CN112100756A - Double-crane system statics uncertainty analysis method based on fuzzy theory - Google Patents

Double-crane system statics uncertainty analysis method based on fuzzy theory Download PDF

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CN112100756A
CN112100756A CN202010810844.8A CN202010810844A CN112100756A CN 112100756 A CN112100756 A CN 112100756A CN 202010810844 A CN202010810844 A CN 202010810844A CN 112100756 A CN112100756 A CN 112100756A
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周斌
訾斌
曾亿山
石柯
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Abstract

The invention discloses a double-crane system statics uncertainty analysis method based on a fuzzy theory, which comprises the following steps of: establishing a static response equation of a double-crane system; establishing a fuzzy parameter model; establishing a fuzzy statics response equivalent equation, a fuzzy kinematics Jacobian matrix, a fuzzy first virtual work vector and a fuzzy second virtual work vector of the double-crane system with a fuzzy parameter model; forming a multi-fuzzy variable model; … …, the approximate expansion expression is brought into a static response equivalent equation of a double-crane system interval with a corresponding truncation vector to obtain a midpoint value and a change interval of a driving moment interval vector; obtaining the upper bound value and the lower bound value of the driving torque interval vector by the midpoint value and the change interval of the driving torque interval vector; and obtaining the fuzzy driving moment vector interval with the corresponding truncated vector from the upper and lower boundaries of the driving moment vector of the interval.

Description

Double-crane system statics uncertainty analysis method based on fuzzy theory
Technical Field
The invention relates to the technical field of reliability, in particular to a statics uncertainty analysis method of a double-crane system based on a fuzzy theory.
Background
Because the response of the dual-crane system exceeds the safety threshold value to cause safety accidents due to the influence of uncertain factors such as mechanical errors, environmental excitation and the like, the research on the reliability technology of the dual-crane system is urgently needed to improve the safety performance. The premise of the reliability analysis of the double-crane system is to carry out the statics response analysis of the double-crane system with uncertain parameters. Therefore, how to carry out the uncertainty quantitative analysis of the double-crane system according to the uncertainty theory and by combining the existing statics modeling method is a key link. The patent application number 'CN 201710019654.2' designs a variable amplitude angle response modeling algorithm and a random response domain prediction method of a double-crane system, and can solve the prediction problem of a variable amplitude angle response domain with random parameters. The patent application number 'CN 201710772385.7' designs a method for acquiring the variable amplitude angle response domain of a crane system under the condition of inter-cell parameters, and can solve the problem of analysis of the variable amplitude angle response domain containing the inter-cell structural parameters in the crane system. The patent application number "CN 201811011899.1" designs a load distribution method in a double-crane cooperative hoisting system, and improves the feasibility of path planning in the actual hoisting process by calculating the load distributed by each crane. The patent application number 'CN 201910988450.9' designs a real-time scheduling-control cascade system and a method of a double crane, which can solve the problems of poor real-time performance and low efficiency of a scheduling plan and poor accuracy and stability of a control system. However, there are problems such as the following: firstly, random parameters require a large number of samples to accurately establish probability distribution characteristics of uncertain parameters, such as expectation, variance and the like, so that the random parameters are difficult to apply in engineering; secondly, the interval parameters lose probability statistical information; and thirdly, researches such as statics analysis, path planning and control of a double-crane system in the traditional sense do not consider the existence of uncertain parameters, and the method has limitation.
Disclosure of Invention
The invention aims to make up for the defects of the prior art, and provides a statics uncertainty analysis method of a double-crane system based on a fuzzy theory, so as to solve the problem of how to quickly and efficiently obtain statics response of the double-crane system under uncertain parameters in the prior art.
In order to achieve the purpose, the technical scheme adopted by the invention is as follows:
a double-crane system statics uncertainty analysis method based on a fuzzy theory is carried out according to the following steps:
the method comprises the following steps: establishing a statics response equation of the double-crane system;
step two: establishing a fuzzy parameter model of a double-crane system;
step three: combining the statics response equation in the first step and the fuzzy parameter model in the second step, establishing a fuzzy statics response equivalent equation, a fuzzy kinematics Jacobian matrix, a fuzzy first virtual work vector and a fuzzy second virtual work vector of the double-crane system with the fuzzy parameter model;
step four: constructing a multi-fuzzy variable model;
step five: establishing a static response equivalent equation of a double-crane system interval with a corresponding truncation vector, an interval kinematic Jacobian matrix, an interval first virtual work vector and an interval second virtual work vector;
step six: converting the parameters obtained in the fifth step to obtain an approximate expansion expression;
step seven: substituting the approximate expansion expression into a dual-crane system interval statics response equivalent equation with a corresponding truncation vector to obtain a midpoint value and an interval radius of a driving torque interval vector;
step eight: converting the midpoint of the interval driving moment vector and the interval radius to obtain the upper and lower limits of the interval driving moment vector;
step nine: and converting the upper and lower boundaries of the interval driving moment vector to obtain a fuzzy driving moment vector with a corresponding truncated vector, wherein the driving moment vector is the statics uncertainty analysis conclusion of the double-crane system.
Further, in the step one, the dual crane system static response equation is established according to the geometric model of the dual crane system and by combining the virtual work principle.
Furthermore, the fuzzy parameter model in the step two is established based on uncertain parameters of the double-crane system.
Furthermore, in the fourth step, the multi-fuzzy-variable model is an interval variable with a corresponding truncation domain, which is obtained by converting the fuzzy parameter model obtained in the second step according to an alpha truncation strategy;
and fifthly, the obtained interval statics response equivalent equation, the interval kinematic Jacobian matrix, the interval first virtual work vector and the interval second virtual work vector of the double-crane system are all based on a multi-mode fuzzy variable model, a double-crane system fuzzy statics response equivalent equation with a fuzzy parameter model, a fuzzy kinematic Jacobian matrix, a fuzzy first virtual work vector and a fuzzy second virtual work vector.
Further, the approximate expansion expression in the sixth step is obtained by performing approximate expansion on the interval kinematic jacobian matrix, the interval first virtual work vector and the interval second virtual work vector in the fifth step.
Further, an interval algorithm is adopted in the process of acquiring the midpoint value and the interval radius of the driving torque interval vector.
Further, in step one:
the geometric model of the double-crane system refers to that two cranes carry out cooperative hoisting operation on the same heavy object;
the virtual work principle means that the necessary and sufficient condition that the deformation system is in balance is that for any virtual displacement, the sum of the virtual work done by the external force is equal to the sum of the virtual work done by the external force on each micro-segment on the corresponding deformation, namely the external force virtual work is equal to the deformation virtual work;
according to a geometric model of the double-crane system, a statics response equation of the double-crane system is established by combining a virtual work principle, wherein the statics response equation comprises the following steps:
Figure BDA0002630922350000031
wherein, W refers to virtual work, A is a hinge point of the crane jib and the rotary table, B is a hinge point of the crane jib and the lifting rope, i is a crane serial number, 1 or 2; tau is a static response vector of the double-crane system, in particular to a driving moment vector of the double-crane system.
Figure BDA0002630922350000032
Is a kinematic Jacobian matrix J of a dual crane systemDACSThe transpose of (a) is performed,
Figure BDA0002630922350000033
acting on the suspension arm AiBiThe first virtual work of
Figure BDA0002630922350000034
The transpose of (a) is performed,
Figure BDA0002630922350000035
acting on a weight C1C2The second virtual work of
Figure BDA0002630922350000036
Respectively, as:
Figure BDA0002630922350000037
Figure BDA0002630922350000038
Figure BDA0002630922350000039
wherein the content of the first and second substances,
Figure BDA00026309223500000310
Figure BDA00026309223500000311
Figure BDA00026309223500000312
Figure BDA00026309223500000313
Figure BDA00026309223500000314
Figure BDA00026309223500000315
Figure BDA00026309223500000316
Figure BDA0002630922350000041
wherein the base coordinate system { B }, O-YZ is located at A1A2The center of the connection point, a moving coordinate system { P }: O }p-YpZpIs located at C1C2The centers of the connecting points, D and D are respectively the crane spacing A1A2And a load C1C2Length of (L)iIs the i-th crane jib AiBiY and z are respectively the load C1C2Center OpTheta represents the rotation angle of the moving coordinate system { P } with respect to the base coordinate system { B }, mpIs the mass of the heavy object, gammaiIs the argument of the ith crane,
Figure BDA0002630922350000042
is the first derivative of the rotation angle theta with respect to time,
Figure BDA0002630922350000043
are respectively y, z, theta, gammaiThe second derivative with respect to time, g, is the gravitational acceleration.
Further, the specific steps of the second step are as follows: step two: in the hoisting operation of the crane, system parameters have uncertainty due to the influence of uncertainty factors such as mechanical errors, environmental excitation and the like; therefore, n fuzzy parameters are introduced to quantitatively represent uncertain system parameters, and a fuzzy parameter model is established as follows:
Figure BDA0002630922350000044
where f is a blur vector consisting of n blur parameters, fmIs the mth blur parameter; f. of1Refers to the crane spacing A1A2Length D, f of2Is referred to as a load C1C2Length d of (d), other blurring parameters (f)m,…,fn) Depending on the particular uncertainty factor or source; t refers to the transpose operation of the vector inside the linear algebra.
Further, the third step comprises the following specific steps: combining the static response equation of the two-step crane system and the two-step fuzzy parameter model to establish the fuzzy static response equivalent equation of the two-step crane system with the fuzzy parameter model
Figure BDA0002630922350000045
Wherein the content of the first and second substances,
Figure BDA0002630922350000046
is a fuzzy kinematics Jacobian matrix,
Figure BDA0002630922350000047
is to blur the first imaginary work vector,
Figure BDA0002630922350000048
is the blurred second imaginary work vector and τ (f) is the blurred drive moment vector.
Further, the specific steps of the fourth step are as follows: according to the alpha truncation strategy, the fuzzy parameter model can be converted into interval variable composition with corresponding truncation domains, and a multi-fuzzy variable model is formed as follows:
Figure BDA0002630922350000051
wherein the content of the first and second substances,
Figure BDA0002630922350000052
is a blurring parameter fmAt the corresponding cutoff value alphau,mThe variables of the interval under (a) are,
Figure BDA0002630922350000053
is the blur vector f at the corresponding truncated vector alphauA lower interval vector, F is a multi-fuzzy variable model; the alpha truncation strategy is a basic algorithm of a fuzzy theory, and the fuzzy parameters can be decomposed into a set consisting of a series of interval parameters with the truncation level of alpha (alpha is more than or equal to 0 and less than or equal to 1) through the alpha truncation strategy;
in the above formula, superscript I refers to the meaning of the interval variable; the subscript u refers to the u-th parameter;
Figure BDA0002630922350000054
the method comprises the steps of obtaining a u-th interval vector consisting of interval parameters by truncating n fuzzy parameters according to an alpha truncation strategy; alpha is alphauThe method comprises the steps of obtaining a u-th truncation vector consisting of truncation values obtained by truncating n fuzzy parameters according to an alpha truncation strategy; v refers to the total number of vectors formed by interval parameters obtained by truncating n fuzzy parameters according to an alpha truncation strategy.
Further, the specific steps of step 5 are:
establishing a fuzzy statics response equivalent equation with a corresponding truncation vector alpha by combining the three-step double crane system and the four-step multi-mode fuzzy variable modeluThe static response equivalent equation of the section of the double-crane system is as follows:
Figure BDA0002630922350000055
wherein the content of the first and second substances,
Figure BDA0002630922350000056
is an interval kinematics Jacobian matrix,
Figure BDA0002630922350000057
is the first virtual work vector of the interval,
Figure BDA0002630922350000058
is the second virtual work vector of the interval,
Figure BDA0002630922350000059
is the range drive torque vector.
Further, the specific steps of step 6 are:
according to the first-order Taylor series expansion, the kinematic Jacobian matrix among the areas in the step five is processed
Figure BDA00026309223500000510
Interval first virtual work vector
Figure BDA00026309223500000511
Interval second virtual work vector
Figure BDA00026309223500000512
Performing approximate expansion to obtain
Figure BDA00026309223500000513
And
Figure BDA00026309223500000514
approximation ofAnd expanding the expression.
First, the interval kinematics Jacobian matrix
Figure BDA00026309223500000515
Vector in interval
Figure BDA00026309223500000517
The first order taylor series expansion at the midpoint may be expressed as:
Figure BDA00026309223500000518
wherein the content of the first and second substances,
Figure BDA0002630922350000061
Figure BDA0002630922350000062
wherein, JcAnd Δ J are respectively the interval kinematic Jacobian matrix
Figure BDA0002630922350000063
The midpoint and the span radius.
Second, interval first imaginary work vector
Figure BDA0002630922350000064
Vector in interval
Figure BDA0002630922350000065
The first order taylor series expansion at the midpoint may be expressed as:
Figure BDA0002630922350000066
wherein the content of the first and second substances,
Figure BDA0002630922350000067
Figure BDA0002630922350000068
wherein the content of the first and second substances,
Figure BDA0002630922350000069
and
Figure BDA00026309223500000610
respectively, the first virtual work vector of the interval
Figure BDA00026309223500000611
The midpoint and the span radius.
Finally, the second imaginary work vector of the interval
Figure BDA00026309223500000612
Vector in interval
Figure BDA00026309223500000613
The first order taylor series expansion at the midpoint may be expressed as:
Figure BDA00026309223500000614
wherein the content of the first and second substances,
Figure BDA00026309223500000615
Figure BDA0002630922350000071
wherein the content of the first and second substances,
Figure BDA0002630922350000072
and
Figure BDA0002630922350000073
respectively, the second virtual work vector of the interval
Figure BDA0002630922350000074
The midpoint and the span radius.
Further, the specific steps of step 7 are:
the product obtained in the step six
Figure BDA0002630922350000075
And
Figure BDA0002630922350000076
substituting the approximate expansion expression into the vector alpha with corresponding truncation in the step fiveuThe interval statics of the double-crane system responds to the equivalent equation, and a midpoint value and an interval radius of a driving torque interval vector are obtained according to an interval algorithm.
Firstly, the product obtained in the step six
Figure BDA0002630922350000077
And
Figure BDA0002630922350000078
substituting the approximate expansion expression into the vector alpha with corresponding truncation in the step fiveuThe static response equivalent equation of the section of the double-crane system is obtained as follows:
Figure BDA0002630922350000079
secondly, according to the first-order Newman series expansion,
Figure BDA00026309223500000710
the approximate expression of (c) is:
(Jc+ΔJ)-1＝(Jc)-1-(Jc)-1ΔJ(Jc)-1
mixing the above (J)c+ΔJ)-1By substituting the approximate expression into the preceding vector with the corresponding truncationαuThe interval statics response equivalent equation of the double-crane system is obtained by neglecting a high-order term:
Figure BDA00026309223500000711
wherein the content of the first and second substances,
Figure BDA00026309223500000712
Figure BDA00026309223500000713
wherein, taucAnd Δ τIRespectively, interval driving moment vector
Figure BDA00026309223500000714
The midpoint and the span.
Figure BDA00026309223500000715
Is a standard unit interval [ -1,1 [)]。
Finally, due to Δ τIAbout standard unit interval
Figure BDA00026309223500000716
Is monotonic, and obtains the interval driving moment vector according to the monotonicity technology
Figure BDA0002630922350000081
Section radius Δ τ:
Figure BDA0002630922350000082
wherein the content of the first and second substances,
Figure BDA0002630922350000083
Figure BDA0002630922350000084
further, the specific steps of step 8 are:
driving moment vector of interval obtained in the seventh step
Figure BDA0002630922350000085
The midpoint and the interval radius of the interval, and an interval driving moment vector is obtained according to an interval algorithm
Figure BDA0002630922350000086
The upper and lower bounds of (c).
Obtaining interval driving moment vector according to interval algorithm
Figure BDA0002630922350000087
Respectively expressed as:
Figure BDA0002630922350000088
wherein the content of the first and second substances,τand
Figure BDA0002630922350000089
are respectively provided with corresponding truncated vectors alphauDouble crane system interval driving moment vector
Figure BDA00026309223500000810
A lower bound value and an upper bound value.
Further, the specific steps of step 9 are:
driving moment vector of interval obtained in step eight
Figure BDA00026309223500000811
The upper and lower boundaries of (a) are obtained according to the fuzzy decomposition theory with corresponding truncated vectors alphauFuzzy driving moment vector τ (f):
Figure BDA00026309223500000812
in the above formula, the symbol $ means the sum of products; τ (f) refers to the blurred drive moment vector;
Figure BDA00026309223500000813
refers to the range drive torque vector.
For a better explanation of the invention, the invention will now be explained in the following with the following alternative angles:
the double-crane system statics uncertainty analysis method based on the fuzzy theory is carried out according to the following steps:
the method comprises the following steps: establishing a statics response equation of the double-crane system by combining a virtual work principle according to a geometric model of the double-crane system;
step two: establishing a fuzzy parameter model according to uncertain parameters of a double-crane system;
step three: combining the first step and the second step, establishing a fuzzy statics response equivalent equation of the double-crane system with a fuzzy parameter model, a fuzzy kinematics Jacobian matrix, a fuzzy first virtual work vector and a fuzzy second virtual work vector;
step four: according to an alpha truncation strategy, the fuzzy parameter model in the step two can be converted into interval variable composition with a corresponding truncation domain to form a multi-fuzzy variable model;
step five: combining the third step and the fourth step, establishing a static response equivalent equation of the interval of the double-crane system with corresponding truncated vectors, an interval kinematic Jacobian matrix, an interval first virtual work vector and an interval second virtual work vector;
step six: performing approximate expansion on the inter-kinematics Jacobian matrix, the first virtual work vector of the interval and the second virtual work vector of the interval in the fifth step to obtain an approximate expansion expression;
step seven: substituting the approximate expansion expression obtained in the sixth step into the interval static response equivalent equation of the double-crane system with the corresponding truncation vector in the fifth step, and obtaining a midpoint value and an interval radius of the driving torque interval vector according to an interval algorithm;
step eight: obtaining the middle point and the interval radius of the interval driving torque vector obtained in the step seven according to an interval algorithm to obtain the upper and lower boundaries of the interval driving torque vector;
step nine: and (5) obtaining fuzzy driving moment vectors with corresponding truncated vectors according to a fuzzy decomposition theory by using the upper and lower boundaries of the interval driving moment vectors obtained in the step eight.
Further, the invention relates to a dual crane system statics uncertainty analysis method based on the fuzzy theory, which comprises the following specific steps:
the method comprises the following steps: according to a geometric model of the double-crane system, a statics response equation of the double-crane system is established by combining a virtual work principle, wherein the statics response equation comprises the following steps:
Figure BDA0002630922350000091
wherein the content of the first and second substances,
Figure BDA0002630922350000092
is a kinematic Jacobian matrix J of a dual crane systemDACSThe transpose of (a) is performed,
Figure BDA0002630922350000093
acting on the suspension arm AiBiThe first virtual work of
Figure BDA0002630922350000094
The transpose of (a) is performed,
Figure BDA0002630922350000095
acting on a weight C1C2The second virtual work of
Figure BDA0002630922350000096
The transposing of (1). Respectively expressed as:
Figure BDA0002630922350000097
Figure BDA0002630922350000101
Figure BDA0002630922350000102
wherein the content of the first and second substances,
Figure BDA0002630922350000103
Figure BDA0002630922350000104
Figure BDA0002630922350000105
Figure BDA0002630922350000106
Figure BDA0002630922350000107
Figure BDA0002630922350000108
Figure BDA0002630922350000109
Figure BDA00026309223500001010
wherein the base coordinate system { B }, O-YZ is located at A1A2Center of connection point, motionCoordinate system { P }: Op-YpZpIs located at C1C2The centers of the connecting points, D and D are respectively the crane spacing A1A2And a load C1C2Length of (L)iIs the i-th crane jib AiBiY and z are respectively the load C1C2Center OpTheta represents the rotation angle of the moving coordinate system { P } with respect to the base coordinate system { B }, mpIs the mass of the heavy object, gammaiIs the argument of the ith crane,
Figure BDA00026309223500001011
is the first derivative of the rotation angle theta with respect to time,
Figure BDA00026309223500001012
are respectively y, z, theta, gammaiThe second derivative with respect to time, g, is the gravitational acceleration.
Step two: in the hoisting operation of the crane, system parameters have uncertainty due to the influence of uncertainty factors such as mechanical errors and environmental excitation. Therefore, n fuzzy parameters are introduced to quantitatively represent uncertain system parameters, and a fuzzy parameter model is established as follows:
f＝{f1,…,fm,…,fn}T,m＝1,2,…,n
where f is a blur vector consisting of n blur parameters, fmIs the mth blur parameter.
Step three: combining the static response equation of the two-step crane system and the two-step fuzzy parameter model to establish the fuzzy static response equivalent equation of the two-step crane system with the fuzzy parameter model
Figure BDA0002630922350000111
Wherein the content of the first and second substances,
Figure BDA0002630922350000112
is a dieThe jacobian matrix of the kinematics is blurred,
Figure BDA0002630922350000113
is to blur the first imaginary work vector,
Figure BDA0002630922350000114
is the blurred second imaginary work vector and τ (f) is the blurred drive moment vector.
Step four: according to the alpha truncation strategy, the fuzzy parameter model can be converted into interval variable composition with corresponding truncation domains, and a multi-fuzzy variable model is formed as follows:
Figure BDA0002630922350000115
wherein the content of the first and second substances,
Figure BDA0002630922350000116
is a blurring parameter fmAt the corresponding cutoff value alphau,mThe variables of the interval under (a) are,
Figure BDA0002630922350000117
is that the blur vector f is alpha under the corresponding truncated vectoruF is a multi-fuzzy variable model.
Step five: establishing a fuzzy statics response equivalent equation with a corresponding truncation vector alpha by combining the three-step double crane system and the four-step multi-mode fuzzy variable modeluThe static response equivalent equation of the section of the double-crane system is as follows:
Figure BDA0002630922350000118
wherein the content of the first and second substances,
Figure BDA0002630922350000119
is an interval kinematics Jacobian matrix,
Figure BDA00026309223500001110
is the first virtual work of the intervalThe vector of the vector is then calculated,
Figure BDA00026309223500001111
is the second virtual work vector of the interval,
Figure BDA00026309223500001112
is the range drive torque vector.
Step six: according to the first-order Taylor series expansion, the kinematic Jacobian matrix among the areas in the step five is processed
Figure BDA00026309223500001113
Interval first virtual work vectorInterval second virtual work vector
Figure BDA00026309223500001115
Performing approximate expansion to obtain
Figure BDA00026309223500001116
And
Figure BDA00026309223500001117
the approximation of (2) expands the expression.
First, the interval kinematics Jacobian matrix
Figure BDA00026309223500001118
Vector in interval
Figure BDA00026309223500001119
The first order taylor series expansion at the midpoint may be expressed as:
Figure BDA00026309223500001120
wherein the content of the first and second substances,
Figure BDA00026309223500001121
Figure BDA0002630922350000121
wherein, JcAnd Δ J are respectively the interval kinematic Jacobian matrix
Figure BDA0002630922350000122
The midpoint and the span radius.
Second, interval first imaginary work vector
Figure BDA0002630922350000123
Vector in interval
Figure BDA0002630922350000124
The first order taylor series expansion at the midpoint may be expressed as:
Figure BDA0002630922350000125
wherein the content of the first and second substances,
Figure BDA0002630922350000126
Figure BDA0002630922350000127
wherein the content of the first and second substances,
Figure BDA0002630922350000128
and
Figure BDA0002630922350000129
respectively, the first virtual work vector of the interval
Figure BDA00026309223500001210
The midpoint and the span radius.
Finally, the second imaginary work vector of the interval
Figure BDA00026309223500001211
Vector in interval
Figure BDA00026309223500001212
The first order taylor series expansion at the midpoint may be expressed as:
Figure BDA00026309223500001213
wherein the content of the first and second substances,
Figure BDA00026309223500001214
Figure BDA0002630922350000131
wherein the content of the first and second substances,
Figure BDA0002630922350000132
and
Figure BDA0002630922350000133
respectively, the second virtual work vector of the interval
Figure BDA0002630922350000134
The midpoint and the span radius.
Step seven: the product obtained in the step six
Figure BDA0002630922350000135
And
Figure BDA0002630922350000136
substituting the approximate expansion expression into the vector alpha with corresponding truncation in the step fiveuThe interval statics of the double-crane system responds to the equivalent equation, and a midpoint value and an interval radius of a driving torque interval vector are obtained according to an interval algorithm.
Firstly, the following components are mixedObtained in the sixth step
Figure BDA0002630922350000137
And
Figure BDA0002630922350000138
substituting the approximate expansion expression into the vector alpha with corresponding truncation in the step fiveuThe static response equivalent equation of the section of the double-crane system is obtained as follows:
Figure BDA0002630922350000139
secondly, according to the first-order Newman series expansion,
Figure BDA00026309223500001310
the approximate expression of (c) is:
(Jc+ΔJ)-1＝(Jc)-1-(Jc)-1ΔJ(Jc)-1
mixing the above (J)c+ΔJ)-1By substituting the approximate expression of (a) into the preceding vector with the corresponding truncated vector alphauThe interval statics response equivalent equation of the double-crane system is obtained by neglecting a high-order term:
Figure BDA00026309223500001311
wherein the content of the first and second substances,
Figure BDA00026309223500001312
Figure BDA00026309223500001313
wherein, taucAnd Δ τIRespectively, interval driving moment vector
Figure BDA00026309223500001314
The midpoint and the span.
Figure BDA00026309223500001315
Is a standard unit interval [ -1,1 [)]。
Finally, due to Δ τIAbout standard unit interval
Figure BDA00026309223500001316
Is monotonic, and obtains the interval driving moment vector according to the monotonicity technology
Figure BDA00026309223500001317
Section radius Δ τ:
Figure BDA0002630922350000141
wherein the content of the first and second substances,
Figure BDA0002630922350000142
Figure BDA0002630922350000143
step eight: driving moment vector of interval obtained in the seventh step
Figure BDA0002630922350000144
The midpoint and the interval radius of the interval, and an interval driving moment vector is obtained according to an interval algorithm
Figure BDA0002630922350000145
The upper and lower bounds of (c).
Obtaining interval driving moment vector according to interval algorithm
Figure BDA0002630922350000146
Respectively expressed as:
Figure BDA0002630922350000147
wherein the content of the first and second substances,τand
Figure BDA0002630922350000148
are respectively provided with corresponding truncated vectors alphauDouble crane system interval driving moment vector
Figure BDA0002630922350000149
A lower bound value and an upper bound value.
Step nine: driving moment vector of interval obtained in step eight
Figure BDA00026309223500001410
The upper and lower boundaries of (a) are obtained according to the fuzzy decomposition theory with corresponding truncated vectors alphauFuzzy driving moment vector τ (f):
Figure BDA00026309223500001411
compared with the prior art, the invention has the advantages that:
[1] the distribution range of the dual-crane system static response is predicted through a fuzzy theory, the fuzzy uncertainty of the dual-crane system parameters is considered for the first time, the fuzzy distribution characteristics of the dual-crane system static response are obtained quantitatively, and the calculation result has important guiding significance for analyzing the uncertainty of the dual-crane system static response.
[2] Aiming at the occasions with small uncertainty, the traditional solution is to adopt a Monte Carlo method, and have the defects of low calculation efficiency, large sample size, difficulty in practical application and the like.
[3] Aiming at the occasions with small uncertainty, the dual-crane system statics uncertainty analysis method based on the fuzzy theory provided by the invention has better precision compared with the traditional method. An engineer can solve the situation of known deterministic parameters and fuzzy parameters with small uncertainty by adopting a traditional method (see the first embodiment in fig. 3 and 4); furthermore, the prediction method provided by the method (see scheme two in fig. 3 and 4) of the invention can fully consider the fuzzy uncertainty of uncertain parameters, and combine the perturbation theory, the fuzzy decomposition theory and the interval algorithm to deduce the distribution characteristics of the fuzzy driving moment vector, thereby fully considering the complexity of engineering problems and ensuring the relative accuracy of the calculation result.
Drawings
FIG. 1 is a flow chart of the present invention.
FIG. 2 is a schematic three-dimensional model of a double truck crane system; the figure shows a transfer platform 1 of a first automobile crane system, a transfer platform 2 of a second automobile crane system and a suspension arm A of the first automobile crane system1B1Suspension arm A in second automobile crane system2B2Lifting rope B in first automobile crane system1C1Lifting rope B in second automobile crane system2C2Load C1C2Load center of gravity OpHinge point A1、A2、B1、B2、C1、C2And their positional relationship.
FIG. 3 is a graph of upper bound value of the interval of the driving torque of the first automobile crane system calculated by the conventional method and the method of the present invention in a computer by using the prediction method of the upper and lower bounds of the statics response under the fuzzy parameters of the two automobile crane systems when the small uncertainty range of the fuzzy parameters provided by the present invention is [0, 0.10% ].
FIG. 4 is a graph of the lower bound value of the interval of the driving torque of the first automobile crane system calculated by the conventional method and the method of the present invention in a computer by using the prediction method of the upper and lower bounds of the statics response under the fuzzy parameters of the two automobile crane systems when the small uncertainty range of the fuzzy parameters provided by the present invention is [0, 0.10% ].
Detailed Description
The structural features and advantages of the present invention will now be described in detail with reference to the accompanying drawings.
As shown in FIG. 1, the dual crane system statics uncertainty analysis method based on the fuzzy theory is carried out according to the following steps:
the method comprises the following steps: establishing a statics response equation of the double-crane system by combining a virtual work principle according to a geometric model of the double-crane system;
step two: establishing a fuzzy parameter model according to uncertain parameters of a double-crane system;
step three: combining the first step and the second step, establishing a fuzzy statics response equivalent equation of the double-crane system with a fuzzy parameter model, a fuzzy kinematics Jacobian matrix, a fuzzy first virtual work vector and a fuzzy second virtual work vector;
step four: according to an alpha truncation strategy, the fuzzy parameter model in the step two can be converted into interval variable composition with a corresponding truncation domain to form a multi-fuzzy variable model;
step five: combining the third step and the fourth step, establishing a static response equivalent equation of the interval of the double-crane system with corresponding truncated vectors, an interval kinematic Jacobian matrix, an interval first virtual work vector and an interval second virtual work vector;
step six: performing approximate expansion on the inter-kinematics Jacobian matrix, the first virtual work vector of the interval and the second virtual work vector of the interval in the fifth step to obtain an approximate expansion expression;
step seven: substituting the approximate expansion expression obtained in the sixth step into the interval static response equivalent equation of the double-crane system with the corresponding truncation vector in the fifth step, and obtaining a midpoint value and an interval radius of the driving torque interval vector according to an interval algorithm;
step eight: obtaining the middle point and the interval radius of the interval driving torque vector obtained in the step seven according to an interval algorithm to obtain the upper and lower boundaries of the interval driving torque vector;
step nine: and (5) obtaining fuzzy driving moment vectors with corresponding truncated vectors according to a fuzzy decomposition theory by using the upper and lower boundaries of the interval driving moment vectors obtained in the step eight.
As shown in fig. 1 and 2, further, in the first step, according to the geometric model of the dual crane system, the virtual work principle is combined to establish a static response equation of the dual crane system as follows:
Figure BDA0002630922350000161
wherein the content of the first and second substances,
Figure BDA0002630922350000162
is a kinematic Jacobian matrix J of a dual crane systemDACSThe transpose of (a) is performed,
Figure BDA0002630922350000163
acting on the suspension arm AiBiThe first virtual work of
Figure BDA0002630922350000164
The transpose of (a) is performed,
Figure BDA0002630922350000165
acting on a weight C1C2The second virtual work of
Figure BDA0002630922350000166
The transposing of (1). Respectively expressed as:
Figure BDA0002630922350000167
Figure BDA0002630922350000171
Figure BDA0002630922350000172
wherein the content of the first and second substances,
Figure BDA0002630922350000173
Figure BDA0002630922350000174
Figure BDA0002630922350000175
Figure BDA0002630922350000176
Figure BDA0002630922350000177
Figure BDA0002630922350000178
Figure BDA0002630922350000179
Figure BDA00026309223500001710
wherein the base coordinate system { B }, O-YZ is located at A1A2The center of the connection point, a moving coordinate system { P }: O }p-YpZpIs located at C1C2The centers of the connecting points, D and D are respectively the crane spacing A1A2And a load C1C2Length of (L)iIs the ith tableCrane jib AiBiY and z are respectively the load C1C2Center OpTheta represents the rotation angle of the moving coordinate system { P } with respect to the base coordinate system { B }, mpIs the mass of the heavy object, gammaiIs the argument of the ith crane,
Figure BDA00026309223500001711
is the first derivative of the rotation angle theta with respect to time,
Figure BDA00026309223500001712
are respectively y, z, theta, gammaiThe second derivative with respect to time, g, is the gravitational acceleration.
As shown in fig. 1 and 2, further, in the second step, in the crane hoisting operation, the system parameters have uncertainty due to the influence of uncertainty factors such as mechanical errors and environmental excitation. Therefore, n fuzzy parameters are introduced to quantitatively represent uncertain system parameters, and a fuzzy parameter model is established as follows:
f＝{f1,…,fm,…,fn}T,m＝1,2,…,n
where f is a blur vector consisting of n blur parameters, fmIs the mth blur parameter.
As shown in fig. 1 and 2, further, in the third step, a fuzzy static response equivalent equation of the double-crane system with the fuzzy parameter model is established by combining the static response equation of the double-crane system in the first step and the fuzzy parameter model in the second step
Figure BDA0002630922350000181
Wherein the content of the first and second substances,
Figure BDA0002630922350000182
is a fuzzy kinematics Jacobian matrix,
Figure BDA0002630922350000183
is to blur the first imaginary work vector,
Figure BDA0002630922350000184
is the blurred second imaginary work vector and τ (f) is the blurred drive moment vector.
As shown in fig. 1 and 2, further, in step four, according to the α truncation strategy, the fuzzy parameter model can be converted into interval variable composition with corresponding truncation domains, and a multi-fuzzy variable model is formed as follows:
Figure BDA0002630922350000185
wherein the content of the first and second substances,
Figure BDA0002630922350000186
is a blurring parameter fmAt the corresponding cutoff value alphau,mThe variables of the interval under (a) are,
Figure BDA0002630922350000187
is that the blur vector f is alpha under the corresponding truncated vectoruF is a multi-fuzzy variable model.
As shown in fig. 1 and 2, further, in the fifth step, a fuzzy statics response equivalent equation with a corresponding truncation vector alpha is established by combining the fuzzy statics response equivalent equation of the system of the three-step double crane and the fuzzy variable model in the fourth stepuThe static response equivalent equation of the section of the double-crane system is as follows:
Figure BDA0002630922350000188
wherein the content of the first and second substances,
Figure BDA0002630922350000189
is an interval kinematics Jacobian matrix,
Figure BDA00026309223500001810
is the first virtual work vector of the interval,
Figure BDA00026309223500001811
is the second virtual work vector of the interval,
Figure BDA00026309223500001812
is the range drive torque vector.
As shown in fig. 1 and 2, further, in step six, the kinematic jacobian matrix in step five is processed according to the first-order taylor series expansion
Figure BDA00026309223500001813
Interval first virtual work vector
Figure BDA00026309223500001814
Interval second virtual work vector
Figure BDA00026309223500001815
Performing approximate expansion to obtain
Figure BDA00026309223500001816
And
Figure BDA00026309223500001817
the approximation of (2) expands the expression.
First, the interval kinematics Jacobian matrix
Figure BDA00026309223500001818
Vector in interval
Figure BDA00026309223500001819
The first order taylor series expansion at the midpoint may be expressed as:
Figure BDA0002630922350000191
wherein the content of the first and second substances,
Figure BDA0002630922350000192
Figure BDA0002630922350000193
wherein, JcAnd Δ J are respectively the interval kinematic Jacobian matrix
Figure BDA0002630922350000194
The midpoint and the span radius.
Second, interval first imaginary work vector
Figure BDA0002630922350000195
Vector in interval
Figure BDA0002630922350000196
The first order taylor series expansion at the midpoint may be expressed as:
Figure BDA0002630922350000197
wherein the content of the first and second substances,
Figure BDA0002630922350000198
Figure BDA0002630922350000199
wherein the content of the first and second substances,
Figure BDA00026309223500001910
and
Figure BDA00026309223500001911
respectively, the first virtual work vector of the interval
Figure BDA00026309223500001912
The midpoint and the span radius.
Finally, the second imaginary work vector of the interval
Figure BDA00026309223500001913
Vector in interval
Figure BDA00026309223500001914
The first order taylor series expansion at the midpoint may be expressed as:
Figure BDA00026309223500001915
wherein the content of the first and second substances,
Figure BDA0002630922350000201
Figure BDA0002630922350000202
wherein the content of the first and second substances,
Figure BDA0002630922350000203
and
Figure BDA0002630922350000204
respectively, the second virtual work vector of the interval
Figure BDA0002630922350000205
The midpoint and the span radius.
As shown in fig. 1 and 2, further, in the seventh step, the product obtained in the sixth step
Figure BDA0002630922350000206
And
Figure BDA0002630922350000207
substituting the approximate expansion expression into the vector alpha with corresponding truncation in the step fiveuThe interval statics response equivalent equation of the double-crane system is obtained, and a midpoint value and an interval radius of a driving moment response interval vector are obtained according to an interval algorithm.
Firstly, the following components are mixedObtained in the sixth step
Figure BDA0002630922350000208
And
Figure BDA0002630922350000209
substituting the approximate expansion expression into the vector alpha with corresponding truncation in the step fiveuThe static response equivalent equation of the section of the double-crane system is obtained as follows:
Figure BDA00026309223500002010
secondly, according to the first-order Newman series expansion,
Figure BDA00026309223500002011
the approximate expression of (c) is:
(Jc+ΔJ)-1＝(Jc)-1-(Jc)-1ΔJ(Jc)-1
mixing the above (J)c+ΔJ)-1By substituting the approximate expression of (a) into the preceding vector with the corresponding truncated vector alphauThe interval statics response equivalent equation of the double-crane system is obtained by neglecting a high-order term:
Figure BDA00026309223500002012
wherein the content of the first and second substances,
Figure BDA00026309223500002013
Figure BDA00026309223500002014
wherein, taucAnd Δ τIRespectively, interval driving moment vector
Figure BDA00026309223500002015
The midpoint and the span.
Figure BDA00026309223500002016
Is a standard unit interval [ -1,1 [)]。
Finally, due to
Figure BDA0002630922350000211
About standard unit interval
Figure BDA0002630922350000212
Is monotonic, and obtains the interval driving moment vector according to the monotonicity technology
Figure BDA0002630922350000213
Section radius Δ τ:
Figure BDA0002630922350000214
wherein the content of the first and second substances,
Figure BDA0002630922350000215
Figure BDA0002630922350000216
as shown in fig. 1 and 2, further, in step eight, the interval driving torque vector obtained in step seven is used
Figure BDA0002630922350000217
The midpoint and the interval radius of the interval, and an interval driving moment vector is obtained according to an interval algorithm
Figure BDA0002630922350000218
The upper and lower bounds of (c).
Obtaining interval driving moment vector according to interval algorithm
Figure BDA0002630922350000219
Respectively expressed as:
Figure BDA00026309223500002110
wherein the content of the first and second substances,τand
Figure BDA00026309223500002111
are respectively provided with corresponding truncated vectors alphauDouble crane system interval driving moment vector
Figure BDA00026309223500002112
A lower bound value and an upper bound value.
As shown in fig. 1 and 2, further, in step nine, the interval driving torque vector obtained in step eight is used
Figure BDA00026309223500002113
The upper and lower boundaries of (a) are obtained according to the fuzzy decomposition theory with corresponding truncated vectors alphauFuzzy driving moment vector τ (f):
Figure BDA00026309223500002114
FIG. 2 is a schematic diagram of a three-dimensional model of a dual truck crane system corresponding to this embodiment, including a turntable 1 of a first truck crane system, a turntable 2 of a second truck crane system, and a boom A of the first truck crane system1B1Boom A of second automobile crane system2B2Lifting rope B of first automobile crane system1C1Lifting rope B of second automobile crane system2C2Load C1C2Load center of gravity OpHinge point A1、A2、B1、B2、C1、C2. Hydraulic cylinder D1E1(amplitude variable cylinder D2E2) One end of the rotary table is hinged with the rotary table 1 (the rotary table 2),the other end is connected with the suspension arm A1B1(boom A)2B2) Hinged by adjusting the amplitude-variable oil cylinder D in the amplitude-variable mechanism1E1(amplitude variable cylinder D2E2) Further realizing the suspension arm A1B1(boom A)2B2) In the vertical plane around the variable-amplitude oil cylinder D1E1(amplitude variable cylinder D2E2) Rotating at the hinged point of the rotary table 1 (the rotary table 2) to change the suspension arm A1B1(boom A)2B2) A change in elevation angle. For the above-mentioned two-crane system, the prediction method of the statics response under the fuzzy parameter of the two-crane system provided by the present invention is described below.
The implementation steps of the method for solving the statics response under the fuzzy parameters of the double-crane system in the computer are further described as follows:
determining the determined values of all deterministic parameters and the distribution characteristics of fuzzy parameters according to the design parameters and the working condition requirements of the crane;
and sequentially substituting the determined values of the determined parameters and the distribution characteristics of the fuzzy parameters into formulas of an upper bound value and a lower bound value of the driving moment vector of the corresponding lower interval of the truncated vector by using MATLAB programming.
Therefore, the distribution characteristics of the fuzzy driving moment vector of the corresponding truncated vector under the fuzzy parameters are obtained.
The distribution characteristic of the hydrostatic response includes an upper bound and a lower bound of the drive torque at the respective truncated vector.
For solving the static response of the double-crane system under the fuzzy parameters by the traditional method (Monte Carlo method), the implementation steps in the computer are further described as follows:
determining the determination value of each deterministic parameter and the distribution characteristics of system fuzzy parameters according to the design parameters and the working condition requirements of the crane;
on the premise that the deterministic values of all deterministic parameters and the distribution characteristics of fuzzy parameters are obtained, a random value is selected from the random distribution values of all fuzzy parameters and is input into an MATLAB program;
and sequentially substituting the determined values of all deterministic parameters and the random values of all fuzzy parameters into a dual crane system statics response equation by using MATLAB programming.
Therefore, the driving torque of the double-crane system under a certain fuzzy parameter is obtained.
Repeating the above process until the times i is 10000 times, outputting a distribution curve of the driving moment of the double-crane system under the fuzzy parameter, and outputting the mathematical characteristics of the driving moment of the double-crane system under the fuzzy parameter according to a computer instruction.
Referring to fig. 3 and 4, when the uncertainty range of the fuzzy parameter provided by the present invention is [0, 0.10% ], the interval upper bound value and lower bound value graphs of the driving torque of the first automobile crane system in the dual automobile crane system shown in fig. 2 are predicted in a computer by using the conventional method (monte carlo method) and the method of the present invention.
The specific values of the upper and lower boundary values of the interval of the driving torque of the first automobile crane system in the two automobile crane systems are respectively calculated by adopting a traditional method and the method disclosed by the invention, and are shown in table 1. Respectively calculating a curve chart of the upper bound value of the interval of the driving torque of the first automobile crane system in the two automobile crane systems by adopting a traditional method and the method disclosed by the invention, as shown in a figure 3; the traditional method and the method of the invention are adopted to respectively calculate the lower bound value curve of the interval of the driving torque of the first automobile crane system in the double automobile crane systems, as shown in figure 4. The abscissa represents uncertainty, the ordinate represents an upper bound value and a lower bound value of the interval of the driving torque, and the solid line and the dotted line represent results calculated by the conventional method and the method of the invention, respectively.
By taking the first automobile crane system as a research object, as can be seen from fig. 3 and 4, when the fuzzy parameter is in small uncertainty, the calculation results of the static response prediction method under the fuzzy parameter of the double automobile crane systems in the computer by the traditional method and the method of the invention are basically consistent, but after the method is adopted, the calculation time is obviously shortened, namely the calculation time is shortened by 5 orders of magnitude compared with the original method, so that the method has the advantages of high calculation efficiency (less calculation time), high solution precision and particular suitability for the engineering problem of less uncertain parameter samples.
Table 1 upper and lower limits of the range of the drive torque of the first mobile crane system when the cutoff α is 0.8
Figure BDA0002630922350000231
In conclusion, the invention can solve the problem of prediction of the statics response distribution of double or even multiple automobile crane systems, fixed cranes, mobile lift trucks or conveyances with hooks under small uncertain fuzzy parameters. The above-described embodiments are merely exemplary embodiments of the present invention, and the present invention is not limited to the above-described embodiments, and all modifications made within the principle and content of the present invention should be included in the protection scope of the present invention.

Claims (10)

1. A double-crane system statics uncertainty analysis method based on a fuzzy theory is characterized by comprising the following steps:
the method comprises the following steps: establishing a statics response equation of the double-crane system;
step two: establishing a fuzzy parameter model of a double-crane system;
step three: combining the statics response equation in the first step and the fuzzy parameter model in the second step, establishing a fuzzy statics response equivalent equation, a fuzzy kinematics Jacobian matrix, a fuzzy first virtual work vector and a fuzzy second virtual work vector of the double-crane system with the fuzzy parameter model;
step four: constructing a multi-fuzzy variable model;
step five: establishing a static response equivalent equation of a double-crane system interval with a corresponding truncation vector, an interval kinematic Jacobian matrix, an interval first virtual work vector and an interval second virtual work vector;
step six: converting the parameters obtained in the fifth step to obtain an approximate expansion expression;
step seven: substituting the approximate expansion expression into a dual-crane system interval statics response equivalent equation with a corresponding truncation vector to obtain a midpoint value and an interval radius of a driving torque interval vector;
step eight: converting the midpoint of the interval driving moment vector and the interval radius to obtain the upper and lower limits of the interval driving moment vector;
step nine: and converting the upper and lower boundaries of the interval driving moment vector to obtain a fuzzy driving moment vector with a corresponding truncated vector, wherein the driving moment vector is the statics uncertainty analysis conclusion of the double-crane system.
2. The method for analyzing the statics uncertainty of the dual crane system based on the fuzzy theory as claimed in claim 1, wherein in the step one, the statics response equation of the dual crane system is established according to the geometric model of the dual crane system and by combining the virtual work principle.
3. The method for analyzing the statics uncertainty of the dual crane system based on the fuzzy theory as claimed in claim 1, wherein the fuzzy parameter model in the second step is established based on the uncertainty parameters of the dual crane system.
4. The method for analyzing the statics uncertainty of the double-crane system based on the fuzzy theory as claimed in claim 1, wherein in the fourth step, the multi-fuzzy variable model is an interval variable with a corresponding truncation domain obtained by converting the fuzzy parameter model obtained in the second step according to an alpha truncation strategy;
and fifthly, the obtained interval statics response equivalent equation, the interval kinematic Jacobian matrix, the interval first virtual work vector and the interval second virtual work vector of the double-crane system are all based on a multi-mode fuzzy variable model, a double-crane system fuzzy statics response equivalent equation with a fuzzy parameter model, a fuzzy kinematic Jacobian matrix, a fuzzy first virtual work vector and a fuzzy second virtual work vector.
5. The method for analyzing uncertainty of statics of a dual crane system based on fuzzy theory as claimed in claim 1, wherein said approximate expansion expression of step six is obtained by performing approximate expansion on interval kinematic Jacobian matrix, interval first virtual work vector and interval second virtual work vector in step five.
6. The dual crane system statics uncertainty analysis method based on fuzzy theory as claimed in claim 1, characterized in that in the process of obtaining the midpoint value and the interval radius of the driving moment interval vector, an interval algorithm is adopted.
7. The dual crane system statics uncertainty analysis method based on the fuzzy theory as claimed in claim 1, characterized in that in step one:
the geometric model of the double-crane system refers to that two cranes carry out cooperative hoisting operation on the same heavy object;
the virtual work principle means that the necessary and sufficient condition that the deformation system is in balance is that for any virtual displacement, the sum of the virtual work done by the external force is equal to the sum of the virtual work done by the external force on each micro-segment on the corresponding deformation, namely the external force virtual work is equal to the deformation virtual work;
according to a geometric model of the double-crane system, a statics response equation of the double-crane system is established by combining a virtual work principle, wherein the statics response equation comprises the following steps:
Figure FDA0002630922340000021
wherein, W refers to virtual work, A is a hinge point of the crane jib and the rotary table, B is a hinge point of the crane jib and the lifting rope, i is a crane serial number, 1 or 2; tau is a statics response vector of the double-crane system, in particular to a driving moment vector of the double-crane system;
Figure FDA0002630922340000022
is a kinematic Jacobian matrix J of a dual crane systemDACSThe transpose of (a) is performed,
Figure FDA0002630922340000023
acting on the suspension arm AiBiThe first virtual work of
Figure FDA0002630922340000024
The transpose of (a) is performed,
Figure FDA0002630922340000025
acting on a weight C1C2The second virtual work of
Figure FDA0002630922340000026
Respectively, as:
Figure FDA0002630922340000027
Figure FDA0002630922340000028
Figure FDA0002630922340000029
wherein the content of the first and second substances,
Figure FDA00026309223400000210
Figure FDA00026309223400000211
Figure FDA0002630922340000031
Figure FDA0002630922340000032
Figure FDA0002630922340000033
Figure FDA0002630922340000034
Figure FDA0002630922340000035
Figure FDA0002630922340000036
wherein the base coordinate system { B }, O-YZ is located at A1A2The center of the connection point, a moving coordinate system { P }: O }p-YpZpIs located at C1C2The centers of the connecting points, D and D are respectively the crane spacing A1A2And a load C1C2Length of (L)iIs the i-th crane jib AiBiY and z are respectively the load C1C2Center OpTheta represents the rotation angle of the moving coordinate system { P } with respect to the base coordinate system { B }, mpIs the mass of the heavy object, gammaiIs the argument of the ith crane,
Figure FDA0002630922340000037
is the first derivative of the rotation angle theta with respect to time,
Figure FDA0002630922340000038
are respectively y, z, theta, gammaiThe second derivative with respect to time, g, is the gravitational acceleration.
8. The dual crane system statics uncertainty analysis method based on the fuzzy theory as claimed in claim 1, wherein the second step comprises the following specific steps: step two: in the hoisting operation of the crane, system parameters have uncertainty due to the influence of uncertainty factors such as mechanical errors, environmental excitation and the like; therefore, n fuzzy parameters are introduced to quantitatively represent uncertain system parameters, and a fuzzy parameter model is established as follows:
f＝{f1,…,fm,…,fn}T,m＝1,2,…,n
where f is a blur vector consisting of n blur parameters, fmIs the mth blur parameter; f. of1Refers to the crane spacing A1A2Length D, f of2Is referred to as a load C1C2Length d of (d), other blurring parameters (f)m,…,fn) Depending on the particular uncertainty factor or source; t refers to the transpose operation of the vector inside the linear algebra.
9. The dual crane system statics uncertainty analysis method based on the fuzzy theory as claimed in claim 1, wherein the third step comprises the following specific steps: combining the static response equation of the two-step crane system and the two-step fuzzy parameter model to establish the fuzzy static response equivalent equation of the two-step crane system with the fuzzy parameter model
Figure FDA0002630922340000041
Wherein the content of the first and second substances,
Figure FDA0002630922340000042
is a fuzzy kinematics Jacobian matrix,
Figure FDA0002630922340000043
is to blur the first imaginary work vector,
Figure FDA0002630922340000044
is the blurred second imaginary work vector and τ (f) is the blurred drive moment vector.
10. The dual crane system statics uncertainty analysis method based on the fuzzy theory as claimed in claim 1, wherein the concrete steps of the fourth step are: according to the alpha truncation strategy, the fuzzy parameter model can be converted into interval variable composition with corresponding truncation domains, and a multi-fuzzy variable model is formed as follows:
Figure FDA0002630922340000045
wherein the content of the first and second substances,
Figure FDA0002630922340000046
is a blurring parameter fmAt the corresponding cutoff value alphau,mThe variables of the interval under (a) are,
Figure FDA0002630922340000047
is the blur vector f at the corresponding truncated vector alphauA lower interval vector, F is a multi-fuzzy variable model; the alpha truncation strategy is a basic algorithm of a fuzzy theory, fuzzy parameters can be decomposed into a set consisting of a series of interval parameters with the truncation level of alpha through the alpha truncation strategy, and alpha is more than or equal to 0 and less than or equal to 1;
in the above formula, superscript I refers to the meaning of the interval variable; the subscript u refers to the u-th parameter;
Figure FDA0002630922340000048
the method comprises the steps of obtaining a u-th interval vector consisting of interval parameters by truncating n fuzzy parameters according to an alpha truncation strategy; alpha is alphauRefers to truncation according to alphaThe strategy is used for truncating n fuzzy parameters to obtain a u-th truncated vector consisting of truncation values; v refers to the total number of vectors formed by interval parameters obtained by truncating n fuzzy parameters according to an alpha truncation strategy.
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