JPS61270636A - Vibration analysing system - Google Patents

Abstract

PURPOSE:To obtain values having high accuracy with not only displacement but stress as well by preparing such generalized coordinates where there are no missing parts for the external force terms and making calculation. CONSTITUTION:A mode expressing the deformation of the entire structure and a mode expressing the deformation of respective partial structures are used. The mode taking the influence of external force into consideration is added in addition to the mode of normal vibration as the mode expressing the deformation of the partial structures. Processings 1-7 are executed with, for example, respective elements and the hyper-element matrix is made and is outputted to a file A. The result is read as the element data by element matrix formation (coordinate transformation) 9, by which the calculation over the entire part is made. The parts of reference mode calculation 5 and eigenvalue calculation 12 are rewritten to the calculation of the generalized coordinates taking the external force terms into consideration. The construction of the program is not disturbed even if such generalized coordinates are incorporated into the conventional program. The values having high accuracy are thus obtd.

Description

[Detailed description of the invention]

[Field of Application of the Invention] The present invention relates to a system for analyzing the vibration response (i-analysis) of various types of equipment. BACKGROUND OF THE INVENTION When trying to predict the vibration characteristics of a structure by t-it calculation, there is a method of modeling a high-rise pill or a large ship using beam elements and performing vibration calculations. In this method, when modeling, it is necessary to have knowledge of the degree of contraction and experience with regard to structural vibration.Therefore, actual structures and machines are modeled using three-dimensional shell elements and solid elements using the finite element method. A method is being adopted in which the shape of the VC is modeled as accurately as possible and all computer simulations are performed.
Since the equations to be solved are large in scale, they may be difficult to solve even with today's large computers. For this,
A method called substructure mode synthesis method (for example,
Published by Science and Technology Publishing, September 4th, K, J-Bathe
, written by E-L-Wilson, translated by Fumio Kikuchi, Numerical Calculations Using the Finite Element Method, disclosed on pages 456 to 458) can compensate for the lack of computer capacity. However, with this method, although the natural frequency and displacement response can be determined with relatively high precision, there are cases where the vibration stress due to forced force cannot be determined with high precision. [Object of the Invention] An object of the present invention is to provide a complete vibration analysis system using a partial structural mode synthesis method that accurately predicts vibration stress for large or complex structures. [Summary of the Invention J The feature of the present invention is that in the substructure mode synthesis method, in addition to the constraint mode and reference mode used to express the deformation of the entire system, the deformation mode caused by the action of external force is also considered. , to improve accuracy. [Embodiments of the Invention] Hereinafter, embodiments of the present invention will be described with reference to all drawings. It is a diagram showing generalized coordinates used in comparison with a conventional method. Prior to these detailed explanations, let us consider the substructure mode synthesis method and external force term used in the present invention, and explain the generalized locus 4B. Many methods have been proposed for calculating vibrations, but the present invention is based on Craig-Bampton (1).
It is based on the method. This method will be explained below. When modeled using finite elements as shown in Figure 3 of the continuum, the equation of motion is. LM) (10[K]in= (fl・・・・・・・・・
・・・・・・・・・・・・・・・(1) Here, (Ml is the mass matrix, [K] is the stiffness matrix, (l is the acceleration vector, (ul is the displacement vector, (f
l is an external force vector. The relevant term is omitted. (1
), introduce -film displacement (Q) to the equation, lJ=[F)(QJ・・・・・・・・・・・・・・・・・・・
・・・・・・・・・・・・・・・・・・ Song・Song (2
) and multiplying by [F]Ti from the left, we get [
r]T[MlI:F)(4j +(F)T[K][F)
(q)−[γ)”(fl・・・・・・・・・・・・・・・
・(8) Furthermore, CM']=CF] "[Ml(F)...Song...
・Concave curve... f4) [K” ]=(!/']T[K
][F]...Song...Song...Listening (5)
irl = (+/')T(f)・・・・・・・・・
・・Song・・Beat・Song・・・・・・・・・・If we set (6), the formula (8) becomes [M″ month old + [K’) (q) −(r) ・・・・・・・・
I can swear (7). here,(%')
is a square matrix considering the transformation money that has a one-to-one correspondence with respect to ('') and (qI.
'e is determined, and the solution (u) to be determined can be obtained by converting the variables in equation (2) back to the original state. (2) Formula is just a variable conversion, and moreover, it is a one-to-one conversion, and from the left [%I']'
Multiplying r means multiplying the original simultaneous equations by coefficients and adding them together to create another set of simultaneous equations.Instead of solving equation (1),
Even if you solve equation ('r) and put it back together using equation (2), you will get exactly the same solution. Therefore, the question is what should be used as [F] to be advantageous in terms of calculation accuracy and efficiency. The nodes in the finite element model shown in Figure 3 are divided into two types. One is the point indicated by a mark, and is called the nt boundary point. The other point is the point indicated by a * mark, and is called an internal point. The entire model is delimited by boundary points, and in this example is divided into three parts. As [v], a shape as shown in FIG. 4 is used. [ΦG] is the constraint mode, which is the deformation mode when all degrees of freedom of the boundary point other than that degree of freedom are constrained for each degree of freedom of the boundary point, and the forced displacement of the unit site is considered in that degree of freedom. , there are as many degrees of freedom as the boundary points. [ψN]
is the standard mode, which is the natural vibration mode when all boundary points are constrained, and exists as many as the degrees of freedom of the internal points. [F] is a square matrix. In the overall model shown in Figure 3, if the boundary points and interior points are grouped together as shown in ■ to ■, and the corresponding constraint modes and reference modes are arranged together, then the matrix will be created as shown in Figure 4. A submatrix with all components zero is created. Next, if we divide the equation into an equation for the boundary point and an equation for the interior point, and express it by adding a subscript B to the former and a ■ to the latter, we get (1)
The equation of motion in equation (2) is as follows.The variable conversion in equation (2) is as follows. uB on the right side should be expressed as qB, but since it has the same value as 1+B on the left side, we used the same symbol sound. qN is the generalized displacement corresponding to the reference mode. EV), but it is clear from the definitions of the constraint mode and the reference mode that the upper half is ■B, 0. However, ■'' is a unit matrix. [M''] in equation (4) is CM'l = (FlTI:M] u'3...
・・・・・・・・・・・・・・・・・・・・・・・・
・[K″] in αυ (5) formula is [K′]=[F)T[K] (F) ・・・・・・・・・・・・・・・・・・・・・...
...03. However, here the restraint mode [
ΦG] is (K"J[Φo" ten (KIB) - [0]...
・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・. The constraint mode is Equation 01, which is calculated by giving (K'J and (KlBJ).The generalized load vector (qJ of Equation (6) is (VJ = CF)T(f). The above results are used to illustrate the form of the matrix of the equation of motion in equation (7).Since the reference mode is the natural vibration mode when the boundary point is fixed, from the orthogonality of the natural mode, [ΦN )TCM"3[, ΦN] and [ΦN]T[K")[ΦN] are diagonal matrices.Furthermore, normalize with [ΦN3 k CM") and [ΦN]T[K")[ΦN]
If Φs3T(M")CΦN) is made into a unit matrix, the diagonal components of [Φs]"(KXI][ΦN] will be the square of the natural angular frequency corresponding to each standard mode. In Fig. 5, the parts where each component of IJ socks does not become zero are indicated by hatching. Rigidity Ma) Regarding the IJ socks, the achievement part of the internal point equation and the boundary point equation is zero, as found using equation (b), but for the mass matrix, as found using equation 09, it becomes zero. It is not. If all the equations are considered using this formula, it is the same as solving the original formula (1), so the results of both will match, but the efficiency of calculation will not improve. So, Craig
-Baml)tOn's method reduces the degree of freedom r by leaving only the low-order standard modes and omitting the high-order standard modes. Regarding the stiffness matrix, the equations corresponding to higher-order normal modes shown in FIG. 5 are not coupled with other equations. On the other hand, regarding the mass matrix, it is thought that the influence of the quality term is small in higher-order modes, so even if the part where the higher-order modes are coupled with the constraint mode is ignored, the influence is small. However, regarding the load term, the part related to the higher-order normal mode of Φ/(I in equation 01 is omitted, but since this part is not zero, when trying to find the forced vibration response, the modal The same problem as when high-order modes are omitted in the analysis remains.The purpose of the present invention is to solve this problem.Next, generalized coordinates that take external force terms into consideration will be explained.Conventional In order to improve the calculation accuracy of the response to forced vibration in the modal analysis of
, C'/r3, [Fb] color structure. [ψt] is a matrix whose column vectors are p low-order 1ml vectors. CvIr) and [V,] are
It is a trick that is set to zero and that the stiffness term is not achieved. [%'r) consists of a number of column vectors equal to the number of load vectors LK. to the mouth] consists of %(np-t) column vectors. Here, n is the number e of degrees of freedom of the system, that is, the number of equations. Defined as such, [Φt] is (v
/r ) and [Fh ] are no longer coupled. Actual NF
In the screen, there is no need to calculate 〔re''. FIG. 7 shows a method for calculating general 1H coordinates as described above. It has been mentioned that when calculating forced vibration using the substructure mode synthesis method mentioned above, there are problems similar to those of conventional modal analysis. Regarding this point, an explanation will be added using FIG. 2. Figure 2 is part 11 of #1 in Figure 3.
It is a diagram showing only the parts related to one structure. In the conventional method, even though the generalized load yhN for the higher-order reference mode is not zero, the higher-order reference mode is omitted, so the calculated value of stress especially when a load is applied to the inside of the super element is There is a fear that this will be inaccurate. In contrast to this 7'15, in the present invention, in addition to the lower-order normal mode (ΦNl), instead of the sieve-order normal mode, generalized coordinates consisting of [ψNr] and CFNh) are set to 1-0. stipulate. This is a generalized coordinate such as -t shown in Figure 2, and in the dt calculation at Fugiwa, [
(1"rib) is omitted. By using this general ratio image, all the drawbacks of Kamime's conventional method can be overcome. Next, what are r, WN, J and [FNh)? Let me explain in detail.The element loading vector of the superelement we are considering now is tr.
Power load vector gold (Fe+) (+ = 1 +
2+..., 1). For this 1FeiJ, all boundary points are fixed and a static analysis is performed. That is, (K”) l'I J-(Fei J...
・・・・・・・・・・・・・・・・・・・・・ CJQ This is solved and obtained (ujl(i-1,2+・・・・・・
, 1) Total [ΦN4. ] and [K”).Furthermore, the t pieces are made orthogonal to each other with respect to [K11), and C
The matrix of vector gold sequence vector obtained by normalizing with M"] is C%'N").Therefore, (
FN,] is CFN, )"CK"]CΦNz)-(01...
・・・・・・・・・・・・・・・・・・・・・・・・QQ(
Fsr)T[KrI] (Fs,) −[ΩNr2]・
・・・・・・・・・・・・・・・== 071? be satisfied. [ΩN,′] is a diagonal matrix. [ΦNtI is the eigenvector when the boundary point gold is fixed [Fs[) is [Φ
This is because [K11] is orthogonal to Nt). ('l'Hr 3 "CM") (ΦNz) =
[01・・・・・・・・・・・・・・・・・・・・・
・Q→(!l: lx le+y ('N,) ノL column hectors do not necessarily have to be orthogonal to (MII), but the method in Figure 7 makes them orthogonal. ['J'Nh ) K Id(, [K") lc is orthogonal to [ΦNtII[v/N,"], and they are also mutually orthogonal to (Kllll). All vectors can be prepared. , (FNh) is (FNb)te (K'summer) [ΦNt) −
[01””””””””””””” (Fsh) ding (
K")C'J'Nr] -[0)...
・・・・・・・・・・・・・・・) [WNb)”
[K”] [PNi) = (ΩN♂)・・・・・・
・・・・・・・・・・・・・・・(c) Satisfy. Here, [ΩN, 2] is a diagonal matrix. (For FN- obtained as above. ('sb)T(Fe+ J = (0) (i=1
+ 2 + −1z>・ea. Therefore, [rh] in Figure 2(b) is

It becomes [0]. (For MlJ, (’i’Nb 3T[MI”] [ΦNt]-

[0]
・・・・・・・・・・・・・・・・・・・・・ @ becomes [FshlT(M11][vIsrl-[0]...
・・・・・・・・・・・・・・・・・・(to). As described above, the coordinate transformation matrix [ψ0ΦNt 'i'
Nr FNh ) exists. However, what actually needs to be calculated is Φ0. Up to ΦNt+'Nf, FN
There is no need to calculate h. The generalized coordinate calculation method described below can be carried out by using the conventional flow shown in FIG. 7 as is. In the first half, p eigenvectors (normative modes) are determined by eigenvalue analysis using the subspace method. Next, a matrix [Ya] is prepared in which (p+t) column vectors are the sum of the p obtained modes and the element weight vector lF'e+)(1=lt21"""1t)k of the QQ formula. And [K”] (Xa] = (Ya) song・1 song...
・・・・・・・・・ Solve the rumors and get [Xa] [Also use 1 gold [MI'], [KII] Reduce to all subspaces. Rigidity in subspace (ma) l) x [circle J
is (Ka) = [xa]TCKll) [Xa)”
``'' song --- song --- @, but considering the % (to) formula, [K, ) - [X, )T [Ya)... song... song...
・One song...Song...Can be calculated by song@. The quality in the subspace tMa) IJ socks Ma) is [Ma) = (Xa)T[M'') (Xa]
・・1 song・・・・・・song・・）c'Ra], [
M, ], the eigenvalue problem in partial quantities [Ka) [Qa ] = [Ma) [Qa] (O
J ] ・= Song = Song (Solve 4 to find [Qa]. Finally, [ΦNt %F,, ] is [ΦNI FNr ) = (XB ) CQB ]
It is calculated as hit, song, egg weight, song, etc. (to). Furthermore, the element load vector (Ge) in the original space is transformed by the element load vector [from Fe3, restraint mode, JltJ mo)' (-), and the element load vector (Ge) in the space is [Ge) = (ΦNt v
/NJ T(Fe)・・・・・・・・・・・・・・・
...c31), this value corresponds to the upper half of the right-hand side of the U◆ expression. The upper half is calculated separately and corresponds to the equal lateral point load of a C5 normal finite element for both entire units. Furthermore, the Sturm sequence method is used to check whether there are any missing low-order standard modes. The error norm as an eigenvector is calculated in the same way. Next, an example of a vibration analysis system using the partial structure mode synthesis method of the present invention will be explained with reference to a schematic flow diagram shown in FIG. All processes 1 to 7 are performed for each super-element to create the entire super-element matrix and output to file A. The results are read as element data in element matrix creation (coordinate transformation) 9, and HF calculation of the entire system is performed. In the present invention, the reference mode 1 calculation 5 and the eigenvalue calculation 12 are replaced by generalized coordinate calculation considering the external force term in FIG. 7. Another feature is that even if the generalized coordinates of the present invention are introduced into a conventional program, the structure of the program will not be disturbed in any way. The effects of the present invention will be shown for a portal frame. Figure 8 shows the specifications of the model. This portal Freno has a longitudinal elastic modulus if 2 x 101'N/ffII + density ρ of 5oo
. Is / -9 judgment l@product A 10 -'n? , the moment of inertia of area is 10-8 m4, and the total length of the frame is 1
Each superelement is composed of one to three superelements. Then, 31 nodes deformed in the x, y, and z directions, 87
It has a degree of freedom. Here, only the deformation within the yz song was considered. Regarding this model, as shown in Table 1, A
, B, C, D, and E. Generalized coordinates were calculated for three load vectors: a human force vector for ground motion acceleration in the y and two directions, and a concentrated load on the zero point. Table 2 shows the calculated values of generalized coordinates for calculation conditions B-E. Vector &, 1 to 5 are eigenvectors, and their eigenfrequency is almost the same under any of the calculation conditions of 13 to E. Error norm of solid vector is 10-3 or less under calculation condition B, calculation conditions C to E
In this case, it is sufficiently small at 10-4 or less. The error norm of the external force term is considerably large under calculation condition B in which the present invention is not adopted, but is sufficiently small under calculation conditions CE. However, under calculation condition l), the element load vector cannot be accurately expressed at the time of super-elementization. The third light is # of the transient response when a concentrated load is applied to the zero point.
'F is a calculation result. Calculation condition A is a case in which all modes are considered, and t'L is considered to be the correct answer. The upper row is the displacement (IT!=
1). Under calculation condition B, the error in displacement at the 0 point is only about 2 inches compared to calculation condition A, and the accuracy is good. The middle row is the bending moment (KNm). In calculation condition B, which does not adopt the present invention, the error is calculated under calculation condition A.
The accuracy is not good, with a maximum of 17%. Calculation condition 1 that does not apply to all of the present invention regarding the standard mode using hyperelements]
In this case, the error is 4% at maximum for calculation condition A. Calculation conditions C and E both have an error of 0 with respect to calculation condition A.
, 4 inches or less. The bottom row is the shear force (KN). Under calculation condition B, there is a maximum error of 49 inches compared to calculation condition A, and under calculation condition there is an error of 24 inches compared to calculation condition A. However, under calculation conditions C and E using the present invention, calculation 1 for condition A
The error is less than . From the above results, when no hyperelement is used, the accuracy is good under calculation condition C, which is adopted throughout the present invention, for the eigenvector. Furthermore, when using super-elements, the accuracy is good under calculation condition E, which employs the present invention, both for the reference mode when defining the super-elements and for the eigenvectors of the entire system. On the other hand, when the superelement metal is not used, the accuracy is poor under calculation condition B in which the present invention is not adopted for the eigenvectors. Furthermore, when hyperelements are used, even if the present invention is adopted for the eigenvectors of the entire system, if the present invention is not adopted for the reference mode, the accuracy will be poor as shown in the calculation conditions. However, the result is better than calculation condition B. [Effects of the Invention] According to the present invention, in the partial structure mode synthesis method, generalized coordinates with no missing parts can be prepared and calculated for the external force term, so not only displacement but also stress can be calculated with high precision. value can be obtained.

[Brief explanation of drawings]

Fig. 1 is a schematic flow diagram explaining an example of the vibration analysis system of the present invention, and Fig. 2 is a diagram illustrating the entire substructure mode synthesis method used in the present invention. Figure 4 is a diagram showing the square matrix IF+ used in the finite element model shown in Figure 3, Figure 5 is a diagram showing the shape of the matrix of the transformed equation of motion, and Figure 6 is a diagram showing the external force. Fig. 7 is a schematic flow diagram illustrating an example of a method for calculating generalized coordinates considering all external force terms, and Fig. 8 is a model of a calculation example to explain the present invention in detail. It is a diagram. 14...Natural vibration mode, 15...Mode considering the influence of external force. O 2 Figure 2 Ronnie = 5-22 ko ○ e (c) ■ ((c) [・(- - ・ sai ■ Circle talent δ prisoner vt

Claims (1)

[Claims] 1. In a vibration analysis system that divides the entire structure into several partial structures, determines the dynamic characteristics of each partial structure, and then synthesizes them to perform vibration analysis of the entire structure. A vibration analysis characterized in that a mode representing the deformation of the structure and a mode representing the deformation of each partial structure are used, and a mode considering the influence of external force is added in addition to the natural vibration mode as the mode representing the deformation of each partial structure. system. 2. The vibration analysis system according to claim 1, characterized in that, in addition to the natural vibration mode, a mode that takes into account the influence of external force is added as a mode representing the deformation of the entire structure.