JP2675648B2 - Vibration analysis method - Google Patents

Description

The present invention relates to a method for performing vibration analysis by combining experimental data with a finite element method.

<Prior art> In order to perform vibration analysis of a mechanical structure by the finite element method, the analysis target is modeled as a continuous body of an appropriate type of finite element, and data regarding its shape and material,
For example, the coordinate values of the nodes, the material constants, the continuous node numbers of the finite element method, etc. may be input to the electronic computer to perform a predetermined calculation. However, for complex, large-scale machine-built objects,
Direct application of the finite element method requires a large number of nodes and finite elements, so the degree of freedom of the characteristic matrix (mass matrix and stiffness matrix) becomes large, and the capacity of the computer is insufficient and the calculation time is too long. Has been pointed out.

Therefore, a mode synthesis method has been studied as a method for solving such a problem. The research results of the mode synthesis method have been reported extensively in recent years, but prior to the design of the present invention, an example thereof (Masaaki Okuma et al. “Vibration analysis by the segmented mode synthesis method”, The Japan Society of Mechanical Engineers, Volume 50, 450 Issue (Sho 59
-2)) will be described with reference to FIG.

First, the whole system is divided into several subsystems that are complex enough to be formulated by the finite element method. The displacement of the whole system is divided into the displacement of the joint region where the subsystems join each other and the displacement of the internal region (region excluding the joint region) of each subsystem. Then, we consider the entire coupled region as one subsystem, and call it the coupled subsystem. That is, in the calculation, this combined system is added to the number of divided subsystems. Then, a mass matrix and a stiffness matrix expressing this coupled subsystem are created from the mass matrix and the stiffness matrix of the subsystem, respectively. In other words, the mass matrix and stiffness matrix of all subsystems are reduced by Guyan's static reduction to the degrees of freedom of the joint region only, and they are created by superimposing them. The displacement of the entire coupling region is expressed by the primary coupling of the eigenmodes obtained from the mass matrix and stiffness matrix of the coupling subsystem thus obtained. The displacement of the internal region of each subsystem is represented by the sum of the displacement caused by the displacement of the coupling region and the primary coupling of the eigenmodes of only the internal region with the coupling region fixed. Using these eigenmodes as generalized coordinates, the dynamic characteristics of the whole system are expressed and the vibration behavior is analyzed. Since the calculation process for the bond region is automatically performed, division into subsystems can be freely performed, and a general-purpose analysis program can be easily created. The greatest feature of this method is that the degree of freedom of the statutory formula for the whole system can be made much smaller than that of conventional analysis methods such as the finite element method.

<Problems to be Solved by the Invention> However, in the conventional method, the internal node degree of freedom {u ₀ } of the partial structure (division system) is approximated by the following equation.

{U _0} = [Φ] eigenmodes vector of _{{u g} + [G]} {u t} ... (1) where [[Phi] partial structure,
[G] It represents a deformation mode of internal degrees of freedom obtained by forcibly displacing the boundary node for each degree of freedom, and is called a constraint mode vector. {U _g } is the mode coordinate,
{U _t } is the boundary point degree of freedom.

The internal node degrees of freedom {u ₀ } and the boundary node degrees of freedom are translational degrees of freedom {u _x , u _y , u _z } and rotational degrees of freedom {θ _x , θ _y , at each node.
θ _z } is included, and in order to improve the conversion accuracy of the equation (1), the experimental mode vector including the rotational displacement mode is also included for the component of the eigenvalue mode vector [Φ] input as the experimental mode vector. Need to use.

Therefore, the conventional experimental mode synthesis method has the following problems.

Using the experimental data, the internal node degrees of freedom {u ₀ } (translation components u _x , u _y , u _z and rotation components θ _x , θ _y , θ _z ) of the substructure are _defined as boundary nodes using the experimental data as shown in equation (1). In order to directly convert the degrees of freedom and the eigenmode coordinates, it is necessary to simultaneously measure and input not only the translation component but also the rotation component as experimental data.

As a method of measuring the rotation component, for example, as shown in FIG. 7, two sensors 6 capable of measuring the translational component are arranged close to each other, and the outputs from these outputs u _z ¹ and u _z ^{2 are} rotated by a computer 15. A method for obtaining the component θ _x has also been proposed. However, in order to detect the rotation component with high accuracy by this method, it is necessary to provide the sensors 6 as close as possible, but there is a physical limit. Further, it is practically impossible to arrange a large number of two sensors.

For these reasons, the mode synthesis method using experimental data has not been put to practical use, although it is strongly recognized as a method that can efficiently perform structural analysis of complex structures. .

The present invention provides a method that does not require measurement of a rotational component mode, which is the largest bottleneck of the experimental mode synthesis method, and that can obtain a practically accurate mode synthesis solution using a measurement mode vector of only a translational component. The purpose is.

<Means for Solving the Problems> First, in the present invention, the entire system (overall structure) is divided into several subsystems (partial structures), and the divided subsystems are provided by means such as a finite element method (FEM). , The discretized rigid matrix,
Create a mass matrix.

Next, the displacement of the subsystem is expressed by the coupling boundary degrees of freedom and the primary coupling of the measurement direction component of the measurement point at which the natural vibration mode is measured by an experiment, and the stiffness matrix and mass matrix are statically coupled. Is a reduction matrix reduced to the measurement direction degree of freedom component.

Then, the eigenmode vector obtained by the experiment represents the displacement in the measurement direction of the measurement point as the primary combination of the mode coordinates and the joint boundary degrees of freedom, and the reduction matrix is further represented by the joint boundary degrees of freedom and the mode coordinate components. Reduce to shape.

Furthermore, the dynamic characteristics of the subsystem represented only by the joint boundary degrees of freedom and the mode coordinates are combined with other subsystems by considering the continuity of the joint boundary.

The biggest difference between the present invention and the prior art is that the internal nodal degrees of freedom of the subsystem are once reduced to the measurement direction component and the boundary degrees of freedom (first-stage reduction), and then this reduction matrix and experiment are performed. This is to combine the data into a matrix reduced to the boundary degrees of freedom and the eigenmode coordinates (second reduction).

In the first stage reduction, the rotation component of the measurement point is reduced to the measurement direction component as the internal nodal degree of freedom, and the reduction matrix by this first stage reduction becomes a matrix that can consider the rotation component using the translation component of the measurement point. <Action> The equation of motion of the system is generally expressed as follows. However, attenuation was ignored for simplicity.

[M _gg ] { _g } + [K _gg ] [u _g ] = {F _g } (2) where [u _g ] is the displacement vector of the _subsystem and [M _gg ] is the mass (g × g) Matrix, [K _gg ] is a stiffness (g × g) matrix, and {F _g } is an external force vector.

In general, the number of displacement coordinates is much larger than the degree of freedom u _e measured in the experiment of subsystems. Therefore, the equation (2) is divided as follows.

Here, {u _e } is the sum ({u _e } = {u _t , u _m } ^T ) of the joint part degree of freedom {u _t } and the measurement direction degree of freedom {u _m } of the measurement point, and {u _e } u ₀ } is the degree of freedom of the other subsystems. The following conversion is performed as an approximate expression of {u ₀ }.

{U ₀ } = [G ₁ ] {u _e } (4) Here, [G ₁ ] is a solution when the inertial term and the external force are zero in the equation (3), and is represented by the following equation.

[G ₁ ] =-[K _oo ] ^-1 [K _oe ] (5) Further, from the equation (4), the displacement of the subsystem is expressed as follows.

Substituting equation (6) into equation (2) or equation (3) of the subsystem, the following equation of motion for only {u _e } is obtained.

[ _Mee ^* ] { _e ^* } + [ _Kee ^* ] = { _Fe ^* } (7) Here, the following relationship is recognized.

Equation (7) is a reduced equation of motion that expresses the equation of motion of the subsystem by only the joint direction nodes and the measurement direction degrees of freedom components of the measurement points.

Expression (7) can be divided as follows with respect to the coupling degree of freedom {u _t } and the measurement direction degree of freedom component {u _m } of the measurement point.

The following equation can be considered as an approximate expression for u _m .

{U _m } = [φ _mq ] {u _q } + [G ₂ ] {u _t } (10) In the expression (10), [G ₂ ] is a static expression similar to the expressions (4) and (5). A transformation matrix, which is represented by the following equation.

[G ₂ ] =-[K _mm ] ^-1 [K _me ] ... (11) Further, [φ _mq ] is the experimental mode of the division system, and {u _q } represents the mode coordinates. If {u _e } is expressed using equation (10),
It looks like this:

However, the following equation was used as a condition.

{U _a } = {u _q , u _t } ^T (13) Substituting Eq. (12) into the reduced motion equation (7) of the subsystem, we obtain the following equation of motion only for {u _a }.

[M _aa ^* ] { _a } + [K _aa ] {u _a } = {F _a ^* } (15) However, the following equation was used as a condition.

[M _aa ^* ] = [T ₂ ] ^T [M _ee ^* ] [T ₂ ]… (16) [K _aa ^* ] = [T ₂ ] ^T [K _ee ^* ] [T ₂ ]… (17) {F _a ^* } = [T ₂ ] ^T {F _e } (18) Equation (15) is a motion equation expressed using only the coupling boundary degrees of freedom {u _t } and mode coordinates {u _q }, and others. With respect to the coupling with the subsystem, it is possible to easily construct a motion equation coupled with another subsystem by considering the displacement matching condition and the force balance condition at the coupling boundary.

<Examples> Hereinafter, the present invention will be described in detail based on Examples.

FIG. 1 shows an explanatory diagram of an analysis procedure according to an embodiment of the present invention. This analysis procedure applies the present invention to the out-of-plane vibration problem of the cantilevered rectangular plate 1 whose one end is completely restrained as shown in FIG. That is, the supporting rectangular plate 1 is the whole system A, and the rectangular plate 1 is divided into two partial structures 2 and 3 in the lateral direction at the joint boundary 24, and the divided system shown in FIGS. 3 and 4 respectively. A ₁ and A ₂ . As shown in FIG. 3, for the subsystem A ₁ which is the partial structure 2 on the base end side,
A finite element method is used to model with a large number of nodes 4 and finite elements 5, and data such as the coordinate values of the nodes and material constants are obtained as before, and the central processing circuit (hereinafter referred to as CPU) 1 from the terminal device 1
If you enter in 1, the stiffness matrix and the mass matrix will be calculated by the specified program as before. The CPU 11 has a memory that stores necessary data and programs in advance.
14 connected. The same applies to the subsystem A _2, which is the partial structure 3 on the tip side shown in FIG. Further, with respect to the subsystem A ₂ which is the partial structure 3, a vibration experiment is performed by the vibration exciter 7 while being constrained by the coupling boundary 24. In the vibration experiment, acceleration mode sensors 6 provided at 6 points of the substructure 3 measure the vibration mode vector in the u _z direction, and as shown in FIG. Together with an amplifier (hereinafter referred to as AMP) 9, a fast Fourier transformer (hereinafter referred to as F
It is performed by inputting to the CPU 11 via the (ET) 10 and obtaining data such as vibration eigenvalue and vibration data.

Next, the rigidity matrix and the mass matrix of the substructure 3 are converted into the substructure 2
After being reduced to the measurement direction degree of freedom at the point where the coupling boundary 24 with and the acceleration measuring sensor 6 are provided, the vibration eigenvalue and vibration mode data of the substructure 3 are input and the measurement direction degree of freedom is converted to the eigenmode coordinates. to shrink.

That is, the structural characteristics such as the stiffness matrix and mass matrix of the subsystem A ₂ are
6-degree-of-freedom components (3 translations, 3 rotations) of nodes provided arbitrarily in the subsystem and at the connection boundary 24 are discretized and expressed. Since the number of nodes provided in the subsystem is usually much larger than the number of points that can be measured during the experiment, in order to make this correspond to the measurement point, the degree of freedom of the bonding boundary is set in a statically equivalent form. And convert to the form expressed by the degree of freedom in the measurement direction of the measurement point. After that, by using the natural vibration mode vector to express the displacement of the measurement direction DOF of the measurement point by the primary combination of this mode coordinate and the degree of freedom of the connection boundary, the previously obtained connection boundary DOF and measurement point The structural characteristics of the sentence system expressed in the degree of freedom in the measurement direction are expressed only by the mode coordinate component and the joint boundary degree of freedom component. The connection with other subsystems is made by considering the continuous condition at the connection boundary, that is, the displacement matching condition and the force balance condition,
Finally, the equation of motion of the whole system expressed only by the coupled boundary degrees of freedom and the modal coordinate components of each subsystem is obtained.
By solving this, it becomes possible to grasp the dynamic behavior when the subsystems are connected.

After all, the most important point in the present invention is to reduce the mass matrix, stiffness matrix, etc. created for the subsystem to the measurement direction component (and bond boundary degrees of freedom) of the measurement point. The reduction is performed accurately according to the measurement direction.

The result of the equation of motion of the whole system A thus analyzed is displayed on the display device 13.

Table 1 shows the results of performing the eigenvalue analysis on the whole system A in the above-mentioned example. In Table 1, the calculation result of the natural frequency is shown together with the exact solution and the result by the conventional method. As is clear from the results shown in Table-1, the conventional method has an error of 100% or more, which is not practical. In contrast to the problem, in the present embodiment, an error of up to 10% was generated in the fourth mode, which was very practical as an experimental mode synthesis method for a rigidly coupled partial structure. However, as the translation mode, the mode up to April of subsystem A ₁ was used.

<Effects of the Invention> As described above in detail with reference to the embodiments, the present invention has the following effects.

(1) Coupling analysis of substructures becomes possible using translational mode vectors that can be easily measured in vibration experiments, etc., and the range of application to practical structures is expanded.

(2) It is applicable and has a very wide range of application even under the connection condition that the connection between the partial structures transmits the moment.

(3) Since the dynamic characteristics of the whole system can be expressed only by the coupling degrees of freedom of the partial structure and the modal coordinate system obtained in the experiment, the degree of freedom is greatly reduced, and the dynamic characteristics of complex structures can be examined at the PC level. Can be implemented in.

BRIEF DESCRIPTION OF THE DRAWINGS FIG. 1 is an analysis procedure diagram according to an embodiment of the present invention, FIG. 2 is a schematic diagram of an entire machine structure in an embodiment of the present invention before division, and FIG. finite analysis model diagram of the partial system a ₁ in an embodiment of FIG. 4 is a perspective view illustrating a vibration test conditions of partial system a ₂ in one embodiment of the present invention, Fig. 5 an embodiment of the present invention FIG. 6 is an explanatory diagram showing a vibration test of the subsystem A _{2 in} the example, FIG. 6 is an analysis procedure diagram showing a conventional mode combining method, and FIG. 7 is an explanatory diagram showing an example of a rotational component measuring method by the conventional mode combining method. Is. In the drawing, 1 is a cantilevered rectangular plate (whole system A), 2 and 3 are substructures (division systems A ₁ and A ₂ ), 4 are nodes of the finite element method, 5 is a finite element, and 6 is acceleration (or displacement). ) Sensor, 7 is an exciter, 8 is a load cell, 9 is an AMP, 10 is an FFT, 11 is a CPU, 12 is a terminal, 13 is a display, 14 is a memory, 15 is a calculator, and 24 is a coupling boundary. .

Claims (1)

(57) [Claims] 1. The whole structure is divided into a plurality of partial structures, the dynamic characteristics of these partial structures are formulated, and then recombined,
In the vibration analysis method adapted to obtain the vibration behavior of the entire structure, a step of reducing the composition and mass matrix of the partial structures into a joint boundary degree of freedom and a measurement direction degree of freedom component of a measurement point, and a reduction in the same step. A method for vibration analysis, further comprising the step of reducing an object to a combined boundary degree of freedom and a mode coordinate component.