are shown in the fifth column of Table 6. The eigenvalues corresponding to structure $\left({\mathbf{M}}_{\mathbf{G}},{\mathbf{K}}_{\mathbf{G}}\right)$ are shown in the fifth column of Table 7. Column (c) of Table 8 shows the parameter intervals of the optimal structural parameters. The eigenvalues of structure $\left(\mathbf{M},\mathbf{K}\right)$ are shown in the eighth column of Table 7.

Table 6.

The mass-spring system’s mass and stiffness parameter values.

Parameter	Original value	Values obtained by multi-population genetic algorithm			Values obtained by radial basis function neural network
		(a)	(b)	(c)	(a)	(b)	(c)
${\mathit{m}}_{\mathbf{1}}\left[\text{kg}\right]$	1	1.9992	9.1920	39.3504	1.9979	9.1975	39.3113
${\mathit{m}}_{\mathbf{2}}\left[\text{kg}\right]$	1	1.0592	8.2115	24.6856	1.0577	8.2174	24.6694
${\mathit{m}}_{\mathbf{3}}\left[\text{kg}\right]$	1	0.7218	7.5690	23.6253	0.7206	7.5601	23.7108
${\mathit{m}}_{\mathbf{4}}\left[\text{kg}\right]$	1	0.7280	4.3559	18.8588	0.7303	4.3542	18.8251
${\mathit{m}}_{\mathbf{5}}\left[\text{kg}\right]$	1	0.3702	1.5661	6.8212	0.3708	1.5664	6.8328
${\mathit{k}}_{\mathbf{1}}\left[\mathbf{N}/\mathbf{m}\right]$	1	3.4969	11.5537	47.5533	3.4970	11.5526	47.5725
${\mathit{k}}_{\mathbf{2}}\left[\mathbf{N}/\mathbf{m}\right]$	1	1.7200	7.8495	29.4220	1.7212	7.8456	29.4841
${\mathit{k}}_{\mathbf{3}}\left[\mathbf{N}/\mathbf{m}\right]$	1	0.5390	6.8992	21.0208	0.5415	6.9032	21.0064
${\mathit{k}}_{\mathbf{4}}\left[\mathbf{N}/\mathbf{m}\right]$	1	0.6187	7.1199	25.8260	0.6165	7.1163	25.7974
${\mathit{k}}_{\mathbf{5}}\left[\mathbf{N}/\mathbf{m}\right]$	1	0.5231	2.2286	9.9031	0.5230	2.2302	9.9157
Norm of modifications		2.9813	20.1472	83.4742	2.9809	20.1466	83.4897

Table 7.

The mass-spring system’s eigenvalue parameter values.

		$\left({\mathbf{M}}_{\mathbf{G}},{\mathbf{K}}_{\mathbf{G}}\right)$			$\left(\mathbf{M},\mathbf{K}\right)$
	$\left({\mathbf{M}}_{\mathbf{0}},{\mathbf{K}}_{\mathbf{0}}\right)$	(a)	(b)	(c)	(a)	(b)	(c)
Eigenvalues	$8.100\times 10^{-2}$	$1.500\times 10^{-1}$	$1.500\times 10^{-1}$	$1.500\times 10^{-1}$	$1.500\times 10^{-1}$	$1.500\times 10^{-1}$	$1.500\times 10^{-1}$
	$6.903\times 10^{-1}$	$9.502\times 10^{-1}$	$9.502\times 10^{-1}$	$9.498\times 10^{-1}$	$9.500\times 10^{-1}$	$9.500\times 10^{-1}$	$9.500\times 10^{-1}$
	$1.715\times 10^0$	$1.713\times 10^0$	$1.715\times 10^0$	$1.717\times 10^0$	$1.716\times 10^0$	$1.715\times 10^0$	$1.715\times 10^0$
	$2.831\times 10^0$	$2.839\times 10^0$	$2.833\times 10^0$	$2.826\times 10^0$	$2.831\times 10^0$	$2.831\times 10^0$	$2.831\times 10^0$
	$3.683\times 10^0$	$3.676\times 10^0$	$3.681\times 10^0$	$3.687\times 10^0$	$3.683\times 10^0$	$3.683\times 10^0$	$3.683\times 10^0$
Relative errors in %	$0.4600$	$0.0000$	$0.0000$	$0.0000$	$0.0000$	$0.0000$	$0.0000$
	$0.2734$	$0.0002$	$0.0002$	$0.0002$	$0.0000$	$0.0000$	$0.0000$
	—	$0.0012$	$0.0000$	$0.0012$	$0.0006$	$0.0000$	$0.0000$
	—	$0.0028$	$0.0007$	$0.0018$	$0.0000$	$0.0000$	$0.0000$
	—	$0.0019$	$0.0005$	$0.0011$	$0.0000$	$0.0000$	$0.0000$

Table 8.

The mass parameter and stiffness parameter adjustment interval of the mass-spring system.

Parameter	(a)		(b)		(c)
	Lower bound	Upper bound	Lower bound	Upper bound	Lower bound	Upper bound
${\mathit{m}}_{\mathbf{1}}\left[\text{kg}\right]$	1.9942	2.0042	9.1870	9.1970	39.3454	39.3554
${\mathit{m}}_{\mathbf{2}}\left[\text{kg}\right]$	1.0542	1.0642	8.2065	8.2165	24.6806	24.6906
${\mathit{m}}_{\mathbf{3}}\left[\text{kg}\right]$	0.7168	0.7268	7.5640	7.5740	23.6203	23.6303
${\mathit{m}}_{\mathbf{4}}\left[\text{kg}\right]$	0.7230	0.7330	4.3509	4.3609	18.8538	18.8638
${\mathit{m}}_{\mathbf{5}}\left[\text{kg}\right]$	0.3652	0.3752	1.5611	1.5711	6.8162	6.8262
${\mathit{k}}_{\mathbf{1}}\left[\mathbf{N}/\mathbf{m}\right]$	3.4919	3.5019	11.5487	11.5587	47.5483	47.5583
${\mathit{k}}_{\mathbf{2}}\left[\mathbf{N}/\mathbf{m}\right]$	1.7150	1.7250	7.8445	7.8545	29.4170	29.4270
${\mathit{k}}_{\mathbf{3}}\left[\mathbf{N}/\mathbf{m}\right]$	0.5340	0.5440	6.8942	6.9042	21.0158	21.0258
${\mathit{k}}_{\mathbf{4}}\left[\mathbf{N}/\mathbf{m}\right]$	0.6137	0.6237	7.1149	7.1249	28.8210	28.8310
${\mathit{k}}_{\mathbf{5}}\left[\mathbf{N}/\mathbf{m}\right]$	0.5181	0.5281	2.2236	2.2336	9.8981	9.9081
Norm of modifications		0.0316		0.0316		0.0316

The three groups of data in (a), (b) and (c) show that there can be a variety of schemes of structural modifications to achieve the required partial eigenvalue assignment, and all of them differ from those determined in Ouyang, ⁴⁵ indicating the non-uniqueness of partial eigenvalue assignment parameter modifications. It can be concluded from Table 7 that this method can achieve accurate assignment of the target frequency without frequency spillover for the partial frequency assignment of the mass-spring system, like the method proposed by Ouyang and Zhang. ⁴⁵ It should also be pointed out that the solution of Ouyang and Zhang ⁴⁵ can also be found by the presented approach (when a suitable constraint interval is specified). Since the accuracy of the results obtained using the authors’ method are the same as that of Ouyang and Zhang, ⁴⁵ they are not shown in this paper. Three other sets of solutions are listed in this paper, which also prove the method’s effectiveness in the partial eigenvalue assignment of discrete structure.

It can be observed from Table 6 and Table 8 that the parameter values for eigenvalue assignment are increasing from sets (a) to (b) to (c). Usually, bigger modifications tend to be more costly and thus undesirable. The ‘effort’ of modifications can be measured by the norms of the vector of structural modifications, which are given in the last row of Table 6 and Table 8.

Cantilever beam

The performance of the proposed method for partial eigenvalue assignment of discretised continuous structures is demonstrated using an Euler–Bernoulli cantilever beam. Belotti et al. ²⁵ studied the problem of partial eigenvalue assignment of an Euler–Bernoulli cantilever beam.

For an Euler–Bernoulli cantilever beam with equal section length $L$ , the frequency equation of the cantilever beam can be obtained from Irvine ⁵³ as shown in equation (9), and the eigenvalue solution equation is shown in equation (10).

$\mathit{cos}\left({\beta }_{n}L\right)\mathit{cos}h\left({\beta }_{n}L\right)=-1,n=\mathrm{1,2,3},\cdots$

(9)

$w_n^2={\left({\beta }_{n}L\right)}^4\left(\frac{EJ}{\rho A{L}^4}\right)$

(10)

From equation (9) and equation (10), the linear mass density $\rho A=1.4\text{kg}/\mathrm{m}$ , the product $EJ=27\mathrm{N}{\mathrm{m}}^2$ of Young’s modulus and the moment of inertia of the cross section about the neutral axis and the first six eigenvalues of the Euler–Bernoulli cantilever beam of length $L=9m$ (the paper’s author may have a typo) are shown in Table 9.

Table 9.

The first six eigenvalues of the cantilever beam length $L=9m$ .

n	β n L	Eigenvalues
1	1.8751	$3.634\times 10^{-2}$
2	4.6941	$1.427\times 10^0$
3	7.8548	$1.119\times 10^1$
4	10.9955	$4.297\times 10^1$
5	14.1372	$1.174\times 10^2$
6	17.2788	$2.620\times 10^2$

The Euler–Bernoulli cantilever beam in this paper is made up of 12 Euler–Bernoulli beam elements of length $l=0.75m$ , as shown in Figure 6. The cantilever beam in Figure 6 was created in ABAQUS with $\rho A_i\left(i=\mathrm{1,2,3}\right)=1.4\text{kg}/\mathrm{m}$ and $E{J}_i\left(i=\mathrm{1,2,3}\right)=27\mathrm{N}{\mathrm{m}}^2$ . The cantilever beam’s modal analysis is performed in ABAQUS, and the resulting eigenvalues are shown in the third column of Table 10. The errors between the six eigenvalues of the cantilever beam calculated by ABAQUS and the first six eigenvalues given by the analytical solution are all less than 0.6%, as shown in Table 10. So, it is appropriate to use 12 Euler–Bernoulli beam elements of length $l=0.75m$ to form an Euler–Bernoulli beam.

Figure 6.

Cantilever beam.

Table 10.

Analytical and finite element solutions for cantilever beam eigenvalues.

	Analytical solutions	Solution of ABAQUS	Relative errors in %
Eigenvalues	$3.634\times 10^{-2}$	$3.634\times 10^{-2}$	$0.0000$
	$1.427\times 10^0$	$1.427\times 10^0$	$0.0000$
	$1.119\times 10^1$	$1.119\times 10^1$	$0.0000$
	$4.297\times 10^1$	$4.301\times 10^1$	$0.0009$
	$1.174\times 10^2$	$1.177\times 10^2$	$0.0026$
	$2.620\times 10^2$	$2.635\times 10^2$	$0.0057$

In order to be consistent with Belotti et al.’s paper, ²⁵ the three lowest eigenvalues ${\mu }_1=0.05$ , ${\mu }_2=1.5$ and ${\mu }_3=11.00$ of the cantilever beam are required in this paper, while the cantilever beam’s subset of eigenvalues $\left\{{\lambda }_4,{\lambda }_5,{\lambda }_6\right\}$ is left unchanged, and partial eigenvalue assignment for the cantilever beam. Three scenarios (a), (b) and (c) are described as follows and their results are listed in Tables 11–13.

(a) The constraint interval of the linear mass density $\rho A_i\left(i=\mathrm{1,2,3}\right)$ of the cantilever beam is set to $\left[\mathrm{1,15}\right]$ , and the constraint interval of the product $E{J}_i\left(i=\mathrm{1,2,3}\right)$ of Young’s modulus and the moment of inertia of the cross section about the neutral axis is set to $\left[\mathrm{1,150}\right]$ , in the MPGA for partial eigenvalue assignment. MPGA is run to obtain the optimal value of $\rho A_i\left(i=\mathrm{1,2,3}\right)$ and $E{J}_i\left(i=\mathrm{1,2,3}\right)$ , as shown in the second column of Table 11, and determine the structure $\left({\mathbf{M}}_{\mathbf{G}},{\mathbf{K}}_{\mathbf{G}}\right)$ . The eigenvalues of structure $\left({\mathbf{M}}_{\mathbf{G}},{\mathbf{K}}_{\mathbf{G}}\right)$ are shown in the third column of Table 12. The structural parameter intervals of the optimal values of $\rho A_i\left(i=\mathrm{1,2,3}\right)$ and $E{J}_i\left(i=\mathrm{1,2,3}\right)$ determined are shown in column (a) of Table 13. In the optimization, 5000 sets of structural data were generated, and the eigenvalues corresponding to these data sets are read into the MATLAB-PYTHON-ABAQUS data interaction system to produce 5000 sets of training data for the RBF NN. Eigenvalues ${\lambda }_1^{*}=0.05$ , ${\lambda }_2^{*}=1.49$ , ${\lambda }_3^{*}=11.01$ , ${\lambda }_4^{*}=46.8$ , ${\lambda }_5^{*}=127.1$ and ${\lambda }_6^{*}=271.2$ are taken by the RBF NN to obtain the exact optimal values of the structure parameters, shown in the fifth column of Table 11, and form the structure $\left(\mathbf{M},\mathbf{K}\right)$ . Finally, the eigenvalues of modified structure $\left(\mathbf{M},\mathbf{K}\right)$ are shown in the sixth column of Table 12.

(b) The constraint intervals of $\rho A_i\left(i=\mathrm{1,2,3}\right)$ and $E{J}_i\left(i=\mathrm{1,2,3}\right)$ are $\left[\mathrm{2,30}\right]$ and $\left[\mathrm{60,300}\right]$ , respectively. The same solution procedure as in (a) is used. The optimal values of the obtained structural parameters are shown in the third column of Table 11. The eigenvalues corresponding to structure $\left({\mathbf{M}}_{\mathbf{G}},{\mathbf{K}}_{\mathbf{G}}\right)$ are shown in the fourth column of Table 12. Column (b) of Table 13 shows the parameter intervals of the optimal structural parameters. The eigenvalues of modified structure $\left(\mathbf{M},\mathbf{K}\right)$ are shown in the seventh column of Table 12.

(c) The constraint intervals of $\rho A_i\left(i=\mathrm{1,2,3}\right)$ and $E{J}_i\left(i=\mathrm{1,2,3}\right)$ are $\left[\mathrm{4,50}\right]$ and $\left[\mathrm{120,500}\right]$ respectively. The same solution procedure as in (a) is used. The optimal values of the obtained structural parameters are shown in the fourth column of Table 11. The eigenvalues corresponding to structure $\left({\mathbf{M}}_{\mathbf{G}},{\mathbf{K}}_{\mathbf{G}}\right)$ are shown in the fifth column of Table 12. Column (c) of Table 13 shows the parameter intervals of the optimal structural parameters. The eigenvalues of modified structure $\left(\mathbf{M},\mathbf{K}\right)$ are shown in the eighth column of Table 12.

Table 11.

The linear mass density of the cantilever beam, and the product of Young’s modulus and the moment of inertia of the cross section about the neutral axis of the cantilever beam.

Parameter	Values obtained by multi-population genetic algorithm			Values obtained by radial basis function neural network
	(a)	(b)	(c)	(a)	(b)	(c)
$\rho A_1\left[\text{kg}/\mathrm{m}\right]$	4.7272	9.4547	14.1818	4.7302	9.4544	14.1745
$\rho A_2\left[\text{kg}/\mathrm{m}\right]$	1.8494	3.6986	5.5486	1.6881	3.4262	5.1568
$\rho A_1\left[\text{kg}/\mathrm{m}\right]$	10.9581	21.9168	32.8744	11.8436	23.5206	35.4454
$E{J}_1\left[\mathrm{N}{\mathrm{m}}^2\right]$	121.8532	243.7066	365.5594	122.8219	245.3255	368.4276
$E{J}_2\left[\mathrm{N}{\mathrm{m}}^2\right]$	55.5134	111.0269	166.5408	56.2601	112.4956	168.6701
$E{J}_3\left[\mathrm{N}{\mathrm{m}}^2\right]$	148.4728	296.9457	445.4183	149.3240	298.5152	447.8646
Norm of modifications	157.0618	356.9090	557.0810	158.4973	359.5196	561.3436

Table 12.

The cantilever beam’s eigenvalue parameter values.

		$\left({\mathrm{M}}_{\mathrm{G}},{\mathrm{K}}_{\mathrm{G}}\right)$			$\left(\mathrm{M},\mathrm{K}\right)$
	$\left({\mathrm{M}}_0,{\mathrm{K}}_0\right)$	(a)	(b)	(c)	(a)	(b)	(c)
Eigenvalues	$3.634\times 10^{-2}$	$4.939\times 10^{-2}$	$4.938\times 10^{-2}$	$4.938\times 10^{-2}$	$5.000\times 10^{-2}$	$4.997\times 10^{-2}$	$4.998\times 10^{-2}$
	$1.427\times 10^0$	$1.484\times 10^0$	$1.484\times 10^0$	$1.484\times 10^0$	$1.474\times 10^0$	$1.476\times 10^0$	$1.476\times 10^0$
	$1.119\times 10^1$	$1.101\times 10^1$	$1.101\times 10^1$	$1.101\times 10^1$	$1.099\times 10^1$	$1.099\times 10^1$	$1.099\times 10^1$
	$4.301\times 10^1$	$4.687\times 10^1$	$4.687\times 10^1$	$4.687\times 10^1$	$4.680\times 10^1$	$4.681\times 10^1$	$4.682\times 10^1$
	$1.177\times 10^2$	$1.273\times 10^2$	$1.273\times 10^2$	$1.273\times 10^2$	$1.271\times 10^2$	$1.269\times 10^2$	$1.266\times 10^2$
	$2.635\times 10^2$	$2.712\times 10^2$	$2.712\times 10^2$	$2.712\times 10^2$	$2.712\times 10^2$	$2.712\times 10^2$	$2.712\times 10^2$
Relative errors in %	0.2732	0.0122	0.0124	0.0124	0.0000	0.0006	0.0004
	0.0487	0.0107	0.0107	0.0107	0.0173	0.0160	0.0160
	0.0173	0.0009	0.0009	0.0009	0.0009	0.0009	0.0009
	—	0.0897	0.0897	0.0897	0.0881	0.0884	0.0886
	—	0.0816	0.0816	0.0816	0.0799	0.0782	0.0756
	—	0.0292	0.0292	0.0292	0.0292	0.0292	0.0292

Table 13.

The parameter adjustment interval of the linear mass density of the cantilever beam, and the parameter adjustment interval of the product of Young’s modulus and the moment of inertia of the cross section about the neutral axis of the cantilever beam.

Parameter	(a)		(b)		(c)
	Lower bound	Upper bound	Lower bound	Upper bound	Lower bound	Upper bound
$\rho A_1\left[\text{kg}/\mathrm{m}\right]$	4.7172	4.7372	7.4544	11.4544	11.1816	17.1816
$\rho A_2\left[\text{kg}/\mathrm{m}\right]$	1.8394	1.8594	1.6988	5.6988	2.5482	8.5482
$\rho A_3\left[\text{kg}/\mathrm{m}\right]$	10.9481	10.9681	19.9162	23.9162	29.8743	35.8743
$E{J}_1\left[\mathrm{N}{\mathrm{m}}^2\right]$	121.8432	121.8632	241.7064	245.7064	362.5596	368.5596
$E{J}_2\left[\mathrm{N}{\mathrm{m}}^2\right]$	55.5034	55.5234	109.0268	113.0268	163.5402	169.5402
$E{J}_3\left[\mathrm{N}{\mathrm{m}}^2\right]$	148.4628	148.4828	294.9456	298.9456	442.4184	448.4184
Norm of modifications		0.0490		9.7980		14.6969

From the three sets of data in (a), (b) and (c), there can be various parameter modifications for the partial eigenvalue assignment of discretised continuous structures. These results show that the three smallest eigenvalues assigned to discretised continuous structures are all close to the target eigenvalues. At the same time, the amount of spillover of the remaining unassigned eigenvalues is low. This method’s ability to assign low-frequency eigenvalues of discretised continuous structures is demonstrated. At the same time, when the accuracy of assigning the target frequency and the degree of spillover are considered, this method performs better than the method proposed by Belotti et al. ²⁵

It can be observed from Table 11 and Table 13 that the parameter values are increasing from sets (a) to (b) to (c). The ‘effort’ of modifications can be measured by the norms of the vector of structural modifications, which are given in the last row of Table 11 and Table 13. A lower norm is usually desirable.

Door frame

As shown in Figure 7, (a) door frame is made up of three Euler–Bernoulli beam elements with lengths $l_1=2m$ , $l_2=1.2m$ and $l_3=2m$ . Because ABAQUS uses the B23 element, the door frame has 12 active degrees of freedom. The linear mass density $\rho A_i\left(i=\mathrm{1,2,3}\right)=1.4\text{kg}/\mathrm{m}$ of the beam element, the product $E{J}_i\left(i=\mathrm{1,2,3}\right)=27\mathrm{N}{\mathrm{m}}^2$ of Young’s modulus and the moment of inertia of the cross section about the neutral axis and the corresponding eigenvalues are shown in the second column of Table 15.

Figure 7.

Door frame.

The ability to make partial eigenvalue assignment to discrete structures and discretised continuous structures is shown in section 3.1 and section 3.2, respectively, and the required structural modifications to low-frequency eigenvalues are also shown. Usually, it is more difficult to assign high-frequency eigenvalues. In section 3.3, modifications required to assign high-frequency eigenvalues are determined. It is intended that the door frame’s high-frequency eigenvalues should be ${\mu }_6=19.050$ , ${\mu }_8=22.600$ , ${\mu }_9=66.500$ , ${\mu }_{10}=209.00$ , ${\mu }_{11}=3015.00$ and ${\mu }_{12}=6385.00$ , while the remaining eigenvalues remain unchanged. Three scenarios (a), (b) and (c) are described as follows and their results are listed in Tables 14–16.

(a) The constraint interval of the linear mass density $\rho A_i\left(i=\mathrm{1,2,3}\right)$ of the beam element is set to $\left[\mathrm{1,10}\right]$ , and the constraint interval of the product $E{J}_i\left(i=\mathrm{1,2,3}\right)$ of Young’s modulus and the moment of inertia of the cross section about the neutral axis is set to $\left[\mathrm{1,150}\right]$ , in the MPGA for partial eigenvalue assignment. MPGA is run to obtain the optimal value of $\rho A_i\left(i=\mathrm{1,2,3}\right)$ and $E{J}_i\left(i=\mathrm{1,2,3}\right)$ , as shown in the second column of Table 14, and determine the structure $\left({\mathbf{M}}_{\mathbf{G}},{\mathbf{K}}_{\mathbf{G}}\right)$ . The eigenvalues of structure $\left({\mathbf{M}}_{\mathbf{G}},{\mathbf{K}}_{\mathbf{G}}\right)$ are shown in the third column of the Table 15. The structural parameter intervals of the optimal values of $\rho A_i\left(i=\mathrm{1,2,3}\right)$ and $E{J}_i\left(i=\mathrm{1,2,3}\right)$ determined are shown in column (a) of Table 16. In the optimization, 5000 sets of structural data were generated, and the eigenvalues corresponding to these data sets are read into the MATLAB-PYTHON-ABAQUS data interaction system to produce 5000 sets of training data for the RBF NN. Eigenvalues ${\lambda }_1^{*}=0.3721$ , ${\lambda }_2^{*}=2.345$ , ${\lambda }_3^{*}=3.838$ , ${\lambda }_4^{*}=4.120$ , ${\lambda }_5^{*}=13.52$ , ${\lambda }_6^{*}=19.06$ , ${\lambda }_7^{*}=19.74$ , ${\lambda }_8^{*}=22.58$ , ${\lambda }_9^{*}=66.47$ , ${\lambda }_{10}^{*}=209.3$ , ${\lambda }_{11}^{*}=3013.0$ and ${\lambda }_{12}^{*}=6386.0$ are taken by the RBF NN to obtain the optimal exact values of the structure parameters, shown in the fifth column of Table 14, and form the structure $\left(\mathbf{M},\mathbf{K}\right)$ . Finally, the eigenvalues of modified structure $\left(\mathbf{M},\mathbf{K}\right)$ are shown in the sixth column of Table 15.

(b) The constraint intervals of $\rho A_i\left(i=\mathrm{1,2,3}\right)$ and $E{J}_i\left(i=\mathrm{1,2,3}\right)$ are $\left[\mathrm{5,15}\right]$ and $\left[\mathrm{100,300}\right]$ , respectively. The same solution procedure as in (a) is used. The optimal values of the obtained structural parameters are shown in the third column of Table 14. The eigenvalues corresponding to structure $\left({\mathbf{M}}_{\mathbf{G}},{\mathbf{K}}_{\mathbf{G}}\right)$ are shown in the fourth column of Table 15. Column (b) of Table 16 shows the parameter intervals of the optimal structural parameters. The eigenvalues of modified structure $\left(\mathbf{M},\mathbf{K}\right)$ are shown in the seventh column of Table 15.

(c) The constraint intervals of $\rho A_i\left(i=1,2,3\right)$ and $E{J}_i\left(i=1,2,3\right)$ are $\left[15,25\right]$ and $\left[200,500\right]$ , respectively. The same solution procedure as in (a) is used. The optimal values of the obtained structural parameters are shown in the fourth column of Table 14. The eigenvalues corresponding to structure $\left({\mathbf{M}}_{\mathbf{G}},{\mathbf{K}}_{\mathbf{G}}\right)$ are shown in the fifth column of Table 15. Column (c) of Table 16 shows the parameter intervals of the optimal structural parameters. The eigenvalues of modified structure $\left(\mathbf{M},\mathbf{K}\right)$ are shown in the eighth column of Table 15.

Table 14.

The linear mass density of the door frame, and the product of Young’s modulus and the moment of inertia of the cross section about the neutral axis of the door frame.

Parameter	Values obtained by multi-population genetic algorithm			Values obtained by radial basis function neural network
	(a)	(b)	(c)	(a)	(b)	(c)
$\rho A_1\left[\text{kg}/\mathrm{m}\right]$	5.2114	10.4228	15.6342	5.1888	10.3873	15.5846
$\rho A_2\left[\text{kg}/\mathrm{m}\right]$	5.1586	10.3172	15.4758	5.1818	10.3655	15.5195
$\rho A_1\left[\text{kg}/\mathrm{m}\right]$	5.2771	10.5544	15.8315	5.2845	10.5671	15.8251
$E{J}_1\left[\mathrm{N}{\mathrm{m}}^2\right]$	84.2573	168.5146	252.7719	84.0848	168.4821	252.6047
$E{J}_2\left[\mathrm{N}{\mathrm{m}}^2\right]$	140.2503	280.5006	420.7509	140.3448	280.7867	420.6743
$E{J}_3\left[\mathrm{N}{\mathrm{m}}^2\right]$	117.4180	234.8360	352.2540	117.7152	235.0160	352.2610
Norm of modifications	155.9587	357.3926	558.9382	156.1368	357.6880	558.8205

Table 15.

The door frame’s eigenvalue parameter values.

		$\left({\mathrm{M}}_{\mathrm{G}},{\mathrm{K}}_{\mathrm{G}}\right)$			$\left(\mathrm{M},\mathrm{K}\right)$
	$\left({\mathrm{M}}_0,{\mathrm{K}}_0\right)$	(a)	(b)	(c)	(a)	(b)	(c)
Eigenvalues	$3.721\times 10^{-1}$	$3.746\times 10^{-1}$	$3.746\times 10^{-1}$	$3.746\times 10^{-1}$	$3.737\times 10^{-1}$	$3.737\times 10^{-1}$	$3.739\times 10^{-1}$
	$2.345\times 10^0$	$2.346\times 10^0$	$2.346\times 10^0$	$2.346\times 10^0$	$2.346\times 10^0$	$2.346\times 10^0$	$2.346\times 10^0$
	$3.838\times 10^0$	$3.854\times 10^0$	$3.854\times 10^0$	$3.854\times 10^0$	$3.849\times 10^0$	$3.849\times 10^0$	$3.850\times 10^0$
	$4.120\times 10^0$	$4.115\times 10^0$	$4.115\times 10^0$	$4.115\times 10^0$	$4.110\times 10^0$	$4.110\times 10^0$	$4.112\times 10^0$
	$1.352\times 10^1$	$1.353\times 10^1$	$1.353\times 10^1$	$1.353\times 10^1$	$1.354\times 10^1$	$1.354\times 10^1$	$1.354\times 10^1$
	$1.884\times 10^1$	$1.906\times 10^1$	$1.906\times 10^1$	$1.906\times 10^1$	$1.905\times 10^1$	$1.905\times 10^1$	$1.905\times 10^1$
	$1.974\times 10^1$	$1.974\times 10^1$	$1.974\times 10^1$	$1.974\times 10^1$	$1.974\times 10^1$	$1.975\times 10^1$	$1.975\times 10^1$
	$2.193\times 10^1$	$2.258\times 10^1$	$2.258\times 10^1$	$2.258\times 10^1$	$2.260\times 10^1$	$2.258\times 10^1$	$2.259\times 10^1$
	$6.665\times 10^1$	$6.647\times 10^1$	$6.647\times 10^1$	$6.647\times 10^1$	$6.661\times 10^1$	$6.660\times 10^1$	$6.658\times 10^1$
	$1.980\times 10^2$	$2.093\times 10^2$	$2.093\times 10^2$	$2.093\times 10^2$	$2.090\times 10^2$	$2.090\times 10^2$	$2.091\times 10^2$
	$2.481\times 10^3$	$3.013\times 10^3$	$3.013\times 10^3$	$3.013\times 10^3$	$3.016\times 10^3$	$3.015\times 10^3$	$3.015\times 10^3$
	$4.854\times 10^3$	$6.386\times 10^3$	$6.386\times 10^3$	$6.386\times 10^3$	$6.385\times 10^3$	$6.384\times 10^3$	$6.385\times 10^3$
Relative errors in %	—	0.0067	0.0067	0.0067	0.0043	0.0043	0.0048
	—	0.0004	0.0004	0.0004	0.0004	0.0004	0.0004
	—	0.0042	0.0042	0.0042	0.0029	0.0029	0.0031
	—	0.0012	0.0012	0.0012	0.0024	0.0024	0.0019
	—	0.0007	0.0007	0.0007	0.0015	0.0015	0.0015
	0.0110	0.0005	0.0005	0.0005	0.0000	0.0000	0.0000
	—	0.0000	0.0000	0.0000	0.0000	0.0005	0.0005
	0.0296	0.0009	0.0009	0.0009	0.0000	0.0009	0.0004
	0.0023	0.0005	0.0005	0.0005	0.0017	0.0015	0.0012
	0.0526	0.0014	0.0014	0.0014	0.0000	0.0000	0.0005
	0.1771	0.0007	0.0007	0.0007	0.0003	0.0000	0.0000
	0.2398	0.0002	0.0002	0.0002	0.0000	0.0002	0.0000

Table 16.

The parameter adjustment interval of the linear mass density of the door frame, the parameter adjustment interval of the product of Young’s modulus and the moment of inertia of the cross section about the neutral axis of the door frame.

Parameter	(a)		(b)		(c)
	Lower bound	Upper bound	Lower bound	Upper bound	Lower bound	Upper bound
$\rho A_1\left[\text{kg}/\mathrm{m}\right]$	5.2064	5.2164	10.4178	10.4278	15.6292	15.6392
$\rho A_2\left[\text{kg}/\mathrm{m}\right]$	5.1536	5.1636	10.3122	10.3222	15.4708	15.4808
$\rho A_3\left[\text{kg}/\mathrm{m}\right]$	5.2721	5.2821	10.5494	10.5594	15.8265	15.8365
$E{J}_1\left[\mathrm{N}{\mathrm{m}}^2\right]$	84.2523	84.2623	168.5096	168.5196	252.7669	252.7769
$E{J}_2\left[\mathrm{N}{\mathrm{m}}^2\right]$	140.2453	140.2553	280.4956	280.5056	420.7459	420.7559
$E{J}_3\left[\mathrm{N}{\mathrm{m}}^2\right]$	117.4130	117.4230	234.8310	234.8410	352.2490	352.2590
Norm of modifications		0.0245		0.0245		0.0245

From the three sets of data in (a), (b) and (c), it can be seen that the six high-frequency eigenvalues assigned to the structure are all close to the target eigenvalues, and the remaining unassigned eigenvalues have a very low level of spillover. Therefore, the method’s ability to perform partial eigenvalue assignments for high-frequency eigenvalues is demonstrated. Because assigning high frequencies to structures is difficult, most existing methods focus on assigning low frequencies to structures. This method is shown to be capable of high-frequency assignment of structures.

Again, the norms of modifications are increasing from sets (a) to (b) to (c). Therefore, the modifications (a) are more desirable.

Conclusions

A method for making eigenvalue assignment with low spillover for undamped structures is proposed in this paper. It combines a multi-population genetic algorithm and an RBF NN and implemented on a MATLAB-PYTHON-ABAQUS data interaction system. It has the advantage of being easy in determining the structural modifications required to make partial eigenvalue assignment, in comparison with other methods. Three numerical examples (a lumped mass-spring chain, a discretised cantilever beam and a discretised door frame) are used to show the capability and the procedure of the method. The above three numerical examples show that the method has good applicability. The results indicate that low-frequency eigenvalues and high-frequency eigenvalues can be assigned while all the other eigenvalues undergo very small changes (very low level of spillover), which is virtually equally useful as a proper partial eigenvalue assignment whose solution are much more difficult to find or often do not exist.

Because the method proposed in this paper uses the finite element method, the calculation time is determined by the computer’s performance. Its greatest advantages include universal applicability to both discrete structures and discretised continuous structures. The main shortcoming is that it requires construction of a detailed finite element model, while some other methods (those receptance-based methods) use measured frequency response functions and do not require a theoretical model of the structure.