CN107679363B - Method for determining stability critical force of n-order telescopic boom of crane - Google Patents

Method for determining stability critical force of n-order telescopic boom of crane Download PDF

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CN107679363B
CN107679363B CN201710928290.XA CN201710928290A CN107679363B CN 107679363 B CN107679363 B CN 107679363B CN 201710928290 A CN201710928290 A CN 201710928290A CN 107679363 B CN107679363 B CN 107679363B
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critical force
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姚峰林
孟文俊
赵婕
张文军
佘占蛟
石国善
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Taiyuan University of Science and Technology
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Abstract

The invention discloses a method for determining stability critical force of an n-order telescopic boom of a crane, relates to a recursion formula and a numerical solution of the critical force of an arm support of a telescopic boom type engineering crane, and belongs to the field of structural design of large-scale mechanical equipment. The invention comprises the following steps: step 1, establishing an n-order ladder column model and deducing an instability characteristic equation; step 2, deducing a recursion transcendental equation of the instability characteristic equation in the step 1; step 3, solving a numerical solution of the transcendental equation in the step 2; step 4, solving the critical force P according to the numerical solution; step 5, obtaining the length coefficient mu of the n-step ladder column by using the critical force determined in the step 42And the practical engineering problem is convenient to solve. The invention aims to provide a method for determining the critical force of a telescopic arm of an n-step ladder column crane, which can accurately determine the maximum lifting capacity of the telescopic arm type crane, and can reduce the use of materials and the self weight, thereby realizing the lightweight design of the telescopic arm.

Description

Method for determining stability critical force of n-order telescopic boom of crane
Technical Field
The invention relates to a recurrence formula and a numerical solution of a critical force of an arm support of a telescopic arm type engineering crane, in particular to a method for determining a stability critical force of an n-order telescopic arm of the crane, and belongs to the field of structural design of large-scale mechanical equipment.
Background
In recent years, with the use of a large amount of high-strength steel and the maturity of an automatic single-cylinder bolt technology, the multi-section box-shaped telescopic boom is widely applied to a large engineering crane. With the continuous increase of the height of the steel structure, the sectional area is necessarily correspondingly increased, and in order to reasonably utilize materials and reduce weight, the section of the beam column must be changed along the axial direction; the ladder column model is widely applied to various mechanical equipment such as engineering cranes, high-altitude rescue vehicles and the like due to the advantages of easy telescopic change of the length, high bearing capacity, high material utilization rate and the like, and the stability of the ladder column model becomes the key point of study of scholars. The Timoshenko carries out more in-depth research and analysis on the step column model, and calculates the critical force of the high-order step column by using an energy method. In China, researchers also use an energy method and a litz method to research the stair columns by means of a preset approximate deflection curve, but the method is equivalent to introduce additional constraint, and particularly, larger errors can be generated for more than four-order stair columns. Liuqingtan and the like use a transfer matrix method to research the critical force of the ladder columns, but the method can only be used for the ladder columns with fewer orders at present, and the method can also generate larger errors when used for the ladder columns with higher orders. With the development and the improvement of a finite element theory, the continental thought force and the like use an accurate finite element method to research the critical force of the ladder column and form the currently used national standard GB3811-2008 (hereinafter abbreviated as GB3811), but the algorithm considers the axial force action of the ladder column, so that an axial force coefficient and a rigidity matrix are introduced in the calculation, and for the ladder column with a higher order, the solution of an equation is very difficult or even can not be solved.
The existing national standard of China adopts a precise computation method for stability analysis of the ladder column based on a precise finite element method, but for a multi-step ladder column, the method has quite large rigidity matrix and complex characteristic equation, and for the stability precise numerical solution of the ladder column in a common support form, the method is usually obtained by a trial and error method, so that the computation amount is huge, and certain difficulty is brought to practical application. In the current design specifications, a graph is used to show the length coefficient in the case of a part of special combinations of the staircase columns of 2 to 5 steps. When the order is more than 5, the method cannot be applied, and the linear interpolation method is used in non-specific combination, but the error of the linear interpolation is not given. And if the product exceeds 5 grades, manufacturers can only use commercial software to carry out design calculation, and the risk is extremely high. As is known, the orders of telescopic arm supports of the models including Liebherr LTM 11200 in Germany, Xuxu God QAY1200, SANY SAC12000 in the third and fourth industries and QAY2000 in the middle and fourth families reach 8 orders. At this time, the use of the specification cannot meet the requirements of practical engineering, so that a method for conveniently and rapidly solving the critical force of the step column with 5 steps or more is urgently needed.
Disclosure of Invention
The invention aims to provide a method for determining the critical force of a telescopic arm of an n-step ladder column crane, which can determine the maximum lifting capacity of the telescopic arm type crane.
The purpose of the invention is realized by the following technical scheme.
The invention discloses a method for determining stability critical force of an n-order telescopic arm of a crane, which comprises the following steps of:
step 1, establishing an n-order ladder column model and deducing an instability characteristic equation.
The n-step telescopic arm ladder column model is formed by combining n equal-section columns, and is formed by fixing and restricting a bottom surface to a top end free end, namely the 1 st, 2 nd, 3 th step ladder column; and the cross-sectional area of the column also decreases in turn. The length from the bottom fixed constraint end to the top end of the 1 st step ladder column is l1The length from the bottom fixed constraint end to the top of the 2 nd step pillar is l2And so on, the length from the bottom surface fixing and restraining end to the top end of the ith step pillar is liThe length from the bottom fixed constraint end to the top of the nth step ladder column is lnWherein the first stepped pillar is referred to as a base arm.
An XOY coordinate system is established on a plane where the telescopic arm is pressed to bend, the bottom fixed and restrained end is an O point, the direction from the bottom fixed and restrained end to the free end of the top end is the positive direction of an X axis, and the direction where the telescopic arm bends laterally is the positive direction of a Y axis. P is the axial pressure born by the top end of the telescopic boom, the direction of P is vertical downward and is kept unchanged, the P is the lateral displacement of the top end of the telescopic boom, and the axial force and the bending moment are all born by the telescopic boom. The method is characterized in that a differential equation of the deflection of each telescopic arm is established based on the longitudinal and transverse bending theory and is shown as a formula (1)
Figure BDA0001428103240000021
In the formula (I), the compound is shown in the specification,
Iithe section moment of inertia (m) of the ith step column4)，(i＝1,2,3,...n)；
li-fixing the restraining end from the bottom surface to the length (m), (i ═ 1,2, 3.. n) of the top end of the ith step ladder;
yi-lateral displacement (m) of the ith step column at x, (i ═ 1,2, 3.. n);
E-Young's modulus of elasticity (Pa) of the material of the telescopic arm;
formula (1) is uniformly represented as
Figure BDA0001428103240000031
Figure BDA0001428103240000032
The general solution of differential equation (2) is
yi＝Aisin(kix)+Bicos(kix)+ (4)
The relationship among all the integral constants is solved by the deflection displacement boundary condition
Figure BDA0001428103240000033
Figure BDA0001428103240000034
Using a matrix expression method;
Figure BDA0001428103240000035
Figure BDA0001428103240000036
the formula (6) is represented by
Figure BDA0001428103240000037
Recursive expression between integration constants from equation (9)
Figure BDA0001428103240000038
Note the book
Qi＝Ui -1Ti(11)
Thus obtaining the coefficient AnAnd BnExpression (2)
Figure BDA0001428103240000041
The boundary condition of the top of the telescopic boom is x ═ lnWhen y isnGet when becoming
Ansin(knln)+Bncos(knln)＝0 (13)
The instability characteristic equation of the telescopic boom is obtained by substituting the formula (12) into the formula (13)
Figure BDA0001428103240000042
Substituting into the geometric condition l of the box-shaped telescopic arm of the crane1，l2，…，lnIn order (14) to k1，k2，…，knHas n unknowns transcendental equations. Therefore, for any model with an n-step ladder column, the instability characteristic equation is
Figure BDA0001428103240000043
Equation (15) is the instability characteristic equation of the n-step ladder column model.
Step 2, deducing a recursion transcendental equation of the instability characteristic equation in the step 1;
using the formula (15),
when n is 1, obtain
cosk1l1＝0 (16)
When n is 2, obtain
Figure BDA0001428103240000044
When n is 3, obtain
Figure BDA0001428103240000045
Wherein the content of the first and second substances,
Figure BDA0001428103240000051
the instability characteristic equation of the n-step ladder column is obtained according to the mathematical induction method as follows
f(P)＝sin(knln)A(n)+cos(knln)B(n)＝0 (20)
Wherein the content of the first and second substances,
Figure BDA0001428103240000052
Figure BDA0001428103240000053
B(1)＝1，A(1)＝0，(n＝1,2,3,…) (23)
the equations (20), (21), (22) and (23) are recursion rules of the instability characteristic equation of the n-order ladder column model, and the transcendental equation with n unknowns can be conveniently written out by obtaining the recursion rules, so that the subsequent solution is convenient.
Step 3, solving a numerical solution of the transcendental equation in the step 2;
when the order of the stepped column is n, the transcendental equation comprises n unknowns, the solution is possible only by building an equation system by increasing the constraint limiting equation, and (n-1) constraint equations are increased according to the formula (3).
Figure BDA0001428103240000054
The established n-order transcendental equation set is shown as a formula (24), the nonlinear transcendental equation does not have analytical solution and only has numerical solution, and thus the nonlinear transcendental equation is solved by a numerical method.
Common numerical solutions include the Eulerian method, the Longge Kutta method, the Gauss-Newton method, the gradient descent method, and the Levenberg-Marquardt algorithm; however, in the methods such as the eulerian method, the longge lattice tower method, the Gauss-Newton method, the gradient descent method and the like, when the order of the ladder column is higher, a numerical solution cannot be obtained due to the singularity of the solution matrix.
In the step 3, the Levenberg-Marquardt algorithm is preferably used for solving the nonlinear transcendental equation set through a numerical method, the Levenberg-Marquardt algorithm has the advantages of a Newton method and a gradient method, the speed is improved by dozens of times or even hundreds of times compared with the gradient descent method, and the optimal numerical solution can be always obtained for the equation set constructed in the step (24).
Step 4, solving the critical force P according to the numerical solution;
by solving for k by equation (24)1，k2，…，knAnd (3) obtaining the critical force P of each section of the ladder column by combining the n unknowns and a formula (3), wherein the critical force is the critical force P of the whole ladder column model, and the maximum lifting capacity of the telescopic boom crane can be calculated through the critical force P and the multiplying power of the lifting mechanism.
Further comprising a step 5 of solving the length coefficient mu of the n-step ladder column by using the critical force determined in the step 42And the practical engineering problem is convenient to solve.
For convenience of engineering use, the critical force determined in step 4 is compared with the critical Euler force of a column with uniform section formed by the sections of the basic arms with the same length as the ladder column to obtain the length coefficient mu of the ladder column with n steps2. For the same step pillar with the same length ratio and section inertia moment ratio, the length coefficient mu is used2Can directly obtain the critical force and avoid the number of complex transcendental equation setsAnd the value solving process is convenient for solving the actual engineering problem.
The sections of the n-order telescopic arm support of the crane are fixedly connected with each other in a working state.
Has the advantages that:
1. the method for determining the stability critical force of the n-order telescopic boom of the crane disclosed by the invention is used for calculating the stability critical force of the n-order telescopic boom through establishing a recursion transcendental equation, is a breakthrough compared with the current design specification of 5-order and below algorithms, and can be used for conveniently and rapidly solving the critical force of 5-order and above ladder columns so as to meet the requirements of actual engineering.
2. The invention discloses a method for determining the stability critical force of an n-order telescopic boom of a crane, which realizes the arbitrary effective combination of the length and the section of the boom in the process of calculating the stability critical force of the n-order telescopic boom, always obtains the optimal numerical solution, and has the length coefficient mu of an n-order ladder column2The method has certain nonlinearity, and by using the calculation method, the error caused by using a linear interpolation method in the past is avoided.
3. The invention discloses a method for determining stability critical force of an n-step telescopic arm of a crane2Compared with the national standard, the calculated partial length coefficient is smaller than that of the national standard and is closer to the value of commercial software, and the calculated telescopic arm can reduce the section, reduce the use of materials and reduce the self weight, thereby realizing the lightweight design of the telescopic arm.
4. The method for determining the stability critical force of the n-order telescopic arm of the crane preferably uses a Levenberg-Marquardt algorithm to solve a nonlinear transcendental equation, has the advantages of a Newton method and a gradient method, improves the speed by dozens or even hundreds of times compared with a gradient descent method, and can always obtain an optimal numerical solution for a constructed equation set.
Drawings
FIG. 1 is a flow chart of a method for determining stability critical force of n-order telescopic boom of crane
FIG. 2 is a schematic diagram of a two-step pillar model and stress
FIG. 3 is a polynomial of the instability characteristic equation and k for a second order ladder column1Relationships between
FIG. 4 shows a length coefficient model of a two, three, four, and five step ladder in the national standard
FIG. 5 Fine solution of Length coefficient of second order ladder column compared with ANSYS results
FIG. 6 five step pillar calculation length factor and comparison (β)1＝1.3，β2＝1.3)
FIG. 7 fifth order ladder column calculated length coefficients and comparison (β)1＝1.3，β2＝1.6)
FIG. 8 five step column calculation length factor and comparison (β)1＝1.3，β2＝1.9)
FIG. 9 calculation of length coefficients for the fifth-order staircase columns and comparison (β)1＝1.3，β2＝2.2)
FIG. 10 five step pillar calculation length factor and comparison (β)1＝1.3，β2＝2.5)
FIG. 11 fifth order ladder column calculated length coefficients and comparisons (β)1＝1.6，β2＝1.3)
FIG. 12 fifth order ladder column calculation length factor and comparison (β)1＝1.6，β2＝1.6)
FIG. 13 five step column calculation length factor and comparison (β)1＝1.6，β2＝1.9)
FIG. 14 fifth order ladder column calculation length factor and comparison (β)1＝1.6，β2＝2.2)
FIG. 15 five step column calculation length factor and comparison (β)1＝1.6，β2＝2.5)
FIG. 16 five step column calculation length factor and comparison (β)1＝1.9，β2＝1.3)
FIG. 17 five step column calculation length factor and comparison (β)1＝1.9，β2＝1.6)
FIG. 18 five step column calculation length factor and comparison (β)1＝1.9，β2＝1.9)
FIG. 19 fifth stepThe ladder bars are calculated length coefficients and compared (β)1＝1.9，β2＝2.2)
FIG. 20 five step strut calculation length factor and comparison (β)1＝1.9，β2＝2.5)
FIG. 21 calculation of length coefficients for the fifth order staircase columns and comparison (β)1＝2.2，β2＝1.3)
FIG. 22 five step column calculation length factor and comparison (β)1＝2.2，β2＝1.6)
FIG. 23 calculation of length coefficients for the five-step ladder columns and comparison (β)1＝2.2，β2＝1.9)
FIG. 24 five step column calculation length factor and comparison (β)1＝2.2，β2＝2.2)
FIG. 25 five step column calculation length factor and comparison (β)1＝2.2，β2＝2.5)
FIG. 26 calculation of length coefficients for the fifth step bars and comparison (β)1＝2.5，β2＝1.3)
FIG. 27 fifth order ladder column calculated length coefficients and comparison (β)1＝2.5，β2＝1.6)
FIG. 28 five step strut calculation length factor and comparison (β)1＝2.5，β2＝1.9)
FIG. 29 five step bar calculation length factor and comparison (β)1＝2.5，β2＝2.2)
FIG. 30 five step column calculation length factor and comparison (β)1＝2.5，β2＝2.5)
Detailed Description
For a better understanding of the objects and advantages of the present invention, reference should be made to the following detailed description taken in conjunction with the accompanying drawings and examples.
Example 1: it is known that the second order staircase model shown in FIG. 2 calculates its critical force,/1＝15.5m，l2＝29.9m，I1＝43.26×10-3m4，I2＝26.09×10-3m4
The method for determining the stability critical force of the n-order telescopic arm of the crane disclosed by the embodiment comprises the following steps of:
step 1, establishing an n-order ladder column model and deducing an instability characteristic equation.
As shown in fig. 2, the n-step telescopic arm ladder column model is formed by combining n equal-section columns, and the number of the columns from a bottom surface fixing and restraining end to a top end free end is 1,2,3,... cndot.; and the cross-sectional area of the column also decreases in turn. The length from the bottom fixed constraint end to the top end of the 1 st step ladder column is l1The length from the bottom fixed constraint end to the top of the 2 nd step pillar is l2And so on, the length from the bottom surface fixing and restraining end to the top end of the ith step pillar is liThe length from the bottom fixed constraint end to the top of the nth step ladder column is lnWherein, the first step ladder column is called basic arm.
An XOY coordinate system is established on a plane where the telescopic arm is pressed to bend, the bottom fixed and restrained end is an O point, the direction from the bottom fixed and restrained end to the free end of the top end is the positive direction of an X axis, and the direction where the telescopic arm bends laterally is the positive direction of a Y axis. P is the axial pressure born by the top end of the telescopic boom, the direction of P is vertical downward and is kept unchanged, the P is the lateral displacement of the top end of the telescopic boom, and the axial force and the bending moment are all born by the telescopic boom. The method is characterized in that a differential equation of the deflection of each telescopic arm is established based on the longitudinal and transverse bending theory and is shown as a formula (1)
Figure BDA0001428103240000081
In the formula (I), the compound is shown in the specification,
Iithe section moment of inertia (m) of the ith step column4)，(i＝1,2,3,...n)；
li-fixing the restraining end from the bottom surface to the length (m), (i ═ 1,2, 3.. n) of the top end of the ith step ladder;
yi-lateral displacement (m) of the ith step column at x, (i ═ 1,2, 3.. n);
E-Young's modulus of elasticity (Pa) of the material of the telescopic arm;
formula (1) is uniformly represented as
Figure BDA0001428103240000091
Figure BDA0001428103240000092
The general solution of differential equation (2) is
yi＝Aisin(kix)+Bicos(kix)+ (4)
The relationship among all the integral constants is solved by the deflection displacement boundary condition
Figure BDA0001428103240000093
Figure BDA0001428103240000094
Using matrix representation
Figure BDA0001428103240000095
Figure BDA0001428103240000096
The formula (6) is represented by
Figure BDA0001428103240000097
Recursive expression between integration constants from equation (9)
Figure BDA0001428103240000101
Note the book
Qi＝Ui -1Ti(11)
Thus obtaining the coefficient AnAnd BnExpression (2)
Figure BDA0001428103240000102
The boundary condition of the top of the telescopic boom is x ═ lnWhen y isnGet when becoming
Ansin(knln)+Bncos(knln)＝0 (13)
The instability characteristic equation of the telescopic boom obtained by substituting the formula (12) into the formula (13) is as follows,
Figure BDA0001428103240000103
substituting into the geometric condition l of the box-shaped telescopic arm of the crane1，l2，…，lnIn order (14) to k1，k2，…，knWith n unknowns overriding the system of equations. Therefore, for any model with an n-step ladder column, the characteristic equation of unbalance instability is
Figure BDA0001428103240000104
Equation (15) is the instability characteristic equation of the n-step ladder column model.
And 2, deriving a transcendental equation according to the instability characteristic equation in the step 1.
Figure BDA0001428103240000105
The curve drawn according to the transcendental equation is a nonlinear non-periodic function satisfying f (k)1,k2) There are many points of the condition 0, such as points a, b, c, d, e, f, and g shown in fig. 3, which can satisfy the above equations, but the critical force of the step pillar is only one, so that the constraint equation needs to be added.
And step 3: solving numerical values;
Figure BDA0001428103240000111
and 4, step 4: solving numerical values;
k1＝0.0480，k2＝0.0617
obtaining the critical force, Pcr＝2.0491×107N
And 5: a length factor;
μ2＝1.06012
example 2:
for a second order ladder column, for comparison with the length factor in the national standard GB3811, the effective combination of the length and cross-section of the boom using the model shown in fig. 4 (α)1＝0.51，0.52，0.53，0.54，0.55，0.56，0.57，0.58，0.59，0.60，0.61，0.62，0.63，0.64，0.65，0.66，0.67，0.68，0.69，0.70)(β21.1, 1.2, 1.3, 1.6, 1.9, 2.2, 2.5) and compared to the results calculated using ANSYS17.0, the results are shown in fig. 5.
The length coefficient value μ obtained by the method of this example2Very close to the difference calculated using ANSYS17.0, the maximum relative error is 1.7 × 10-4. Following the second step pillar (I)2) Increase of moment of inertia, length coefficient mu2Decreasing, namely gradually increasing the critical force of the step pillar; following the second step pillar (I)2) Reduction of relative length by the factor of mu2The non-linearity is reduced. Therefore, for GB3811, when combinations that are not in the selection table are used, more accurate results can be obtained by linear interpolation using relatively close values, and the linear interpolation is erroneous.
Example 3: compared with the national standard GB3811
Table 3 is a comparison of the length factor of a third step using the present invention,
TABLE 1 calculation of the Length coefficient μ for the third-order step2And comparing
Figure BDA0001428103240000112
Figure BDA0001428103240000121
Table 1 shows the length coefficient values μ for the three-step pillars2Very close to the difference calculated using ANSYS17.0, the maximum relative error is 1.8 × 10-5And the lack of μ in GB3811 was found2The value is calculated according to an interpolation method, the linearity of the value is high in a small range, and the interpolation error is small.
TABLE 2 calculation of the Length coefficient μ for the four step ladder2And comparing
Figure BDA0001428103240000122
Figure BDA0001428103240000131
Figure BDA0001428103240000141
Figure BDA0001428103240000151
Table 2 shows the length coefficient values μ of the four-step pillars2Mu missing in GB38112The value can also be calculated according to an interpolation method, the linearity of the value is higher in a small range, and the interpolation error is small.
FIGS. 6-30 show the length coefficient values μ for various combinations of step pillars based on the ratios in GB38112The values are substantially linear, but as shown in FIG. 11, when β2＝1.6，β3＝1.3，β4＝2.5，β5Length coefficient mu when 1.62The values are clearly non-linear. In addition, in various combinations of the five-step pillars recommended in GB3811, the relationship is basically linear, and the length coefficient mu is shown2The use of linear interpolation may result in large errors, which are currently unpredictable.
Example 4: calculating the critical force l of the eight-step ladder column model1＝15.5m，l2＝27.125m，l3＝38.75m，l4＝50.375m，l5＝62m，l6＝73.625m，l7＝85.25m，l8＝96.875m，I1＝43.26×10-3m4，I2＝33.2769×10-3m4，I3＝25.5976×10-3m4，I4＝19.69×10-3m4，I5＝15.1465×10-3m4，I6＝11.6512×10-3m4，I7＝8.96244×10-3m4，I8＝6.89419×10-3m4
Steps 1,2 and 3 are similar to the implementation method of the embodiment 1,
and 4, step 4: solving numerical values;
k1＝0.0121，k2＝0.0138，k3＝0.0157，k4＝0.0179，k5＝0.0205，k6＝0.0233，k7＝0.0266，k8＝0.0303
obtaining the critical force, Pcr＝1.3049×106N
And 5: a length factor;
μ2＝2.6800
TABLE 3 eight-step ladder column calculated Length coefficients and comparison
Figure BDA0001428103240000161
The calculated values of the length coefficients of the partial eight-step ladder columns are shown in the table 3, the obtained results of the method are compared with those of the document [1], the document [2] and ANSYS17.0, and the length coefficients obtained by the method have extremely high precision and have strong practical significance on the stability of the telescopic boom of the large crane.
Literature [1] continental thought, landmass, white birch [ J ] accurate theoretical solution of crane box telescopic arm stability analysis [ proceedings of harlbine university of construction, 2000, 33 (2): 89-93.
Document [2] continental idea, all bright, accurate analysis of lateral stiffness and axial compression critical force of a multistage stepped column and practical formula [ J ] engineering mechanics, 2015, 32 (8): 217-222.
The above detailed description is intended to illustrate the objects, aspects and advantages of the present invention, and it should be understood that the above detailed description is only exemplary of the present invention and is not intended to limit the scope of the present invention, and any modifications, equivalents, improvements and the like made within the spirit and principle of the present invention should be included in the scope of the present invention.

Claims (5)

1. A method for determining stability critical force of an n-order telescopic boom of a crane is characterized by comprising the following steps: comprises the following steps of (a) carrying out,
step 1, establishing an n-order ladder column model and deducing an instability characteristic equation;
the n-step telescopic arm ladder column model is formed by combining n equal-section columns, and is formed by fixing and restricting a bottom surface to a top end free end, namely the 1 st, 2 nd, 3 th step ladder column; and the cross-sectional area of the column also decreases in turn; the length from the bottom fixed constraint end to the top end of the 1 st step ladder column is l1The length from the bottom fixed constraint end to the top of the 2 nd step pillar is l2And so on, the length from the bottom surface fixing and restraining end to the top end of the ith step pillar is liThe length from the bottom fixed constraint end to the top of the nth step ladder column is lnWherein, the first step ladder column is called as a basic arm;
establishing an XOY coordinate system on a plane where the telescopic arm is pressed to bend, wherein the bottom fixed and restrained end is an O point, the direction from the bottom fixed and restrained end to the free end of the top end is the positive direction of an X axis, and the direction where the telescopic arm bends laterally is the positive direction of a Y axis; p is the axial pressure borne by the top end of the telescopic boom, the direction of P is vertical downward and is kept unchanged, the P is the lateral displacement of the top end of the telescopic boom, and the axial force and the bending moment are borne by the telescopic boom; the method is characterized in that a differential equation of the deflection of each telescopic arm is established based on the longitudinal and transverse bending theory and is shown as a formula (1)
Figure FDA0002615653840000011
In the formula (I), the compound is shown in the specification,
Iithe section moment of inertia (m) of the ith step column4)，(i＝1,2,3,...n)；
li-fixing the restraining end from the bottom surface to the length (m), (i ═ 1,2, 3.. n) of the top end of the ith step ladder;
yi-lateral displacement (m) of the ith step column at x, (i ═ 1,2, 3.. n);
E-Young's modulus of elasticity (Pa) of the material of the telescopic arm;
formula (1) is uniformly represented as
Figure FDA0002615653840000012
Figure FDA0002615653840000021
The general solution of differential equation (2) is
yi＝Aisin(kix)+Bicos(kix)+ (4)
The relationship among all the integral constants is solved by the deflection displacement boundary condition
Figure FDA0002615653840000022
Figure FDA0002615653840000023
Using matrix representation
Figure FDA0002615653840000024
Figure FDA0002615653840000025
The formula (6) is represented by
Figure FDA0002615653840000026
Recursive expression between integration constants from equation (9)
Figure FDA0002615653840000027
Note the book
Qi＝Ui -1Ti(11)
Thus obtaining the coefficient AnAnd BnExpression (2)
Figure FDA0002615653840000028
From the telescopic boom top boundary conditions: x ═ lnWhen y isnGet when becoming
Ansin(knln)+Bncos(knln)＝0 (13)
The instability characteristic equation of the telescopic boom is obtained by substituting the formula (12) into the formula (13)
Figure FDA0002615653840000031
Substituting into the geometric condition l of the box-shaped telescopic arm of the crane1，l2，…，lnIn order (14) to k1，k2，…，knHas n unknown transcendental equations; therefore, for any model with an n-step ladder column, the instability characteristic equation is
Figure FDA0002615653840000032
Equation (15) is the instability characteristic equation of the n-step ladder column model;
step 2, deducing a recursion transcendental equation of the instability characteristic equation in the step 1;
using the formula (15),
when n is 1, obtain
cosk1l1＝0 (16)
When n is 2, obtain
Figure FDA0002615653840000033
When n is 3, obtain
Figure FDA0002615653840000034
Wherein the content of the first and second substances,
Figure FDA0002615653840000035
the instability characteristic equation of the n-step ladder column is obtained according to the mathematical induction method as follows
f(P)＝sin(knln)A(n)+cos(knln)B(n)＝0 (20)
Wherein the content of the first and second substances,
Figure FDA0002615653840000041
Figure FDA0002615653840000042
B(1)＝1，A(1)＝0，(n＝1,2,3,…) (23)
the equations (20), (21), (22) and (23) are recursion rules of the instability characteristic equation of the n-order ladder column model, and the transcendental equation with n unknown quantities can be conveniently written out by obtaining the recursion rules, so that the subsequent solution is convenient;
step 3, solving a numerical solution of the transcendental equation in the step 2;
when the order of the stepped column is n, the transcendental equation comprises n unknowns, the solution is possible only by adding the constraint limiting equation to establish an equation set, and (n-1) constraint equations are added according to the formula (3);
Figure FDA0002615653840000043
the established n-order transcendental equation set is shown as a formula (24), and the nonlinear transcendental equation set does not have analytical solution and only has numerical value solution; therefore, the nonlinear transcendental equation is solved by a numerical method;
step 4, solving the critical force P according to the numerical solution;
by solving for k by equation (24)1，k2，…，knAnd n unknowns, and combining a formula (3) to obtain the critical force P of each section of the ladder column, wherein the critical force is the critical force P of the whole ladder column model, and the maximum lifting capacity of the telescopic boom crane can be calculated through the critical force P and the multiplying power of the lifting mechanism.
2. The method for determining the stability critical force of the n-order telescopic boom of the crane as claimed in claim 1, wherein: further comprising a step 5 of solving the length coefficient mu of the n-step ladder column by using the critical force determined in the step 42The practical engineering problem is convenient to solve;
for convenience of engineering use, the critical force determined in step 4 is compared with the critical Euler force of a column with uniform section formed by the sections of the basic arms with the same length as the ladder column to obtain the length coefficient mu of the ladder column with n steps2(ii) a For the same step pillar with the same length ratio and section inertia moment ratio, the length coefficient mu is used2The critical force can be directly obtained, the complex numerical solving process of an transcendental equation set is avoided, and the practical engineering problem is conveniently solved.
3. The method for determining the stability critical force of the n-order telescopic boom of the crane as claimed in claim 2, wherein: the section of the telescopic arm support is fixedly connected with the section of the n-order telescopic arm support in a working state.
4. The method for determining the stability critical force of the n-order telescopic boom of the crane as claimed in claim 1,2 or 3, wherein: the numerical solution commonly used in the step 3 comprises an Eulerian method, a Rungestota method, a Gauss-Newton method, a gradient descent method and a Levenberg-Marquardt algorithm.
5. The method for determining the stability critical force of the n-order telescopic boom of the crane as claimed in claim 4, wherein: in the step 3, the transcendental equation is solved by a numerical method, a Levenberg-Marquardt algorithm is used, the advantages of a Newton method and a gradient method are achieved, the speed is improved by dozens or even hundreds of times compared with that of a gradient descent method, and the optimal numerical solution can be always obtained for the equation set constructed by the formula (24).
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Citations (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN104528552A (en) * 2014-12-31 2015-04-22 中联重科股份有限公司 Super hoisting device, operation method thereof and crane
CN106503399A (en) * 2016-11-19 2017-03-15 东北石油大学 Peupendicular hole hangs the determination method of tubing string Helical Buckling Critical Load

Family Cites Families (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US6991119B2 (en) * 2002-03-18 2006-01-31 Jlg Industries, Inc. Measurement system and method for assessing lift vehicle stability
US20080217279A1 (en) * 2005-10-18 2008-09-11 Putzmeister Concrete Pumps Gmbh Working Boom, Especially for Large Manipulators and Mobile Concrete Pumps

Patent Citations (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN104528552A (en) * 2014-12-31 2015-04-22 中联重科股份有限公司 Super hoisting device, operation method thereof and crane
CN106503399A (en) * 2016-11-19 2017-03-15 东北石油大学 Peupendicular hole hangs the determination method of tubing string Helical Buckling Critical Load

Non-Patent Citations (4)

* Cited by examiner, † Cited by third party
Title
A numerical study of the overall stability of flexible giant crane booms;Wenjun Li et al.;《Journal of Constructional Steel Research》;20141119;第12-27页 *
一种n阶变截面压杆稳定性计算方法的研究;王欣 等;《中国机械工程》;20140715;第25卷(第13期);第1744-1748页 *
变截面阶梯压杆精确失稳特征方程及其稳定型计算实用方法--《塔式起重机设计规范》中阶梯柱计算长度的快捷计算及精度分析;陆念力 等;《建筑机械》;20140630;第76-81页 *
多级阶梯柱侧向刚度与轴压临界力的精确分析及其实用算式;陆念力，都亮;《工程力学》;20150831;第32卷(第8期);第217-222页 *

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