CN108763674A - A kind of method for solving of elastic boundary condition downhaul beam frequency - Google Patents
A kind of method for solving of elastic boundary condition downhaul beam frequency Download PDFInfo
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Abstract
The invention discloses a kind of method for solving of elastic boundary condition downhaul beam frequency, the performance parameter of drag-line to be detected is obtained first and inhaul cable vibration partial differential equation are built based on the performance parameter of the drag-line got, then the separation of variable is used to carry out abbreviation to inhaul cable vibration partial differential equation, obtain the inhaul cable vibration differential equation, establish the elastic boundary condition of inhaul cable vibration, the model function of vibration of drag-line is unfolded in the form of Chebyshev series, then the model function of vibration based on drag-line and elastic boundary condition obtain the characteristic equation of inhaul cable vibration, the characteristic equation for being finally based on inhaul cable vibration obtains the frequency equation of drag-line, it solves the frequency equation and obtains the vibration frequency of drag-line；Advantage is that accuracy is higher.
Description
Technical field
The present invention relates to a kind of method for solving of drag-line beam frequency, more particularly, under a kind of elastic boundary condition
The method for solving of drag-line beam frequency.
Background technology
Currently, in order to ensure the safe handling of bridge, it usually needs be monitored to the indices of bridge.And drag-line
Important monitoring index one of of the bending vibration as science of bridge building usually assumes that the boundary condition at drag-line both ends is ideal clamped
Or freely-supported condition is calculated the vibration frequency of drag-line.However, in fact, the boundary condition at drag-line both ends and nonideal letter
Branch and clamped, but between freely-supported and it is clamped between, such as the end of cable-stayed bridge cable is by anchorage and girder and Sarasota phase
Connection, anchorage mode is complex in engineering, and the boundary condition at drag-line both ends is assumed to clamped or simple boundary, this
It is assumed that being constrained there are larger error with actual boundary, there are larger mistakes for the drag-line beam frequency thus caused
Difference, accuracy be not high.
Invention content
Technical problem to be solved by the invention is to provide a kind of higher elastic boundary condition downhaul bendings of accuracy
The method for solving of vibration frequency.
Technical solution is used by the present invention solves above-mentioned technical problem：A kind of bending of elastic boundary condition downhaul is shaken
The method for solving of dynamic frequency, includes the following steps：
(1) performance parameter of drag-line to be detected and inclined based on the performance parameter of the drag-line got structure inhaul cable vibration is obtained
The differential equation：
In formula (1), x indicates that the vibration displacement of drag-line to be detected, x ∈ [0, L], L indicate the length of drag-line to be detected, T tables
Show that the design value of the Suo Li of drag-line to be detected, E indicate that the elasticity modulus of drag-line to be detected, I indicate that the section of drag-line to be detected is used
Property square, ρ indicate that the density of drag-line to be detected, A indicate that the sectional area of drag-line to be detected, t indicate the time of vibration of drag-line to be detected,
U (x, t) indicates each point displacement function that t changes at any time of drag-line to be detected,Indicate that u (x, t) asks two to time t
Rank partial derivative；Indicate that u (x, t) seeks second-order partial differential coefficient to x,Indicate that u (x, t) seeks quadravalence partial derivative to x；
(2) separation of variable is used to carry out abbreviation, specific abbreviation mistake to the inhaul cable vibration partial differential equation that step (1) is built
Cheng Wei：
A. u (x, t) is decomposed into the multiplication relationship between the function of time and spatial function, obtained：
Y (x) is the model function of vibration of drag-line in formula (2), and sin (*) is SIN function, and ω is the vibration frequency of drag-line,To draw
The phase angle of Suo Zhendong；
B. formula (2) is substituted into formula (1) and obtains the inhaul cable vibration differential equation：
EIY(4)(x)-TY(2)(x)-ρAω2Y (x)=0 (3)
Wherein, Y(4)(x) indicate that Y (x) asks Fourth-Derivative, Y to x(2)(x) indicate that Y (x) seeks second dervative to x；
(3) elastic boundary condition of inhaul cable vibration is established：
C. it is K to assume that the left end of drag-line is connected separately with elastic stiffness1Extension spring and elastic stiffness be K2Torsion bullet
Spring, it is K that the right end of drag-line, which is connected separately with elastic stiffness,3Extension spring and elastic stiffness be K4Torsionspring, thus
It arrives：
In formula (4), u " ' |X=0Indicate that u (x, t) seeks three rank partial derivatives to x at x=0, u |X=0Indicate at x=0 u (x,
T) functional value, u " |X=0Indicate that u (x, t) asks second-order partial differential coefficient, u'| to x at x=0X=0Indicate that u (x, t) is to x at x=0
First-order partial derivative is sought, u " ' |X=LIndicate that u (x, t) seeks three rank partial derivatives to x at x=L, u |X=LIndicate u (x, t) at x=L
Functional value, u " |X=LIndicate that u (x, t) asks second-order partial differential coefficient, u'| to x at x=LX=LIndicate that u (x, t) seeks x at x=L
First-order partial derivative, k1It is K for elastic stiffness1Extension spring rigidity nondimensionalization coefficient, k2It is K for elastic stiffness2Torsion bullet
Spring rigidity nondimensionalization coefficient, k3It is K for elastic stiffness3Extension spring rigidity nondimensionalization coefficient, k4It is K for elastic stiffness4
Torsionspring rigidity nondimensionalization coefficient, k1, k2, k3And k4Value be respectively：
In formula (5), symbol "/" is division operation symbol；k1, k2, k3And k4Value range be [0,104], and it is full simultaneously
It is enough lower three conditions：One, k1, k2, k3And k4Value cannot simultaneously be 0；Two, k1, k2, k3And k4Value cannot be simultaneously
104；Three, work as k1And k3Value simultaneously be 104, and k2Value be 0 when, k4Value cannot be 104, work as k1And k3Value
It is 10 simultaneously4, k4Value be 0 when, k2Value cannot be 104；
D. it is by the elastic boundary condition for obtaining drag-line in (2) formula substitution (4) formula：
In formula (6)：Y"'|X=0Indicate that Y (x) seeks three order derivatives to x at x=0, Y |X=0Indicate the letter of the Y (x) at x=0
Numerical value, Y " |X=0Indicate that Y (x) asks second dervative, Y'| to x at x=0X=0Indicate that Y (x) seeks first derivative to x at x=0,
Y”'|X=LIndicate that Y (x) seeks three order derivatives to x at x=L, Y |X=LIndicate the functional value of the Y (x) at x=L, Y " |X=LIt indicates
Y (x) asks second dervative, Y'| to x at x=LX=LIndicate that Y (x) seeks first derivative to x at x=L；
(4) the model function of vibration Y (x) of drag-line is unfolded in the form of Chebyshev series, obtains the expansion of Y (x)
Formula：
In formula (7)：∑ (*) expression sums to *, and n is the most high-order of Chebyshev series, and n is whole more than or equal to 1
Number, i=1,2 ..., n；Φi(x) it is to move letter using the i-th component level for meeting boundary condition formula (6) that Chebyshev series construct
Number, aiFunction phi is moved for the i-th component leveli(x) coefficient is undetermined coefficient, wherein
Φi(x)=h (x) Ti(x)+g(x) (8)
Wherein, Ti(x) it is the i-th rank Chebyshev series, h (x) and g (x) are the auxiliary function of construction, Ti(x) expression formula
For：
In formula (9)：Cos (*) indicates that cosine function, arccos (*) indicate inverse cosine function；
The expression formula of h (x) and g (x) indicates as follows respectively：
(5) according to the principle of virtual displacement enable drag-line on virtual displacement δ Y (x) made by work(be zero, wherein δ indicate variation, δ Y
(x) expression formula is：
Drag-line on virtual displacement δ Y (x) made by work(：
According to formula (12) and formula (13), can obtain：
Wherein Φi (4)(x) Φ is indicatedi(x) Fourth-Derivative, Φ are asked to xi (2)(x) Φ is indicatedi(x) second dervative, Φ are asked to xj
(x) jth rank displacement function, j=1,2 ..., n are indicated；Arrangement formula can obtain equation after (14)：
In formula (15)：
Write formula (15) as matrix form, the as characteristic equation of inhaul cable vibration：
(D-ω2M) (16) H=0
In formula (16), D indicates that stiffness matrix, M indicate that mass matrix, H indicate coefficient matrix, dijThe i-th row of representing matrix D
Jth column element, mijThe i-th row jth column element of representing matrix M, i.e.,：
(6) characteristic equation (16) of inhaul cable vibration is solved, obtaining frequency equation is：
|D-ω2M |=0 (17)
Wherein | * | the determinant of representing matrix * is solved to obtain under different boundary constraints to frequency equation (17)
The vibration frequency ω of drag-line.
Compared with the prior art, the advantages of the present invention are as follows the performance parameters and base first by obtaining drag-line to be detected
Inhaul cable vibration partial differential equation are built in the performance parameter of the drag-line got, then use the separation of variable inclined to inhaul cable vibration
The differential equation carries out abbreviation, obtains the inhaul cable vibration differential equation, establishes the elastic boundary condition of inhaul cable vibration, the elastic boundary item
The boundary condition that drag-line left and right ends are eliminated in part is clamped (k1, k2, k3And k4Value simultaneously be 104) the case where, drawing
Rope left and right ends are in free state (k1, k2, k3And k4Value simultaneously for 0) the case where, drag-line left and right ends boundary condition
It is that freely-supported (works as k1And k3Value simultaneously be 104When, k2And k4Value simultaneously for 0) the case where, drag-line left and right ends side
Boundary's condition is that clamped-freely-supported (works as k1And k3Value simultaneously be 104, k2Value be 104, k4Value be 0) the case where and
The boundary condition of drag-line left and right ends, which is that freely-supported-is clamped, (works as k1And k3Value simultaneously be 104, k2Value be 0, k4Value is
104) the case where, make the left and right ends boundary condition of drag-line be：Freely-supported-elastic boundary, elasticity-simple boundary, clamped-elastic edge
Then the model function of vibration of drag-line is used Chebyshev series by one kind in boundary, elasticity-built-in boundary and inelastic-elastic boundary
Form be unfolded, then the model function of vibration based on drag-line and elastic boundary condition obtain the characteristic equation of inhaul cable vibration, most
The characteristic equation based on inhaul cable vibration obtains the frequency equation of drag-line afterwards, solves the frequency equation and obtains the vibration frequency of drag-line,
In the method for the present invention, the frequency equation of drag-line is the model of elastic boundary constraints downhaul bending vibration, is considered simultaneously
Bending stiffness and different boundary constrain the influence to drag-line beam frequency, and use Chebyshev polynomial sides
Method (i.e. Chebyshev series) approaches drag-line shaping function, further increases the accuracy and convergence of measurement result,
Get the higher inhaul cable vibration frequency of accuracy.
Description of the drawings
Fig. 1 is the realization structure for the elastic boundary condition that the present invention is built.
Specific implementation mode
Below in conjunction with attached drawing embodiment, present invention is further described in detail.
Embodiment：A kind of method for solving of elastic boundary condition downhaul beam frequency, includes the following steps：
(1) performance parameter of drag-line to be detected and inclined based on the performance parameter of the drag-line got structure inhaul cable vibration is obtained
The differential equation：
In formula (1), x indicates that the vibration displacement of drag-line to be detected, x ∈ [0, L], L indicate the length of drag-line to be detected, T tables
Show that the design value of the Suo Li of drag-line to be detected, E indicate that the elasticity modulus of drag-line to be detected, I indicate that the section of drag-line to be detected is used
Property square, ρ indicate that the density of drag-line to be detected, A indicate that the sectional area of drag-line to be detected, t indicate the time of vibration of drag-line to be detected,
U (x, t) indicates each point displacement function that t changes at any time of drag-line to be detected,Indicate that u (x, t) asks two to time t
Rank partial derivative；Indicate that u (x, t) seeks second-order partial differential coefficient to x,Indicate that u (x, t) seeks quadravalence partial derivative to x；
(2) separation of variable is used to carry out abbreviation, specific abbreviation mistake to the inhaul cable vibration partial differential equation that step (1) is built
Cheng Wei：
A. u (x, t) is decomposed into the multiplication relationship between the function of time and spatial function, obtained：
Y (x) is the model function of vibration of drag-line in formula (2), and sin (*) is SIN function, and ω is the vibration frequency of drag-line,To draw
The phase angle of Suo Zhendong；
B. formula (2) is substituted into formula (1) and obtains the inhaul cable vibration differential equation：
EIY(4)(x)-TY(2)(x)-ρAω2Y (x)=0 (3)
Wherein, Y(4)(x) indicate that Y (x) asks Fourth-Derivative, Y to x(2)(x) indicate that Y (x) seeks second dervative to x；
(3) elastic boundary condition of inhaul cable vibration is established, as shown in Figure 1：
C. it is K to assume that the left end of drag-line is connected separately with elastic stiffness1Extension spring and elastic stiffness be K2Torsion bullet
Spring, it is K that the right end of drag-line, which is connected separately with elastic stiffness,3Extension spring and elastic stiffness be K4Torsionspring, thus
It arrives：
In formula (4), u " ' |X=0Indicate that u (x, t) seeks three rank partial derivatives to x at x=0, u |X=0Indicate at x=0 u (x,
T) functional value, u " |X=0Indicate that u (x, t) asks second-order partial differential coefficient, u'| to x at x=0X=0Indicate that u (x, t) is to x at x=0
First-order partial derivative is sought, u " ' |X=LIndicate that u (x, t) seeks three rank partial derivatives to x at x=L, u |X=LIndicate u (x, t) at x=L
Functional value, u " |X=LIndicate that u (x, t) asks second-order partial differential coefficient, u'| to x at x=LX=LIndicate that u (x, t) seeks x at x=L
First-order partial derivative, k1It is K for elastic stiffness1Extension spring rigidity nondimensionalization coefficient, k2It is K for elastic stiffness2Torsion bullet
Spring rigidity nondimensionalization coefficient, k3It is K for elastic stiffness3Extension spring rigidity nondimensionalization coefficient, k4It is K for elastic stiffness4
Torsionspring rigidity nondimensionalization coefficient, k1, k2, k3And k4Value be respectively：
In formula (5), symbol "/" is division operation symbol；k1, k2, k3And k4Value range be [0,104], and it is full simultaneously
It is enough lower three conditions：One, k1, k2, k3And k4Value cannot simultaneously be 0；Two, k1, k2, k3And k4Value cannot be simultaneously
104；Three, work as k1And k3Value simultaneously be 104, and k2Value be 0 when, k4Value cannot be 104, work as k1And k3Value
It is 10 simultaneously4, k4Value be 0 when, k2Value cannot be 104；D. (2) formula is substituted into (4) formula and obtains the elastic edge of drag-line
Boundary's condition is：
In formula (6)：Y"'|X=0Indicate that Y (x) seeks three order derivatives to x at x=0, Y |X=0Indicate the letter of the Y (x) at x=0
Numerical value, Y " |X=0Indicate that Y (x) asks second dervative, Y'| to x at x=0X=0Indicate that Y (x) seeks first derivative to x at x=0,
Y”'|X=LIndicate that Y (x) seeks three order derivatives to x at x=L, Y |X=LIndicate the functional value of the Y (x) at x=L, Y " |X=LIt indicates
Y (x) asks second dervative, Y'| to x at x=LX=LIndicate that Y (x) seeks first derivative to x at x=L；
(4) the model function of vibration Y (x) of drag-line is unfolded in the form of Chebyshev series, obtains the expansion of Y (x)
Formula：
In formula (7)：∑ (*) expression sums to *, and n is the most high-order of Chebyshev series, and n is whole more than or equal to 1
Number, i=1,2 ..., n；Φi(x) it is to move letter using the i-th component level for meeting boundary condition formula (6) that Chebyshev series construct
Number, aiFunction phi is moved for the i-th component leveli(x) coefficient is undetermined coefficient, wherein
Φi(x)=h (x) Ti(x)+g(x) (8)
Wherein, Ti(x) it is the i-th rank Chebyshev series, h (x) and g (x) are the auxiliary function of construction, Ti(x) expression formula
For：
In formula (9)：Cos (*) indicates that cosine function, arccos (*) indicate inverse cosine function；
The expression formula of h (x) and g (x) indicates as follows respectively：
(5) according to the principle of virtual displacement enable drag-line on virtual displacement δ Y (x) made by work(be zero, wherein δ indicate variation, δ Y
(x) expression formula is：
Drag-line on virtual displacement δ Y (x) made by work(：
According to formula (12) and formula (13), can obtain：
Wherein Φi (4)(x) Φ is indicatedi(x) Fourth-Derivative, Φ are asked to xi (2)(x) Φ is indicatedi(x) second dervative, Φ are asked to xj
(x) jth rank displacement function, j=1,2 ..., n are indicated；Arrangement formula can obtain equation after (14)：
In formula (15)：
Write formula (15) as matrix form, the as characteristic equation of inhaul cable vibration：
(D-ω2M) (16) H=0
In formula (16), D indicates that stiffness matrix, M indicate that mass matrix, H indicate coefficient matrix, dijThe i-th row of representing matrix D
Jth column element, mijThe i-th row jth column element of representing matrix M, i.e.,：
(6) characteristic equation (16) of inhaul cable vibration is solved, obtaining frequency equation is：
|D-ω2M |=0 (17)
Wherein | * | the determinant of representing matrix * is solved to obtain under different boundary constraints to frequency equation (17)
The vibration frequency ω of drag-line.
Claims (1)
1. a kind of method for solving of elastic boundary condition downhaul beam frequency, it is characterised in that include the following steps：
(1) it obtains the performance parameter of drag-line to be detected and inhaul cable vibration partial differential is built based on the performance parameter of the drag-line got
Equation：
In formula (1), x indicates that the vibration displacement of drag-line to be detected, x ∈ [0, L], L indicate that the length of drag-line to be detected, T expressions wait for
The design value of the Suo Li of drag-line is detected, E indicates that the elasticity modulus of drag-line to be detected, I indicate the cross sectional moment of inertia of drag-line to be detected,
ρ indicates the density of drag-line to be detected, and A indicates the sectional area of drag-line to be detected, and t indicates the time of vibration of drag-line to be detected, u (x,
T) each point displacement function that t changes at any time of drag-line to be detected is indicated,Indicate that u (x, t) seeks Second Order Partial to time t
Derivative；Indicate that u (x, t) seeks second-order partial differential coefficient to x,Indicate that u (x, t) seeks quadravalence partial derivative to x；
(2) separation of variable is used to carry out abbreviation, specific abbreviation process to the inhaul cable vibration partial differential equation that step (1) is built
For：
A. u (x, t) is decomposed into the multiplication relationship between the function of time and spatial function, obtained：
Y (x) is the model function of vibration of drag-line in formula (2), and sin (*) is SIN function, and ω is the vibration frequency of drag-line,It shakes for drag-line
Dynamic phase angle；
B. formula (2) is substituted into formula (1) and obtains the inhaul cable vibration differential equation：
EIY(4)(x)-TY(2)(x)-ρAω2Y (x)=0 (3)
Wherein, Y(4)(x) indicate that Y (x) asks Fourth-Derivative, Y to x(2)(x) indicate that Y (x) seeks second dervative to x；
(3) elastic boundary condition of inhaul cable vibration is established：
C. it is K to assume that the left end of drag-line is connected separately with elastic stiffness1Extension spring and elastic stiffness be K2Torsionspring,
It is K that the right end of drag-line, which is connected separately with elastic stiffness,3Extension spring and elastic stiffness be K4Torsionspring, thus obtain：
In formula (4), u " ' |X=0Indicate that u (x, t) seeks three rank partial derivatives to x at x=0, u |X=0Indicate the u (x, t) at x=0
Functional value, u " |X=0Indicate that u (x, t) asks second-order partial differential coefficient, u'| to x at x=0X=0Indicate that u (x, t) asks one to x at x=0
Rank partial derivative, u " ' |X=LIndicate that u (x, t) seeks three rank partial derivatives to x at x=L, u |X=LIndicate the letter of the u (x, t) at x=L
Numerical value, u " |X=LIndicate that u (x, t) asks second-order partial differential coefficient, u'| to x at x=LX=LIndicate that u (x, t) seeks single order to x at x=L
Partial derivative, k1It is K for elastic stiffness1Extension spring rigidity nondimensionalization coefficient, k2It is K for elastic stiffness2Torsionspring it is rigid
Spend nondimensionalization coefficient, k3It is K for elastic stiffness3Extension spring rigidity nondimensionalization coefficient, k4It is K for elastic stiffness4Torsion
Turn spring rate nondimensionalization coefficient, k1, k2, k3And k4Value be respectively：
In formula (5), symbol "/" is division operation symbol；k1, k2, k3And k4Value range be [0,104], and meet simultaneously with
Lower three conditions：One, k1, k2, k3And k4Value cannot simultaneously be 0；Two, k1, k2, k3And k4Value cannot simultaneously be 104；
Three, work as k1And k3Value simultaneously be 104, and k2Value be 0 when, k4Value cannot be 104, work as k1And k3Value simultaneously
It is 104, k4Value be 0 when, k2Value cannot be 104；D. (2) formula is substituted into (4) formula and obtains the elastic boundary item of drag-line
Part is：
In formula (6)：Y"'|X=0Indicate that Y (x) seeks three order derivatives to x at x=0, Y |X=0Indicate the functional value of the Y (x) at x=0,
Y”|X=0Indicate that Y (x) asks second dervative, Y'| to x at x=0X=0Indicate that Y (x) seeks first derivative to x at x=0, Y " ' |X=L
Indicate that Y (x) seeks three order derivatives to x at x=L, Y |X=LIndicate the functional value of the Y (x) at x=L, Y " |X=LIt indicates at x=L
Y (x) asks second dervative, Y'| to xX=LIndicate that Y (x) seeks first derivative to x at x=L；
(4) the model function of vibration Y (x) of drag-line is unfolded in the form of Chebyshev series, obtains the expansion of Y (x)：
In formula (7)：∑ (*) expression sums to *, and n is the most high-order of Chebyshev series, and n is the integer more than or equal to 1, i
=1,2 ..., n；Φi(x) it is the i-th rank displacement function for meeting boundary condition formula (6) constructed using Chebyshev series, aiFor
I-th component level moves function phii(x) coefficient is undetermined coefficient, wherein
Φi(x)=h (x) Ti(x)+g(x) (8)
Wherein, Ti(x) it is the i-th rank Chebyshev series, h (x) and g (x) are the auxiliary function of construction, Ti(x) expression formula is：
In formula (9)：Cos (*) indicates that cosine function, arccos (*) indicate inverse cosine function；
The expression formula of h (x) and g (x) indicates as follows respectively：
(5) according to the principle of virtual displacement enable drag-line on virtual displacement δ Y (x) made by work(be zero, wherein δ indicate variation, δ Y's (x)
Expression formula is：
Drag-line on virtual displacement δ Y (x) made by work(：
According to formula (12) and formula (13), can obtain：
Wherein Φi (4)(x) Φ is indicatedi(x) Fourth-Derivative, Φ are asked to xi (2)(x) Φ is indicatedi(x) second dervative, Φ are asked to xj(x)
Indicate jth rank displacement function, j=1,2 ..., n；Arrangement formula can obtain equation after (14)：
In formula (15)：
Write formula (15) as matrix form, the as characteristic equation of inhaul cable vibration：
(D-ω2M) (16) H=0
In formula (16), D indicates that stiffness matrix, M indicate that mass matrix, H indicate coefficient matrix, dijThe i-th row jth of representing matrix D
Column element, mijThe i-th row jth column element of representing matrix M, i.e.,：
(6) characteristic equation (16) of inhaul cable vibration is solved, obtaining frequency equation is：
|D-ω2M |=0 (17)
Wherein | * | the determinant of representing matrix * solves frequency equation (17) to obtain different boundary constraints downhaul
Vibration frequency ω.
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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 |
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