CN109948174B - Mass distribution method for calculating natural frequency of frame structure by centralized mass method - Google Patents
Mass distribution method for calculating natural frequency of frame structure by centralized mass method Download PDFInfo
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Abstract
A mass distribution method for calculating the natural frequency of a frame structure based on a concentrated mass method comprises the steps of measuring the horizontal displacement of cross beams of each layer of a three-layer frame structure under the action of static load, deriving a flexibility matrix according to a flexibility theory, further solving the natural frequency value of the frame structure in the common mass distribution method, and analyzing the relative error between the natural frequency value and each frequency of a free vibration test result; then, on the basis of the traditional distribution method, the equivalent mass of each layer is quantitatively changed respectively, and the influence of the change of the equivalent mass of each layer on the theoretical calculation natural frequency precision is researched; then, the optimal distribution proportion of the mass of each layer under the constraint of a certain error is accurately calculated by using a cyclic nesting theory; and finally, changing the initial mass of each cross beam, and performing test verification on the mass distribution proportion on a three-layer frame vibration test device. The method has the advantages of small calculated amount, simplicity, intuition, high calculation precision and the like, and improves the precision of calculating the natural frequency of the frame structure by the centralized mass method.
Description
Technical Field
The invention belongs to the calculation of the natural frequency and the vibration mode of a frame structure in the field of vibration, and particularly relates to a mass distribution method for calculating the natural frequency of the frame structure by adopting a centralized mass method.
Background
The frame structure is universally existed in production and life, such as factory buildings, high-rise buildings and the like, the layers of various frame structures are different, the quality is different, the analysis and the research of the natural frequency and the vibration mode of the frame structure are beneficial to avoiding the resonance phenomenon caused by external excitation in the production, and the frame structure has great practical significance for the production benefit and the service life of the device.
At present, the research on the natural frequency and the vibration mode of the frame structure is mostly focused on three aspects: calculation method, optimization design and device fabrication. The existing method for theoretically calculating the natural frequency of the frame structure mainly comprises a return matrix method, a transmission matrix method, a centralized quality method and the like, most of the methods adopt finite element methods such as the transmission matrix method and the like to calculate the natural frequency of the frame structure, and the method has large calculation amount and is troublesome. When the natural frequency of the frame structure is calculated by adopting a concentrated mass method, the mass of the side plates between the cross beams of each layer is ignored, the side plates are simply equivalent to springs with certain rigidity, the distribution research on the mass of each layer is less, the calculation error of the traditional mass distribution method is larger, and the influence of the related mass distribution on the precision of the theoretical calculation of the natural frequency of the frame structure is less.
Disclosure of Invention
The invention provides a reliable and effective frame structure quality distribution method, and aims to obtain the qualitative influence of the equivalent mass change of each layer by analyzing the influence of the equivalent mass change of each layer on the theoretical calculation natural frequency precision, accurately calculate the optimal distribution proportion of equivalent mass under certain error constraint by using a cyclic nesting theory, and improve the precision of the frame structure natural frequency calculated by a centralized mass method.
A mass distribution method for calculating the natural frequency of a frame structure by a centralized mass method is characterized in that the natural frequency of the frame structure in the traditional mass distribution method is calculated through a flexibility matrix, the relative error between the natural frequency of the frame structure theoretically calculated by a common mass distribution method and each frequency of a free vibration test result is analyzed, the mass of each layer is quantitatively changed on the basis of the traditional mass distribution method, the influence of the change of equivalent mass of each layer on the precision of the theoretically calculated natural frequency is analyzed, the distribution condition of the equivalent mass under the constraint of a certain error is further calculated by utilizing a cyclic nesting theory, and the optimal distribution proportion is obtained.
The mass distribution method for calculating the natural frequency of the frame structure by the centralized mass method is characterized by comprising the following steps of:
(1) respectively applying static loads with the same size to the shackle of each layer of cross beams of the three-layer frame structure, simultaneously measuring the horizontal displacement of each layer of cross beams by using a dial indicator, and solving a flexibility matrix according to a flexibility theory;
(2) the undamped free vibration differential equation based on the concentrated mass method is written in a row, the natural frequency of the lower frame structure in the common mass distribution method of 'equal distribution' and '2-4-4 distribution' is solved, and the relative error between the natural frequency and the free vibration test result is analyzed;
(3) respectively and quantitatively changing equivalent mass of each layer on the basis of traditional equal distribution, and carrying out qualitative research on the influence of equivalent mass change of each layer;
(4) accurately calculating the distribution condition of equivalent mass of each layer under the constraint of a certain error by using a cyclic nesting theory to obtain an optimal distribution proportion;
(5) and changing the mass of each layer of cross beams of the frame structure, solving the natural frequency of the frame structure by using the distribution proportion and the free vibration test respectively, and verifying the accuracy of the distribution proportion.
Compared with the existing distribution method, the invention has the following advantages:
1. the method for quantitatively changing the equivalent mass of each layer is adopted, so that the qualitative influence of the change of the equivalent mass of each layer on the accuracy of the natural frequency of the theoretical calculation frame structure can be conveniently and accurately researched, and the method has important guiding significance for locally adjusting the mass distribution of a certain layer in actual production so as to improve the accuracy of theoretical calculation;
2. the method adopts the cycle nesting theory to calculate the distribution of the equivalent mass of the frame structure, realizes the quick automatic calculation of the mass distribution under the constraint of certain errors, has the advantages of simplicity, quickness and the like compared with the finite element method, greatly improves the precision of calculating the natural frequency of the frame structure by the centralized mass method, can replace the finite element method under certain conditions, and is beneficial to quickly and accurately calculating the natural frequency of the frame structure in practice;
3. the method can be popularized to the frame structure quality distribution calculation of four layers and higher layers, and a simple and effective thought and method can be provided for the method.
Drawings
FIG. 1 is a schematic view of compliance measurement;
FIG. 2 is a three degree-of-freedom mechanical model;
FIG. 3 is a technical roadmap;
FIG. 4 is a graph showing the effect of equivalent mass changes of layers on frequency errors;
FIG. 5 is a flow chart for calculating equivalent mass of each layer based on a loop nesting theory;
FIG. 6 is a frequency error diagram of each order under the three-component method;
FIG. 7 is a graph showing the results of the free vibration test of example 1;
FIG. 8 is a graph showing the results of the free vibration test of example 2.
Detailed Description
The embodiments of the present invention will be described in detail below with reference to the accompanying drawings: the embodiment is implemented on the premise of the distribution scheme of the invention, and a detailed implementation mode and a specific operation process are given.
Example (b):
1. in this embodiment, a three-layer frame structure is taken as an example, static loads F equal to 15N are respectively applied to shackle positions of beams of each layer of a three-layer frame device, and horizontal displacements of the beams of each layer when each layer of the beams is stressed are loaded and measured simultaneously by using a dial indicator to form a displacement matrix X1The measurement schematic diagram is shown in fig. 1. In fig. 1: 1 represents a first-layer beam, 2 represents a second-layer beam, and 3 represents a third-layer beam; m represents the beam mass, and m is in kilograms. The matrix X1Is an n-order square matrix, in this example, a 3-order square matrix, which specifically includes:
displacement matrix X1The first row indicates the horizontal displacement distance x of the top cross member when the top cross member is subjected to a static load F of 15N11Horizontal displacement distance x of middle-layer beam12Horizontal displacement distance x of bottom layer beam13The units are mm;
the second row shows the horizontal displacement distance x of the top cross beam when the middle cross beam is subjected to the static load F-15N21Horizontal displacement distance x of middle-layer beam22Horizontal displacement distance x of bottom layer beam23The units are mm;
the third row shows the top cross member horizontally displaced by a distance x when the bottom cross member is subjected to a static load of 15N, F31Horizontal displacement distance x of middle-layer beam32Horizontal displacement distance x of bottom layer beam33The units are mm;
solving a flexibility matrix H ═ X according to a flexibility theory1/F。
2. Column writing is carried out on a three-degree-of-freedom undamped free vibration differential equation based on a concentrated mass method, the natural frequency of the lower frame structure in a common mass distribution method of equal distribution and 2-4-4 distribution is solved, and the natural frequency is compared and analyzed with a free vibration test result.
The solving steps of the natural frequency of the three-layer frame structure under the common mass distribution method of equal distribution and 2-4-4 distribution comprise:
(1) the three-layer frame structure is equivalent to a three-degree-of-freedom mechanical model shown in figure 2 according to the common mass distribution method of 'equal distribution' and '2-4-4 distribution', wherein k is shown in figure 21、k2、k3Respectively representing the spring stiffness of the side plate of the top layer cross beam, the spring stiffness of the side plate of the middle layer cross beam and the spring stiffness of the side plate of the bottom layer cross beam.
(2) Column write three-degree-of-freedom undamped free vibration differential equation (damping effect is not considered):
(K-ω2M){X}＝0 1)
the formula 1) is as follows: k is a stiffness matrix, K ═ H-1(ii) a Omega is the natural circular frequency; m is a mass diagonal matrix, the order of which is related to the number of the mass blocks, in this example, a 3-order diagonal matrix; x is a third order mode matrix corresponding to the natural circular frequency ω.
(3) Column-write the characteristic equation of the above equation 1):
|λ2I-D|＝0 2)
the formula 2) is as follows: lambda [ alpha ]2＝1/ω2D is HM, D is a power matrix, and I is a 3-order identity matrix.
(4) The natural frequency f of the frame structure is determined using the relationship f between the natural circular frequency ω and the natural frequency f as ω/2 pii(i＝1,2,3)。
The distribution of 2-4-4 is as follows: as shown in fig. 1, the side plates on both sides between the floors are bisected, the mass of the side plates on both sides above the dotted line is concentrated on the upper-floor beam, the mass of the side plates below the dotted line is concentrated on the lower-floor beam, and the mass of the side plates below the dotted line contacting the ground is concentrated on the ground.
The equal distribution is as follows: and (4) enabling the mass of the side plates between the cross beams of each layer to be equivalent to that of the cross beam of the upper layer.
Table 1 shows the theoretical results, test results and relative errors between the theoretical results and the free vibration test results of the calculation of the natural frequency of the three-layer frame structure according to the conventional mass distribution method.
TABLE 1 theoretical and test results and comparison thereof
And (3) data analysis: it can be seen from the relative errors shown in table 1 that the errors of the calculation results of the conventional mass distribution method are large, and the error of the third-order natural frequency is as high as more than 30%.
3. On the basis of equal distribution, the equivalent mass of each layer is quantitatively changed respectively, and the influence of the change of the equivalent mass of each layer on theoretical calculation natural frequency is researched.
The effect of increasing the equivalent mass of each layer on the theoretical calculation natural frequency error is opposite to the effect of reducing the equivalent mass of each layer, and therefore, the quantitative reduction of the mass of each layer by 100g is taken as an example. Fig. 4 is a graph showing the influence of the equivalent mass change of each layer on the error of the theoretically calculated natural frequency.
And (3) data analysis: it can be seen from fig. 4 that at the first order frequency f, compared to the "equal division" curve1In the influence of the error of (2), the top layer equivalent mass m1To a first order frequency f1Has the largest influence on the middle layer equivalent mass m2Affecting the next, bottom layer equivalent mass m3The influence is minimal, and when the top layer has equivalent mass m1After a mass reduction of 100g, f1The error of (2) is increased by about 8%. At a second order frequency f2In the influence of the error of (2), the top layer equivalent mass m1Bottom layer equivalent mass m3Is relatively large and close, wherein the top layer equivalent mass m1Has a slightly larger influence than the bottom layer equivalent mass m3Middle layer equivalent mass m2Has the least influence, and reduces the equivalent mass m of the top layer1After mass, f2The error of (2) is reduced by about 4%. At third order frequency f3In the influence of the error of (2), the middle layer equivalent mass m2Bottom layer equivalent mass m3Has basically the same influence, reduces the error by 5 percent, and has the equivalent mass m of the top layer1The effect of (c) is minimal. Therefore, the temperature of the molten metal is controlled,reducing the equivalent mass m of the bottom layer3Will contribute to a large increase of the second order frequency f2Third order frequency f3But at the same time, the first order frequency f is increased slightly1So that to improve the accuracy of the "equal division" method, the frequency f to the first order is changed1Top layer equivalent mass m with the greatest influence1The quality of (c).
Based on the above analysis results, the method adopts the steps of reducing the bottom layer equivalent mass m3Increasing the top layer equivalent mass m1"improved distribution method, the error of each order frequency is: 2.35%, 15.38% and 7.49%, compared with the traditional allocation method of equal division allocation, the accuracy of each order of natural frequency is respectively improved to a certain extent.
Said "reducing the equivalent mass m of the bottom layer3Increasing the top layer equivalent mass m1"improved dispensing method: based on the qualitative influence analysis of the equivalent mass change of each layer, the equivalent mass m of the bottom layer is quantitatively reduced3A mass of 200g and adding it to the top layer equivalent mass m1。
4. And calculating the distribution result of equivalent mass of each layer under certain error constraint by using a circular nesting theory to obtain the optimal distribution proportion, and further improving the precision of calculating the natural frequency of the frame structure by using a concentrated mass method.
FIG. 5 is a flow chart for calculating the equivalent mass distribution of each layer using the loop nesting theory, where eiThe relative error of each order frequency is shown, i is 1,2, 3.
Calculating the top equivalent mass m in the program1Middle layer equivalent mass m2Bottom layer equivalent mass m3Respectively carrying out average calculation to obtain the top equivalent mass m as the optimal distribution result of the equivalent mass of each layer under the error constraint1Middle layer equivalent mass m2Bottom layer equivalent mass m3The weight percentages of 218g, 974g and 157g in the total mass are respectively as follows: 16%, 72% and 12% (all rounded).
TABLE 2 for "aliquoting", "subtracting m3Increase m1Distribution condition of quality of each layer under three distribution methods of ' sum ' and ' optimal distributionFig. 6 is a frequency error map of each step corresponding to table 2.
Table 2 equivalent mass distribution table for each layer under three methods
| Distribution method | m1(g) | m2(g) | m3(g) |
| Equal distribution | 449.67 | 449.67 | 449.67 |
| Decreasing m3Increase m1 | 699.34 | 449.67 | 200 |
| Optimal distribution | 218 | 974 | 157 |
And (3) data analysis: as can be seen from fig. 6, the errors of the respective orders of frequency based on the "optimal allocation" are 3.73%, 0.38% and 1.60%, respectively, and the accuracy of the respective orders of frequency is improved by 6.88%, 16.14% and 29.55% compared to the "equal allocation".
5. And changing the mass of each layer of cross beams of the three-layer frame structure, respectively solving the natural frequency of the frame structure by using the optimal distribution proportion and the free vibration test, and verifying the accuracy of the distribution proportion.
Example 1: the weight of each layer of beam is increased by 76g, the total mass of the corresponding three layers of frames is 1577g, and the free vibration test result is shown in figure 7.
Example 2: the mass of each layer of the beam is increased by 152g, the total mass of the corresponding three layers of frames is 1805g, and the free vibration test result is shown in FIG. 8.
Table 3 shows the theoretical values of natural frequency and the error results of comparison with the experimental values, which are calculated according to the optimal distribution ratio, for the two examples.
Table 3 shows the theoretical results, the experimental results and the relative errors of the two examples
And (3) data analysis: as can be seen from table 3, the relative error of the second and third frequencies in both the two examples is much lower than 4%, and the error of the first frequency is 4.58% and 4.63%, respectively, and slightly higher than 4%, but does not affect the optimality of the distribution ratio.
By integrating the analysis of the embodiment, the method can effectively improve the calculation precision of the inherent frequency of the frame structure calculated by the centralized mass method; the inherent frequency of the frame structure is calculated by adopting a centralized mass method, and compared with a finite element method, the method is simpler and more intuitive; the qualitative influence of the equivalent mass change of each layer on the theoretical calculation natural frequency can be obtained by adopting a method of quantitatively changing the equivalent mass of each layer; the mass distribution under certain error constraint is programmed and calculated by adopting a loop nesting theory, so that the quick and accurate calculation of the natural frequency of the lower frame structure under the error constraint can be realized. The method can be applied to the calculation of the natural frequency of the four-layer and higher-layer frame structure through proper modification.
Claims (3)
1. The mass distribution method for calculating the natural frequency of the frame structure by the concentrated mass method is characterized in that the natural frequency of the frame structure in the traditional mass distribution method is calculated by a flexibility matrix, the relative error between the natural frequency and each frequency of a free vibration test result is analyzed, the mass of each layer is quantitatively changed on the basis of the traditional mass distribution method, the influence of the change of the equivalent mass of each layer on the precision of the theoretically calculated natural frequency is analyzed, the distribution condition of the equivalent mass under the constraint of a certain error is further calculated by utilizing a cyclic nesting theory, and the optimal distribution example is finally obtained;
on a three-layer frame vibration test device, respectively and quantitatively reducing the equivalent mass of each layer by 0.1kg, and obtaining the following conclusion by calculation and analysis:
(1) at a first order frequency f1In the influence of the error of (2), the top layer equivalent mass m1The change in (c) has the greatest effect;
(2) at a second order frequency f2In the influence of the error of (2), the top layer equivalent mass m1And the equivalent mass m of the bottom layer3The change influence is large and the equivalent mass m of the top layer1Influence greater than the bottom equivalent mass m3(ii) an effect;
(3) at third order frequency f3In the influence of the error of (2), the middle layer equivalent mass m2And the equivalent mass m of the bottom layer3Has large variation influence and middle layer equivalent mass m2Variation influence and bottom layer equivalent mass m3The same effect is achieved by the change in (c);
(4) reducing the equivalent mass m of the bottom layer3Will contribute to a large increase of the second order frequency f2Third order frequency f3The theoretical calculation accuracy of (2);
the traditional quality allocation method is equal allocation or 2-4-4 allocation; the equal distribution is as follows: the mass of the side plate between each layer of cross beams is all equivalent to that of the upper layer of cross beams;
the distribution of 2-4-4 is as follows: the side plates on two sides between layers are divided into an upper part and a lower part, the mass of the upper parts of the side plates on two sides between layers is concentrated on the upper layer cross beam, the mass of the lower parts of the side plates on two sides between layers is concentrated on the lower layer cross beam, and the mass of the lower parts of the side plates between layers connected with the ground is concentrated on the ground.
2. The method of mass distribution for calculating the natural frequency of a framework structure according to the lumped mass method as set forth in claim 1, comprising the steps of:
(1) respectively applying static loads with the same size to the shackle of each layer of cross beams of the three-layer frame structure, simultaneously measuring the horizontal displacement of each layer of cross beams by using a dial indicator, and solving a flexibility matrix according to a flexibility theory;
(2) the undamped free vibration differential equation based on the concentrated mass method is written in a row, the natural frequency of the lower frame structure in the traditional mass distribution method of 'equal distribution' and '2-4-4 distribution' is solved, and the relative error between the natural frequency and each frequency of the free vibration test result is analyzed;
(3) respectively and quantitatively changing equivalent mass of each layer on the basis of equal distribution, and carrying out qualitative research on the influence of equivalent mass change of each layer;
(4) accurately calculating the distribution condition of equivalent mass of each layer under the constraint of a certain error by using a cyclic nesting theory to obtain an optimal distribution proportion;
(5) and changing the mass of each layer of cross beams of the frame structure, solving the natural frequency of the frame structure by using the distribution proportion and the free vibration test respectively, and verifying the accuracy of the distribution proportion.
3. The method according to claim 1, wherein the equivalent mass of each layer of the three-layer frame structure is the equivalent mass m of the top layer in percentage of the total mass under the constraint of 4% of frequency error1Middle layer equivalent mass m2Bottom layer equivalent mass m3When the frequency is not less than 16% and not more than 72% and not more than 12%, the relative errors between the inherent frequency value of the three-layer frame structure and the free vibration test result calculated by the lumped mass method are respectively as follows: 3.73 percent, 0.38 percent and 1.60 percent, and improves the accuracy of calculating the natural frequency of the frame structure by a centralized mass method.
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