JP7690844B2 - Method for identifying modal parameters, device for identifying modal parameters, and program - Google Patents

Description

This disclosure relates to a method for identifying modal parameters, and an apparatus and program for identifying modal parameters.

Methods for determining the dynamic characteristics of a physical model of a structure include the following technologies. For example, Non-Patent Document 1 describes a technology for identifying complex eigenvalues in a general viscous damping system model by using the impulse response of a structure determined by a single-point excitation, multi-point reference (SIMO: Single-Input Multi-Output) method as a reference. In addition, Non-Patent Document 2 describes a technology for identifying complex eigenvalues and complex eigenmodes in a general viscous damping system model by using the frequency response of a structure determined by a multi-point excitation, multi-point reference (MIMO: Multi-Input Multi-Output) method as a reference.

Kerem Ege, Xavier Boutillon, Bertrand David. High-resolution modal analysis. Journal of Sound and Vibration, Elsevier, 2009, 325 (4-5), p.852-869. Acoustical Society of Japan, ed., "Modal Analysis and Control of Sound and Vibration," Corona Publishing, 1996, pp.110-114.

However, the technology described in Non-Patent Document 1 uses a single-point excitation and multi-point reference method, making it difficult to identify complex eigenmodes with high accuracy. The technology described in Non-Patent Document 2 uses a multi-point excitation and multi-point reference method, so although the accuracy of identifying complex eigenmodes is higher than when using the technology described in Non-Patent Document 1, it is difficult to identify complex eigenvalues with high accuracy, and since complex eigenvalues and complex eigenmodes are identified simultaneously, it is difficult to apply this technology to large-scale structures.

A method for identifying modal parameters according to one aspect of the present disclosure is a method for identifying modal parameters in a physical model of a structure when it exists in the air, or in a physical model of the air when the structure is attached, by a computer, and includes a first step in which the computer calculates a frequency response of the structure when it exists in the air, or a frequency response of the air when the structure is attached, a second step in which the computer identifies complex eigenvalues using an inverse Fourier transform of the frequency response as a reference, and a third step in which the computer identifies a complex eigenmode of a vibration point in the structure, or a complex eigenmode of a vibration point in the air, by using the frequency response as a reference.

FIG. 1 is a diagram for explaining a physical model. FIG. 13 illustrates a computer that executes a process for identifying modal parameters. FIG. 1 is a block diagram showing a functional configuration of a computer. FIG. 13 illustrates a computer that executes a process for identifying modal parameters. FIG. 13 illustrates a computer that executes a process for identifying modal parameters. 11 is a diagram showing that the reference FR and the FR obtained by mode synthesis are almost identical to each other. 11 is a diagram showing that the reference FR and the FR obtained by mode synthesis are almost identical to each other. 11 is a diagram showing that the reference FR and the FR obtained by mode synthesis are almost identical to each other. A figure showing that the reference FR and the IR due to mode synthesis are almost the same. A figure showing that the reference FR and the IR due to mode synthesis are almost the same. A figure showing that the reference FR and the IR due to mode synthesis are almost the same. FIG. 2 shows frequency response in a vacuum and frequency response in air.

Hereinafter, a method for identifying modal parameters according to an embodiment of the present disclosure will be described with reference to the drawings.
In each drawing, the dimensions and scale of each part are appropriately different from the actual ones. In addition, since the embodiments described below are preferred specific examples, various technically preferable limitations are attached, but the scope of the present invention is not limited to these embodiments unless otherwise specified in the following description to the effect that the present invention is limited.

First, when performing a time-domain simulation of a structure that can be assumed to be linear, the "modal synthesis method," which expresses the displacement of a structural model as a superposition of eigenmodes, is known as the most cost-effective analysis method. For a structural model placed in a vacuum, a method has already been established to calculate the modal parameters of the structure, i.e., the real eigenvalues and real eigenmodes, with high accuracy. Eigenvalue analysis using the finite element method (FEM) is known as the most powerful method for this purpose.

On the other hand, it must be said that it is extremely difficult to calculate highly accurate modal parameters for a large-scale structural model placed in air, specifically, in a room of a certain size or larger, in an acoustically silent room, or outdoors. The reason for this is that when identifying the eigenmodes of a structural model placed in air while considering the interaction between the air and the structure using the finite element method, the scale of calculations becomes significantly larger as the model becomes larger or the analysis frequency becomes higher, resulting in unrealistically high calculation costs. The Fast Multipole Boundary Element Method (FMBEM) is known as a cost-effective numerical analysis method to replace the finite element method for acoustic-structural interaction problems, but this method determines frequency responses, not eigenmodes. In short, no method has been proposed to calculate the eigenmodes of a large-scale "structure placed in air" with high accuracy at a realistic calculation cost.

12 is a diagram showing the results of calculating the frequency response in a vacuum and in air for the soundboard of a certain musical instrument as an example of a structure. As this example shows, depending on the structure, there may be a non-negligible difference between the frequency response in a vacuum and the frequency response in air. The main reason for this difference is that in air, the natural frequency is lower than in a vacuum due to the mass effect of air, and in addition, the energy of vibration is consumed for acoustic radiation, resulting in greater attenuation.
12, the frequency response of the structure in a vacuum is a response calculated as a superposition of eigenmodes obtained as a result of a finite element eigenvalue analysis of the structure. The frequency response in air is a response calculated by performing an acoustic-structural coupling analysis by the fast multipole boundary element method using the results of the finite element eigenvalue analysis of the above-mentioned vacuum structure, assuming that the structure is placed in an anechoic chamber.

Now, many methods have been proposed so far for experimentally identifying the eigenmodes of a structure, including the methods described in Non-Patent Document 1 or Non-Patent Document 2, and these have also been commercialized as experimental mode identification software.
However, because it is difficult to obtain a large number of high-quality frequency responses through experiments, conventional methods developed for experimental modal identification are only designed to identify a few to a few dozen eigenmodes at most, and are therefore unable to handle large-scale models.

Figure 1 is a conceptual diagram showing the input and response of a structure placed in air, or the air to which the structure is attached.

In this embodiment, firstly, the coupled system between the air and the structure is particularly represented as a linear coupled system Lc, and the linear coupled system Lc is expressed by a general viscous damping system model.
The theoretical basis for being able to express a coupled system consisting of a body and air, i.e., an acoustic-structural coupled system, with a general viscous damping model is that the acoustic-structural coupled equations of motion calculated using the finite element method, where the structural displacement and air velocity potential are "variables that depend on time and space," can be formally expressed using real symmetric mass matrices, damping matrix, and stiffness matrix. The form of the equations of motion consisting of these three real symmetric matrices is mathematically the same as the form of the equations of motion calculated using the finite element method for a model of a structure with general viscous damping (not proportional viscous damping), i.e., a general viscous damping model.
The advantage of expressing an acoustic-structural interaction system with a general viscous damping model is that the eigenmodes of this model are orthogonal in a broad sense, which allows for fast time-domain simulations using the mode synthesis method.

また、一般粘性減衰系モデルにおける周波数応答（Frequency Response）は、次式（２）のように示される。

The impulse response in the general viscous damping system model is expressed as the following equation (1).

Moreover, the frequency response in the general viscous damping system model is expressed as in the following equation (2).

In formula (1) or formula (2), j is the imaginary unit, * is the complex conjugate, r is the mode number, i is the index of the excitation point, and l is the index of the response point.
λ _r is a complex eigenvalue, and is expressed as λ _r =-σ _r +jω _dr .
σ _r is the modal damping ratio and ω _dr is the damped natural angular frequency.

なお、式（３）において、
ｘ_ｒｉは、加振点ｉにおける複素固有モードの実部であり、
ｙ_ｒｉは、加振点ｉにおける複素固有モードの虚部である。 R _ril is the modal residue and is given by the following equation (3).

In addition, in the formula (3),
x _ri is the real part of the complex eigenmode at excitation point i,
y _ri is the imaginary part of the complex eigenmode at excitation point i.

When expressing the linear coupled system Lc by a general viscous damping system model, it is important to determine with high accuracy the modal parameters of the general viscous damping system model.
For this reason, in this embodiment, secondly, the “frequency response of a linear coupled system calculated by the multi-point excitation multi-point reference (MIMO) method using the fast multipole boundary element method” is used as a reference to identify the complex eigenvalues and complex eigenmodes, which are modal parameters, so as to fit the general viscous damping system model.

2 is a block diagram of a computer 10 that executes the method for identifying modal parameters according to this embodiment. The method for identifying modal parameters according to this embodiment is realized by cooperation between a computer and software.
The computer 10 includes a control device 11 and a storage device 12 .

The control device 11 is composed of one or more processing circuits, such as a CPU (Central Processing Unit), and controls each element of the computer 10 in an integrated manner. The control device 11 may be composed of circuits such as a DSP (Digital Signal Processor) and an ASIC (Application Specific Integrated Circuit) in addition to a CPU.

The storage device 12 is a single or multiple memories configured with known recording media such as magnetic recording media or semiconductor recording media, and stores the programs executed by the control device 11 and various data used by the control device 11. The storage device 12 may be configured by combining multiple types of recording media. Also, a portable recording medium that can be attached to and detached from the computer 10, or an external recording medium (e.g., online storage) with which the computer 10 can communicate via a communication network may be used as the storage device 12.

3 is a block diagram illustrating an example of a functional configuration of the control device 11. The control device 11 executes a program stored in the storage device 12 to function as a plurality of elements (a processing control unit 110, a preprocessing unit 111, a calculation unit 112, a first identification unit 115, a second identification unit 116, and a third identification unit 117) for identifying modal parameters.
It should be noted that some of the functions of the control device 11 may be implemented by another device, for example, a server device connected via a network.

Next, a procedure for identifying modal parameters in the computer 10 will be described.
FIG. 4 is a flow chart illustrating a procedure for identifying modal parameters.

First, the processing control unit 110 in the computer 10 causes the preprocessing unit 111 to execute the processes from the next step S11 to step S14, thereby making the preparations required for identifying the modal parameters of the general viscous damping system model representing the linear coupled system Lc.
Specifically, the pre-processing unit 111 creates a three-dimensional finite element model of the structure, taking into account the three-dimensional orthotropy of the elastic coefficient and hysteresis damping coefficient of the material constituting the structure, and calculates the mass matrix, damping matrix, and stiffness matrix of the three-dimensional finite element model (step S11).
The pre-processing unit 111 performs a nonlinear static analysis on the three-dimensional finite element model, taking into consideration contact nonlinearity and geometric nonlinearity associated with manufacturing processes such as bonding and bolting, and corrects the stiffness matrix calculated in step S11 (step S12). Note that step S12 may be skipped to simplify the process.

The pre-processing unit 111 uses as input the mass matrix and stiffness matrix calculated in step S11 or modified in step S12, and calculates the real eigenvalues and real eigenmodes of the three-dimensional finite element model when the three-dimensional finite element model exists in a vacuum, i.e., when no air is present, by eigenvalue analysis (step S13).
The pre-processing unit 111 uses the damping matrix calculated in step S11 and the real eigenmode calculated in step S13 as input, and calculates a modal damping matrix for the case in which the three-dimensional finite element model of the structure exists in a vacuum (step S14).

Next, the processing control unit 110 causes the calculation unit 112, the first identification unit 115, the second identification unit 116, and the third identification unit 117 to execute the following processing to identify the modal parameters (complex eigenvalues and complex eigenmodes) of the general viscous damping system model (step S15).

FIG. 5 is a flowchart illustrating in detail the procedure of step S15 in FIG.
The calculation unit 112 analyzes a three-dimensional model of the linear coupled system Lc and calculates the frequency response of the linear coupled system Lc by using, for example, the fast multipole boundary element method or the finite element method (step S151). The frequency response calculated by this analysis is the response of the displacement of the response point l to the force at the excitation point i in the linear coupled system Lc.

Here, the total number of excitation points i is m, and the total number of response points l is n. When there are multiple excitation directions at a given excitation position, such as the x-direction, y-direction, and z-direction, each is distinguished as a single excitation point. The same applies to response points l. The set of response points l includes the set of excitation points i. m and n are integers that satisfy n ≥ m ≥ 2.

The first identification unit 115 identifies a complex eigenvalue λ _r (=-σ _r +jω _dr ) in the linear coupled system using an inverse fast Fourier transform of the calculated frequency response as a reference, for example, by ESPRIT (Estimation of Signal Parameters via Rotational Invariance Techniques) (step S152).
In addition, when identifying the complex eigenvalues, for example, the auto-impulse response at the m points is referred to. In addition, the Prony method may be used to identify the complex eigenvalues, but the identification accuracy tends to be inferior to that of ESPRIT.

The second identification unit 116 uses the calculated frequency response as a reference to identify the complex eigenmode at the excitation point i of the linear coupled system Lc using a nonlinear optimization method (step S153). Examples of the nonlinear optimization method include a quasi-Newton method with a line search method using the BFGS algorithm (Broyden Fletcher Goldfarb Shanno algorithm), or a modified Gauss-Newton method (also called the Levenberg-Marquardt method) with a trust region method. For large-scale problems, the former method is more powerful. The real and imaginary parts of the complex eigenmode at the excitation point i are specifically represented by x _ri and y _ri in the first term on the right side of Equation (5) described later. In addition, when identifying the complex eigenmode at the excitation point i, for example, the self- and cross-frequency responses at m·(m+1)/2 points are referenced.

The third identification unit 117 uses the frequency response as a reference to identify a complex eigenmode at a response point l other than the excitation point i in the linear coupled system Lc by using a linear optimization method (step S154). Examples of the linear optimization method include the QR decomposition method and the singular value decomposition method. Specifically, the real part and the imaginary part of the complex eigenmode at the response point l are represented by x _rl and y _rl in the first term on the right side of Equation (5) described later. Note that, when identifying the complex eigenmode at the response point l, for example, the cross frequency response at the m·(n−m) point is referenced.

If the scale of the linear coupled system Lc is not very large, or if there are not many points for which you want to output eigenmodes, the computational cost of the frequency response analysis in step S151 will not be very high, so n may be set to m. In this case, step S154 is skipped.

The modal parameters (complex eigenvalues and complex eigenmodes) of the general viscous damping system model are identified in steps S151 to S154.

The identification will be explained with reference to the following equations (4) and (5).

Identification means finding the value (real part, imaginary part) that minimizes the sum of squares of the residual, as shown on the right side of equation (4), when the left side of equation (5) is expressed as the residual obtained by subtracting the "reference frequency response" in the second term on the right side from the "frequency response resynthesized using the model" in the first term on the right side.

Next, we will explain to what extent the modal parameters identified in the general viscous damping system model in this embodiment approach the reference.

6 to 8 are diagrams showing frequency responses (circles) due to mode synthesis at three different points (points 1 to 3) relative to a reference frequency response (solid line). In each of the figures, the left column shows phase and amplitude characteristics relative to frequency, and the right column shows real and imaginary characteristics relative to frequency.
As shown in FIG. 6 to FIG. 8, the frequency response due to mode synthesis almost coincides with the reference frequency response, and it is understood that identification is established with high accuracy.

9 to 11 are diagrams showing impulse responses (circles) due to mode synthesis against impulse responses (solid lines) as IFFT of frequency responses used as references at the above three points (points 1 to 3).
As shown in FIG. 9 to FIG. 11, the impulse response by mode synthesis almost coincides with the reference impulse response, and it is understood that the identification is established with high accuracy.

In the embodiment, the modal parameters in the physical model of the "structure in the air" are identified, but the modal parameters in the physical model of the "air accompanied by the structure" can also be identified in a similar manner.
The only difference is that in the former, "frequency response analysis of airborne structure" is performed in step S151, whereas in the latter, "frequency response analysis of air when a structure is attached" is performed. However, while the frequency response to be calculated in the former is "Fourier transform of structural displacement"/"Fourier transform of excitation force applied to structure", the frequency response to be calculated in the latter is "Fourier transform of air pressure"/"Fourier transform of air flow rate".

According to the method for identifying modal parameters of this embodiment, the identification of complex eigenvalues and the identification of complex eigenmodes are performed separately, so that it is easy to apply the method to a physical model of a "large-scale structure existing in the air" or a physical model of "air accompanied by a large-scale structure" with a large number of excitation points and response points. This makes it possible to efficiently predict the dynamic characteristics of a "large-scale structure placed in the air" or "air accompanied by a large-scale structure", which was difficult to do with conventional methods, by simulation.

As described above, according to this embodiment, for products whose value depends on vibration or acoustic characteristics, or whose performance depends on the coupling between the structure and the air, it is possible to predict the dynamic characteristics by simulation using a physical model before prototyping the product, and the causal relationship between cause and effect is clarified, thereby significantly improving the development efficiency of the product.
An example of such a product is an acoustic musical instrument. When manufacturing the main body (soundboard, tube, etc.) of such an acoustic musical instrument, the design of the physical model of the main body is modified, and the modified physical model is used to synthesize musical tone signals, allowing the musical tones of the musical instrument to be heard by simulation before being prototyped. Therefore, for example, new musical instruments can be developed efficiently.

<Modification>
The above-described exemplary embodiment may be modified in various ways. Specific modified aspects that may be applied to the embodiment are exemplified below. Two or more aspects selected from the following examples may be combined to the extent that they are not mutually contradictory.

As described above, the function of the computer 10 to synthesize musical tone signals is realized by the cooperation of one or more processors constituting the control device 11 and the programs stored in the storage device 12 .
This program can be provided in a form stored in a recording medium readable by the computer 10 and installed in the computer. The recording medium is, for example, a non-transitory recording medium, and is exemplified by an optical recording medium (optical disk) such as a CD-ROM, but also includes any known type of recording medium such as a semiconductor recording medium or a magnetic recording medium. Note that a non-transitory recording medium includes any recording medium other than a transient, propagating signal, and does not exclude volatile recording media. In a configuration in which a distribution device distributes a program via a communication network, the storage device 12 that stores the program in the distribution device corresponds to the non-transitory recording medium described above.

<Additional Notes>
From the above description, for example, preferred embodiments of the present invention can be understood as follows. In order to facilitate understanding of each embodiment, reference numerals in the drawings are written in parentheses for convenience, but the present invention is not limited to the illustrated embodiments.

A musical sound synthesis method according to one aspect (aspect 1) of the present disclosure is a modal parameter identification method in which a computer (10) identifies modal parameters in a physical model of a "structure when present in the air" or a physical model of "the air when the structure is attached," and includes a first step in which the computer (10) calculates a frequency response of the structure when present in the air, or a frequency response of the air when the structure is attached, a second step in which the computer (10) identifies complex eigenvalues using an inverse Fourier transform of the frequency response as a reference, and a third step in which the computer (10) identifies a complex eigenmode of a vibration point in the structure, or a complex eigenmode of a vibration point in the air, using the frequency response as a reference.
According to this aspect 1, the identification of the complex eigenvalues and the identification of the complex eigenmodes are performed separately, rather than simultaneously, so that it becomes easy to apply the present invention to a physical model of a large-scale structure existing in the air with a large number of excitation points and response points, or a physical model of the air accompanied by a large-scale structure.

In a specific example (Aspect 2) of Aspect 1, after the third step, a fourth step is included in which the computer (10) identifies, using the frequency response as a reference, a complex eigenmode of a response point in the structure other than the excitation point, or a complex eigenmode of a response point in the air other than the excitation point.
According to this aspect 2, the number of response points excluding the excitation points can be increased, making it possible to apply the method to a larger-scale physical model.

The method for identifying modal parameters according to aspect 1 can be understood as a program according to aspect 3. That is, the program according to aspect 34 is a program that causes a computer (10) to identify modal parameters in a physical model of a "structure in the air" or a physical model of the "air with an associated structure", and causes the computer (10) to function as a calculation unit (112) that calculates the frequency response of the "structure in the air" or the frequency response of the "air with an associated structure", a first identification unit (115) that identifies a complex eigenvalue using the inverse Fourier transform of the frequency response as a reference, and a second identification unit (116) that identifies a complex eigenmode of a vibration point in the structure or a complex eigenmode of a vibration point in the air using the frequency response as a reference.

The method for identifying modal parameters according to aspect 1 can be understood as an apparatus for identifying modal parameters according to aspect 4. That is, the apparatus for identifying modal parameters according to aspect 4 is an apparatus for identifying modal parameters in a physical model of a "structure in the air" or a physical model of the "air with an associated structure", and includes a calculation unit (112) for calculating a frequency response of the "structure in the air" or a frequency response of the "air with an associated structure", a first identification unit (115) for identifying a complex eigenvalue using an inverse Fourier transform of the frequency response as a reference, and a second identification unit (116) for identifying a complex eigenmode of a vibration point in the structure or a complex eigenmode of a vibration point in the air using the frequency response as a reference.

10...computer, 11...control device, 12...storage device, 13...performance information output device, 14...sound emission device, 110...processing control unit, 111...preprocessing unit, 112...calculation unit, 115...first identification unit, 116...second identification unit, 117...third identification unit, Lc...linear coupled system.

Claims (4)

1. A method for identifying modal parameters in a physical model of a structure when in air, or in a physical model of air when the structure is associated with the structure, comprising:
a first step in which the computer calculates a frequency response of the structure when present in the air or a frequency response of the air when associated with the structure;
a second step in which the computer identifies complex eigenvalues using an inverse Fourier transform of the frequency response as a reference;
a third step in which the computer identifies a complex eigenmode of an excitation point in the structure or a complex eigenmode of an excitation point in the air using the frequency response as a reference;
A method for identifying modal parameters in a physical model including:
After the third step,
a fourth step in which the computer identifies a complex eigenmode of a response point in the structure different from the excitation point, or a complex eigenmode of a response point in the air different from the excitation point, using the frequency response as a reference;
2. A method for identifying modal parameters in a physical model according to claim 1, comprising:
A program for causing a computer to identify modal parameters of a physical model of a structure in air or a physical model of air in the presence of a structure, the program comprising:
The computer includes:
A calculation unit that calculates a frequency response of the structure when present in the air, or a frequency response of the air when the structure is present;
a first identification unit that identifies a complex eigenvalue using an inverse Fourier transform of the frequency response as a reference; and
a second identification unit that identifies a complex eigenmode of an excitation point in the structure or a complex eigenmode of an excitation point in the air using the frequency response as a reference;
A program that functions as a
A modal parameter identification device that identifies modal parameters in a physical model of a structure when it exists in air, or in a physical model of air when a structure is associated with the structure, comprising:
A calculation unit that calculates a frequency response of the structure when present in the air, or a frequency response of the air when the structure is present;
a first identification unit that identifies a complex eigenvalue using an inverse Fourier transform of the frequency response as a reference;
a second identification unit that identifies a complex eigenmode of an excitation point in the structure or a complex eigenmode of an excitation point in the air using the frequency response as a reference;
A modal parameter identifier including:

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