CN111783199A - Refined rapid solving method for dynamic characteristics of multi-section cable structure - Google Patents

Refined rapid solving method for dynamic characteristics of multi-section cable structure Download PDF

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CN111783199A
CN111783199A CN202010570240.0A CN202010570240A CN111783199A CN 111783199 A CN111783199 A CN 111783199A CN 202010570240 A CN202010570240 A CN 202010570240A CN 111783199 A CN111783199 A CN 111783199A
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韩飞
邓子辰
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Northwestern Polytechnical University
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Abstract

The invention relates to a refined rapid solving method for the dynamic characteristics of a multi-section cable structure, wherein the solving process is in a closed form, so that the method has higher calculation efficiency and precision compared with the traditional time domain solution. The accuracy and the effectiveness of the method are verified by testing the vibration characteristics of a solid cable with the length of 20 meters in a cable plant and comparing the vibration characteristics with a finite element solution. The maximum error of the calculation result of the invention is 2.6% compared with the actual measurement frequency, and the maximum error is not more than 0.67% compared with the finite element solution, thereby explaining the calculation precision of the invention. In addition, the invention is suitable for the scene that the transverse support is an elastic support, a damper, a centralized mass and any combination of the three, thereby having good universality and being further used for cable force monitoring, damage identification and the like of the structure in practical engineering.

Description

Refined rapid solving method for dynamic characteristics of multi-section cable structure
Technical Field
The invention belongs to the technical field of structural engineering, relates to dynamic characteristic analysis and cable force monitoring of cable bearing members such as main cables, stay cables, suspenders and the like of a large-span bridge structure, and particularly relates to a refined rapid solving method for dynamic characteristics of a multi-section cable structure
Background
In recent thirty years, with the high-speed growth of the economy and transportation industry in China, the span of the bridge structure is continuously broken through. The guy cable is one of the main bearing structures of a large-span bridge, and the structural form of the guy cable is also becoming more complex. The study on the multi-section cable structure with a plurality of transverse supports, such as a stayed cable-damper system, a main cable-suspender system of a suspension cable, a cable net system and the like, has important significance and engineering value.
Due to the characteristics of high flexibility, small damping and the like, the cable structure is easy to vibrate under the action of external load, so that the research on the dynamic characteristics of the cable structure has important theoretical value on the optimization design, the health monitoring, the vibration control and the like of the structure. However, the difficulty in accurately solving the dynamic characteristics of the multi-section cable structure is great, and the main problems are that:
(1) the accurate description of the structure motion configuration is a premise of fine power analysis and is also a starting point. For a multi-section cable system, under the influence of transverse support, each cable section divided by the cable system follows different motion configurations in the vibration process, and how to accurately calculate the motion configurations of the cable sections is a big difficulty;
(2) the additional cable force can be caused by the elastic extension in the cable vibration process, the domestic and foreign researches can only provide an analytical expression of the additional cable force of one-section or two-section cables so far, and no relevant research work is available for a plurality of cable sections;
(3) when the factors such as the bending rigidity, the sag, the inclination angle and the damping of the inhaul cable are considered, the refined and rapid analysis of the dynamic characteristics of the system is very difficult, and the traditional dynamic theory and numerical method are difficult to simultaneously consider the calculation precision and the efficiency.
In view of the above difficulties, there is no general analytic or semi-analytic dynamic analysis theory at present, which can solve the difficulties described in (1), (2), and (3) at the same time, so it is necessary to develop a universal dynamic analysis method for multi-segment cable system, so as to realize a fine and fast analysis of structural dynamic characteristics.
Disclosure of Invention
The technical problem solved by the invention is as follows: in view of the limitations of the conventional analysis methods in addressing the above problems, the present invention provides a method for fine and fast analysis of dynamic characteristics of a multi-segment cable system, and aims to:
(1) the problem of fine power modeling of a multi-section cable system is solved;
(2) giving an explicit solution of the additional cable force of the multi-section cable system;
(3) and giving an accurate and rapid solution scheme of the frequency equation of the multi-section cable system.
The technical scheme of the invention is as follows: a refined rapid solving method for the dynamic characteristics of a multi-section cable structure comprises the following steps:
the method comprises the following steps: based on multi-section cable structure dynamics modeling, a system motion differential equation set is established, and the method comprises the following substeps:
the first substep: defining the dip angle of a stayed cable as theta, the initial sag as d and the chord length as l0，(x0Y) is an overall coordinate system of the cable; the ends A, B of the cables are elastically constrained at two points, and the rotational support stiffness and the vertical support stiffness at the end A are respectively expressed as
Figure BDA0002549338010000021
And
Figure BDA0002549338010000022
the rotational and vertical bearing stiffness at the end point B are respectively indicated as
Figure BDA0002549338010000023
And
Figure BDA0002549338010000024
the cable is subjected to a horizontal tension H, and its initial configuration under the combined action of tension and dead weight is represented by y0(x0) Represents; supposing that n-1 transverse components are installed on the cable, the cable is divided into n cable sections by the n-1 transverse components, and the chord length, the local coordinate system and the displacement function of the jth cable section are respectively recorded as lj、xjAnd uj(xj,t)， (j＝1,2,...n)；
And a second substep: and (3) establishing a motion differential equation of the cable section j in a local coordinate system, wherein the establishment and solution methods of the motion differential equations of other cable sections are completely the same.
Figure BDA0002549338010000025
EI, m and H are respectively bending rigidity, mass per linear meter and initial cable force of the stay cable; u. ofj、yjRespectively the motion and initial configuration of the jth cable segment, hjFor the additional cable force value of the cable segment caused by elastic elongation during vibration, which is equal to the dynamic strain of the cable segmentj(t) and the cable axial stiffness EA, i.e.:
hj＝EAj(t) (2)
in the formula, E and A are respectively a cable elastic model and a section effective area, and the general expression for obtaining the multi-section cable additional force according to the formula is
Figure BDA0002549338010000026
In the formula
Figure BDA0002549338010000031
Is the vertical span ratio, g is the acceleration of gravity;
Figure BDA0002549338010000032
is the curve length of the jth rope segment; lsjRepresenting the sum of the chordwise lengths of the first j cord segments.
Step two: and calculating the mode shape function of each cable segment. Applying the separation variable method to the formulas (8) and (2) and solving the general solution to determine the mode shape function of each cable segment, wherein the dimensionless mode shape function of the jth cable segment
Figure BDA0002549338010000033
Comprises the following steps:
Figure BDA0002549338010000034
wherein A is(j)＝{A1 (j)A2 (j)A3 (j)A4 (j)}TThe undetermined coefficient vector can be determined by the boundary conditions of nodes at two ends of the cable segment;
Figure BDA0002549338010000035
is a vibration pattern vector, mui＝li/l0，
Figure BDA0002549338010000036
B(j)Is a sag matrix, which is determined by the special solution of equation (8) as follows:
Figure BDA0002549338010000037
wherein
Figure BDA0002549338010000038
Figure BDA0002549338010000039
Figure BDA00025493380100000310
Figure BDA00025493380100000311
Step three: calculating a unit dynamic stiffness matrix K of each cable section(j)The method comprises the following substeps:
the first substep: node displacement U according to jth cable segment(j)And its mode shape function
Figure BDA00025493380100000312
By shifting the node by U(j)Uniformly expressed as:
Figure BDA00025493380100000313
wherein
Figure BDA00025493380100000314
And
Figure BDA00025493380100000315
respectively showing the displacement and the rotation angle of the left end node of the jth cable segment,
Figure BDA00025493380100000316
and
Figure BDA00025493380100000317
respectively representing the displacement and the rotation angle of a node at the right end of the jth cable segment;
Figure BDA00025493380100000318
symbol ()' represents pair ξjAnd (6) derivation.
And a second substep: the cable segment S can be adjusted by the node force balance conditionjNodal force vector F at both ends(j)Is shown as
Figure BDA0002549338010000041
Wherein
Figure BDA0002549338010000042
And
Figure BDA0002549338010000043
respectively representing node shearing forces of the left end and the right end of the cable section;
Figure BDA0002549338010000044
and
Figure BDA0002549338010000045
respectively representing node bending moments of the left end and the right end of the cable section;
Figure BDA0002549338010000046
formula (7) can be further written as
F(j)＝K(j)·U(j)(8)
Figure BDA0002549338010000047
Namely the unit dynamic stiffness matrix of the j cable segment.
Step four: dynamic stiffness matrix K for each cable section(j)Grouping is carried out, and a dynamic stiffness matrix K of the cable system under the overall coordinate system is calculated(0)The method comprises the following substeps:
the first substep: calculating the equivalent support stiffness kappa of each component;
and a second substep: to find K(j)After K, the contribution of each cable segment, transverse element and boundary elastic support to the system rigidity can be superposed in the same way as the finite element method unit rigidity matrix grouping process, and the integral east rigidity matrix K can be solved(0)。
Step five: and solving the frequency and the mode shape. Obtaining an overall stiffness matrix K(0)Then, the characteristic equation det (K) of the system can be solved by applying a numerical iteration algorithm(0)(ω)) -0, and then the modal frequencies and modes of the system of each order are obtained.
The further technical scheme of the invention is as follows: the transverse component in the first step is a spring, a mass block, a damper or any combination of the spring, the mass block and the damper.
Effects of the invention
The invention has the technical effects that: the method provided by the invention is a frequency domain solution, and the solving process is in a closed form, so that the method has higher calculation efficiency and precision compared with the traditional time domain solution. The accuracy and the effectiveness of the method are verified by testing the vibration characteristics of a solid cable with the length of 20 meters in a cable plant and comparing the vibration characteristics with a finite element solution. The maximum error of the calculation result of the invention is 2.6% compared with the actual measurement frequency, and the maximum error is not more than 0.67% compared with the finite element solution, thereby illustrating the calculation precision of the invention (see tables 1 and 2). In addition, the invention is suitable for the scenes that the transverse support is elastic support (a suspension rod or an auxiliary cable), a damper, centralized mass and any combination of the three, thereby having good universality and being further used for cable force monitoring, damage identification and the like of the structure in practical engineering.
Drawings
FIG. 1 is a mechanical model diagram of a multi-section cable system
FIG. 2 is a stiffness equivalence mechanism diagram of different lateral supports
Detailed Description
Referring to fig. 1-2, the multi-section cable system dynamic model shown in fig. 1 is established based on the euler beam theory, and the model can simultaneously consider the influences of factors such as the bending rigidity, sag, inclination angle and additional cable force of the cable. According to the principle of segmented modeling, a node is established at the transverse support, and then a motion differential equation of each cable segment under a local coordinate system is established. The kinetic modeling steps are given below:
step 1, calculating the additional cable force of each cable segment divided by the transverse support;
and Step 2, establishing a motion differential equation of each cable segment under a local coordinate system.
According to the established cable section motion differential equation, the dynamic stiffness method can be applied to realize the accurate solution of the dynamic characteristics of the system, and the specific steps are as follows:
step 1, introducing simple harmonic motion assumption to perform variable separation on a displacement function, and solving a vibration mode function of each cable section;
step 2, calculating a unit dynamic stiffness matrix of each cable section according to the displacement function obtained in the first Step by combining a node force balance condition and a continuous equation;
step 3, determining the equivalent support stiffness of the transverse support according to the specific form of the transverse support (see figure 2 for an equivalent way), grouping the dynamic stiffness matrix of each unit and the transverse support obtained in the third Step, and combining boundary conditions to obtain a dynamic stiffness matrix under a general coordinate system;
and Step 4, listing a system frequency equation according to the dynamic stiffness matrix obtained in the fourth Step, and solving a characteristic root of the frequency equation by using a numerical iteration algorithm to further determine the modal frequency and the mode shape of the system.
Further comprises the following steps:
the first step is as follows: establishing a dynamic model of the multi-section cable system, and determining the additional cable force h of each cable sectionjAnd a differential equation of motion;
the second step is that: obtaining a motion differential equation in a frequency domain by applying a separation variable method, and determining a mode shape function of each cable section by solving the motion differential equation of each cable section
Figure BDA0002549338010000061
The third step: combining continuous conditions and force balance condition elimination
Figure BDA0002549338010000062
Obtaining a unit dynamic stiffness matrix K of each cable section according to undetermined constant(j)；
The fourth step: determining equivalent rigidity k according to the type of the transverse support of the multi-section cable systemeqThen, grouping the dynamic stiffness arrays of the units to obtain an overall dynamic stiffness matrix K of the system;
the fifth step: solving the frequency equation det (K (omega)) of the system to be 0 by using a numerical iterative algorithm, and solving the ith order modal frequency omega of the systemiThen, the modal shape of the system can be further determined by the boundary conditions.
The following detailed description is made with reference to the accompanying drawings:
kinetic modeling
According to the multi-section cable system dynamic model shown in fig. 1, the motion differential equation of the jth cable section can be listed as follows:
Figure BDA0002549338010000063
EI, m and H are respectively bending rigidity, mass per linear meter and initial cable force of the stay cable; u. ofj、yjRespectively the motion and initial configuration of the jth cable segment, hjThe additional cable force value caused by elastic extension of the cable segment in the vibration process is obtained; x is the number ofjAs local coordinates of the individual cable sections,/0Is a stay cableThe overall length in the chord direction.
To solve equation (8), it is first necessary to determine the additional cable force h for each degree of cablejThe analytical expression of (2). Additional cable force h of the jth cable sectionjIs defined as: additional strain induced by elastic elongation of the cable segments during vibration due to deviation of the dynamic configuration from the static configurationjThe product of (t) and axial stiffness EA, i.e.:
hj＝EAj(t) (9)
the general expression of the multi-section additional cable force of the inhaul cable obtained according to the above formula is
Figure BDA0002549338010000064
In the formula
Figure BDA0002549338010000065
Is the vertical span ratio, g is the acceleration of gravity;
Figure BDA0002549338010000066
is the curve length of the jth rope segment; lsjRepresenting the sum of the chordwise lengths of the first j cord segments.
2. Dynamic characteristic solving
The invention solves the problem of the free vibration of the stay cable expressed by the formula (8) by applying the dynamic stiffness theory:
(1) mode shape function of each cable section
Figure BDA0002549338010000071
Is solved for
Will be provided with
Figure BDA0002549338010000072
And (10) substituting the formula into the formula (8) can obtain:
Figure BDA0002549338010000073
ξ introduction of dimensionless parametersj＝xj/l0，
Figure BDA0002549338010000074
And
Figure BDA0002549338010000075
the differential equation of motion after the system dimensionless can be obtained as follows
Figure BDA0002549338010000076
Wherein
Figure BDA0002549338010000077
From equation (12), the mode shape function of each cable segment can be determined
Figure BDA0002549338010000078
Is composed of
Figure BDA0002549338010000079
Wherein
Figure BDA00025493380100000710
Is the mode vector.
Figure BDA00025493380100000711
Wherein
Figure BDA00025493380100000712
ljIs the chord length of the jth cell (see FIG. 1), l0Is the length of the cable. Substituting (13) into (14) can
Figure BDA00025493380100000713
Further expressed in matrix form as follows:
Figure BDA00025493380100000714
wherein A is(j)＝{A1 (j)A2 (j)A3 (j)A4 (j)}TThe undetermined coefficient vector can be determined by the boundary conditions of nodes at two ends of the cable segment; b is(j)Is a sag matrix, which is determined by the special solution of equation (8) as follows:
Figure BDA00025493380100000715
wherein
Figure BDA00025493380100000716
Figure BDA00025493380100000717
Figure BDA0002549338010000081
Figure BDA0002549338010000082
(2) Unit dynamic stiffness matrix K(j)Is calculated by
According to the displacement continuous condition and the force balance condition
Figure BDA0002549338010000083
Substituting equation (13) into equation (17) and eliminating the predetermined coefficient A1 (j)A2 (j)A3 (j)A4 (j)The dynamic stiffness matrix K of the multi-section system cable section can be obtained(j)。
Figure BDA0002549338010000084
(3) Set of global stiffness matrices K
According to the equivalent shown in fig. 2, different types of lateral supports can be equated with a stiffness ofkCAnd then grouping the vertical springs according to a finite element method unit stiffness matrix grouping rule. Taking the beam section AB as an example, assume that there is an equivalent stiffness k at a point C in the middleCThe overall dynamic stiffness matrix K of the system is:
Figure BDA0002549338010000085
the superscript of each stiffness coefficient represents the number of the cable section where the dynamic stiffness coefficient is located, AC is No. 1, and CB is No. 2; the subscript represents the position of the coefficient in the dynamic stiffness matrix of the cell.
(4) Solving for frequency and mode shape
After the overall stiffness matrix K is obtained, solving the characteristic equation det (K (ω)) -0 to obtain the modal frequency of each order of the system. The equation is usually a transcendental equation, and can be solved by using a numerical iterative algorithm, such as a Newton method, a Muller method and the like, so as to obtain the modal frequency omegai. Thereafter, ω will beiSubstituting into formula (13) to obtain the vibration mode function of cable segment j
Figure BDA0002549338010000091
The mode function further comprises four waiting coefficients A1 (j)A2 (j)A3 (j)A4 (j)They can be determined by the following formula:
Figure BDA0002549338010000092
wherein
Figure BDA0002549338010000093
Cj＝cos(qμj),Sj＝sin(qμj)；
Figure BDA0002549338010000094
Is given by the formula (16).
TABLE 1 comparison of measured results and relative error (Hz)
Figure BDA0002549338010000095
TABLE 2 comparative error with finite element solution (Hz)
Figure BDA0002549338010000096

Claims (2)

1. A refined rapid solving method for the dynamic characteristics of a multi-section cable structure is characterized by comprising the following steps: based on multi-section cable structure dynamics modeling, a system motion differential equation set is established, and the method comprises the following substeps:
the first substep: defining the dip angle of a stayed cable as theta, the initial sag as d and the chord length as l0，(x0Y) is an overall coordinate system of the cable; the ends A, B of the cables are elastically constrained at two points, and the rotational support stiffness and the vertical support stiffness at the end A are respectively expressed as
Figure FDA0002549337000000011
And
Figure FDA0002549337000000012
the rotational and vertical bearing stiffness at the end point B are respectively indicated as
Figure FDA0002549337000000013
And
Figure FDA0002549337000000014
the cable is subjected to a horizontal tension H, and its initial configuration under the combined action of tension and dead weight is represented by y0(x0) Represents; supposing that n-1 transverse components are installed on the cable, the cable is divided into n cable sections by the n-1 transverse components, and the chord length, the local coordinate system and the displacement function of the jth cable section are respectively recorded as lj、xjAnd uj(xj,t)，(j＝1,2,...n)；
And a second substep: and (3) establishing a motion differential equation of the cable section j in a local coordinate system, wherein the establishment and solution methods of the motion differential equations of other cable sections are completely the same.
Figure FDA0002549337000000015
EI, m and H are respectively bending rigidity, mass per linear meter and initial cable force of the stay cable; u. ofj、yjRespectively the motion and initial configuration of the jth cable segment, hjFor the additional cable force value of the cable segment caused by elastic elongation during vibration, which is equal to the dynamic strain of the cable segmentj(t) and the cable axial stiffness EA, i.e.:
hj＝EAj(t) (2)
in the formula, E and A are respectively a cable elastic model and a section effective area, and the general expression for obtaining the multi-section cable additional force according to the formula is
Figure FDA0002549337000000016
In the formula
Figure FDA0002549337000000017
Is the vertical span ratio, g is the acceleration of gravity;
Figure FDA0002549337000000018
is the curve length of the jth rope segment; lsjRepresenting the sum of the chordwise lengths of the first j cord segments.
Step two: and calculating the mode shape function of each cable segment. Applying the separation variable method to the formulas (1) and (2) and solving the general solution to determine the mode shape function of each cable segment, wherein the dimensionless mode shape function of the jth cable segment
Figure FDA0002549337000000021
Comprises the following steps:
Figure FDA0002549337000000022
wherein A is(j)＝{A1 (j)A2 (j)A3 (j)A4 (j)}TThe undetermined coefficient vector can be determined by the boundary conditions of nodes at two ends of the cable segment;
Figure FDA0002549337000000023
is a vibration pattern vector, mui＝li/l0，
Figure FDA0002549337000000024
B(j)Is a sag matrix, which is determined from the special solution of equation (1) as follows:
Figure FDA0002549337000000025
wherein
Figure FDA0002549337000000026
Figure FDA0002549337000000027
Figure FDA0002549337000000028
Figure FDA0002549337000000029
μsj＝lsj/l0
Step three: calculating a unit dynamic stiffness matrix K of each cable section(j)The method comprises the following substeps:
the first substep: node displacement U according to jth cable segment(j)And its mode shape function
Figure FDA00025493370000000210
By shifting the node by U(j)Uniformly expressed as:
Figure FDA00025493370000000211
wherein
Figure FDA00025493370000000212
Figure FDA00025493370000000213
And
Figure FDA00025493370000000214
respectively showing the displacement and the rotation angle of the left end node of the jth cable segment,
Figure FDA00025493370000000215
and
Figure FDA00025493370000000216
respectively representing the displacement and the rotation angle of a node at the right end of the jth cable segment;
Figure FDA00025493370000000217
symbol ()' represents pair ξjAnd (6) derivation.
And a second substep: the cable segment S can be adjusted by the node force balance conditionjNodal force vector F at both ends(j)Is shown as
Figure FDA00025493370000000218
Wherein
Figure FDA0002549337000000031
Figure FDA0002549337000000032
And
Figure FDA0002549337000000033
respectively representing node shearing forces of the left end and the right end of the cable section;
Figure FDA0002549337000000034
and
Figure FDA0002549337000000035
respectively representing node bending moments of the left end and the right end of the cable section;
Figure FDA0002549337000000036
formula (7) can be further written as
F(j)＝K(j)·U(j)(8)
Figure FDA0002549337000000037
Namely the unit dynamic stiffness matrix of the j cable segment.
Step four: dynamic stiffness matrix K for each cable section(j)Grouping is carried out, and a dynamic stiffness matrix K of the cable system under the overall coordinate system is calculated(0)The method comprises the following substeps:
the first substep: calculating the equivalent support stiffness kappa of each component;
and a second substep: to find K(j)After K, the contribution of each cable segment, transverse element and boundary elastic support to the system rigidity can be superposed in the same way as the finite element method unit rigidity matrix grouping process, and the integral east rigidity matrix K can be solved(0)。
Step five: and solving the frequency and the mode shape. Obtaining an overall stiffness matrix K(0)Then, the characteristic equation det (K) of the system can be solved by applying a numerical iteration algorithm(0)(ω)) -0, and then the modal frequencies and modes of the system of each order are obtained.
2. The method for refining and rapidly solving the dynamic characteristics of the multi-section cable structure according to claim 1, wherein the transverse component in the first step is a spring, a mass block, a damper or any combination of the spring, the mass block and the damper.
CN202010570240.0A 2020-06-21 2020-06-21 Refined rapid solving method for dynamic characteristics of multi-section cable structure Pending CN111783199A (en)

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CN115017681A (en) * 2022-05-10 2022-09-06 西北工业大学 Fine analysis method for dynamic characteristics of small-sag stay cable
CN115452226A (en) * 2022-10-05 2022-12-09 大连理工大学 Inhaul cable force identification algorithm considering semi-rigid constraints at two ends

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Publication number Priority date Publication date Assignee Title
CN112651072A (en) * 2021-01-06 2021-04-13 华南理工大学 Suspension bridge double-sling parameter identification method based on cable net model
CN112651072B (en) * 2021-01-06 2023-08-22 华南理工大学 Suspension bridge double-sling parameter identification method based on cable network model
CN115017681A (en) * 2022-05-10 2022-09-06 西北工业大学 Fine analysis method for dynamic characteristics of small-sag stay cable
CN115452226A (en) * 2022-10-05 2022-12-09 大连理工大学 Inhaul cable force identification algorithm considering semi-rigid constraints at two ends
CN115452226B (en) * 2022-10-05 2023-12-19 大连理工大学 Cable force recognition algorithm considering semi-rigid constraint at two ends

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