CN112100756B - 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 PDFInfo
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Abstract
The invention discloses a method for analyzing the statics uncertainty of a double-crane system based on a fuzzy theory, which comprises the following steps: establishing a hydrostatic response equation of the 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 a double crane system with a fuzzy parameter model; constructing a multi-fuzzy variable model; … … introducing the approximate expansion expression into a double crane system interval statics response equivalent equation with corresponding truncated vector to obtain a midpoint value and a change interval of a driving moment interval vector; the midpoint value and the change interval of the driving moment interval vector are used for obtaining an upper limit value and a lower limit value of the driving moment interval vector; and obtaining the upper and lower boundaries of the interval driving moment vector into a fuzzy driving moment vector interval with corresponding truncated vector.
Description
Technical Field
The invention relates to the technical field of reliability, in particular to a double-crane system statics uncertainty analysis method based on a fuzzy theory.
Background
Because of the influence of uncertainty factors such as mechanical errors, environmental excitation and the like, the response of the dual-crane system exceeds a safety threshold to cause safety accidents, and therefore, research on the reliability technology of the dual-crane system is needed to improve the safety performance. The precondition of the reliability analysis of the double crane system is to develop the hydrostatic response analysis of the double crane system containing uncertain parameters. Therefore, how to conduct the quantitative analysis of the uncertainty of the double-crane system according to the uncertainty theory and combining the existing statics modeling method is a key link. The patent application number CN201710019654.2 designs a double crane system luffing angle response modeling algorithm and a random response domain prediction method, and can solve the problem of luffing angle response domain prediction under random parameters. The patent application number CN201710772385.7 designs a crane system amplitude-changing angle response domain acquisition method applicable to inter-cell parameters, and can solve the problem of analysis of amplitude-changing angle response domains of crane systems containing inter-cell structural parameters. The patent application number CN201811011899.1 designs a load distribution method in the double-crane collaborative hoisting system, and the feasibility of path planning in the actual hoisting process is improved by calculating the load distributed by each crane. The patent application number CN201910988450.9 designs a real-time dispatching-control cascade system and method of a double crane, which can solve the problems of poor real-time property, low efficiency and poor accuracy and stability of a control system of a dispatching plan. However, there are also problems such as: 1. random parameters require a large number of samples to accurately establish probability distribution characteristics of uncertain parameters, such as expectations, variances and the like, so that the random parameters are difficult to apply in engineering; 2. the interval parameters lose probability statistical information; 3. the traditional researches of hydrostatic analysis, path planning, control and the like of the double crane system do not consider the existence of uncertain parameters, and have limitations.
Disclosure of Invention
The invention aims to overcome the defects of the prior art, and provides a method for analyzing the statics uncertainty of a double-crane system based on a fuzzy theory, so as to solve the problem of how to quickly and efficiently obtain the statics response of the double-crane system under the condition of containing uncertain parameters in the prior art.
In order to achieve the above purpose, the technical scheme adopted by the invention is as follows:
a method for analyzing the statics uncertainty of a double crane system based on a fuzzy theory comprises the following steps:
Step one: 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 of the first step and the fuzzy parameter model of the second step, and establishing a fuzzy statics response equivalent equation, a fuzzy kinematic 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 an interval statics response equivalent equation, an interval kinematics Jacobian matrix, an interval first virtual work vector and an interval second virtual work vector of the double crane system with corresponding truncated vectors;
step six: converting the parameters obtained in the step five to obtain an approximate expansion expression;
Step seven: substituting the approximate expansion expression into a double crane system interval statics response equivalent equation with corresponding truncated vector to obtain a midpoint value and an interval radius of a driving moment interval vector;
step eight: the middle point of the interval driving moment vector and the interval radius are converted to obtain the upper and lower boundaries 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 corresponding truncated vector, wherein the driving moment vector is the analysis conclusion of the statics uncertainty of the double crane system.
Further, in step one, the statics response equation of the dual crane system is established according to the geometric model of the dual crane system and combined with the virtual work principle.
Further, the fuzzy parameter model in the second step is established based on uncertain parameters of the double crane system.
Further, in the fourth step, the fuzzy variable model is a section variable with a corresponding truncated domain obtained by converting the fuzzy parameter model obtained in the second step according to an α -truncated strategy;
And in the fifth step, the obtained equivalent equation of the interval statics response of the double crane system, the interval kinematic Jacobian matrix, the interval first virtual work vector and the interval second virtual work vector are all based on a fuzzy variable model, the equivalent equation of the fuzzy statics response of the double crane system with a fuzzy parameter model, the fuzzy kinematic Jacobian matrix, the fuzzy first virtual work vector and the fuzzy second virtual work vector.
Further, the approximate expansion expression in the step six is obtained by performing approximate expansion on the inter-section kinematic jacobian matrix, the inter-section first virtual work vector and the inter-section second virtual work vector in the step five.
Further, in the process of obtaining the midpoint value and the interval radius of the driving moment interval vector, an interval algorithm is adopted.
Further, in step one:
the geometric model of the double-crane system refers to that two cranes cooperatively hoist the same weight;
The principle of virtual work refers to the necessary and sufficient condition that the deformation system is in balance, and for any virtual displacement, the sum of virtual work done by external force is equal to the sum of virtual work done by 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, combining with a virtual work principle to establish a hydrostatic response equation of the double crane system, wherein the hydrostatic response equation is as follows:
wherein W refers to virtual work, A refers to a hinge point of a crane boom and a turntable, B refers to a hinge point of the crane boom and a lifting rope, and i refers to a crane serial number, 1 or 2; τ is the hydrostatic response vector of the dual crane system, specifically the drive torque vector of the dual crane system. Is the transpose of the kinematic Jacobian matrix J DACS of a double crane system,/>Is the first virtual work/>, acting on the boom A iBi Transpose of/>Is the second virtual work/>, acting on weight C 1C2 Is denoted as:
wherein,
Wherein the base coordinate system { B }. O-YZ is located at the center of the connection point of A 1A2, the movable coordinate system { P }. O p-YpZp is located at the center of the connection point of C 1C2, D and D are the length of the crane spacing A 1A2 and the load C 1C2 respectively, L i is the length of the i-th crane boom A iBi, Y and Z are Cartesian coordinate values of the center O p of the load C 1C2 along the Y axis and the Z axis respectively, θ represents the rotation angle of the movable coordinate system { P } relative to the base coordinate system { B }, m p is the mass of the heavy object, gamma i is the luffing angle of the i-th crane,Is the first derivative of rotation angle θ with respect to time,/>And the second derivatives of y, z, theta and gamma i with respect to time are respectively shown, and g is the gravitational acceleration.
Further, the specific steps of the second step are as follows: step two: in the hoisting operation of a crane, system parameters have uncertainty due to the influence of uncertainty factors such as mechanical errors, environmental excitation and the like; thus, n fuzzy parameters are introduced to quantitatively represent the uncertain system parameters, and a fuzzy parameter model is built as follows:
Wherein f is a blur vector composed of n blur parameters, and f m is an mth blur parameter; f 1 refers to the length D of the crane spacing a 1A2, f 2 refers to the length D of the load C 1C2, and other fuzzy parameters (f m,…,fn) are dependent on specific uncertainty factors or sources; t refers to the transpose operation of the vector inside the linear algebra.
Further, the specific steps of the third step are as follows: combining the step one double crane system statics response equation and the step two fuzzy parameter model to establish a double crane system fuzzy statics response equivalent equation with a fuzzy parameter model
Wherein,Is a fuzzy kinematics Jacobian matrix,/>Is the blurred first virtual work vector,Is the fuzzy second virtual work vector and τ (f) is the fuzzy driving torque 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 compositions with corresponding truncation domains, and the multi-fuzzy variable model is formed by the following steps:
wherein, Is the interval variable of the fuzzy parameter f m under the corresponding cut-off value alpha u,m,/>Is the interval vector of the fuzzy vector F under the corresponding truncated vector alpha u, and 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 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, the superscript I refers to the meaning of the interval variable; subscript u refers to the u-th parameter; The method comprises the steps of intercepting n fuzzy parameters according to an alpha interception strategy to obtain an interval parameter which is an nth interval vector; alpha u refers to a nth truncated vector formed by truncated values obtained by truncating n fuzzy parameters according to an alpha truncated strategy; v refers to the total number of vectors composed of interval parameters obtained by truncating n fuzzy parameters according to an alpha truncating strategy.
Further, the specific steps of the step 5 are as follows:
Combining the step three double crane system fuzzy statics response equivalent equation and the step four multi-fuzzy variable model to establish a double crane system interval statics response equivalent equation with a corresponding truncated vector alpha u:
wherein, Is an interval kinematic Jacobian matrix,/>Is the first virtual work vector of the interval,Is the second virtual work vector of the interval,/>Is an interval drive torque vector.
Further, the specific steps of the step 6 are as follows:
According to the first-order Taylor series expansion, the method comprises the following steps of Interval first virtual work vector/>Interval second virtual work vector/>Performing approximate expansion to obtain/>And/>Is a similar expansion expression of (a).
First, an inter-range kinematic Jacobian matrixIn-interval vector/>The first order taylor series expansion at the midpoint of (a) can be expressed as:
wherein,
Wherein J c and ΔJ are respectively the inter-range kinematic Jacobian matricesIs defined as the mid-point and interval radius.
Second, the first virtual work vector is dividedIn-interval vector/>The first order taylor series expansion at the midpoint of (a) can be expressed as:
wherein,
Wherein,And/>The first virtual work vector/>, respectivelyIs defined as the mid-point and interval radius.
Finally, the second virtual work vector is dividedIn-interval vector/>The first order taylor series expansion at the midpoint of (a) can be expressed as:
wherein,
Wherein,And/>The interval second virtual work vector/>, respectivelyIs defined as the mid-point and interval radius.
Further, the specific steps of the step 7 are as follows:
and step six, obtaining And/>Substituting the approximate expansion expression of (a) into the interval statics response equivalent equation of the double crane system with the corresponding truncated vector alpha u in the step five, and obtaining the midpoint value and the interval radius of the driving moment interval vector according to the interval algorithm.
First, the step six is performedAnd/>Substituting the approximate expansion expression of the (3) into the equivalent equation of the static response of the double crane system interval with the corresponding truncated vector alpha u in the step five to obtain:
next, according to the first order Newman series expansion, Is expressed as:
(Jc+ΔJ)-1＝(Jc)-1-(Jc)-1ΔJ(Jc)-1
substituting the approximate expression (J c+ΔJ)-1) into the equivalent equation of the hydrostatic response of the double crane system interval with the corresponding truncated vector alpha u, and neglecting a high-order term to obtain the following equation:
wherein,
Wherein τ c and Δτ I are respectively interval driving torque vectorsMid-point and span range of (c). /(I)Is the standard unit interval [ -1,1].
Finally, since Δτ I relates to the standard unit intervalIs monotonic, and obtains interval driving moment vector/>, according to monotonic technologyInterval radius Δτ:
wherein,
Further, the specific steps of the step 8 are as follows:
and D, driving the interval driving moment vector obtained in the step seven The middle point and the interval radius of the (4) are used for obtaining an interval driving moment vector/>, according to an interval algorithmUpper and lower bounds of (2).
Obtaining an interval driving moment vector according to an interval algorithmUpper and lower bounds of (2) are expressed as:
Wherein τ and The driving moment vector/>, respectively, of the section of the double crane system with the corresponding cut-off vector alpha u Lower and upper bounds of (2).
Further, the specific steps of the step 9 are as follows:
Driving moment vector of interval obtained in the step eight According to the fuzzy decomposition theory, obtaining a fuzzy driving moment vector tau (f) with a corresponding truncated vector alpha u:
In the above formula, the symbol U refers to the sum of products; τ (f) refers to the fuzzy drive torque vector; Refers to an interval drive torque vector.
For a better explanation of the invention, the present invention will now be described with the following changing angles:
the method for analyzing the statics uncertainty of the double crane system based on the fuzzy theory comprises the following steps:
step one: establishing a statics response equation of the double-crane system according to a geometric model of the double-crane system and by combining a virtual work principle;
step two: establishing a fuzzy parameter model by uncertain parameters of the double crane system;
Step three: combining the first step and the second step, and establishing a fuzzy static response equivalent equation, a fuzzy kinematic 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;
Step four: according to the alpha truncation strategy, the fuzzy parameter model in the second step can be converted into interval variable components with corresponding truncation domains to form a multi-fuzzy variable model;
Step five: establishing an interval statics response equivalent equation, an interval kinematics Jacobian matrix, an interval first virtual work vector and an interval second virtual work vector of the double crane system with corresponding truncated vectors by combining the third step with the fourth step;
Step six: performing approximate expansion on the inter-region kinematic Jacobian matrix, the interval first virtual work vector and the interval second virtual work vector in the fifth step to obtain an approximate expansion expression;
Step seven: substituting the approximate expansion expression obtained in the step six into the equivalent equation of the interval statics response of the double crane system with the corresponding truncated vector in the step five, and obtaining the midpoint value and the interval radius of the driving moment interval vector according to an interval algorithm;
Step eight: the middle point and the section radius of the section driving moment vector obtained in the step seven are used for obtaining the upper and lower boundaries of the section driving moment vector according to the section algorithm;
step nine: and D, obtaining the upper and lower boundaries of the interval driving moment vector obtained in the step eight, and obtaining the fuzzy driving moment vector with the corresponding truncated vector according to a fuzzy decomposition theory.
Further, the method for analyzing the statics uncertainty of the double-crane system based on the fuzzy theory comprises the following specific steps:
Step one: according to a geometric model of the double crane system, combining with a virtual work principle to establish a hydrostatic response equation of the double crane system, wherein the hydrostatic response equation is as follows:
wherein, Is the transpose of the kinematic Jacobian matrix J DACS of a double crane system,/>Is the first virtual work/>, acting on the boom A iBi Transpose of/>Is the second virtual work/>, acting on weight C 1C2 Is a transpose of (a). Expressed as:
wherein,
Wherein the base coordinate system { B }. O-YZ is located at the center of the connection point of A 1A2, the movable coordinate system { P }. O p-YpZp is located at the center of the connection point of C 1C2, D and D are the length of the crane spacing A 1A2 and the load C 1C2 respectively, L i is the length of the i-th crane boom A iBi, Y and Z are Cartesian coordinate values of the center O p of the load C 1C2 along the Y axis and the Z axis respectively, θ represents the rotation angle of the movable coordinate system { P } relative to the base coordinate system { B }, m p is the mass of the heavy object, gamma i is the luffing angle of the i-th crane,Is the first derivative of rotation angle θ with respect to time,/>And the second derivatives of y, z, theta and gamma i with respect to time are respectively shown, and g is the gravitational acceleration.
Step two: in the crane hoisting operation, system parameters have uncertainty due to the influence of uncertainty factors such as mechanical errors, environmental excitation and the like. Thus, n fuzzy parameters are introduced to quantitatively represent the uncertain system parameters, and a fuzzy parameter model is built as follows:
f＝{f1,…,fm,…,fn}T,m＝1,2,…,n
where f is a blur vector composed of n blur parameters, and f m is the mth blur parameter.
Step three: combining the step one double crane system statics response equation and the step two fuzzy parameter model to establish a double crane system fuzzy statics response equivalent equation with a fuzzy parameter model
Wherein,Is a fuzzy kinematics Jacobian matrix,/>Is the blurred first virtual work vector,Is the fuzzy second virtual work vector and τ (f) is the fuzzy driving torque vector.
Step four: according to the alpha truncation strategy, the fuzzy parameter model can be converted into interval variable compositions with corresponding truncation domains, and the multi-fuzzy variable model is formed by the following steps:
wherein, Is the interval variable of the fuzzy parameter f m under the corresponding cut-off value alpha u,m,/>Is the interval vector of the fuzzy vector F alpha u under the corresponding truncated vector, and F is the fuzzy variable model.
Step five: combining the step three double crane system fuzzy statics response equivalent equation and the step four multi-fuzzy variable model to establish a double crane system interval statics response equivalent equation with a corresponding truncated vector alpha u:
wherein, Is an interval kinematic Jacobian matrix,/>Is the first virtual work vector of the interval,Is the second virtual work vector of the interval,/>Is an interval drive torque vector.
Step six: according to the first-order Taylor series expansion, the method comprises the following steps ofInterval first virtual work vector/>Interval second virtual work vector/>Performing approximate expansion to obtain/>And/>Is a similar expansion expression of (a).
First, an inter-range kinematic Jacobian matrixIn-interval vector/>The first order taylor series expansion at the midpoint of (a) can be expressed as:
wherein,
Wherein J c and ΔJ are respectively the inter-range kinematic Jacobian matricesIs defined as the mid-point and interval radius.
Second, the first virtual work vector is dividedIn-interval vector/>The first order taylor series expansion at the midpoint of (a) can be expressed as:
wherein,
Wherein,And/>The first virtual work vector/>, respectivelyIs defined as the mid-point and interval radius.
Finally, the second virtual work vector is dividedIn-interval vector/>The first order taylor series expansion at the midpoint of (a) can be expressed as:
wherein,
Wherein,And/>The interval second virtual work vector/>, respectivelyIs defined as the mid-point and interval radius.
Step seven: and step six, obtainingAnd/>Substituting the approximate expansion expression of (a) into the interval statics response equivalent equation of the double crane system with the corresponding truncated vector alpha u in the step five, and obtaining the midpoint value and the interval radius of the driving moment interval vector according to the interval algorithm.
First, the step six is performedAnd/>Substituting the approximate expansion expression of the (3) into the equivalent equation of the static response of the double crane system interval with the corresponding truncated vector alpha u in the step five to obtain:
next, according to the first order Newman series expansion, Is expressed as:
(Jc+ΔJ)-1＝(Jc)-1-(Jc)-1ΔJ(Jc)-1
substituting the approximate expression (J c+ΔJ)-1) into the equivalent equation of the hydrostatic response of the double crane system interval with the corresponding truncated vector alpha u, and neglecting a high-order term to obtain the following equation:
wherein,
Wherein τ c and Δτ I are respectively interval driving torque vectorsMid-point and span range of (c). /(I)Is the standard unit interval [ -1,1].
Finally, since Δτ I relates to the standard unit intervalIs monotonic, and obtains interval driving moment vector/>, according to monotonic technologyInterval radius Δτ:
wherein,
Step eight: and D, driving the interval driving moment vector obtained in the step sevenThe middle point and the interval radius of the (4) are used for obtaining an interval driving moment vector/>, according to an interval algorithmUpper and lower bounds of (2).
Obtaining an interval driving moment vector according to an interval algorithmUpper and lower bounds of (2) are expressed as:
Wherein τ and The driving moment vector/>, respectively, of the section of the double crane system with the corresponding cut-off vector alpha u Lower and upper bounds of (2).
Step nine: driving moment vector of interval obtained in the step eightAccording to the fuzzy decomposition theory, obtaining a fuzzy driving moment vector tau (f) with a corresponding truncated vector alpha u: /(I)
Compared with the prior art, the invention has the advantages that:
[1] According to the invention, the distribution range of the static response of the double-crane system is predicted by the fuzzy theory, the fuzzy uncertainty of the parameters of the double-crane system is considered for the first time, the fuzzy distribution characteristics of the static response of the double-crane system are quantitatively obtained, and the calculation result has important guiding significance for the analysis of the uncertainty of the static response of the double-crane system.
[2] Aiming at the occasion with small uncertainty, the existing solution is to adopt a Monte Carlo method, so that the method has the defects of low calculation efficiency, large sample size, difficulty in practical application and the like, and the method for analyzing the statics of the double crane system based on the fuzzy theory provided by the invention has the characteristics of small required sample size and high calculation efficiency by utilizing the first-order Taylor series expansion and the first-order Newman series expansion and combining the fuzzy decomposition theory and the interval algorithm to analyze the statics response of the double crane system under the fuzzy parameters.
[3] Aiming at the occasion with small uncertainty, the method for analyzing the statics uncertainty of the double-crane system based on the fuzzy theory has better precision compared with the traditional method. The engineering technician can solve the problem by adopting the traditional method (see the first embodiment in fig. 3 and 4) aiming at the occasions of known deterministic parameters and fuzzy parameters with smaller uncertainty; the prediction method provided by the method (see scheme II in fig. 3 and 4) can be implemented to fully consider the fuzzy uncertainty of the uncertain parameters, and deduce the distribution characteristics of the fuzzy driving moment vector by combining the perturbation theory, the fuzzy decomposition theory and the interval algorithm, so that the complexity of engineering problems is fully considered, the relative accuracy of calculation results is ensured, and notably, the calculation time can be greatly shortened by the method provided by the invention.
Drawings
Fig. 1 is a flow chart of the present invention.
FIG. 2 is a schematic diagram of a three-dimensional model of a dual truck crane system; the figure shows a turntable 1 in a first automobile crane system, a turntable 2 in a second automobile crane system, a suspension arm A 1B1 in the first automobile crane system, a suspension arm A 2B2 in the second automobile crane system, a suspension rope B 1C1 in the first automobile crane system, a suspension rope B 2C2 in the second automobile crane system, a load C 1C2, a load gravity center O p, a hinge point A 1、A2、B1、B2、C1、C2 and the positional relationship thereof.
Fig. 3 is a graph of interval upper bound values of driving moment of a first truck crane system calculated by a conventional method and the method of the invention respectively in a computer by adopting a prediction method of upper and lower bounds of static response under fuzzy parameters of the two truck crane systems when the small uncertainty range of the fuzzy parameters provided by the invention is [0,0.10% ].
Fig. 4 is a graph of interval lower bound values of driving moment of a first truck crane system calculated by a conventional method and the method of the invention respectively in a computer by adopting a prediction method of upper and lower bounds of static response of the fuzzy parameter of the two truck crane systems when the small uncertainty range of the fuzzy parameter provided by the 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 method for analyzing the statics uncertainty of the double crane system based on the fuzzy theory comprises the following steps:
step one: establishing a statics response equation of the double-crane system according to a geometric model of the double-crane system and by combining a virtual work principle;
step two: establishing a fuzzy parameter model by uncertain parameters of the double crane system;
Step three: combining the first step and the second step, and establishing a fuzzy static response equivalent equation, a fuzzy kinematic 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;
Step four: according to the alpha truncation strategy, the fuzzy parameter model in the second step can be converted into interval variable components with corresponding truncation domains to form a multi-fuzzy variable model;
Step five: establishing an interval statics response equivalent equation, an interval kinematics Jacobian matrix, an interval first virtual work vector and an interval second virtual work vector of the double crane system with corresponding truncated vectors by combining the third step with the fourth step;
Step six: performing approximate expansion on the inter-region kinematic Jacobian matrix, the interval first virtual work vector and the interval second virtual work vector in the fifth step to obtain an approximate expansion expression;
Step seven: substituting the approximate expansion expression obtained in the step six into the equivalent equation of the interval statics response of the double crane system with the corresponding truncated vector in the step five, and obtaining the midpoint value and the interval radius of the driving moment interval vector according to an interval algorithm;
Step eight: the middle point and the section radius of the section driving moment vector obtained in the step seven are used for obtaining the upper and lower boundaries of the section driving moment vector according to the section algorithm;
step nine: and D, obtaining the upper and lower boundaries of the interval driving moment vector obtained in the step eight, and obtaining the fuzzy driving moment vector with the corresponding truncated vector according to a fuzzy decomposition theory.
As shown in fig. 1 and 2, in a first step, according to a geometric model of the dual crane system, a static response equation of the dual crane system is established by combining a virtual work principle:
wherein, Is the transpose of the kinematic Jacobian matrix J DACS of a double crane system,/>Is the first virtual work/>, acting on the boom A iBi Transpose of/>Is the second virtual work/>, acting on weight C 1C2 Is a transpose of (a). Expressed as:
wherein,
/>
Wherein the base coordinate system { B }. O-YZ is located at the center of the connection point of A 1A2, the movable coordinate system { P }. O p-YpZp is located at the center of the connection point of C 1C2, D and D are the length of the crane spacing A 1A2 and the load C 1C2 respectively, L i is the length of the i-th crane boom A iBi, Y and Z are Cartesian coordinate values of the center O p of the load C 1C2 along the Y axis and the Z axis respectively, θ represents the rotation angle of the movable coordinate system { P } relative to the base coordinate system { B }, m p is the mass of the heavy object, gamma i is the luffing angle of the i-th crane,Is the first derivative of rotation angle θ with respect to time,/>And the second derivatives of y, z, theta and gamma i with respect to time are respectively shown, and g is the gravitational acceleration.
As shown in fig. 1 and 2, further, in the second step, in the crane lifting operation, the system parameters have uncertainty due to the influence of uncertainty factors such as mechanical errors and environmental excitation. Thus, n fuzzy parameters are introduced to quantitatively represent the uncertain system parameters, and a fuzzy parameter model is built as follows:
f＝{f1,…,fm,…,fn}T,m＝1,2,…,n
where f is a blur vector composed of n blur parameters, and f m is the mth blur parameter.
As shown in fig. 1 and 2, in a third step, a step one dual crane system statics response equation and a step two fuzzy parameter model are combined to establish a dual crane system statics response equivalent equation with a fuzzy parameter model
Wherein,Is a fuzzy kinematics Jacobian matrix,/>Is the blurred first virtual work vector,Is the fuzzy second virtual work vector and τ (f) is the fuzzy driving torque vector.
As shown in fig. 1 and 2, in a fourth step, according to the α -truncated strategy, the fuzzy parameter model may be converted into a section variable composition with a corresponding truncated domain, so as to form a multi-fuzzy variable model as follows:
wherein, Is the interval variable of the fuzzy parameter f m under the corresponding cut-off value alpha u,m,/>Is the interval vector of the fuzzy vector F alpha u under the corresponding truncated vector, and F is the fuzzy variable model.
As shown in fig. 1 and 2, in a fifth step, a double crane system interval statics response equivalent equation with a corresponding truncated vector α u is established by combining the double crane system ambiguity statics response equivalent equation in the third step and the multi-ambiguity variable model in the fourth step:
wherein, Is an interval kinematic Jacobian matrix,/>Is the first virtual work vector of the interval,Is the second virtual work vector of the interval,/>Is an interval drive torque vector.
As shown in fig. 1 and 2, further, in step six, the inter-regional kinematic jacobian matrix is developed according to the taylor series of the first orderInterval first virtual work vector/>Interval second virtual work vector/>Performing approximate expansion to obtain/>And/>Is a similar expansion expression of (a).
First, an inter-range kinematic Jacobian matrixIn-interval vector/>The first order taylor series expansion at the midpoint of (a) can be expressed as:
wherein,
Wherein J c and ΔJ are respectively the inter-range kinematic Jacobian matricesIs defined as the mid-point and interval radius.
Second, the first virtual work vector is dividedIn-interval vector/>The first order taylor series expansion at the midpoint of (a) can be expressed as:
wherein,
Wherein,And/>The first virtual work vector/>, respectivelyIs defined as the mid-point and interval radius.
Finally, the second virtual work vector is dividedIn-interval vector/>The first order taylor series expansion at the midpoint of (a) can be expressed as:
/>
wherein,
Wherein,And/>The interval second virtual work vector/>, respectivelyIs defined as the mid-point and interval radius.
As shown in fig. 1 and 2, in a seventh step, the method of step six is performedAndSubstituting the approximate expansion expression of (a) into the interval statics response equivalent equation of the double crane system with the corresponding truncated vector alpha u in the step five, and obtaining the midpoint value and the interval radius of the driving moment response interval vector according to the interval algorithm.
First, the step six is performedAnd/>Substituting the approximate expansion expression of the (3) into the equivalent equation of the static response of the double crane system interval with the corresponding truncated vector alpha u in the step five to obtain:
next, according to the first order Newman series expansion, Is expressed as:
(Jc+ΔJ)-1＝(Jc)-1-(Jc)-1ΔJ(Jc)-1
substituting the approximate expression (J c+ΔJ)-1) into the equivalent equation of the hydrostatic response of the double crane system interval with the corresponding truncated vector alpha u, and neglecting a high-order term to obtain the following equation:
wherein,
Wherein τ c and Δτ I are respectively interval driving torque vectorsMid-point and span range of (c). /(I)Is the standard unit interval [ -1,1].
Finally, due toWith respect to standard unit interval/>Is monotonic, and obtains interval driving moment vector/>, according to monotonic technologyInterval radius Δτ:
wherein,
As shown in fig. 1 and 2, in step eight, the section driving torque vector obtained in step seven is further describedThe middle point and the interval radius of the (4) are used for obtaining an interval driving moment vector/>, according to an interval algorithmUpper and lower bounds of (2).
Obtaining an interval driving moment vector according to an interval algorithmUpper and lower bounds of (2) are expressed as:
Wherein τ and The driving moment vector/>, respectively, of the section of the double crane system with the corresponding cut-off vector alpha u Lower and upper bounds of (2).
As shown in fig. 1 and 2, in step nine, the interval driving moment vector obtained in step eight is further describedAccording to the fuzzy decomposition theory, obtaining a fuzzy driving moment vector tau (f) with a corresponding truncated vector alpha u:
Fig. 2 is a schematic diagram of a three-dimensional model of a two-truck crane system according to the present embodiment, which includes a turntable 1 of a first truck crane system, a turntable 2 of a second truck crane system, a boom a 1B1 of the first truck crane system, a boom a 2B2 of the second truck crane system, a hoist rope B 1C1 of the first truck crane system, a hoist rope B 2C2 of the second truck crane system, a load C 1C2, a load center of gravity O p, and a hinge point a 1、A2、B1、B2、C1、C2. One end of a hydraulic oil cylinder D 1E1 (amplitude variation oil cylinder D 2E2) is hinged with the turntable 1 (turntable 2), the other end of the hydraulic oil cylinder D 1E1 is hinged with a suspension arm A 1B1 (suspension arm A 2B2), and the rotation movement of the suspension arm A 1B1 (suspension arm A 2B2) around the hinge point of the amplitude variation oil cylinder D 1E1 (amplitude variation oil cylinder D 2E2) and the turntable 1 (turntable 2) in a vertical plane is further realized by adjusting the length of the amplitude variation oil cylinder D 1E1 (amplitude variation oil cylinder D 2E2) in the amplitude variation mechanism so as to change the elevation angle of the suspension arm A 1B1 (suspension arm A 2B2). For the above-mentioned two truck crane systems, the following describes the prediction method of the statics response under the fuzzy parameters of the two truck crane systems provided by the invention.
For the method of the invention, the implementation steps of solving the statics response under the fuzzy parameters of the double truck crane systems in a computer are further described as follows:
determining the determined value of each deterministic parameter and the distribution characteristics of fuzzy parameters according to the design parameters and the working condition requirements of the crane;
and carrying the determined values of the determined parameters and the distribution characteristics of the fuzzy parameters into formulas of an upper limit value and a lower limit value of the interval driving moment vector under the corresponding truncated vector in sequence by utilizing MATLAB programming.
Thus, the distribution characteristics of the fuzzy driving moment vector of the corresponding truncated vector under the fuzzy parameters are obtained.
The distribution characteristics of the hydrostatic response include upper and lower limits of the drive torque under the corresponding cutoff vectors.
For the traditional method (Monte Carlo method), the steps of solving the statics response under the fuzzy parameters of the double truck crane system are implemented in a computer, and are further described as follows:
determining the determined value of each deterministic parameter and the distribution characteristics of the fuzzy parameters of the system according to the design parameters and the working condition requirements of the crane;
On the premise that the determined values of the deterministic parameters and the distribution characteristics of the fuzzy parameters are obtained, selecting a random value from random distribution values of each fuzzy parameter, and inputting the random value into a MATLAB program;
and carrying the determined values of all the deterministic parameters and the random values of the fuzzy parameters into a hydrostatic response equation of the double crane system in sequence by utilizing MATLAB programming.
Therefore, the driving moment of the double crane system under a certain fuzzy parameter is obtained.
Repeating the above process for the times of i=10000 times, outputting a driving moment distribution curve of the double crane system under the fuzzy parameter, and outputting mathematical characteristics of the driving moment of the double crane system under the fuzzy parameter according to the computer instruction.
Referring to fig. 3 and 4, when the uncertainty range of the fuzzy parameter provided by the invention is [0,0.10% ], the interval upper limit value and the interval lower limit value of the driving moment of the first automobile crane system in the double automobile crane system shown in fig. 2 are predicted in a computer by adopting a traditional method (monte carlo method) and the method of the invention.
Specific numerical values of an upper boundary value and a lower boundary value of a section of a driving torque of a first truck crane system in the two truck crane systems are calculated respectively by adopting a traditional method and the method, and are shown in a table 1. Respectively calculating a graph of an interval upper limit value of a driving moment of a first automobile crane system in the two automobile crane systems by adopting a traditional method and the method of the invention, as shown in figure 3; the graphs of the interval lower limit values of the driving moment of the first automobile crane system in the two automobile crane systems are calculated respectively by adopting the traditional method and the method of the invention, as shown in fig. 4. The abscissa represents uncertainty, the ordinate represents the upper and lower boundary values of the interval of the driving torque, and the solid and broken lines represent the results of calculation by the conventional method and the method of the present invention, respectively.
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 a small uncertainty, the result calculated by the traditional method and the method in the invention in the computer is basically consistent by the prediction method of the static response under the fuzzy parameter of the two automobile crane systems, but after the invention is adopted, the operation time is obviously shortened, the calculation time is shortened by 5 orders of magnitude compared with the original method, so that the method has the engineering problems of high calculation efficiency (less calculation time), high solving precision and less uncertain parameter samples.
Table 1 upper and lower limits for the drive torque of the first crane truck system when the cut-off α=0.8
In summary, the invention can solve the problem of predicting the distribution of the static response of the double or more truck crane systems, fixed cranes, mobile cranes or transportation means with hooks under the condition of 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, but can be modified within the spirit and scope of the invention.
Claims (10)
1. A method for analyzing the statics uncertainty of a double crane system based on a fuzzy theory is characterized by comprising the following steps:
Step one: 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 of the first step and the fuzzy parameter model of the second step, and establishing a fuzzy statics response equivalent equation, a fuzzy kinematic 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 an interval statics response equivalent equation, an interval kinematics Jacobian matrix, an interval first virtual work vector and an interval second virtual work vector of the double crane system with corresponding truncated vectors;
step six: converting the parameters obtained in the step five to obtain an approximate expansion expression;
Step seven: substituting the approximate expansion expression into a double crane system interval statics response equivalent equation with corresponding truncated vector to obtain a midpoint value and an interval radius of a driving moment interval vector;
step eight: the middle point of the interval driving moment vector and the interval radius are converted to obtain the upper and lower boundaries 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 corresponding truncated vector, wherein the driving moment vector is the analysis conclusion of the statics uncertainty of the double crane system.
2. The method of claim 1, wherein in the first step, the statics response equation of the dual crane system is established according to a geometric model of the dual crane system and combined with a virtual work principle.
3. The method for analyzing the statics uncertainty of the double-crane system based on the fuzzy theory according to claim 1, wherein the fuzzy parameter model in the second step is established based on uncertain parameters of the double-crane system.
4. The method for analyzing the statics uncertainty of the double crane system based on the fuzzy theory according to claim 1, wherein in the fourth step, the fuzzy variable model is obtained by converting the fuzzy parameter model obtained in the second step according to an alpha truncation strategy to obtain interval variables with corresponding truncation domains;
And in the fifth step, the obtained equivalent equation of the interval statics response of the double crane system, the interval kinematic Jacobian matrix, the interval first virtual work vector and the interval second virtual work vector are all based on a fuzzy variable model, the equivalent equation of the fuzzy statics response of the double crane system with a fuzzy parameter model, the fuzzy kinematic Jacobian matrix, the fuzzy first virtual work vector and the fuzzy second virtual work vector.
5. The method for analyzing the statics uncertainty of the double crane system based on the fuzzy theory according to claim 1, wherein the approximate expansion expression in the step six is obtained by approximately expanding an interval kinematics jacobian matrix, an interval first virtual work vector and an interval second virtual work vector in the step five.
6. The method for analyzing the statics uncertainty of the double crane system based on the fuzzy theory according to claim 1, wherein an interval algorithm is adopted in the process of obtaining the midpoint value and the interval radius of the driving moment interval vector.
7. The method for analyzing the statics uncertainty of the double crane system based on the fuzzy theory according to claim 1, wherein in the step one:
the geometric model of the double-crane system refers to that two cranes cooperatively hoist the same weight;
The principle of virtual work refers to the necessary and sufficient condition that the deformation system is in balance, and for any virtual displacement, the sum of virtual work done by external force is equal to the sum of virtual work done by 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, combining with a virtual work principle to establish a hydrostatic response equation of the double crane system, wherein the hydrostatic response equation is as follows:
Wherein W refers to virtual work, A refers to a hinge point of a crane boom and a turntable, B refers to a hinge point of the crane boom and a lifting rope, and i refers to a crane serial number, 1 or 2; τ is the hydrostatic response vector of the dual crane system, specifically the drive torque vector of the dual crane system; Is the transpose of the kinematic Jacobian matrix J DACS of a double crane system,/> Is the first virtual work/>, acting on the boom A iBi Transpose of/>Is the second virtual work/>, acting on weight C 1C2 Is denoted as:
wherein,
Wherein the base coordinate system { B }. O-YZ is located at the center of the connection point of A 1A2, the movable coordinate system { P }. O p-YpZp is located at the center of the connection point of C 1C2, D and D are the length of the crane spacing A 1A2 and the load C 1C2 respectively, L i is the length of the i-th crane boom A iBi, Y and Z are Cartesian coordinate values of the center O p of the load C 1C2 along the Y axis and the Z axis respectively, θ represents the rotation angle of the movable coordinate system { P } relative to the base coordinate system { B }, m p is the mass of the heavy object, gamma i is the luffing angle of the i-th crane,Is the first derivative of rotation angle θ with respect to time,/>And the second derivatives of y, z, theta and gamma i with respect to time are respectively shown, and g is the gravitational acceleration.
8. The method for analyzing the statics uncertainty of the double crane system based on the fuzzy theory according to claim 1, wherein the specific steps of the second step are as follows: step two: in the hoisting operation of a crane, system parameters have uncertainty due to the influence of uncertainty factors such as mechanical errors, environmental excitation and the like; thus, n fuzzy parameters are introduced to quantitatively represent the uncertain system parameters, and a fuzzy parameter model is built as follows:
f＝{f1,…,fm,…,fn}T,m＝1,2,…,n
Wherein f is a blur vector composed of n blur parameters, and f m is an mth blur parameter; f 1 refers to the length D of the crane spacing a 1A2, f 2 refers to the length D of the load C 1C2, and other fuzzy parameters (f m,…,fn) are dependent on specific uncertainty factors or sources; t refers to the transpose operation of the vector inside the linear algebra.
9. The method for analyzing the statics uncertainty of the double crane system based on the fuzzy theory according to claim 1, wherein the specific steps of the third step are as follows: combining the step one double crane system statics response equation and the step two fuzzy parameter model to establish a double crane system fuzzy statics response equivalent equation with a fuzzy parameter model
Wherein,Is a fuzzy kinematics Jacobian matrix,/>Is a fuzzy first imaginary power vector,/>Is the fuzzy second virtual work vector and τ (f) is the fuzzy driving torque vector.
10. The method for analyzing the statics uncertainty of the double crane system based on the fuzzy theory according to claim 1, wherein 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 compositions with corresponding truncation domains, and the multi-fuzzy variable model is formed by the following steps:
wherein, Is the interval variable of the fuzzy parameter f m under the corresponding cut-off value alpha u,m,/>Is the interval vector of the fuzzy vector F under the corresponding truncated vector alpha u, and 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 through the alpha truncation strategy, wherein alpha is more than or equal to 0 and less than or equal to 1;
In the above formula, the superscript I refers to the meaning of the interval variable; subscript u refers to the u-th parameter; The method comprises the steps of intercepting n fuzzy parameters according to an alpha interception strategy to obtain an interval parameter which is an nth interval vector; alpha u refers to a nth truncated vector formed by truncated values obtained by truncating n fuzzy parameters according to an alpha truncated strategy; v refers to the total number of vectors composed of interval parameters obtained by truncating n fuzzy parameters according to an alpha truncating strategy.
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