JPH0883303A - Method for optimizing oscillation response - Google Patents

Abstract

PURPOSE: To quickly find out a shape having an optimum oscillation response by providing this method with a static analysis process for preparing a rigidity analyzing model from a result obtained by a dynamic analysis process and finding out a design variable in a specific oscillation mode. CONSTITUTION: Calculation for finding out a load vector on each mass point at the time of applying prescribed load to an analytical object from a prescribed direction from an m-th mode shape found out by an inherent value analyzing part 1A and a rigid matrix is executed by a rigidity analyzing part 1B. Then rigidity is analyzed by using the load vector obtained by the analyzing part 1B to find out a deformation vector on each mass point. The sensitivity analysis of the found deformation vector is executed by a sensitivity analyzing part 1C. In the sensitivity analysis, a design variable optimum to a specified condition is found out. Then the convergence of optimization is checked by the analyzing part 1C, and at the time of convergence, an inherent value is analyzed by a response analyzing part 1D by using the design variable and a response displacement vector corresponding to oscillation is found out and outputted.

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a vibration response optimization method, and more particularly to a vibration response optimization method suitable for vibration mode analysis of a structure.

[0002]

2. Description of the Related Art FIG. 7 shows a conventional example. In the conventional example of FIG. 7, input means 20 for inputting various design data, storage means 30 for temporarily storing design data and various calculation results from the input means 20, and design data stored in the storage means 30. Vibration analysis means 10 for performing vibration analysis based on
Output means 40 for outputting the analysis result of the vibration analysis means 10
It has and.

Here, the vibration analysis means 10 is the storage means 3
An eigenvalue analysis unit 10A that creates a FEM model based on the design data stored in 0 and obtains a mode shape matrix by eigenvalue analysis; and a response analysis unit 10B that obtains a response displacement vector from the mode shape obtained by the eigenvalue analysis unit 10A. And a sensitivity analysis unit 10C that obtains the sensitivity of the response displacement vector obtained by the response analysis unit 10B to the design variable and obtains the optimum design variable for the specified condition.

The operation of the above-mentioned conventional example is shown in FIGS.
This will be described with reference to the flowchart of. Here, it is assumed that various design data have already been stored in the storage unit 30 via the input unit 20.

(I). When the value of (σ [K] / σb) is analytically obtained (first operation):

[0006] First, the analysis target is divided into meshes, and F
An EM model is created (step 301 in FIG. 8).

That is, a mass matrix [M] representing the distribution of mass and a rigidity matrix [K] representing the distribution of rigidity.
To decide. The following equation (1) shows the mass matrix [M], and equation (2) shows the rigidity matrix [K].

[0008]

[Equation 1]

[0009]

[Equation 2]

[0010]. Next, the eigenvalue analysis unit 10A
The equation of motion shown in Expression (3) is solved for the EM model.

[0011]

(Equation 3)

Expression (4) shows the acceleration at each mass point. Also, {x} is the displacement of each mass point, which is expressed by the equation (5).

[0013]

[Equation 4]

[0014]

(Equation 5)

Specifically, the equation (1) is Laplace transformed,
Solve the eigen equation to obtain the eigenvalue matrix [λ].

Then, the mode shape matrix [Φ] is obtained from the eigenvalue matrix [λ] (step 302 in FIG. 8).

[0017] The response analysis unit 10B includes the eigenvalue analysis unit 1
Based on the mode shape matrix [Φ] obtained in 0A, the response calculation is performed using the equation (6) to obtain the response displacement vector {x} (step 303 in FIG. 8).

[0018]

(Equation 6)

Here, {f} is a load vector and km is m.
Th is the modal rigidity, ωm is the mth natural frequency, j is an imaginary unit, ζm is the mth damping ratio, and ω is the frequency.

Further, ωm and km are obtained from the equations (7) and (8).

[0021]

(Equation 7)

[0022]

[Equation 8]

However, mm is a mode mass, and is normally normalized to 1.

The part of the molecule in this equation (6) is called modal flexibility.

[0025]. The sensitivity analysis unit 10C includes the response analysis unit 10
Based on the response displacement vector {x} obtained in B, sensitivity analysis for the design variables of natural frequency and modal flexibility is performed (step 304 in FIG. 8).

-1. The sensitivity (σωm / σb) of the natural frequency to the design variable b is obtained by the equation (9).

[0027]

[Equation 9]

Here, {Φm} is the m-th mode shape.

-2. The sensitivity (σFm / σb) of the modal flexibility F with respect to the design variable b is obtained by the equation (10).

[0030]

[Equation 10]

.. Further, the sensitivity analysis unit 10C performs optimization calculation (step 305 in FIG. 8).

.. It is checked whether or not the optimization calculation has converged (step 306 in FIG. 8).
The result is output from the output means 40 and the process ends.

On the other hand, if the optimization calculation has not converged, the model is updated (step 307 in FIG. 8) and the process returns to the above process (step 301 in FIG. 8).

(II) When the value of (σ [K] / σb) cannot be analytically obtained:

.. First, the analysis target is divided into meshes, and F
An EM model is created (step 401 in FIG. 9).

That is, a mass matrix [M] representing the distribution of mass and a rigidity matrix [K] representing the distribution of rigidity.
To decide.

.. Next, the eigenvalue analysis unit 10A performs eigenvalue analysis on this model based on the equation of motion represented by Expression (3) to obtain an eigenvalue matrix [λ] and a mode shape matrix [Φ] (step in FIG. 9). 402).

.. The response analysis unit 10B includes the eigenvalue analysis unit 1
Based on the mode shape matrix [Φ] obtained in 0A, the response calculation is performed using the equation (6) to obtain the response displacement vector {x} (step 403 in FIG. 9).

.. The response analysis unit 10B checks whether all the design variables have been finely modified and the calculation has been completed (step 404 in FIG. 9).

When uncalculated design variables remain,
The design variable is changed (step 405 in FIG. 9) and the process returns to the above.

On the other hand, when the calculation is completed for all the design variables, the next sensitivity analysis process is performed.

.. The sensitivity analysis unit 10C performs sensitivity analysis on the design variables of natural frequency and modal flexibility based on the result of the difference calculation (step 4 in FIG. 9).
06).

-1. The sensitivity (Δωm / Δb) of the natural frequency to the design variable b is obtained by the equation (11).

[0044]

[Equation 11]

Here, {Φm} is the m-th mode shape.

-2. The sensitivity (ΔFm / σΔ) of the modal flexibility F with respect to the design variable b is obtained by the equation (12).

[0047]

[Equation 12]

.. The optimization calculation is performed (step 407 in FIG. 9).

.. The sensitivity analysis unit 10C checks whether the optimization calculation has converged (step 40 in FIG. 9).
8) If it has converged, the output means 40 outputs the result and ends.

On the other hand, if the optimization calculation has not converged, the model is updated (step 409 in FIG. 9) and the process returns to the above process (step 401 in FIG. 9).

[0051]

However, in the above-mentioned conventional example, since the eigenvalue analysis processing and the sensitivity analysis processing, which require a great deal of processing time, are often repeated, the optimum shape for a predetermined vibration mode can be obtained in a short time. There was an inconvenience that it could not be obtained in.

[0052]

SUMMARY OF THE INVENTION It is an object of the present invention to provide a vibration response optimizing method which can improve the disadvantages of the conventional example and can find a shape having an optimum vibration response in a short time especially in a structure. It is in.

[0053]

Therefore, according to the present invention, in a vibration response optimization method for obtaining a design variable b for reducing a vibration response amount, a dynamic analysis step of analyzing an eigenvalue of a specific vibration mode by vibration analysis is performed. And a static analysis step of obtaining a design variable b in a specific vibration mode by creating a stiffness analysis model from the results obtained by this dynamic analysis step. Further, the dynamic analysis step has a first step of obtaining the mode shape vector Φm in the vibration mode m to be analyzed, and the static analysis step loads the load from the rigidity matrix of the analysis target and the mode shape vector Φm. The second step of calculating a vector and the third step of calculating a design variable b that optimizes the vibration response level in the vibration mode m by calculating a design variable b that maximizes rigidity from the load vector It has a configuration that includes the process. This aims to achieve the above-mentioned object.

[0054]

[Operation] The load value in the static analysis is inversely calculated in the stiffness analysis section from the result of the mode shape by the eigenvalue analysis in the eigenvalue analysis section, and the response is based on the result of the optimization in the static analysis in the sensitivity analysis section. The analysis department solves the dynamic problem.

Therefore, the eigenvalue analysis section and the response analysis section, which require a great deal of processing time, need only be processed once.

[0056]

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT An embodiment of the present invention will be described below with reference to FIGS.

In the embodiment shown in FIG. 1, the input means 2 for inputting various design data, the storage means 3 for temporarily storing the design data and various calculation results from the input means 2, and the storage means 3 are stored. The vibration analysis means 1 performs a vibration analysis based on existing design data, and the output means 4 outputs the analysis result of the vibration analysis means 1.

The vibration analysis means 1 creates an FEM model based on the design data stored in the storage means 3 and obtains a mode shape matrix by eigenvalue analysis, an eigenvalue analysis section 1A, a stiffness matrix and eigenvalue analysis section 1A.
A stiffness analysis unit 1B that obtains the amount of deformation from the mode shape obtained in step 1, and a sensitivity analysis unit 1C that obtains the sensitivity of the amount of deformation obtained by the stiffness analysis unit 1B to the design variables and finds the optimum design variable for the specified conditions. And a response analysis unit 1D that obtains a response displacement vector with the design variable obtained by the sensitivity analysis unit 1C.

Next, the operation of this embodiment will be described with reference to the flowcharts of FIGS. Here, it is assumed that various design data have already been stored in the storage means 3 via the input means 2.

(I) When the value of (σ [K] / σb) is analytically obtained (first operation):

.. The eigenvalue analysis unit 1A analyzes the FE based on various design data stored in the storage unit 3.
An M model is created (step 101 in FIG. 2).

That is, as shown in FIG. 4, the analysis target is divided into meshes in the same manner as in the case of a normal FEM analysis, and a mass matrix [M] representing a mass distribution and a rigidity matrix [K] representing a rigidity distribution. To decide. FIG. 4 shows a simple cantilever beam as an example for the sake of clarity.

.. The eigenvalue analysis unit 1A performs eigenvalue analysis on this FEM model based on the equation of motion represented by the equation (3) to obtain an eigenvalue matrix [λ] and a mode shape matrix [Φ] (step 102 in FIG. 2). .

.. Further, the eigenvalue analysis unit 1A extracts the eigenvector of the mode m in question from the mode shape matrix [Φ] (step 103 in FIG. 2) and obtains the mth mode shape {Φm}.

.. The rigidity analysis unit 1B is the eigenvalue analysis unit 1A.
The load vector {f} at each mass point when a predetermined load is applied to the analysis target from the predetermined direction using the equation (13) from the m-th mode shape {Φm} and the stiffness matrix [K] obtained in Is calculated (step 10 in FIG. 2)
4). In the example of the cantilever beam shown in FIG. 5, the load is applied in the plate thickness direction (-Z direction) of the free tip.

[0066]

[Equation 13]

.. Further, the stiffness analysis unit 1B uses the obtained load vector {f} to perform the stiffness analysis according to the equation (14) (step 105 in FIG. 2) to obtain the deformation vector {x} at each mass point.

[0068]

[Equation 14]

.. The sensitivity analysis unit 1C performs a sensitivity analysis of the deformation vector {x} obtained by the rigidity analysis unit 1B (step 106 in FIG. 2). That is, the sensitivity (σx / σb) of the deformation amount x with respect to the design variable b is calculated using the equation (15).

[0070]

(Equation 15)

Here, the design variable b is a factor that is a target of structural change, and the plate thickness, beam cross-sectional shape, area, etc. are used. In the cantilever example shown in FIG. 5, the design variable b
Corresponds to the plate thickness.

.. Then, the sensitivity analysis unit 1C obtains the optimum design variable for the specified condition (step 10 in FIG. 2).
7).

In the cantilever example shown in FIG. 5, the plate thickness that minimizes the amount of deformation at the load point without increasing the weight is obtained as an optimization condition.

.. Further, the sensitivity analysis unit 1C checks whether the optimization calculation has converged (step 1 in FIG. 2).
08), if not converged, the model is updated (step 109 in FIG. 2) and the above processing (step 10 in FIG. 2) is performed.
Return to 5).

On the other hand, if the optimization calculation has converged, the response analysis unit 1D performs eigenvalue analysis using the design variables at that time (step 110 in FIG. 2), and the eigenvalue matrix [λ] and the mode shape. Find the matrix [Φ]. As shown in FIG. 6A, in the case of a cantilever, a wedge shape having a thick fixed end and a thin free tip is the optimum shape.

.. Then, the response analysis unit 1D performs a response analysis to the vibration using the equation (6), and obtains the response displacement vector {x} (step 111 in FIG. 2).

The result is output via the output means 4.

For example, as shown in FIG. 6 (b), the vibration level of the dynamic response at the tip of the cantilever beam optimized by the above method is about before the optimization (when the plate thickness is uniform). It is 65%.

As described above, by maximizing the static rigidity, the vibration level can be reduced for the vibration modes having the same deformation state.

(II) When the value of (σ [K] / σb) cannot be analytically obtained (second operation):

.. The eigenvalue analysis unit 1A analyzes the FE based on various design data stored in the storage unit 3.
An M model is created (step 201 in FIG. 3).

.. Next, the eigenvalue analysis unit 1A performs eigenvalue analysis on this model, and the eigenvalue matrix [λ]
And the mode shape matrix [Φ] are obtained (step 202 in FIG. 3).

.. The eigenvalue analysis unit 1A extracts the eigenvector of the mode m in question (step 20 in FIG. 3).
3).

.. The rigidity analysis unit 1B is the eigenvalue analysis unit 1A.
The load vector {f} is calculated from the m-th mode shape {Φm} and the stiffness matrix [K] obtained in step 1 using equation (7) (step 204 in FIG. 3).

.. Further, the rigidity analysis unit 1B performs FEM rigidity analysis (step 205 in FIG. 3) and obtains the deformation vector {x} using the equation (8).

.. The rigidity analysis unit 1B checks whether or not the calculation is completed by changing the smiles of all the design variables (step 206 in FIG. 3).

If there are uncalculated design variables,
The design variable is changed (step 207 in FIG. 3) and the process returns to the above process (step 205 in FIG. 3).

On the other hand, when the calculation is completed for all the design variables, the sensitivity analysis section 1C performs the sensitivity analysis based on the result of the difference calculation (step 208 in FIG. 3) and uses the equation (16). Sensitivity of design variable to deformation amount (Δx
/ Δb) is calculated.

[0089]

[Equation 16]

.. The sensitivity analysis unit 1C performs optimization calculation (step 209 in FIG. 3).

.. The sensitivity analysis unit 1C checks whether the optimization calculation has converged (step 21 in FIG. 3).
0), if not converged, the model is updated (step 211 in FIG. 3) and the above processing (step 20 in FIG. 3) is performed.
Return to 5).

On the other hand, if the optimization calculation has converged, the response analysis unit 1D performs eigenvalue analysis using the design variables at that time (step 212 in FIG. 3).

.. Further, the response analysis unit 1D performs a response analysis using the equation (6) to obtain a response displacement vector {x} (step 213 in FIG. 3).

The result is output via the output means 4.

Next, the processing time of the present invention will be compared with the conventional example.

Since the FEM model creation time is the same as that of the conventional example, it is not a comparison target.

The eigenvalue analysis (steps 102 and 110 in FIG. 2, step 302 in FIG. 8) is a process that takes much time compared to other calculations, and as an example, 300 man-hours are set here.

The response analysis process (step 303 in FIG. 8) in the conventional example is one man-hour, the sensitivity analysis process (step 304 in FIG. 8) is 150 man-hours, and the optimization process (step 3 in FIG. 8).
05) as one man-hour, convergence determination process (step 30 in FIG. 8).
If 6) is taken as 1 man-hour, the man-hour required for one loop is 4
53. Since all processes are repeated, assuming that the number of repetitions is n, the total man-hour is (300 + 1 + 150 +
1 + 1) × n, that is, (453 × n).

On the other hand, in the method according to the present invention, the eigenvector extraction process (step 103 in FIG. 2) is one man-hour, the load vector calculation process (step 104 in FIG. 2) is one man-hour, and the rigidity analysis process (step in FIG. 2). 105) is 100 times in 1/3 of the eigenvalue analysis processing, 50 steps in sensitivity analysis processing (step 106 in FIG. 2), 1 step in optimization calculation processing (step 107 in FIG. 2), convergence determination processing (in FIG. 2). Step 108) is one man-hour, response calculation process (Step 1 of FIG. 2)
If 11) is set as one man-hour, the rigidity analysis process and the convergence determination process are repeated. Therefore, if the number of repetitions is n, the total man-hour is (300 + 1 + 1 + (100 + 50 + 1 +).
1) × n + 300 + 1), that is, (603 + 152 ×)
n).

When n = 7, the conventional example is 31.
In contrast to 71 man-hours, the present invention is 1667 man-hours, which is about half the processing time.

Although the above description was calculated based on the first operation, the same applies to the second operation.

Next, the amount of memory used in the storage means 3 will be compared with the conventional example.

In order to obtain the optimization calculation result, in the conventional example, the stiffness matrix [K], the mass matrix [M], and the differential stiffness matrix (σ [K] / σb)
And a plurality of differential mass matrices ([sigma] [M] / [sigma] b) had to be stored respectively, but in the present invention, the stiffness matrix [K] and the differential stiffness matrix ([sigma] [K] / [sigma] b). Since it is only necessary to store a plurality of files, it is possible to significantly reduce the memory usage.

Because of this, the restrictions on the hardware for carrying out the present invention are greatly relaxed.

As described above, in the present invention, the dynamic problem is solved by the inverse calculation of the load value in the static analysis from the mode shape result of the eigenvalue analysis and the optimization result in the static analysis.

Therefore, if the optimization processing by the static analysis has already been performed, the optimization processing of the vibration response can be easily performed by making it consistent with the present invention.

[0107]

EFFECTS OF THE INVENTION Since the present invention is configured and functions as described above, according to this, the number of times of processing in the eigenvalue analysis section and the response analysis section, which requires a great deal of processing time, can be done once, which is Therefore, it is possible to provide an unprecedented excellent vibration response optimization method that can obtain an optimum shape for vibration in a short time.

[Brief description of drawings]

FIG. 1 is a block diagram showing an embodiment of the present invention.

FIG. 2 is a flow chart for explaining a first operation of the embodiment shown in FIG.

FIG. 3 is a flow chart for explaining a second operation of the embodiment shown in FIG.

FIG. 4 is an explanatory diagram for explaining an FEM model in a cantilever.

5 is an explanatory diagram for explaining rigidity analysis in the cantilever of FIG.

6 is an explanatory diagram for explaining a response analysis in the cantilever of FIG. 4, FIG. 6 (a) is a diagram showing an optimum shape calculated by the embodiment shown in FIG. 1, and FIG. ) Is a figure which shows the phase information of this analysis result.

FIG. 7 is a block diagram showing a conventional example.

FIG. 8 is a flow chart for explaining a first operation of the conventional example of FIG.

9 is a flow chart for explaining a second operation of the conventional example of FIG.

[Explanation of symbols]

 1 Vibration Analysis Means 1A Eigenvalue Analysis Section 1B Rigidity Analysis Section 1C Sensitivity Analysis Section 1D Response Analysis Section 2 Input Means 3 Storage Means 4 Output Means

Claims (2)

[Claims] 1. A vibration response optimization method for obtaining a design variable b for reducing a vibration response amount, wherein a dynamic analysis step of analyzing an eigenvalue of a specific vibration mode by vibration analysis and a dynamic analysis step obtained by this dynamic analysis step. And a static analysis step of obtaining the design variable b in the specific vibration mode by creating a stiffness analysis model from the result. 2. The dynamic analysis step includes a first step of obtaining a mode shape vector Φm in a vibration mode m to be analyzed, and the static analysis step includes a stiffness matrix to be analyzed and the mode. The second step of calculating the load vector from the shape vector Φm, and the design variable b that optimizes the vibration response level in the vibration mode m by calculating the design variable b that maximizes the rigidity from the load vector. 3. The vibration response optimization method according to claim 1, further comprising a third step of calculating