JP2017166992A - Simulation device and program - Google Patents

Abstract

PROBLEM TO BE SOLVED: To obtain a frequency response function for each component of wave amplitude.SOLUTION: A simulation device of the present invention calculates, as response to an inputted amplitude to a three-dimensional beam structure, a wave amplitude vector comprising each component of a longitudinal wave forward propagation wave in a direction x, a longitudinal wave backward propagation wave in the direction x, a torsion forward propagation wave, a torsion backward propagation wave, a bending forward propagation wave in a direction y, a bending forward proximity wave in the direction y, a bending backward propagation wave in the direction y, a bending backward proximity wave in the direction y, a bending forward propagation wave in a direction z, a bending forward proximity wave in the direction z, a bending backward propagation wave in the direction z, and a bending backward proximity wave in the direction z.SELECTED DRAWING: Figure 2

Description

  The present invention relates to a simulation apparatus and a program, and more particularly to a simulation apparatus and a program for calculating a wave amplitude as a response to an input amplitude to a three-dimensional beam structure.

  2. Description of the Related Art Conventionally, two methods using a mass matrix and a stiffness matrix are known as methods for obtaining a frequency response function by numerical analysis (Non-Patent Document 1). The first method is called a direct method, and a frequency response function is obtained by directly obtaining an inverse matrix of a combination of a mass matrix and a stiffness matrix. The other method, called theoretical mode analysis, is a method that obtains mode characteristics by applying a matrix that combines a mass matrix and stiffness matrix to eigenvalue analysis, and obtains a frequency response function by superimposing the responses of each mode. It is. In this method, it is possible to evaluate the contribution of each mode in the frequency response function.

  In addition, as an example of conducting a wave propagation analysis of a complex structure using a distributed parameter system model, a case of a time series wave propagation numerical analysis targeting a space structure is known (Non-patent Document 2). .

Akio Nagamatsu, “Introduction to Mode Analysis”, Corona, 1993. Akimi Nishida, “Study on Wave Propagation Characteristics of 3D Frame Structures: Introduction of Tymoshenko Beam Theory”, Structural Engineering Papers B, Vol. 52B, pp.119-124, Architectural Institute of Japan, 2006

  In the analysis technique for obtaining the frequency response function of Non-Patent Document 1, the propagation of each wave inside the structure constituting the vibration is not physically considered, and another physical model that does not consider the wave propagation (discrete) The frequency response function is obtained from the system model.

  In the research and technology development related to wave propagation described in Non-Patent Document 2 above, wave propagation analysis in time series when impact load is applied is the main, and handling of wave propagation in steady state such as frequency response function is handled. There are no research cases on.

  Thus, the wave propagation contribution cannot be obtained from the conventional mode analysis or frequency response function. There is another method to determine the wave propagation contribution of a structure.

  The present invention has been made in view of the above circumstances, and an object of the present invention is to provide a simulation apparatus and a program capable of obtaining a frequency response function for each wave amplitude component.

In order to achieve the above object, a simulation apparatus according to the present invention is a simulation apparatus that calculates a wave amplitude as a response to an input amplitude to a three-dimensional beam structure in which a plurality of beam elements are coupled, and represents the input amplitude. , Longitudinal direction forward propagation wave in the x direction which is the axial direction of the beam element, longitudinal direction backward propagation wave in the x direction, torsional forward propagation wave, torsional backward propagation wave, bending forward propagation wave in the y direction, bending forward proximity in the y direction Wave, y-direction bending backward propagation wave, y-direction bending backward propagation wave, z-direction bending forward propagation wave, z-direction bending forward propagation wave, z-direction bending backward propagation wave, and z-direction bending backward propagation wave Based on a wave amplitude vector a ₀ composed of each component, a scattering matrix S representing the transmissivity of the discontinuous part of the three-dimensional beam structure, and a propagation matrix D representing a phase change accompanying the propagation of each component. And the following formula I represents the wave amplitude as a response, is configured to include a calculating means for calculating wave amplitude vector a of each component.

Here, I is a unit matrix.

A program according to the present invention is a program for calculating a wave amplitude as a response to an input amplitude to a three-dimensional beam structure in which a plurality of beam elements are combined, and the computer represents a beam element axis representing the input amplitude. X direction longitudinal wave forward propagation wave, x direction longitudinal wave backward propagation wave, torsional forward propagation wave, torsional backward propagation wave, y direction bending forward propagation wave, y direction bending forward proximity wave, y direction It consists of bending backward propagation wave, bending backward propagation wave in y direction, bending forward propagation wave in z direction, bending forward proximity wave in z direction, bending backward propagation wave in z direction, and bending backward propagation wave in z direction. Based on the wave amplitude vector a ₀ , the scattering matrix S representing the transmissivity of the discontinuous portion of the three-dimensional beam structure, and the propagation matrix D representing the phase change accompanying the propagation of each component, the above formula According to the wave as a response Represents the width, a program to function as a calculating means for calculating wave amplitude vector a of each component.

According to the present invention, the calculating means represents the input amplitude, the longitudinal wave forward propagation wave in the x direction that is the axial direction of the beam element, the longitudinal wave backward propagation wave in the x direction, the torsional forward propagation wave, the torsional backward propagation wave, bending forward propagation wave in y direction, bending forward proximity wave in y direction, bending backward propagation wave in y direction, bending backward propagation wave in y direction, bending forward propagation wave in z direction, bending forward propagation wave in z direction, z direction Wave bending vector a ₀ composed of the components of the bending backward propagation wave and the bending backward proximity wave in the z direction, the scattering matrix S representing the transmissivity of the discontinuous portion of the three-dimensional beam structure, and the propagation of each component On the basis of the propagation matrix D representing the phase change accompanying the above, the wave amplitude vector a of each component representing the wave amplitude as a response is calculated according to the above formula.

  As described above, as a response to the input amplitude to the three-dimensional beam structure, the longitudinal wave forward propagation wave in the x direction, the longitudinal wave backward propagation wave in the x direction, the torsional forward propagation wave, the torsional backward propagation wave, and the bending forward propagation in the y direction. Wave, bending forward proximity wave in y direction, bending backward propagation wave in y direction, bending backward proximity wave in y direction, bending forward propagation wave in z direction, bending forward proximity wave in z direction, bending backward propagation wave in z direction, By calculating a wave amplitude vector composed of each component of the bending backward receding wave in the z direction, a frequency response function can be obtained for each wave amplitude component.

  The calculation means can calculate the wave amplitude vector a of each component for each frequency, and obtain a frequency response function for each component.

The calculation means is based on the longitudinal elastic modulus, volume mass density, transverse elastic modulus, cross-sectional secondary pole moment, torsional constant, and cross-sectional area of the beam element for each of the plurality of beam elements for each frequency. A wave number calculating means for calculating the wave number k _L of the longitudinal wave in the x direction, the wave number k _{T of the} torsion wave, the wave number k _By of the bending wave in the y direction, and the wave number k _Bz of the bending wave in the z direction; In addition, for each of the plurality of beam elements, the wave number k _L of the longitudinal wave in the x direction, the wave number k _{T of the} torsion wave, the wave number k _By of the bending wave in the y direction, and the wave number of the bending wave in the z direction Based on k _Bz , a displacement wave amplitude conversion matrix Ψ ₁ , Ψ ₂ for each of the forward propagation wave component and the backward propagation wave component, and a propagation matrix representing a phase change according to the propagation distance x for each of the forward wave and the backward wave G ₁ and _{(x), G 2 (x} ), the forward propagation NamiNaru And internal forces wave amplitude transformation matrix [Phi ₁ for each of backward propagating wave components, and the transformation matrix propagation matrix calculation means for calculating the [Phi _2, for each of the beam element, the base vector of the entire coordinate system ei of the beam element and A coordinate transformation matrix calculation means for calculating a coordinate transformation matrix C ⁽ⁱ⁾ for each of the beam elements connected at the discontinuity based on the basis vector Ei of the element coordinate system, and for each of the beam elements, An incident direction calculating means for calculating an incident direction d ⁽ⁱ⁾ of the forward propagation wave of the beam element to the discontinuity, and the displacement wave amplitude conversion matrix Ψ for each of the beam elements for each frequency. ₁ and Ψ ₂ , the propagation matrices G ₁ (x) and G ₂ (x), the internal force wave amplitude conversion matrices Φ ₁ and Φ ₂ , the coordinate conversion matrix C ^(i), and the incident direction d ^{( i)} and each of the beam elements A coefficient calculation means for calculating a reflection coefficient r for each of the beam elements coupled in the discontinuity, and a coefficient of coefficient calculation means for calculating the transmission coefficient t for each pair of beam elements, and for each frequency, the reflection coefficient r for each of the beam elements. And a scattering matrix calculation means for calculating a scattering matrix S representing transmission reflectivity of the discontinuous part of the three-dimensional beam structure based on a transmission coefficient t for each pair of beam elements, and for each frequency, Based on the propagation matrices G ₁ (x), G ₂ (x) for each of the beam elements and the waveguide length L ⁽ⁱ⁾ of each of the beam elements, the phase information of each of the beam elements A propagation matrix calculating means for calculating a propagation matrix D representing the wave amplitude vector a of each component based on the wave amplitude vector a ₀ , the scattering matrix S, and the propagation matrix D for each frequency. Calculate and for each component It may include a frequency response function computation means for obtaining the wave number response function.

  As described above, according to the simulation apparatus and program of the present invention, as a response to the input amplitude to the three-dimensional beam structure, a longitudinal forward propagation wave in the x direction, a longitudinal backward propagation wave in the x direction, and a torsional forward propagation. Wave, torsional backward propagation wave, bending forward propagation wave in y direction, bending forward proximity wave in y direction, bending backward propagation wave in y direction, bending backward propagation wave in y direction, bending forward propagation wave in z direction, z direction By calculating a wave amplitude vector composed of each component of the bending forward proximity wave, the bending backward propagation wave in the z direction, and the bending backward proximity wave in the z direction, a frequency response function can be obtained for each wave amplitude component. An effect is obtained.

It is a block diagram which shows the simulation apparatus which concerns on embodiment of this invention. It is a flowchart which shows the content of the simulation process routine of the simulation apparatus which concerns on embodiment of this invention. It is a figure which shows an example of a three-dimensional beam structure. It is a figure which shows an example of a frequency response function. It is a figure which shows an example of a frequency response function.

  Hereinafter, embodiments of the present invention will be described in detail with reference to the drawings.

<Outline of Embodiment of the Present Invention>
A model called a ray trace model can be configured by the transmission / reflection coefficient, shape information (waveguide length, etc.), and wave amplitude. This model is a model that numerically describes the relationship between the wave amplitude before and after propagation when a wave propagates from one discontinuous part of a structure to another discontinuous part. The wave amplitude when the input is 1N is calculated, and the various wave amplitude contributions in the frequency response function are evaluated by individually looking at the various wave amplitudes at an arbitrary position where the wave propagation has occurred infinitely.

<Principle of Embodiment of the Present Invention>
Next, the principle of wave amplitude analysis by wave propagation will be described.

<Wave amplitude expression in 3D beam element>
First, the wave theory in the three-dimensional beam element is organized. The relationship between wave amplitude, displacement, and internal force for longitudinal waves, torsional waves, and bending waves is organized, and a conversion matrix from wave amplitudes to displacements and internal forces necessary for calculating transmission and reflection coefficients is derived.

<Wave theory for longitudinal waves>
The equation of motion for the longitudinal wave propagating through the beam is as follows. Here, ρ represents a volume mass density [kg / m ³ ], A represents a cross-sectional area [m ² ], and E represents a longitudinal elastic modulus [N / m ² ].

  The general solution of the above equation of motion is:

Here, a _1L represents the wave amplitude of the forward propagation wave propagating in the positive direction of x, and a _2L represents the wave amplitude of the backward propagation wave propagating in the negative direction of x. Further, the internal force F (x) is obtained from the following equation.

k _L is the wave number [rad / m] of the longitudinal wave, and is defined by the following equation.

<Wave theory for torsional waves>
The axial direction of the beam element is the x-axis, and the displacement around the x-axis of the cross section is s (x, t). Also, let h (x, t) be a torsional moment for torsional wave propagation. Here, G is the transverse elastic modulus [N / m ² ], IP is the sectional secondary pole moment [m ⁴ ], and J is the torsional constant. In the linear vibration region, the sectional deformation is small, and the equation of motion for the torsional wave is expressed by the following equation.

  The general solution of the above equation of motion is:

  Further, the internal force H (x) is obtained from the following equation.

Here, a _1T represents the amplitude of the torsional forward propagation wave propagating in the positive x direction, and a _2T represents the amplitude of the torsional backward propagation wave propagating in the negative x direction. At this time, k _T is the wave number [rad / m] of the torsional wave.

<Wave theory for bending waves>
The equation of motion for bending waves can be expressed as:

Here, I is the cross-sectional secondary pole moment [m ⁴ ]. The solution of the equation of motion of this formula is obtained by the following formula.

Here, a _1P represents the amplitude of the bending forward propagation wave, a _2P represents the amplitude of the bending backward propagation wave, a _1N represents the amplitude of the bending forward proximity wave, and a _2N represents the amplitude of the bending backward propagation wave. Here, the wave number k _B of the bending wave is as follows.

  At this time, the relationship between the deflection angle θ (x, t) and the deflection v (x, t) is as follows.

  Further, the moment m (x, t) and the shearing force w (x, t) are obtained by the following equations, respectively.

<Wave amplitude conversion matrix>
When calculating the reflection / transmission coefficient of a beam, a wave amplitude conversion matrix for expressing the displacement and internal force of the beam by the wave amplitude is required. Next, the definition of the wave amplitude conversion matrix will be described.

First, the forward propagation wave component a ₁ and the backward propagation wave component a ₂ of the wave amplitude vector are defined as follows:

Forward formula (2), which is a general solution of displacement due to longitudinal waves, formula (6), which is a general solution of displacement due to torsional waves, and equations (10) and (12), which are general solutions of displacement for bending waves It is divided into a displacement component due to the wave a _{1 and} a displacement component due to the backward wave a ₂ . Longitudinal wave displacement U ₁ , U ₂ , y-axis bending wave deflection V _1y , V _2y , z-axis bending wave deflection V _1z , V _2z , torsion wave displacement S ₁ , S _2. Deflections in the y-axis direction are angles Θ _1y and Θ _2y , and deflection angles in the z-axis direction are Θ _1z and Θ _2z . Here, a vector for a general solution of displacement is defined as U, and displacement vectors due to forward and backward waves are defined as U ₁ and U _{2 by the} following equations.

  From these definitions, the relationship between the wave amplitudes can be expressed by the following equation using equations (2), (6), (10), and (12).

At this time, Ψ ₁ and Ψ ₂ are displacement-wave amplitude conversion matrices for the forward propagation wave component and the backward propagation wave component of the wave amplitude, respectively, and can be expressed by the following two equations, respectively.

G ₁ (x) and G ₂ (x) are propagation matrices representing phase changes depending on the propagation distance x of the forward wave and the backward wave, respectively, and are represented by Expression (20) and Expression (21).

Next, Equation (3), which is the solution of internal force due to longitudinal waves, Equation (7), which is the solution of internal forces due to torsional waves, Equations (13) and (14), which are solutions of internal force due to bending waves, And divide into components due to backward waves. Longitudinal wave internal forces F ₁ , F ₂ , y-axis bending wave shear forces W _1y , W _2y , z-axis bending wave shear forces W _1z , W _2z , torsional wave internal forces H ₁ , H ₂ , The bending moments M _1y and M _{2y in} the _y- axis direction and the bending moments M _1z and M _{2z in the} z-axis direction are assumed. Here, the vector for the general solution of the internal force is defined as F, and the internal force vectors due to the forward and backward waves are defined as F ₁ and F _{2 by the} following equations.

  From these definitions, formula (3), formula (7), formula (13), and formula (14) can be summarized, and the relationship between internal force and wave amplitude can be summarized as the following formula.

The matrices Φ ₁ and Φ _{2 at} this time are internal force-wave amplitude conversion matrices for the forward wave component and the backward wave component of the wave amplitude, respectively, and are expressed by the following two expressions, respectively. Here, I _y represents the sectional secondary pole moment [m ⁴ ] in the y-axis direction, and I _z represents the sectional secondary pole moment [m ⁴ ] in the z-axis direction.

  From the above, general solutions of longitudinal waves, torsional waves, and bending waves were arranged, and a conversion matrix from wave amplitude to displacement and internal force was derived.

<Calculation of transmission / reflection coefficient of discontinuity>
Next, the calculation of the reflection / transmission coefficient in the discontinuous portion where N _c elements are coupled is formulated. a ₁ ⁽ⁱ⁾ and a ₂ ⁽ⁱ⁾ are the forward wave propagating in the positive direction of the x-axis and the backward wave propagating in the negative direction in the element coordinate system of the coupling element i (= 1 to N _c ). This is the wave amplitude vector.

The coordinate transformation matrix C ⁽ⁱ⁾ from the element coordinate system to the global coordinate system of the beam element i (= 1 to N _c ) connected at the discontinuity is e, and the element coordinate system of element i is the base vector of the global coordinate system Let E ^{(i) be} the basis vector of

If the discontinuous portion position is x = 0 in the equation (17), the displacement U ⁽ⁱ⁾ at the discontinuous portion of the beam element i coupled at the discontinuous portion is expressed by the following equation.

  From the continuity of displacement in the discontinuous portion, the following equation is established.

Further, when the position of the discontinuous portion in equation (23) is x = 0, the internal force F ⁽ⁱ⁾ at the discontinuous portion of the beam element i coupled at the discontinuous portion is expressed by the following equation.

  From the balance of force in the discontinuous portion, the following equation is established.

In this equation, d ⁽ⁱ⁾ represents the positive / negative direction of incidence on the discontinuous portion of the forward propagation wave of the element i, and is expressed by the following equation.

Here, w _I ⁽ⁱ⁾ and w _O ⁽ⁱ⁾ are variables that define an incident wave and a reflected wave with respect to the discontinuous portion, and are represented by Expression (32) and Expression (33). X _c is the coordinates of the discontinuity, x ₁ ⁽ⁱ⁾ and x ₂ ⁽ⁱ⁾ are the node coordinates of element i, and the direction from x ₁ ⁽ⁱ⁾ to x ₂ ⁽ⁱ⁾ is the element coordinate The positive direction of the x-axis of the system. w _I ⁽ⁱ⁾ is a variable that determines whether the incident wave amplitude vector for the discontinuity is a forward wave component propagating in the positive direction of the x-axis of the element coordinate system or a backward wave component propagating in the negative direction The element coordinate system is defined so that the forward wave of element i is incident on the discontinuous portion when 1 and the backward wave is incident when 2. On the other hand, w _O ⁽ⁱ⁾ is a variable that determines the wave amplitude vector reflected from the discontinuous part. When 1, the forward wave reflects from the discontinuous part, and when 2, the backward wave reflects from the discontinuous part. The element coordinate system is defined.

  Equation (28), which represents the continuity of displacement at the discontinuity, and Equation (30), which represents the balance of force, are the wave amplitude component of the wave incident on the discontinuity on the left side and the wave amplitude reflected from the discontinuity on the right side. This is expressed in the following equation when arranged in a matrix and expressed as a matrix.

  The change of the wave amplitude of the discontinuity using the transmission / reflection coefficient is expressed by the following equation.

  Therefore, from Equation (34) and Equation (35), (34) reflection coefficient r and transmission coefficient t are expressed as follows using displacement-wave amplitude conversion matrix Ψ, internal force-wave amplitude conversion matrix Φ, and coordinate conversion matrix C: It is possible to calculate with an equation.

<Generalization of ray-trace model>
Next, we formulate a ray-trace model to be analyzed consisting of N elements. A wave amplitude vector a ⁽ⁱ⁾ consisting of the wave amplitude a ₁ ^{(i) of} the forward wave propagating through the beam element i (= 1 to N) and the wave amplitude component a ₂ ^{(i) of the} backward wave is defined as follows: .

  When the transmission / reflection relationship between the beam element i and the beam element j coupled at the discontinuous portion is formulated, it can be expressed as the following expression.

Where R ⁽ⁱ⁾ is the reflection coefficient matrix for the wave incident from beam element i at the discontinuity, and T (i, j) is the transmission coefficient matrix when the wave is transmitted from beam element i to beam element j. is there. The transmission coefficient matrix T and the reflection coefficient matrix R are expressed as the wave vector for the discontinuity when expressed using the transmission and reflection coefficient calculated in Equation (36) in the previous section and w _I and w _O shown in Equation (32). In consideration of the definition of the incident and reflection directions of the first and second reflections, they are expressed by the equations (39) and (40), respectively.

From the above, the relationship between transmission and reflection is formulated. Using the reflection coefficient R ⁽ⁱ⁾ for the beam element i constituting the system and the transmission coefficient matrix T (i, j) between the beam element i and the beam element j connected at the discontinuity, It is possible to construct a scattering matrix S for the entire system as follows:

  Here, a in which the wave amplitude vectors for the elements of the entire system are arranged is defined as follows.

  Further, a propagation matrix D representing that a wave is generated from a discontinuous part and propagates in the beam element to the other discontinuous part is defined as follows.

A wave model in which a certain wave amplitude vector a _n propagates in a beam element, reaches a discontinuity, and changes to the next propagated wave amplitude vector a _{n + 1} is called a ray-trace model, and a propagation matrix D and a scattering matrix Using S, it can be expressed as:

When the ray trace model is used, the relationship between the wave amplitude a ₁ before and after the initial wave propagation is as follows.

Considering the steady state wave amplitude a, it is a situation where wave amplitudes (a ₀ , a ₁ , a ₂ ... A _∞ ) propagated a plurality of times coexist and can be formulated as follows.

  This can be summarized as follows:

If a ₀ is the initial wave amplitude when an excitation force having an amplitude of 1N is applied to an arbitrary location of the structure, the above a is a wave amplitude vector obtained by decomposing the frequency response function according to various wave amplitudes. Represents the wave amplitude contribution. Therefore, if the above equation is used, it is possible to obtain the wave amplitude contribution of the frequency response function.

<System configuration>
As shown in FIG. 1, a simulation apparatus 10 according to an embodiment of the present invention includes a CPU 12, a ROM 14, a RAM 16, an HDD 18, a communication interface 20, and a bus 22 for connecting them together.

  The CPU 12 executes various programs. The ROM 14 stores various programs, parameters, and the like. The RAM 16 is used as a work area when the CPU 12 executes various programs. The HDD 18 as a recording medium stores various programs and various data including a program for executing a later-described differential calculation processing routine.

The simulation apparatus 10 according to the present embodiment has, as inputs, a longitudinal forward wave in the x direction that is an axial direction of the beam element, a longitudinal backward propagation wave in the x direction, a torsional forward propagation wave, and a torsional receding that represent the input amplitude. Propagation wave, bending forward propagation wave in y direction, bending forward proximity wave in y direction, bending backward propagation wave in y direction, bending backward propagation wave in y direction, bending forward propagation wave in z direction, bending forward proximity wave in z direction , A wave amplitude vector a ₀ composed of the components of the bending backward propagation wave in the z direction and the bending backward proximity wave in the z direction is received. Further, the simulation apparatus 10 uses the longitudinal elastic modulus E, the volume mass density ρ, the transverse elastic modulus G, the cross-sectional secondary pole moment I _p for each beam element as shape information of a three-dimensional beam structure in which a plurality of beam elements are coupled. , Torsional constant J, cross-sectional area A, global coordinate system basis vector e, elementary coordinate system basis vector E ⁽ⁱ⁾ , and waveguide length L ⁽ⁱ⁾ .

The simulation apparatus 10 is based on the wave amplitude vector a ₀ , the scattering matrix S representing the transmissivity of the discontinuous part of the three-dimensional beam structure, and the propagation matrix D representing the phase information of each beam element for each frequency. Then, according to the above equation (47), a wave amplitude vector a composed of each component representing the wave amplitude as a response is calculated, and a frequency response function is obtained for each component.

<Operation of simulation device>
Next, the operation of the simulation apparatus 10 according to the present embodiment will be described.

First, when the wave amplitude vector a ₀ and the shape information of the three-dimensional beam structure described above are input to the simulation device 10, the simulation device 10 generates a shape model of the three-dimensional beam structure. Then, the simulation processing routine shown in FIG.

First, in step S100, for each frequency ω, based on the longitudinal elastic modulus E, volume mass density ρ, transverse elastic modulus G, cross-sectional secondary pole moment Ip, torsion constant J, and cross-sectional area A of each beam element, According to the equations (4), (8), (11), for each beam element, the longitudinal wave number k _L in the x direction, the wave number k _{T of the} torsion wave, the wave number k _By of the bending wave in the y direction, and the z direction Calculate the wave number k _{Bz of the} bending wave.

In step S102, for each frequency ω, for each beam element, the wave number k _L of the longitudinal wave in the x direction, the wave number k _{T of the} torsion wave, the wave number k _By of the bending wave in the y direction, and the wave number k of the bending wave in the z direction. Based on _Bz , the displacement wave amplitude transformation matrices Ψ ₁ and Ψ ₂ for each of the forward propagation wave component and the backward propagation wave component according to the above formulas (18) to (21), (24), and (25), and the forward A propagation matrix G ₁ (x), G ₂ (x) representing a phase change according to the propagation distance x for each of the wave and the backward wave, and an internal force wave amplitude conversion matrix Φ for each of the forward propagation wave component and the backward propagation wave component ₁ and Φ ₂ are calculated.

In step S104, based on the base vector e of the overall coordinate system of the beam element and the base vector E ⁽ⁱ⁾ of the element coordinate system for each of the beam elements, the discontinuous portions are combined according to the above equation (26). The coordinate transformation matrix C ⁽ⁱ⁾ for each beam element is calculated.

For each beam element, the incident direction d ⁽ⁱ⁾ of the forward propagation wave of the beam element to the discontinuous portion is calculated according to the above formula (31) based on the shape information of the three-dimensional beam structure. .

In step S106, for each frequency ω, the displacement wave amplitude conversion matrices Ψ ₁ and Ψ ₂ for each of the forward propagation wave component and the backward propagation wave component calculated in step S102 and each of the forward wave and the backward wave are calculated. Propagation matrices G ₁ (x), G ₂ (x) representing the phase change due to the propagation distance x, internal force wave amplitude conversion matrices Φ ₁ , Φ ₂ for each of the forward propagation wave component and the backward propagation wave component, Based on the coordinate transformation matrix C ⁽ⁱ⁾ for each beam element and the incident direction d ⁽ⁱ⁾ for each beam element calculated in step S104, the beam Calculate the reflection coefficient r for each of the elements and the transmission coefficient t for each pair of beam elements connected at the discontinuity.

  In step S108, based on the reflection coefficient r for each beam element and the transmission coefficient t for each beam element pair calculated in step S106 for each frequency ω, the above equation (41) is obtained. Accordingly, the scattering matrix S representing the transmissivity of the discontinuous portion of the three-dimensional beam structure is calculated.

In step S110, for each frequency ω, propagation matrices G ₁ (x) and G ₂ (x) representing the phase change due to the propagation distance x for each of the forward wave and the backward wave calculated in step S102, and beam elements Based on each waveguide length L ⁽ⁱ⁾ , a propagation matrix D representing the phase information of each beam element is calculated according to the above equation (43).

In step S112, the wave amplitude vector a is calculated for each frequency ω based on the wave amplitude vector a ₀ , the scattering matrix S calculated in step S108, and the propagation matrix D calculated in step S110. calculate. In the next step S114, a frequency response function is obtained for each component based on the wave amplitude vector a calculated for each frequency ω in the above step S112, and output by an output unit (not shown) to execute a simulation processing routine. finish.

<Example>
An example of wave amplitude contribution is shown by a three-dimensional beam structure model in which three beam elements having different cross-sectional characteristics are combined.

  As shown in FIG. 3, a harmonic load of 1N is applied to the left end of the three-dimensional beam structure model as an input wave amplitude without any constraint condition. The evaluation points are the excitation point (1) and the center of the beam (2). The specifications of beam elements are shown in Table 1 below.

  The analysis results are shown in FIGS. Since it is complicated to display the wave amplitude contribution of the wave of all components of the three-dimensional beam structure, here we focus on the low-frequency vertical bending vibration, and the forward propagation wave, forward proximity wave, backward propagation wave, backward proximity wave Display the contribution to the frequency response function of. FIG. 4 shows the frequency response function of each component at the evaluation point (1), and FIG. 5 shows the frequency response function of each component at the evaluation point (2).

  As shown in the graphs of FIGS. 4 and 5, it is possible to analyze the contribution of various components to the frequency response function.

  As described above, according to the simulation apparatus according to the embodiment of the present invention, as a response to the input amplitude to the three-dimensional beam structure, the longitudinal wave forward propagation wave in the x direction, the longitudinal wave backward propagation wave in the x direction, Torsional forward propagation wave, torsional backward propagation wave, bending forward propagation wave in y direction, bending forward proximity wave in y direction, bending backward propagation wave in y direction, bending backward propagation wave in y direction, bending forward propagation wave in z direction, A frequency response function can be obtained for each component of the wave amplitude by calculating a wave amplitude vector composed of each component of the bending forward proximity wave in the z direction, the bending backward movement wave in the z direction, and the bending backward proximity wave in the z direction. it can.

  In addition, by evaluating the frequency response function as the contribution of the wave amplitude component in the structure, it is possible to identify the wave that is the main cause of the vibration response, and to study effective countermeasures.

  It is also possible to propose a structure that reduces the vibration response at an arbitrary position by superimposing various wave amplitudes having different phases.

  Note that the present invention is not limited to the above-described embodiment, and various modifications and applications are possible without departing from the gist of the present invention.

  The program of the present invention may be provided by being stored in a storage medium.

10 Simulation device 12 CPU
14 ROM
16 RAM
18 HDD

Claims (4)

A simulation device for calculating a wave amplitude as a response to an input amplitude to a three-dimensional beam structure in which a plurality of beam elements are combined,
Axial longitudinal forward wave, x-direction longitudinal backward wave, torsional forward wave, torsional backward wave, y-direction bending forward wave, y-direction representing the input amplitude. Bending forward proximity wave, y-direction bending backward propagation wave, y-direction bending backward propagation wave, z-direction bending forward propagation wave, z-direction bending forward proximity wave, z-direction bending backward propagation wave, and z-direction A wave amplitude vector a ₀ composed of each component of the bending receding proximity wave, a scattering matrix S representing the transmissivity of the discontinuous portion of the three-dimensional beam structure, and a propagation matrix representing a phase change accompanying the propagation of each component A simulation device including a calculation means for calculating a wave amplitude vector a of each component, which represents a wave amplitude as a response, based on D, according to the following equation.
Here, I is a unit matrix.   The simulation apparatus according to claim 1, wherein the calculation unit calculates a wave amplitude vector a of each component for each frequency and obtains a frequency response function for each component. The calculating means includes
For each of the plurality of beam elements for each frequency, the longitudinal element in the x direction is determined based on the longitudinal elastic modulus, volume mass density, transverse elastic modulus, cross-sectional secondary pole moment, torsion constant, and cross-sectional area of the beam element. A wave number calculating means for calculating the wave number k _{L of} the wave, the wave number k _{T of the} torsion wave, the wave number k _By of the bending wave in the y direction, and the wave number k _Bz of the bending wave in the z direction;
For each of the plurality of beam elements, for each frequency, the longitudinal wave number k _L in the x direction, the wave number k _{T of the} torsional wave, the wave number k _By of the bending wave in the y direction, and the bending in the z direction Based on the wave number k _Bz of the wave, the displacement wave amplitude conversion matrix Ψ ₁ , Ψ ₂ for each of the forward propagation wave component and the backward propagation wave component, and the phase change due to the propagation distance x for each of the forward wave and the backward wave, Transform matrix propagation matrix calculation means for calculating propagation matrices G ₁ (x) and G ₂ (x) representing internal force wave amplitude conversion matrices Φ ₁ and Φ ₂ for each of the forward propagation wave component and the backward propagation wave component; ,
For each of the beam elements, based on the base vector ei of the overall coordinate system of the beam element and the basis vector Ei of the element coordinate system, a coordinate transformation matrix C ⁽ a coordinate transformation matrix calculation means for calculating ⁱ⁾ ;
For each of the beam elements, incident direction calculation means for calculating an incident direction d ⁽ⁱ⁾ of the forward propagation wave of the beam element to the discontinuity;
For each frequency, the displacement wave amplitude transformation matrices Ψ ₁ , Ψ ₂ , the propagation matrices G ₁ (x), G ₂ (x), and the internal force wave amplitude transformation matrix Φ ₁ for each beam element. , Φ ₂ , the coordinate transformation matrix C ^(i), and the incident direction d ⁽ⁱ⁾ , the reflection coefficient r for each of the beam elements and the beam element coupled at the discontinuity portion Coefficient calculating means for calculating a transmission coefficient t for each of the pairs;
For each frequency, a scattering matrix representing the transmissivity of the discontinuity of the three-dimensional beam structure based on the reflection coefficient r for each of the beam elements and the transmission coefficient t for each of the beam element pairs. A scattering matrix calculating means for calculating S;
For each frequency, the beam elements are based on the propagation matrices G ₁ (x), G ₂ (x) for each of the beam elements and the waveguide length L ⁽ⁱ⁾ of each of the beam elements. Propagation matrix calculation means for calculating a propagation matrix D representing each phase information of
For each frequency, based on the wave amplitude vector a ₀ , the scattering matrix S, and the propagation matrix D, a wave amplitude vector a of each component is calculated, and a frequency response for obtaining a frequency response function for each component The simulation apparatus according to claim 2, further comprising function calculation means. A program for calculating a wave amplitude as a response to an input amplitude to a three-dimensional beam structure in which a plurality of beam elements are combined,
Computer
Axial longitudinal forward wave, x-direction longitudinal backward wave, torsional forward wave, torsional backward wave, y-direction bending forward wave, y-direction representing the input amplitude. Bending forward proximity wave, y-direction bending backward propagation wave, y-direction bending backward propagation wave, z-direction bending forward propagation wave, z-direction bending forward proximity wave, z-direction bending backward propagation wave, and z-direction A wave amplitude vector a ₀ composed of each component of the bending receding proximity wave, a scattering matrix S representing the transmissivity of the discontinuous portion of the three-dimensional beam structure, and a propagation matrix representing a phase change accompanying the propagation of each component A program for functioning as a calculation means for calculating a wave amplitude vector a of each component, which represents a wave amplitude as a response, based on D, according to the following equation.
Here, I is a unit matrix.