JPS6267675A - Vibration analyzing device - Google Patents

Abstract

PURPOSE:To find the characteristic value of a motion equation, etc., in a short period by repeating Rayleigh quotient calculation until a Rayleigh quotient converges and approximating the characteristic value of the given motion equation with the converging Rayleigh quotient. CONSTITUTION:A vibration parameter is inputted to a parameter input means 1. Further, an initial angular frequency is inputted to an initial angular frequency setting means 2. One of the angular frequency set by the means 2, a characteristic angular frequency obtained by a characteristic value calculating means 5, and an angular frequency obtained by a Rayleigh quotient calculating means 7 is selected by an angular frequency selecting means 3 and sent to a coefficient selecting means 4. The means 4 obtains signals from the means 1 and 2 to calculate respective coefficient values. Those coefficient values are sent to the means 5 to calculate a characteristic angular frequency, which is stored in a characteristic vector storage means 6. Then this characteristic angular frequency is sent to the means 4 through the means 3 and the means 4 calculates coefficients newly and sends those values to a Rayleigh quotient calculating means 7. The means 7 calculates the Rayleigh quotient. The Rayleigh quotient is calculated repeatedly until it converges and outputted after converging.

Description

DETAILED DESCRIPTION OF THE INVENTION a. Field of Industrial Application The present invention relates to a vibration analysis device. Vibrations in mechanical equipment have a significant impact on the performance of the equipment. Therefore, in the design stage of a machine, it is often necessary to set the natural frequency of the machine within a predetermined angular frequency range or outside the predetermined angular frequency range. For this reason, it is preferable to use a vibration analysis device that analyzes the natural frequency of the machine, etc. at the design stage of the machine. By using the above-mentioned vibration analysis device, for example, in CAD (computer assisted design) design, the natural frequency of a mechanical device can be known in advance from the given dimension values, material constants of the material, etc. Failures caused by vibration can be avoided in advance. It can also be applied to CAM (computer assisted manufacturing).

b. Prior art A vibration analysis device according to the prior art includes: parameter manual means for inputting vibration parameters; initial angular frequency setting means for setting the initial value of the angular frequency; Coefficient calculation means for converting each coefficient of a homogeneous simultaneous linear equation, which is an equation of motion of vibration, into a constant coefficient from the angular frequency; a storage means for storing the output of the coefficient calculation means; and a constant calculated by the coefficient calculation means. an eigenvalue calculation means for calculating the natural angular frequency and eigenvector of the coefficient simultaneous equation 2; and eigenvalue calculation means for evaluating the natural angular frequency according to a predetermined convergence condition, and when the natural angular frequency has not converged, calculating the natural angular frequency as a given value of the coefficient calculation means. Some devices are equipped with a convergence determination/output means that sends the angular frequency as the angular frequency and outputs the eigenangular frequency and the eigenvector when the angular frequency has converged.

Vibration analysis in the vibration analysis device according to the conventional technology is
The process is performed as follows according to the flowchart shown in the figure.

1) An appropriate angular frequency ω is input to the initial angular frequency setting means.

ii) From the angular frequency ω manually input to the initial angular frequency setting means, the coefficient calculation means calculates K(ω
)1 matrix M−'K(ω), etc. are calculated. Furthermore, K(ω)
, M−'K(ω) is generally a matrix that depends on ω.

1) Calculate the eigenvalues of the constant coefficient homogeneous simultaneous linear equations having the coefficients calculated by the coefficient calculation means.

iv) The convergence determination/output means determines whether the eigenvalues calculated by the eigenvalue calculation means have converged;
If it has not converged, the eigenvalue is sent as a given angular frequency to the coefficient calculating means, where the coefficient is calculated anew, and based on that, the following process ii) is repeated.

■) If the eigenvalues are convergent, the eigenvalues.

and output the eigenvectors.

C1 Problems to be Solved by the Invention In the conventional vibration analysis device, the eigenvalue calculation is included in the loop for calculating the eigenangular frequency.
Eigenvalues are calculated each time the loop is repeated. Since the fixed value calculation means itself is a convergence calculation means including a loop, it takes an enormous amount of time to find the eigenvalues of a given equation of motion. Therefore, it has been difficult to perform CAD design using such a vibration analysis device. An object of the present invention is to provide a vibration analysis device that can calculate the eigenvalues of an equation of motion in a short period of time so as to meet such objectives.

Means for solving the d1 problem The above problem is solved by providing a Rayleigh quotient calculation means for calculating the Rayleigh quotient, repeating the Rayleigh quotient calculation until the Rayleigh quotient converges, and using the converged Rayleigh quotient to solve the given equation of motion. It was solved by approximating the eigenvalues. Specifically, a parameter input means for inputting vibration parameters manually, an initial angular frequency setting means for setting the initial value of the angular frequency, and a means for calculating the angular frequency and eigenvalue set by the initial angular frequency setting means. angular frequency selection means for selecting one of the obtained natural frequencies and the square root of the Rayleigh quotient; Based on the coefficient calculation means that calculates the value of each coefficient of the homogeneous simultaneous linear equations that are the equations of motion, and the coefficient values obtained by the coefficient calculation means, the eigenvalue equation accompanying the vibration equation of motion is solved, and the obtained an eigenvalue calculation means for sending the eigenvalue to the angular frequency selection means; an eigenvector storage means for storing the eigenvector obtained by the eigenvalue calculation means; and a Rayleigh quotient calculated based on each coefficient obtained by the coefficient calculation means and the eigenvector. and a Rayleigh quotient calculation means for sending the result to the angular frequency selection means; a convergence determining circuit for determining whether the Rayleigh quotient obtained by the Rayleigh quotient calculation means has converged; and when the Rayleigh quotient has converged. The problem was solved by a vibration analysis device equipped with an output means for outputting the Rayleigh quotient and/or the eigenvector.

In the 61 action finite element method, the vibration problem is formulated as shown in Equation 11.

(K-02M) δ= f -----
--- (11 where: Stiffness matrix (depends on ω) M: Mass matrix ω: Excitation angular frequency δ: Displacement vector f: External force vector Furthermore, in the stiffness matrix, the tffi matrix M is due to its physical symmetry. It is a symmetric matrix, i.e. its matrix elements are K
When r J+ M t a, J=KJ,, M
, J-K j i holds true.

In the absence of damping, the resonant angular frequency is equal to the free vibration angular frequency. In other words, the resonance angular frequency is (1) f-0
It is equivalent to the solution of the eigenvalue equation (2) when

(K−ω”M)δ=O−−+21 Equation (2) is transformed as follows.

Kδ−ω2Mδ −−−−−(31M
Multiply both sides of equation (3) by the inverse matrix M −1 of .

(M−'K)δ=ω2δ −−+41 When the matrices M and K are constant coefficients, equation (4) is a standard eigenvalue problem and can be solved by the QR method or the like.

Matrices M, K, especially if matrix K depends on the angular frequency ω,
In the present invention, the Rayleigh quotient is calculated in order to avoid calculating the eigenvalues many times.

The Rayleigh quotient is a quantity defined by the following equation from the mass matrix M to the stiffness matrix of the eigenvalue equation given by equation (4).

Here, 69 is an arbitrary vector, and δI and T are transposed matrices of δ2.

(4) Let the natural angular frequency and normalized eigenvector of 2 be ω, (i = 0.1.2・−), δ, respectively
, (i-O, 1, 2---), the following equation holds true.

(M-'K)δ, = ωl'δ, -----
−(61 has δ, −ω, ′Mδ, −・−(6a
) Note that ω is the j-th resonant angular frequency.

Let any vector δ2 be the eigenvector δ of equation (4). .

δ1. When expanded by δ2−'−'−, the expansion coefficient of
When Or a I+a2-.-, the following formula is obtained.

δ9=a0δ. +a, δ1+a2δ2+ −・−
(71δ2↑−a0δ.”+a, δlT10a2δ2+
-------(8)(7)2 Substituting (8)2 into equation (5) yields the following equation (the fourth output of the equation is omitted).

When the eigenvector δ2 is close to the j-th eigenvector δ, the other coefficients are very small compared to the coefficient a, so the following equation holds true.

(-> 2< 1 　　　　-(10)J Therefore, ωJ' can be approximated by the Rayleigh quotient R.

ωJ′″= R---(11) (11) Using the value of the resonance angular frequency ω, whose approximate value was obtained in 2, determine the value of each coefficient in the stiffness matrix again, and use that constant matrix to (9) By calculating the Rayleigh quotient R, the accuracy of the approximation of the resonance angular frequency ω is improved.

In the vibration analysis device according to the present invention, the Rayleigh quotient R is repeatedly calculated, and the convergence value thereof is taken as the resonance angular frequency. If an eigenvector for the resonance angular frequency is required, the eigenvector δ2. Solve the eigenvalue equation (4) in which the stiffness matrix K is made into a constant coefficient by δ27 or the above-mentioned convergent resonance angular frequency, and approximate with its eigenvector.

r. Embodiment FIG. 1 is a block diagram of a preferred embodiment of the vibration analysis device according to the present invention, which in this case is applied as an auxiliary device to a CAD design device.

Vibration parameters include the elastic modulus, Poisson's ratio, internal friction coefficient, mass density, and other physical properties that depend on the material of the member, as well as the two dimensions and positions of the member.

Number of divisions, number of restraint nodes, restraint method, number of nodes to which external force is applied,
The magnitude of external force, etc. is input from parameter input means 1 from CAD, etc.
is input. Note that the input to the parameter input means 1 is not limited to input from the CAD, and may also be performed for each parameter by manual operation from a keyboard, for example. Furthermore, it is also possible to manually enter the dimension values of each member in order to attach it to the CAM device and feed forward to the next step.

The initial angular frequency setting means 2 receives an empirically expected resonance angular frequency, a result of the present vibration analysis apparatus that has already been executed, or an initial angular frequency that is a total (arbitrary numerical value).

The angular frequency set by the initial angular frequency setting means, the natural angular frequency obtained by the eigenvalue calculation means, or the angular frequency obtained by the Rayleigh quotient calculation means - is selected by the angular frequency selection means 3 and corresponds to it. A signal is sent to the coefficient calculation means 4.

The coefficient calculating means 4 is provided with a means for calculating each coefficient of a simultaneous linear equation, which is an equation of motion of the system, from vibration parameters and an input angular frequency. The coefficient calculation means 4 obtains not only the signal from the angular frequency selection means 3 but also the signals corresponding to each parameter from the parameter input means 1, and calculates the value of each coefficient from these signals based on the above formula. do.

The value of each coefficient calculated by the coefficient calculation means 4 is sent to the eigenvalue calculation means 5, and a constant coefficient eigenvalue equation based on the coefficient value is calculated using a known method for solving an eigenvalue problem. The natural angular frequency obtained as a solution to the eigenvalue problem is sent to the angular frequency selection means 3, and the eigenvector is sent to the eigenvector storage means 6 and held there.

The natural angular frequency is sent to the coefficient calculation means 4 via the angular frequency selection means. The coefficient calculation means 4 calculates each coefficient anew from the above-mentioned natural angular frequency based on the above formula, and sends the values to the Rayleigh quotient calculation means 7.

The Rayleigh quotient calculation means 7 calculates the Rayleigh quotient from the values of the respective coefficients and the eigenvector obtained from the eigenvector storage means 6.

The Rayleigh quotient obtained by the Rayleigh quotient calculating means 7 is sent to the convergence determining means 8, and it is determined whether the Rayleigh quotient has converged based on a predetermined inequality. If it has not converged, the square root of the Rayleigh quotient is sent to the coefficient calculation means 4 via the angular frequency selection means 3. The coefficient calculation means 4 recalculates the value of each coefficient, and the Rayleigh quotient calculation means recalculates the Rayleigh quotient from that value and the eigenvector held in the eigenvector storage means 6.

Note that the eigenvector does not change in the above loop.

This is because the above loop does not include eigenvalue calculation, and the eigenvector changes only when eigenvalue calculation is performed.

When the Rayleigh quotient converges, the square root of the Rayleigh quotient is output from the output circuit 9 as the resonance angular frequency. If an eigenvector is required, the eigenvalue calculation means calculates an eigenvalue equation based on the eigenvector held in the eigenvector storage circuit 6 or the coefficient value newly calculated from the square root of the convergent Rayleigh quotient and the above parameters. The output is outputted via the output circuit 9.

The vibration analysis device described above can be applied not only as an auxiliary device for a CAD device, but also as an auxiliary device for a CAM device, which particularly requires quick calculations.

In this case, for example, data such as the dimensions of the member is sent from the CAM device to the parameter input means, and the resonance angular frequency is quickly calculated based on that data, and the result is used when feeding forward to the next process. .

Further, the vibration analysis device described above can be used as an auxiliary device for a simulator that simulates machine vibrations and the like in advance. In this case, the various data 4 may be input from, for example, a keyboard, and the output may be displayed on a CRT or a printer.

The output is not limited to the resonance angular frequency, the Rayleigh quotient (the square of the resonance angular frequency), or the displacement of each part during resonance. For example, the damping time constant, the stress of each part during resonance, the reaction force of each fulcrum during resonance, etc. can also be determined from the above-mentioned resonance angular frequency by a known method.

g1 Effects of the Invention When the vibration analysis device according to the present invention is used as an auxiliary device for CAD or the like, the dimensions of each member are determined with reference to the above output. For example, by temporarily determining the dimensions of a member that dislikes resonance, using this device to find the resonance angular frequency corresponding to the dimensions, and feeding that value to the CAD device, the optimum dimensions can be obtained. Furthermore, the resonance angular frequency can be calculated from various data in the CAM device, and the result can be fed forward to the next stage. These applications are made possible by the short time required to calculate the resonance angular frequency in this device.

Next, the convergence time according to the prior art and the convergence time according to the present invention will be compared by showing the time per convergence calculation when the number of iterations is five in the case of complex eigenvalues.

Table-1

[Brief explanation of drawings]

FIG. 1 is a block diagram of a vibration analysis apparatus according to the present invention, and FIG. 2 is a flowchart of a vibration analysis apparatus according to the prior art. 1... Parameter input means, 2... Initial angular frequency input means, 3... Angular frequency selection means, 4... Coefficient calculation means, 5... Eigenvalue calculation means, 6... Eigenvector storage means, 7... Rayleigh calculation means, 8... Convergence determining means, 9... Output means. Figure 1

Claims (1)

[Scope of Claims] 1. Parameter input means for inputting vibration parameters; Initial angular frequency setting means for setting an initial value of angular frequency; and the angular frequency and eigenvalue set by the initial angular frequency setting means. angular frequency selection means for selecting one of the natural angular frequency obtained by the calculation means and the square root of the Rayleigh quotient; and a parameter obtained from the parameter input means and a given angular frequency obtained from the angular frequency selection means. , a coefficient calculation means for calculating the value of each coefficient of the homogeneous simultaneous linear equations which is the equation of motion of vibration, and an eigenvalue equation accompanying the equation of motion of vibration based on the coefficient values obtained by the coefficient calculation means. , eigenvalue calculation means for sending the obtained eigenvalues to the angular frequency selection means, eigenvector storage means for storing the eigenvectors obtained by the eigenvalue calculation means, and based on each coefficient obtained by the coefficient calculation means and the eigenvectors. Rayleigh quotient calculation means for calculating a Rayleigh quotient and sending the result to the angular frequency selection means; a convergence determining circuit for determining whether the Rayleigh quotient obtained by the Rayleigh quotient calculation means has converged; and the Rayleigh quotient A vibration analysis device comprising output means for outputting the Rayleigh quotient and/or the eigenvector when converged. 2. The vibration analysis device according to claim 1, wherein the parameter input means obtains an input signal from a CAD device. 3. The vibration analysis device according to claim 1, wherein the parameter input means obtains an input signal from a CAM device. 4. The vibration analysis device according to claim 1, wherein the parameter input means obtains an input signal from a keyboard. 5. The vibration analysis device according to claim 1, wherein the output means sends an output to a CAD device. 6. The vibration analysis device according to claim 1, wherein the output means sends an output to a CAM device. 7. The vibration analysis device according to claim 1, wherein the output means sends an output to a CRT device.