CN108763674B - Method for solving bending vibration frequency of inhaul cable under elastic boundary condition - Google Patents

Method for solving bending vibration frequency of inhaul cable under elastic boundary condition Download PDF

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CN108763674B
CN108763674B CN201810465820.6A CN201810465820A CN108763674B CN 108763674 B CN108763674 B CN 108763674B CN 201810465820 A CN201810465820 A CN 201810465820A CN 108763674 B CN108763674 B CN 108763674B
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陈跃华
冯志敏
闫伟
张刚
胡海刚
程鹏
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Ningbo University
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Abstract

The invention discloses a method for solving the bending vibration frequency of a stay cable under an elastic boundary condition, which comprises the steps of firstly obtaining the performance parameters of the stay cable to be detected, constructing a stay cable vibration partial differential equation based on the obtained performance parameters of the stay cable, then simplifying the stay cable vibration partial differential equation by adopting a separation variable method to obtain the stay cable vibration differential equation, establishing the elastic boundary condition of the stay cable vibration, expanding the vibration type function of the stay cable in the form of Chebyshev series, then obtaining the characteristic equation of the stay cable vibration based on the vibration type function and the elastic boundary condition of the stay cable, finally obtaining the frequency equation of the stay cable based on the characteristic equation of the stay cable vibration, and solving the frequency equation to obtain the vibration frequency of the stay cable; the advantage is a higher accuracy.

Description

Method for solving bending vibration frequency of inhaul cable under elastic boundary condition
Technical Field
The invention relates to a method for solving the bending vibration frequency of a stay cable, in particular to a method for solving the bending vibration frequency of the stay cable under an elastic boundary condition.
Background
At present, in order to ensure the safe use of the bridge, various indexes of the bridge are generally monitored. The bending vibration of the stay cable is one of important monitoring indexes of bridge engineering, and the vibration frequency of the stay cable is calculated by assuming that boundary conditions at two ends of the stay cable are ideal fixed support or simple support conditions. However, in fact, the boundary conditions at the two ends of the cable are not ideal simple support and solid support, but are between the simple support and the solid support, for example, the ends of the cable-stayed bridge are connected with the main beam and the cable tower through anchors, the anchor mode is complex in engineering, the boundary conditions at the two ends of the cable are assumed to be the solid support or the simple support boundary, the assumption has a large error with the actual boundary constraint, and therefore the bending vibration frequency of the obtained cable has a large error and is not high in accuracy.
Disclosure of Invention
The invention aims to provide a method for solving the bending vibration frequency of a stay cable under the elastic boundary condition with higher accuracy.
The technical scheme adopted by the invention for solving the technical problems is as follows: a method for solving the bending vibration frequency of a stay cable under the elastic boundary condition comprises the following steps:
(1) acquiring performance parameters of a cable to be detected and constructing a cable vibration partial differential equation based on the acquired performance parameters of the cable:
Figure BDA0001662048190000011
in the formula (1), x represents the vibration displacement of the stay cable to be detected, and x belongs to [0, L ]]L represents the length of the stay cable to be detected, T represents the design value of the cable force of the stay cable to be detected, E represents the elastic modulus of the stay cable to be detected, I represents the section inertia moment of the stay cable to be detected, rho represents the density of the stay cable to be detected, A represents the section area of the stay cable to be detected, T represents the vibration time of the stay cable to be detected, u (x, T) represents the displacement function of each point of the stay cable to be detected along with the change of the time T,
Figure BDA0001662048190000021
represents the second partial derivative of u (x, t) over time t;
Figure BDA0001662048190000022
represents u (x, t) taking the second partial derivative of x,
Figure BDA0001662048190000023
represents u (x, t) taking the fourth partial derivative of x;
(2) simplifying the inhaul cable vibration partial differential equation constructed in the step (1) by adopting a separation variable method, wherein the concrete simplification process is as follows:
a. decomposing u (x, t) into a product relation between a time function and a space function to obtain:
Figure BDA0001662048190000024
in the formula (2), Y (x) is a vibration mode function of the inhaul cable, sin (x) is a sine function, omega is the vibration frequency of the inhaul cable,
Figure BDA0001662048190000025
is the phase angle of the stay cable vibration;
b. substituting the formula (2) into the formula (1) to obtain a guy cable vibration differential equation:
EIY(4)(x)-TY(2)(x)-ρAω2Y(x)＝0 (3)
wherein, Y(4)(x) Denotes Y (x) taking the fourth derivative of x, Y(2)(x) Denotes Y (x) second derivative of x;
(3) establishing an elastic boundary condition of stay cable vibration:
c. the left end of the inhaul cable is assumed to be respectively connected with elastic rigidity K1And an elastic stiffness of K2The right end of the inhaul cable is respectively connected with a torsional spring with the elastic rigidity of K3And an elastic stiffness of K4Thereby obtaining:
Figure BDA0001662048190000026
in formula (4), u' gaming neutralx＝0Denotes u (x, t) as a third partial derivative of x at 0, u-x＝0Denotes the function value of u (x, t) at x ═ 0, u' -ray countx＝0Denotes u (x, t) taking the second partial derivative of x where x is 0, u'x＝0Denotes that u (x, t) is the first partial derivative of x at 0, u' ″ counting the number of cellsx＝LDenotes the calculation of the third-order partial derivative of u (x, t) with respect to x at x ═ L, u-x＝LDenotes the function value of u (x, t) at x ═ L, u' -rayx＝LDenotes u (x, t) taking the second partial derivative of x where x is L, u'x＝LDenotes the first partial derivative of x with u (x, t) at x ═ L, k1Has an elastic rigidity of K1K is a dimensionless coefficient of the stiffness of the tension spring2Has an elastic rigidity of K2K is a dimensionless coefficient of the torsional spring stiffness3Has an elastic rigidity of K3K is a dimensionless coefficient of the stiffness of the tension spring4Has an elastic rigidity of K4K is a dimensionless coefficient of the torsional spring stiffness1，k2，k3And k4The values of (A) are respectively as follows:
Figure BDA0001662048190000031
in the formula (5), the symbol "/" is a division operation symbol; k is a radical of1，k2，k3And k4All values of (1) are [0,10 ]4]And simultaneously satisfies the following three conditions: one, k1，k2，k3And k4The value of (a) cannot be 0 at the same time; two, k1，k2，k3And k4Cannot simultaneously take on a value of 104(ii) a III when k1And k3Simultaneously has a value of 104And k is2When the value of (1) is 0, k4Cannot be 104When k is1And k3Simultaneously has a value of 104，k4When the value of (1) is 0, k2Cannot be 104；
d. Substituting the formula (2) into the formula (4) to obtain the elastic boundary condition of the inhaul cable:
Figure BDA0001662048190000032
in formula (6): y' cagex＝0Denotes the third derivative of x by Y (x) at x ═ 0, Y ∞x＝0Denotes a function value of Y (x) at x ═ 0, Y'x＝0Denotes the second derivative of x by Y (x) at x ═ 0, Y'x＝0Denotes the first derivative of x by Y (x) at x ═ 0, Y' ″x＝LDenotes the derivation of the third derivative of x by Y (x) at x ═ L, Y ∞x＝LDenotes the function value of Y (x) at x ═ L, Y'x＝LDenotes the second derivative of x by Y (x) at x ═ L, Y'x＝LDenotes y (x) first derivative of x at x ═ L;
(4) expanding the vibration mode function Y (x) of the inhaul cable in a Chebyshev series mode to obtain an expansion formula of Y (x):
Figure BDA0001662048190000033
in formula (7): Σ (±) denotes summing, n is the highest order of the chebyshev series, n is an integer of 1 or more, i is 1, 2, …, n; phii(x) Is an ith order displacement function satisfying the boundary condition expression (6) constructed by using a Chebyshev series, aiAs a function of the ith order shift phii(x) Is a predetermined coefficient, wherein
Φi(x)＝h(x)Ti(x)+g(x) (8)
Wherein, Ti(x) Is the ith Chebyshev series, h (x) and g (x) are constructed auxiliary functions, Ti(x) The expression of (a) is:
Figure BDA0001662048190000041
in formula (9): cos (×) represents a cosine function, arccos (×) represents an inverse cosine function;
expressions of h (x) and g (x) are respectively as follows:
Figure BDA0001662048190000042
Figure BDA0001662048190000043
Figure BDA0001662048190000044
Figure BDA0001662048190000045
Figure BDA0001662048190000046
Figure BDA0001662048190000047
(5) according to the virtual displacement principle, the work of the guy cable on the virtual displacement delta Y (x) is zero, wherein delta represents variation, and the expression of delta Y (x) is as follows:
Figure BDA0001662048190000048
the work of the cable on the virtual displacement δ y (x):
Figure BDA0001662048190000049
from equations (12) and (13), it is possible to obtain:
Figure BDA00016620481900000410
wherein phii (4)(x) Represents phii(x) Taking the fourth derivative, phi, of xi (2)(x) Represents phii(x) Taking the second derivative, Φ, of xj(x) Represents the j-th order shift function, j is 1, 2, …, n; the equation can be found after working in equation (14):
Figure BDA0001662048190000051
in formula (15):
Figure BDA0001662048190000052
writing the formula (15) into a matrix form, namely, the characteristic equation of the stay cable vibration is:
(D-ω2M)H＝0 (16)
in the formula (16), D represents a stiffness matrix, M represents a mass matrix, H represents a coefficient matrix, and DijRepresenting the ith row and jth column element, m, of the matrix DijThe ith row and jth column elements of matrix M are represented as:
Figure BDA0001662048190000053
Figure BDA0001662048190000054
Figure BDA0001662048190000055
(6) solving a characteristic equation (16) of the stay cable vibration to obtain a frequency equation as follows:
|D-ω2M|＝0 (17)
and | | represents a determinant of the matrix, and the frequency equation (17) is solved to obtain the vibration frequency ω of the cable under different boundary constraint conditions.
Compared with the prior art, the method has the advantages that firstly, the performance parameters of the stay cable to be detected are obtained, the stay cable vibration partial differential equation is constructed on the basis of the obtained performance parameters of the stay cable, then the stay cable vibration partial differential equation is simplified by adopting a separation variable method, the stay cable vibration partial differential equation is obtained, the elastic boundary condition of the stay cable vibration is established, and the boundary conditions excluding the left end and the right end of the stay cable are all solid support (k)1，k2，k3And k4Simultaneously has a value of 104) In the case of (1), both left and right ends of the stay are in a free state (k)1，k2，k3And k4The value of (k) is 0) and the boundary conditions of the left and right ends of the stay are simple (when k is1And k3Simultaneously has a value of 104When k is2And k4The value of (k) is 0) and the boundary condition of the left and right ends of the stay is fixed support-simple support (when k is1And k3Simultaneously has a value of 104，k2Is taken as value of 104，k4Value of (k) is 0) and the boundary condition of the left and right ends of the cable is simply supported-fixedly supported (when k is1And k3Simultaneously has a value of 104，k2Has a value of 0, k4Value of 104) In the case of (1), the boundary conditions of the left and right ends of the cable are as follows: in the method, the frequency equation of the stay is a model of the bending vibration of the stay under the elastic boundary constraint condition, simultaneously the bending rigidity and the influence of different boundary constraints on the bending vibration frequency of the stay are considered, and a Chebyshev polynomial method (namely the Chebyshev series) is adopted to approximate the vibration shape function of the stay, so that the accuracy and the convergence of the measurement result are further improved, and obtaining the inhaul cable vibration frequency with higher accuracy.
Drawings
FIG. 1 shows the implementation structure of elastic boundary condition constructed by the present invention.
Detailed Description
The invention is described in further detail below with reference to the accompanying examples.
Example (b): a method for solving the bending vibration frequency of a stay cable under the elastic boundary condition comprises the following steps:
(1) acquiring performance parameters of a cable to be detected and constructing a cable vibration partial differential equation based on the acquired performance parameters of the cable:
Figure BDA0001662048190000061
in the formula (1), x represents the vibration displacement of the stay cable to be detected, and x belongs to [0, L ]]L represents the length of the stay cable to be detected, T represents the design value of the cable force of the stay cable to be detected, E represents the elastic modulus of the stay cable to be detected, I represents the section inertia moment of the stay cable to be detected, rho represents the density of the stay cable to be detected, A represents the section area of the stay cable to be detected, T represents the vibration time of the stay cable to be detected, and u (x, T) represents the vibration time of the stay cable to be detectedThe displacement function of each point over time t,
Figure BDA0001662048190000062
represents the second partial derivative of u (x, t) over time t;
Figure BDA0001662048190000071
represents u (x, t) taking the second partial derivative of x,
Figure BDA0001662048190000072
represents u (x, t) taking the fourth partial derivative of x;
(2) simplifying the inhaul cable vibration partial differential equation constructed in the step (1) by adopting a separation variable method, wherein the concrete simplification process is as follows:
a. decomposing u (x, t) into a product relation between a time function and a space function to obtain:
Figure BDA0001662048190000073
in the formula (2), Y (x) is a vibration mode function of the inhaul cable, sin (x) is a sine function, omega is the vibration frequency of the inhaul cable,
Figure BDA0001662048190000074
is the phase angle of the stay cable vibration;
b. substituting the formula (2) into the formula (1) to obtain a guy cable vibration differential equation:
EIY(4)(x)-TY(2)(x)-ρAω2Y(x)＝0 (3)
wherein, Y(4)(x) Denotes Y (x) taking the fourth derivative of x, Y(2)(x) Denotes Y (x) second derivative of x;
(3) establishing an elastic boundary condition for the stay cable vibration, as shown in fig. 1:
c. the left end of the inhaul cable is assumed to be respectively connected with elastic rigidity K1And an elastic stiffness of K2The right end of the inhaul cable is respectively connected with a torsional spring with the elastic rigidity of K3And an elastic stiffness of K4Thereby obtaining:
Figure BDA0001662048190000075
in formula (4), u' gaming neutralx＝0Denotes u (x, t) as a third partial derivative of x at 0, u-x＝0Denotes the function value of u (x, t) at x ═ 0, u' -ray countx＝0Denotes u (x, t) taking the second partial derivative of x where x is 0, u'x＝0Denotes that u (x, t) is the first partial derivative of x at 0, u' ″ counting the number of cellsx＝LDenotes the calculation of the third-order partial derivative of u (x, t) with respect to x at x ═ L, u-x＝LDenotes the function value of u (x, t) at x ═ L, u' -rayx＝LDenotes u (x, t) taking the second partial derivative of x where x is L, u'x＝LDenotes the first partial derivative of x with u (x, t) at x ═ L, k1Has an elastic rigidity of K1K is a dimensionless coefficient of the stiffness of the tension spring2Has an elastic rigidity of K2K is a dimensionless coefficient of the torsional spring stiffness3Has an elastic rigidity of K3K is a dimensionless coefficient of the stiffness of the tension spring4Has an elastic rigidity of K4K is a dimensionless coefficient of the torsional spring stiffness1，k2，k3And k4The values of (A) are respectively as follows:
Figure BDA0001662048190000081
in the formula (5), the symbol "/" is a division operation symbol; k is a radical of1，k2，k3And k4All values of (1) are [0,10 ]4]And simultaneously satisfies the following three conditions: one, k1，k2，k3And k4The value of (a) cannot be 0 at the same time; two, k1，k2，k3And k4Cannot simultaneously take on a value of 104(ii) a III when k1And k3Simultaneously has a value of 104And k is2When the value of (1) is 0, k4Cannot be 104When k is1And k3Simultaneously has a value of 104，k4When the value of (1) is 0, k2Cannot be 104(ii) a d. Substituting the formula (2) into the formula (4) to obtain the elastic boundary condition of the inhaul cable:
Figure BDA0001662048190000082
in formula (6): y' cagex＝0Denotes the third derivative of x by Y (x) at x ═ 0, Y ∞x＝0Denotes a function value of Y (x) at x ═ 0, Y'x＝0Denotes the second derivative of x by Y (x) at x ═ 0, Y'x＝0Denotes the first derivative of x by Y (x) at x ═ 0, Y' ″x＝LDenotes the derivation of the third derivative of x by Y (x) at x ═ L, Y ∞x＝LDenotes the function value of Y (x) at x ═ L, Y'x＝LDenotes the second derivative of x by Y (x) at x ═ L, Y'x＝LDenotes y (x) first derivative of x at x ═ L;
(4) expanding the vibration mode function Y (x) of the inhaul cable in a Chebyshev series mode to obtain an expansion formula of Y (x):
Figure BDA0001662048190000083
in formula (7): Σ (±) denotes summing, n is the highest order of the chebyshev series, n is an integer of 1 or more, i is 1, 2, …, n; phii(x) Is an ith order displacement function satisfying the boundary condition expression (6) constructed by using a Chebyshev series, aiAs a function of the ith order shift phii(x) Is a predetermined coefficient, wherein
Φi(x)＝h(x)Ti(x)+g(x) (8)
Wherein, Ti(x) Is the ith Chebyshev series, h (x) and g (x) are constructed auxiliary functions, Ti(x) The expression of (a) is:
Figure BDA0001662048190000084
in formula (9): cos (×) represents a cosine function, arccos (×) represents an inverse cosine function;
expressions of h (x) and g (x) are respectively as follows:
Figure BDA0001662048190000091
Figure BDA0001662048190000092
Figure BDA0001662048190000093
Figure BDA0001662048190000094
Figure BDA0001662048190000095
Figure BDA0001662048190000096
(5) according to the virtual displacement principle, the work of the guy cable on the virtual displacement delta Y (x) is zero, wherein delta represents variation, and the expression of delta Y (x) is as follows:
Figure BDA0001662048190000097
the work of the cable on the virtual displacement δ y (x):
Figure BDA0001662048190000098
from equations (12) and (13), it is possible to obtain:
Figure BDA0001662048190000099
wherein phii (4)(x) Represents phii(x) Taking the fourth derivative, phi, of xi (2)(x) Represents phii(x) Taking the second derivative, Φ, of xj(x) Represents the j-th order shift function, j is 1, 2, …, n; the equation can be found after working in equation (14):
Figure BDA00016620481900000910
in formula (15):
Figure BDA00016620481900000911
Figure BDA0001662048190000101
writing the formula (15) into a matrix form, namely, the characteristic equation of the stay cable vibration is:
(D-ω2M)H＝0 (16)
in the formula (16), D represents a stiffness matrix, M represents a mass matrix, H represents a coefficient matrix, and DijRepresenting the ith row and jth column element, m, of the matrix DijThe ith row and jth column elements of matrix M are represented as:
Figure BDA0001662048190000102
Figure BDA0001662048190000103
Figure BDA0001662048190000104
(6) solving a characteristic equation (16) of the stay cable vibration to obtain a frequency equation as follows:
|D-ω2M|＝0 (17)
and | | represents a determinant of the matrix, and the frequency equation (17) is solved to obtain the vibration frequency ω of the cable under different boundary constraint conditions.

Claims (1)

1. A method for solving the bending vibration frequency of a stay cable under the elastic boundary condition is characterized by comprising the following steps:
(1) acquiring performance parameters of a cable to be detected and constructing a cable vibration partial differential equation based on the acquired performance parameters of the cable:
Figure FDA0001662048180000011
in the formula (1), x represents the vibration displacement of the stay cable to be detected, and x belongs to [0, L ]]L represents the length of the stay cable to be detected, T represents the design value of the cable force of the stay cable to be detected, E represents the elastic modulus of the stay cable to be detected, I represents the section inertia moment of the stay cable to be detected, rho represents the density of the stay cable to be detected, A represents the section area of the stay cable to be detected, T represents the vibration time of the stay cable to be detected, u (x, T) represents the displacement function of each point of the stay cable to be detected along with the change of the time T,
Figure FDA0001662048180000012
represents the second partial derivative of u (x, t) over time t;
Figure FDA0001662048180000013
represents u (x, t) taking the second partial derivative of x,
Figure FDA0001662048180000014
represents u (x, t) taking the fourth partial derivative of x;
(2) simplifying the inhaul cable vibration partial differential equation constructed in the step (1) by adopting a separation variable method, wherein the concrete simplification process is as follows:
a. decomposing u (x, t) into a product relation between a time function and a space function to obtain:
Figure FDA0001662048180000015
in the formula (2), Y (x) is a vibration mode function of the inhaul cable, sin (x) is a sine function, omega is the vibration frequency of the inhaul cable,
Figure FDA0001662048180000016
is the phase angle of the stay cable vibration;
b. substituting the formula (2) into the formula (1) to obtain a guy cable vibration differential equation:
EIY(4)(x)-TY(2)(x)-ρAω2Y(x)＝0 (3)
wherein, Y(4)(x) Denotes Y (x) taking the fourth derivative of x, Y(2)(x) Denotes Y (x) second derivative of x;
(3) establishing an elastic boundary condition of stay cable vibration:
c. the left end of the inhaul cable is assumed to be respectively connected with elastic rigidity K1And an elastic stiffness of K2The right end of the inhaul cable is respectively connected with a torsional spring with the elastic rigidity of K3And an elastic stiffness of K4Thereby obtaining:
Figure FDA0001662048180000021
in formula (4), u' gaming neutralx＝0Denotes u (x, t) as a third partial derivative of x at 0, u-x＝0Denotes the function value of u (x, t) at x ═ 0, u' -ray countx＝0Denotes u (x, t) taking the second partial derivative of x where x is 0, u'x＝0Denotes that u (x, t) is the first partial derivative of x at 0, u' ″ counting the number of cellsx＝LDenotes the calculation of the third-order partial derivative of u (x, t) with respect to x at x ═ L, u-x＝LDenotes the function value of u (x, t) at x ═ L, u' -rayx＝LDenotes u (x, t) taking the second partial derivative of x where x is L, u'x＝LDenotes the first partial derivative of x with u (x, t) at x ═ L, k1Has an elastic rigidity of K1K is a dimensionless coefficient of the stiffness of the tension spring2Has an elastic rigidity of K2K is a dimensionless coefficient of the torsional spring stiffness3Has an elastic rigidity of K3K is a dimensionless coefficient of the stiffness of the tension spring4Has an elastic rigidity of K4K is a dimensionless coefficient of the torsional spring stiffness1，k2，k3And k4The values of (A) are respectively as follows:
Figure FDA0001662048180000022
in the formula (5), the symbol "/" is a division operation symbol; k is a radical of1，k2，k3And k4All values of (1) are [0,10 ]4]And simultaneously satisfies the following three conditions: one, k1，k2，k3And k4The value of (a) cannot be 0 at the same time; two, k1，k2，k3And k4Cannot simultaneously take on a value of 104(ii) a III when k1And k3Simultaneously has a value of 104And k is2When the value of (1) is 0, k4Cannot be 104When k is1And k3Simultaneously has a value of 104，k4When the value of (1) is 0, k2Cannot be 104(ii) a d. Substituting the formula (2) into the formula (4) to obtain the elastic boundary condition of the inhaul cable:
Figure FDA0001662048180000023
in formula (6): y' cagex＝0Denotes the third derivative of x by Y (x) at x ═ 0, Y ∞x＝0Denotes a function value of Y (x) at x ═ 0, Y'x＝0Denotes the second derivative of x by Y (x) at x ═ 0, Y'x＝0Denotes the first derivative of x by Y (x) at x ═ 0, Y' ″x＝LDenotes the derivation of the third derivative of x by Y (x) at x ═ L, Y ∞x＝LDenotes the function value of Y (x) at x ═ L, Y'x＝LDenotes the second derivative of x by Y (x) at x ═ L, Y'x＝LDenotes y (x) first derivative of x at x ═ L;
(4) expanding the vibration mode function Y (x) of the inhaul cable in a Chebyshev series mode to obtain an expansion formula of Y (x):
Figure FDA0001662048180000031
in formula (7): Σ (±) denotes summing, n is the highest order of the chebyshev series, n is an integer of 1 or more, i is 1, 2, …, n; phii(x) Is an ith order displacement function satisfying the boundary condition expression (6) constructed by using a Chebyshev series, aiAs a function of the ith order shift phii(x) Is a predetermined coefficient, wherein
Φi(x)＝h(x)Ti(x)+g(x) (8)
Wherein, Ti(x) Is the ith Chebyshev series, h (x) and g (x) are constructed auxiliary functions, Ti(x) The expression of (a) is:
Figure FDA0001662048180000032
in formula (9): cos (×) represents a cosine function, arccos (×) represents an inverse cosine function;
expressions of h (x) and g (x) are respectively as follows:
Figure FDA0001662048180000033
Figure FDA0001662048180000034
Figure FDA0001662048180000035
Figure FDA0001662048180000036
Figure FDA0001662048180000037
Figure FDA0001662048180000038
(5) according to the virtual displacement principle, the work of the guy cable on the virtual displacement delta Y (x) is zero, wherein delta represents variation, and the expression of delta Y (x) is as follows:
Figure FDA0001662048180000041
the work of the cable on the virtual displacement δ y (x):
Figure FDA0001662048180000042
from equations (12) and (13), it is possible to obtain:
Figure FDA0001662048180000043
wherein phii (4)(x) Represents phii(x) Taking the fourth derivative, phi, of xi (2)(x) Represents phii(x) Taking the second derivative, Φ, of xj(x) Represents the j-th order shift function, j is 1, 2, …, n; the equation can be found after working in equation (14):
Figure FDA0001662048180000044
in formula (15):
Figure FDA0001662048180000045
Figure FDA0001662048180000046
writing the formula (15) into a matrix form, namely, the characteristic equation of the stay cable vibration is:
(D-ω2M)H＝0 (16)
in the formula (16), D represents a stiffness matrix, M represents a mass matrix, H represents a coefficient matrix, and DijRepresenting the ith row and jth column element, m, of the matrix DijThe ith row and jth column elements of matrix M are represented as:
Figure FDA0001662048180000047
Figure FDA0001662048180000048
Figure FDA0001662048180000051
(6) solving a characteristic equation (16) of the stay cable vibration to obtain a frequency equation as follows:
|D-ω2M|＝0 (17)
and | | represents a determinant of the matrix, and the frequency equation (17) is solved to obtain the vibration frequency ω of the cable under different boundary constraint conditions.
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