CN109540349A - A kind of consideration many factors influence sunpender tension recognition methods - Google Patents
A kind of consideration many factors influence sunpender tension recognition methods Download PDFInfo
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- G01L—MEASURING FORCE, STRESS, TORQUE, WORK, MECHANICAL POWER, MECHANICAL EFFICIENCY, OR FLUID PRESSURE
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Abstract
The invention belongs to technical field of building construction, in particular a kind of consideration many factors influence the recognition methods of sunpender tension, the building of sunpender tension expression formula under the conditions of compound boundary, consider bending stiffness, the sunpender dynamic response of rotational restraint and elastic bearing, sunpender useful calculating method derives, the identification of boundary condition, based on Euler-Bernoulli beam theory, establish consideration rotational restraint, sunpender model of vibration under the influence of the boundary conditions such as bending stiffness and elastic bearing, and the analytical expression of sunpender tension Yu oscillation crosswise frequency is derived, since boundary condition specific value has uncertainty, therefore rotational restraint is identified using difference value equation, calculate the embedded coefficient B1 of reflection rotational restraint, B2, and elastic bearing is utilized multistage vibration frequency and is identified, calculate reflection elasticity branch The parameter k held calculates sunpender tension using sunpender tension and the analytical expression of oscillation crosswise frequency according to parameter recognition result.
Description
Technical field
The invention belongs to technical field of building construction, and in particular to a kind of consideration many factors influence sunpender tension identification side
Method.
Background technique
Sunpender is the important component of middle lower bearing arch bridge, assumes responsibility for load transmitting between tie-rod, bridge floor and arch rib
Effect, sunpender is also the position for being easiest to occur damaging in arch bridge, and damage influences greatly, due to hanging bridge overall stability
Bar construction is special, and stress and damage are difficult to detect by appearance, and the method for currently used test tension mainly has pressure gauge survey
Determine method, determination of pressure sensor method, magnetic flux method and vibration frequency method, pressure gauge measuring method and the test of determination of pressure sensor method
Precision is higher, the sunpender tension test suitable for work progress, is not suitable for having completed the sunpender measurement of stretching construction, and
Test equipment is heavy, inconvenient, and magnetic flux method is needed pre-buried and demarcated in advance, is mostly used for small-scale important bridge health prison
It surveys, somewhat expensive, vibration frequency method measuring instrument carries simplicity, and easily operated, efficiency is higher, is current sunpender tonometry
Main method, principle are to pick up vibration signal of the sunpender when being activated using accurate vibration pickup, and sensor records time-histories
Data obtain vibration frequency by filtering, amplification and spectrum analysis, further according to the number between sunpender eigentone and tension
Relationship calculates sunpender tension.
Often assume that sunpender boundary condition is hinged or affixed in existing research, but boom end boundary condition is generally bullet
Property bearing, sunpender rotational restraint is not zero, nor infinitely great, but tends to certain intermediate value, in addition to it is durable to improve sunpender
Property, extend sunpender service life, multi-layer anticorrosion system, including epoxy coating and sticky sheath can be often done outside sunpender, is surveying
When trying sunpender tension, what vibration pickup was surveyed is the fundamental frequency that steel strand wires or steel wire and sticky sheath vibrate jointly, therefore is calculating sunpender
When rigidity and line density, it is necessary to consider the influence of sticky sheath.
Summary of the invention
To solve the problems mentioned above in the background art.The present invention provides a kind of consideration many factors to influence sunpender
Power recognition methods.
To achieve the above object, the invention provides the following technical scheme: the following steps are included:
The building of sunpender tension expression formula under the conditions of S1, compound boundary；
S2, the sunpender dynamic response for considering bending stiffness, rotational restraint and elastic bearing；
S3, sunpender useful calculating method derive；
The identification of S4, boundary condition.
Preferably, V (x, t) is the sunpender lateral displacement function in the S1, and x is longitudinal coordinate, and t is the time, and F is to hang
Bar tension, if the sunpender model function of vibration isAmplitude is indicated by generalized coordinates Z (t).
Preferably, the sunpender model function of vibration may be expressed as:
In formula:
Preferably, the C1、C2、C3、C4For undetermined coefficient, built according to the boundary condition of the sunpender tension computation model
Vertical equation group, to make equation have untrivialo solution, then its undetermined coefficient C1、C2、C3、C4The determinant constituted should be 0.
Preferably, according to the equilibrium condition for acting on the boom end power and torque, mould is calculated by the sunpender tension
Type can derive the sunpender boundary condition equation group are as follows:
It can be obtained by sunpender model function of vibration formula (2) arrangement:
Formula (9) is the sunpender tension and oscillation crosswise when considering that bending stiffness, elastic embedding and elastic bearing influence
Frequency resolution expression formula after determining the sunpender parameter, then tests the sunpender frequency, can calculate the sunpender by the formula
Tension.
Preferably, when considering the sunpender both ends elastic bearing, the sunpender vibration, the support will also shake
It is dynamic, it is assumed that sunpender both ends elastic supporting rigidity is respectively K1And K2, the vibration of supports, which is displaced, to be respectively as follows:
V|X=0=f1(t) (10)
V|X=L=f2(t) (11)
According to sunpender oscillatory differential equation it is found that sunpender boundary condition is converted into inhomogeneous boundary condition, need to carry out letter
Number replacement, it may be assumed that
According to formula (13) it is found that considering the sunpender of elastic bearing vibration, vibrational state is equivalent to external force p (x, t) effect
On the fixed sunpender in both ends.
Boundary condition is homogeneous boundary condition, can establish equation group according to the boundary condition of sunpender tension computation model are as follows:
v|X=0=0 (14)
v|X=L=0 (15)
K3v′|X=0-EIv″|X=0=0 (16)
K4v′|X=L+EIv″|X=L=0 (17)
Since each rank mode of elastomer has orthogonality, therefore the technique study elastomer of model analysis can be used to outside
The response of excitation.Enable the solution of equation (13) for the linear superposition of each rank principal mode of system, it may be assumed that
In formula, viIt (x) is the i-th rank mode of sunpender oscillation crosswise, Zi(t) generalized coordinates to react each rank mode amplitude.
Preferably, if f1And f (t)=02(t)=0, i.e., the described support does not generate vibration, ignores the influence of elastic bearing, this
When research external force p (x, t)=0 the fixed sunpender free vibration in both ends,
Formula (2) are substituted into formula (14)~formula (17), can be obtained:
To carry out no quantization processing to formula (19), introducing dimensionless group convenient for solving:
In formula, fnFor the n-th order vibration frequency of sunpender, It is managed for the n-th order vibration frequency of desired tension string
By value,Kx1With Kx2For embedded coefficient,
Formula (20) substitution formula (19) frequency equation can be converted are as follows:
Formula (21) is sunpender tension analytical expression when considering rotational restraint, bending stiffness and sticky sheath,
When sunpender both ends are hinged boundary, K1=0, K2=0, then the relational expression of sunpender tension and frequency are as follows:
When sunpender both ends are affixed boundary, K1=∞, K2=∞, the then relational expression of sunpender tension and frequency are as follows:
ξsin(αL)sinh(βL)+2nπηn=0 (23) [1-cos (α L) cosh (β L)]
When the orthogonality for considering the system vibration shape, formula (13) can abbreviation are as follows:
In formula, MnFor generalized mass, PnIt (t) is generalized force, ωnFor the sunpender natural frequency of vibration, it is respectively as follows:
For not considering that the sunpender of vibration of supports, each first order mode are desirable are as follows:
Generalized mass and generalized force are as follows:
The Duhamel Integral Solution of equation (27) are as follows:
Analysis is balanced to elastic bearing according to sunpender tension computation model, can be obtained:
It can be obtained by the arrangement of formula (15), formula (32) and formula (33):
Wushu (34) and formula (35) substitute into formula (30) and can obtain:
Work as n=1, when 3,5 ..., formula (36) substitution formula (27) can be obtained:
The vibration frequency of sunpender when can arrange to obtain consideration vibration of supports according to formula (38)
When considering that bending stiffness, elastic embedding and elastic bearing influence, can be obtained according to formula (24):
Formula (40) are substituted into formula (38), and are enabledArrangement can obtain:
Formula (42) can be obtained by equation solution, in which:
Work as n=2, when 4,6 ..., formula (37) substitution formula (27) can similarly be obtained:
The same formula of solution (42) of equation (44), in which:
Formula (42) is sunpender tension practical expression, it is contemplated that the influence of bending stiffness, rotational restraint and elastic bearing, and
It is comparably display expression formula with formula (12), it is practical.
When known to boundary condition, formula (42) can directly be utilized to solve sunpender tension, but Practical Project middle hanger perimeter strip
Part B1, B2, K1, K2 are often unknowable, need the identification of advanced row bound condition.In addition for middle half/through arch bridge suspender and
Speech, sag influence can be ignored, but because the influence of bending stiffness can increase with the raising of frequency order, it is therefore desirable to examine
Consider the influence of bending stiffness.
For K1, K2, that is, consider the influence of elastic bearing, enablesSubstitution formula (42):
Parameter k indicates the influence that elastic bearing calculates sunpender tension, and formula (46) can arrange are as follows:
F2+(k+B2n2λ-3B1bn)F+(2B1n2bnλ-B1bnk+B2n2K λ)=0 (47)
Take 4 order frequency ωn1, ωn2, ωn3, ωn4Substitution formula (47) arrangement can obtain unique solution:
For B1, B2, that is, the influence of rotational restraint is considered, by formula (22) and formula (25) it is found that working as sunpender both ends
When boundary is hinged, B is taken1=1, B2=1；When sunpender both ends boundary is affixed, B is taken1=0.9612, B2=2.8245；But it is real
In the engineering of border, sunpender both ends boundary condition between it is hinged with it is affixed between, can use intermediate value B1=0.9806, B2=1.9123 substitute into
Formula (46) first estimates sunpender tension Fg, enableSubstitution formula (40) arrangement can obtain:
Then by FgSubstitution formula (49) replaces F, calculates separately sunpender both ends boundary and is hinged and is fixed in sunpender tension FgMake
Vibration frequency underAccording to vibration frequency ωn、 It establishes under the influence of interpolation formula calculating rotational restraint
Edge effect coefficient B 1 and B2.If sunpender tension after iteration and estimating sunpender tension FgHave big difference, can carry out secondary
Iteration, but in practical engineering applications, an iteration can be met the requirements, interpolation formula are as follows:
ω in formulanFor sunpender practical frequency.
Using the identification of boundary condition, the numerical value for cleverly having evaded sunpender rotational restraint and elastic bearing is calculated, and is passed through
The influence that parameter expression boundary condition identifies sunpender tension.By formula (48), formula (50) and formula (51) substitute into formula (46) and can obtain
To the sunpender tension value when considering rotational restraint, elastic bearing and bending stiffness.
Compared with prior art, the beneficial effects of the present invention are:
1, it the present invention is based on Euler-Bernoulli beam theory, establishes and considers rotational restraint, bending stiffness and elasticity branch
The sunpender model of vibration under the influence of equal boundary conditions is held, and has derived the analytical expression of sunpender tension Yu oscillation crosswise frequency,
Since boundary condition specific value has uncertainty, therefore rotational restraint is identified using difference value equation, calculates reflection
Embedded coefficient B1, B2 of rotational restraint, and elastic bearing is utilized multistage vibration frequency and is identified, calculates reflection elasticity branch
The parameter k held, according to parameter recognition result, using sunpender tension and the analytical expression of oscillation crosswise frequency to sunpender tension into
Row calculates.
2, the present invention has derived the sunpender for considering that rotational restraint, bending stiffness and elastic bearing influence according to boundary condition
Power implied expression formula, it is complicated due to calculating, therefore derive that sunpender tension shows expression formula using sunpender dynamic response, and carry out pair
Than analysis shows, consider sunpender tension identification error that rotational restraint, bending stiffness and elastic bearing influence less than 1%, it is practical
Property is strong, and precision is higher.
3, the influence that the present invention derives that sunpender both ends rotational restraint and elastic bearing identify sunpender tension be can not ignore, right
For Yu Liujiang booms on large bridge end restraint, in practice between hinged and affixed, but relative to hinged, affixed constraint is more smart
Really.
Detailed description of the invention
Attached drawing is used to provide further understanding of the present invention, and constitutes part of specification, with reality of the invention
It applies example to be used to explain the present invention together, not be construed as limiting the invention.In the accompanying drawings:
Fig. 1 is sunpender section structure diagram.
Specific embodiment
Following will be combined with the drawings in the embodiments of the present invention, and technical solution in the embodiment of the present invention carries out clear, complete
Site preparation description, it is clear that described embodiments are only a part of the embodiments of the present invention, instead of all the embodiments.It is based on
Embodiment in the present invention, it is obtained by those of ordinary skill in the art without making creative efforts every other
Embodiment shall fall within the protection scope of the present invention.
Embodiment
Using Zhengzhou Liu Jiang bridge as background, bridge crane bar model PESC7-091, according to national standard " cable-stayed bridge hot extrusion
Polyethylene dead-hard steel wire drag-line technical conditions " sunpender parameters such as table 1 known to (GB/T18365-2001), steel tendon density
For 7850kg/m3；Bending stiffness EI=217120Nm2；High density polyethylene (HDPE) sheath, the black of use are wrapped with outside sunpender
Internal layer colour outer bilayer structure (c-type), sunpender section are as shown in the figure:
The parameters of 1 sunpender of table
In order to verify the accuracy of sunpender tension calculation expression, using the sunpender of three kinds of different lengths in Liu Jiang bridge,
Using the oscillation crosswise frequency of sunpender first time tensioning and second of tensioning, the sunpender tension of three kinds of different lengths is identified.Sunpender
The preceding 4 rank vibration frequency of first time tensioning and second of tensioning is listed in table 2 and table 3.
Each rank vibration frequency of 2 three kinds of sunpender first time tensioning of table
| Sunpender number | Length | Tension | f1 | f2 | f3 | f4 |
| 1 | 23.458 | 500 | 2.9297 | 5.9766 | 9.1406 | 12.6563 |
| 2 | 22.585 | 500 | 3.0256 | 6.0917 | 9.3430 | 12.9248 |
| 3 | 20.827 | 500 | 3.3203 | 6.4453 | 9.9609 | 13.7500 |
Each rank vibration frequency of table 3 three kinds of sunpenders, second of tensioning
Boundary condition is identified according to table 2 and table 3, in Practical Project, sunpender both ends boundary condition between hingedly with
Between affixed, intermediate value B can use1=0.9806, B2=1.9123, which substitute into formula (48) and formula (46), calculates separately parameter FgAnd k, then
The elastic embedding coefficient and reflection bullet of formula (46), formula (50) and formula (51) calculating different length sunpender are substituted into according to numerical parameter
The parameter k of property supporting constraint, calculated result such as table 4.
4 sunpender parameter recognition result of table
Sunpender tension can be identified according to table 1 and 4 sunpender parameter of table, the results are shown in Table 5 and table 6.
The 1st tensioning tension recognition result of 5 sunpender of table
| Sunpender number | Length | Design value | Formula (22) | Relative error | Formula (25) | Relative error | Formula (46) | Relative error |
| 1 | 23.458 | 500 | 570.436 | 14.09 | 541.047 | 8.21 | 503.963 | 0.79 |
| 2 | 22.585 | 500 | 563.601 | 12.72 | 533.905 | 6.78 | 502.561 | 0.51 |
| 3 | 20.827 | 500 | 576.549 | 15.31 | 544.974 | 8.99 | 501.137 | 0.23 |
The 2nd tensioning tension recognition result of 6 sunpender of table
| Sunpender number | Length | Design value | Formula (22) | Relative error | Formula (25) | Relative error | Formula (46) | Relative error |
| 1 | 23.458 | 1150 | 1143.31 | -0.58 | 1091.70 | -5.07 | 1153.71 | 0.32 |
| 2 | 22.585 | 1150 | 1269.36 | 10.38 | 1212.28 | 5.42 | 1156.79 | 0.59 |
| 3 | 20.827 | 1150 | 1272.83 | 10.68 | 1214.24 | 5.59 | 1156.31 | 0.55 |
From table 5 and table 6 it is found that being calculated for Liu Jiang booms on large bridge tension, when using hinged calculation formula (22), sunpender the
1 time tensioning tension calculating trueness error is larger, and up to 15.31%；The 2nd tensioning tension of sunpender calculates, for long sunpender
For can meet required precision, but in, short steeve computational accuracy error it is relatively large, up to 10.68%.Work as use
When affixed sunpender tension calculation formula (25), the 1st tensioning of sunpender and the 2nd tensioning tension computational accuracy are greater than only hinged
Calculation formula (22) computational accuracy, precision maximum improve 6.32%；When using consideration bending stiffness, rotational restraint and elasticity branch
When holding calculation formula (46) of influence, sunpender tension computational accuracy is significantly improved, and sunpender the 1st time and the 2nd tensioning tension calculate
Trueness error is up to 0.79%, meets requirement of engineering precision, and for hinged and is affixed, maximal accuracy is improved
8.76%, there is stronger accuracy.
Finally, it should be noted that the foregoing is only a preferred embodiment of the present invention, it is not intended to restrict the invention,
Although the present invention is described in detail referring to the foregoing embodiments, for those skilled in the art, still may be used
To modify the technical solutions described in the foregoing embodiments or equivalent replacement of some of the technical features.
All within the spirits and principles of the present invention, any modification, equivalent replacement, improvement and so on should be included in of the invention
Protection scope.
Claims (7)
1. a kind of consideration many factors influence the recognition methods of sunpender tension, it is characterised in that: the following steps are included:
The building of sunpender tension expression formula under the conditions of S1, compound boundary；
S2, the sunpender dynamic response for considering bending stiffness, rotational restraint and elastic bearing；
S3, sunpender useful calculating method derive；
The identification of S4, boundary condition.
2. a kind of consideration many factors according to claim 1 influence the recognition methods of sunpender tension, it is characterised in that: described
V (x, t) is the sunpender lateral displacement function in S1, and x is longitudinal coordinate, and t is the time, and F is sunpender tension, if the sunpender shakes
Type function isAmplitude is indicated by generalized coordinates Z (t).
3. a kind of consideration many factors according to claim 1 influence the recognition methods of sunpender tension, it is characterised in that: described
Sunpender model function of vibration may be expressed as:
In formula:
4. a kind of consideration many factors according to claim 3 influence the recognition methods of sunpender tension, it is characterised in that: described
C1、C2、C3、C4For undetermined coefficient, equation group is established according to the boundary condition of the sunpender tension computation model, to there is equation
Untrivialo solution, then its undetermined coefficient C1、C2、C3、C4The determinant constituted should be 0.
5. a kind of consideration many factors according to claim 1 influence the recognition methods of sunpender tension, it is characterised in that: according to
The equilibrium condition for acting on the boom end power and torque can derive the sunpender side by the sunpender tension computation model
Boundary's conditional equation group are as follows:
It can be obtained by sunpender model function of vibration formula (2) arrangement:
Formula (9) is the sunpender tension and oscillation crosswise frequency when considering that bending stiffness, elastic embedding and elastic bearing influence
Analytical expression after determining the sunpender parameter, then tests the sunpender frequency, can calculate the sunpender by the formula
Power.
6. a kind of consideration many factors according to claim 1 influence the recognition methods of sunpender tension, it is characterised in that: when examining
When considering the sunpender both ends elastic bearing, the sunpender vibration, the support will also vibrate, it is assumed that sunpender both ends bullet
Property support stiffness be respectively K1And K2, the vibration of supports, which is displaced, to be respectively as follows:
V|X=0=f1(t) (10)
V|X=L=f2(t) (11)
According to sunpender oscillatory differential equation it is found that sunpender boundary condition is converted into inhomogeneous boundary condition, need to carry out function generation
It changes, it may be assumed that
According to formula (13) it is found that considering that the sunpender of elastic bearing vibration, vibrational state are equivalent to external force p (x, t) and act on two
It holds on fixed sunpender.
Boundary condition is homogeneous boundary condition, can establish equation group according to the boundary condition of sunpender tension computation model are as follows:
v|X=0=0 (14)
v|X=L=0 (15)
K3v′|X=0-EIv″|X=0=0 (16)
K4v′|X=L+EIv″|X=L=0 (17)
Since each rank mode of elastomer has orthogonality, therefore the technique study elastomer of model analysis can be used to external drive
Response.Enable the solution of equation (13) for the linear superposition of each rank principal mode of system, it may be assumed that
In formula, viIt (x) is the i-th rank mode of sunpender oscillation crosswise, Zi(t) generalized coordinates to react each rank mode amplitude.
7. a kind of consideration many factors according to claim 1 influence the recognition methods of sunpender tension, it is characterised in that: if f1
And f (t)=02(t)=0, i.e., the described support does not generate vibration, ignores the influence of elastic bearing, and research external force p (x, t) at this time=
The fixed sunpender free vibration in 0 both ends,
Formula (2) are substituted into formula (14)~formula (17), can be obtained:
To carry out no quantization processing to formula (19), introducing dimensionless group convenient for solving:
In formula, fnFor the n-th order vibration frequency of sunpender, For the n-th order vibration frequency theoretical value of desired tension string,Kx1With Kx2For embedded coefficient,
Formula (20) substitution formula (19) frequency equation can be converted are as follows:
Formula (21) is sunpender tension analytical expression when considering rotational restraint, bending stiffness and sticky sheath,
When sunpender both ends are hinged boundary, K1=0, K2=0, then the relational expression of sunpender tension and frequency are as follows:
When sunpender both ends are affixed boundary, K1=∞, K2=∞, the then relational expression of sunpender tension and frequency are as follows:
ξsin(αL)sinh(βL)+2nπηn=0 (23) [1-cos (α L) cosh (β L)]
When the orthogonality for considering the system vibration shape, formula (13) can abbreviation are as follows:
In formula, MnFor generalized mass, PnIt (t) is generalized force, ωnFor the sunpender natural frequency of vibration, it is respectively as follows:
For not considering that the sunpender of vibration of supports, each first order mode are desirable are as follows:
Generalized mass and generalized force are as follows:
The Duhamel Integral Solution of equation (27) are as follows:
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Cited By (1)
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