CN116494234A - Novel multi-body dynamics modeling method for flexible hydraulic mechanical arm system - Google Patents
Novel multi-body dynamics modeling method for flexible hydraulic mechanical arm system Download PDFInfo
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- B—PERFORMING OPERATIONS; TRANSPORTING
- B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
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Abstract
The invention discloses a novel multi-rest dynamics modeling method of a flexible hydraulic mechanical arm system. The core comprises: according to different mechanical structures of all arm rods of the mechanical arm, the mechanical arm is respectively and equivalently simplified into a common beam type or a combination thereof; deriving the specific basis functions of the arm frames with different structures according to the Rayleigh-Rizlaw respectively; according to the Hamiltonian principle, calculating the Hamiltonian acting quantity of the arm lever, and calculating to obtain a characteristic vector to be substituted into the basic function set; decomposing the flexible arm into an imaginary rigid part and an imaginary flexible part by adopting the idea of rigid-flexible virtual separation and processing the imaginary rigid part and the imaginary flexible part respectively; deducing and calculating the hydraulic driving force and driving moment of the mechanical arm under the condition of flexible deformation; and integrating the descriptive quantities into a Lagrange equation towel according to a Lagrange dynamics modeling method, and deducing a complete system mathematical model. The method has the advantages of small calculation amount, simple calculation and high efficiency, clear deduction thought and good portability.
Description
Technical Field
The invention relates to the technical field of control of hydraulic mechanical arms, in particular to a novel multi-body dynamics modeling method of a flexible hydraulic mechanical arm system.
Background
The flexible hydraulic mechanical arm system is a liquid-solid coupled series mechanical arm system with flexible deformation driven by a hydraulic cylinder. The hydraulic mechanical arm is suitable for severe working environments with heavy emergency danger in the fields of construction, electric power, nuclear industry, ocean engineering, emergency rescue and the like, and has the outstanding advantages of light weight, large load ratio, high movement speed, low energy consumption and the like.
The actual vibration and control of the mechanical arm support relates to a plurality of intersecting subjects such as machinery, hydraulic pressure, control and the like, a solid-liquid coupling dynamic model and vibration control are very complex, and research works such as an elastic vibration active damping control strategy, track planning and the like are all required to be based on accurate dynamic models. At present, most of many researches only consider dynamic modeling of flexible arms with certain specific degrees of freedom, and the process of actually calculating dynamic models is still complex and has low calculation efficiency.
The mechanical system modeling method is improved on the basis of a traditional mechanical system Lagrange dynamics modeling method applying the thought of a hypothesis mode method, equivalent simplification is carried out on arm rods of different mechanical structures, proper basis functions are selected for the simplified arm rods, and the mechanical system is modeled by using the Lagrange dynamics modeling method.
Disclosure of Invention
The invention aims to provide a novel multi-body dynamics modeling method of a flexible hydraulic mechanical arm system, so as to solve the defects in the prior art, and particularly solve the problem of multi-body dynamics modeling of a liquid-solid coupled planar serial mechanical arm system. After the equivalent simplification of the arm lever, adopting an assumption mode method to select a basic function for the arm frame of the mechanical system after the simplification, and constructing a basic function set; decomposing the flexible arm support into a virtual rigid part and a virtual flexible part by adopting a rigid-flexible virtual separation method, and respectively processing the virtual rigid part and the virtual flexible part so as to simplify calculation and improve efficiency; and (5) finishing the model into a system multi-body dynamics model by using a Lagrangian modeling method. The modeling method has the advantages of small calculation amount, high precision, clear deduction idea and good portability.
In order to achieve the above purpose, the invention provides a novel flexible hydraulic mechanical arm system multi-body dynamics modeling method, which comprises the following steps:
s1, respectively and equivalently simplifying the mechanical structure of each arm rod of the mechanical arm into common beam types such as a cantilever beam, a simply supported beam and the like or the combination of different types of beams;
s2, judging deflection forms and curvature changes under the working state according to the actual structure of each arm lever of the mechanical arm, selecting function types capable of approximately describing deformation of the mechanical arm, and respectively deducing specific basis functions of arm frames with different structures according to the Rayleigh-Retzz method;
s3, calculating the deflection of the obtained arm lever to be a stable value in an instantaneous steady state according to the principle of minimum potential energy when the potential energy is minimum and the system is in a steady state; according to the Hamiltonian principle, calculating the Hamiltonian acting quantity of the arm lever, and calculating to obtain a characteristic vector to be substituted into the basic function set;
s4, calculating modal coordinates of all positions of the arm lever through the common calculation of the basis function and the modal angle equation, further obtaining actual deflection of the arm lever, and performing superposition calculation on the deflection of each arm lever;
s5, decomposing the flexible arm into an imaginary rigid part and an imaginary flexible part by adopting the idea of rigid-flexible virtual separation, and respectively processing the imaginary rigid part and the imaginary flexible part;
s6, deducing and calculating the hydraulic driving force and driving moment of the mechanical arm under the condition of flexible deformation;
s7, deducing a mathematical model of the flexible hydraulic mechanical arm system: according to the Lagrange dynamics modeling method, the description quantities such as the position vector, the speed vector and the like of each arm lever of the hydraulic mechanical arm are integrated into a Lagrange equation, a complete system mathematical model is deduced, and system dynamics modeling and analysis are carried out.
Preferably, the flexible hydraulic mechanical arm is a hydraulically driven planar multi-link serial mechanical arm with flexible deformation and solid-liquid coupling. The invention takes a plane series three-link flexible hydraulic mechanical arm as an example to model the system.
Preferably, in the step S1, according to the actual mechanism of the arm lever, the two types of beam frames are respectively and equivalently simplified, and the simplified equivalent is performed on the example arm lever, wherein the first equivalent of the arm lever is a cantilever beam, the second equivalent of the arm lever is a combination of a simple cantilever beam and a cantilever beam, and the third equivalent of the arm lever is a cantilever beam; in particular, the length of the front end portion of the joint between the first arm and the hydraulic rod is short relative to the actual length of the entire arm, and the influence of the deflection deformation generated on the entire deflection is approximately calculated to generate interference, so that the influence is ignored and is assumed to be a rigid component.
Preferably, in the step S2, for the mechanical arm system, the geometric boundary conditions that the arm base function needs to satisfy are as follows:
，
base function of arm one based on rayleigh-litz method satisfying boundary conditions:
，
selecting a complete frontThe function of the order mode superposition is too complex as a basis function for the overall calculation of the system, and the improvement brought by the accuracy is very little, so the first two-order mode superposition with larger overall influence is selected as an arm lever basis function,
，
wherein the method comprises the steps ofThe length of the arm lever is the length in front of a hinge point;
the last arm rod at the tail end of the whole arm support can be simplified into a typical cantilever structure, and only the superposition of the first two-order modes is selected as a basis function, so that the precision requirement can be met, and the calculation efficiency is greatly improved;
the invention is exemplified by a planar series three-link flexible hydraulic mechanical arm system, the basis function set of which is as follows:
。
preferably, in the step S3, the hamiltonian amount of the first arm lever may be expressed as:
，
the defined variables are:
，
the hamiltonian is represented in matrix form as:
，
solving the approximate solution of the basis function into a solution parameter problem, i.e. solving. The constraint that each parameter has a non-zero solution is that its coefficient determinant be zero,
，
thus, n eigenvalues and corresponding eigenvectors can be obtained as，And will->Substitution into the set of basis functions facilitates subsequent computation substitutions.
Preferably, in the step S5, the arm lever is subjected to a rigid-flexible virtual decomposition. Deflection change of the plane serial mechanical arm system is mainly embodied in the vertical direction, so that the plane serial mechanical arm system is supposed to be in a follow-up coordinate systemIs->The shaft is a flexible arm->Is a virtual rigid part of->The shaft is a flexible portion. Generalized coordinates of the respective virtual rigid movements>And generalized coordinates of flexible deformations ∈>Analysis is carried out to obtain the virtual rigid arm rotary joint coupling item +.>And a coupling item for reflecting the deformation of the flexible arm and the deformation speed +.>、/>；
Rigid joint coupling term:
，
flexible deformation coupling term:
，
wherein the method comprises the steps ofIs arm lever->Inertia matrix of>And->Is arm lever->And the deformation and the coupling effect of the deformation speed.
Preferably, in the step S6, the influence of the valve element and the cylinder leakage on the rigid body is ignored, and the friction loss in the pipe is ignored assuming that the oil pressure in the cylinder is the same. The equation of motion of the hydraulic cylinder piston is:
，
order theThe hydraulic cylinder driving force equation is:
，
is provided with、/>Are respectively arranged as the hinge points of the hydraulic cylinder and the adjacent two arm rods, < >>，/>. The position vector of the hydraulic cylinder 1 between the arm 1 and the base can be expressed as:
，
obtaining a speed vector and an acceleration vector of the hydraulic cylinder 1 after deriving:
，
similarly, the position vectors of the hydraulic cylinders 2 and 3 are respectively:
，
and similarly, the speed vector and the acceleration vector of the second hydraulic cylinder and the third hydraulic cylinder can be obtained. Substituting the hydraulic cylinder position, speed and acceleration vector equation into a hydraulic cylinder driving force equation to obtain the following components:
，
the flexible hydraulic mechanical arm hydraulic cylinder driving moment is expressed as:
。
preferably, in the step S7, the kinetic energy, gravitational potential energy, and elastic potential energy of the flexible arm support are solved, as follows:
，
wherein the kinetic energy isGravitational potential energy is->The elastic potential energy stored by deformation is +.>；
Defining a Lagrangian function of the flexible mechanical arm:
，
selecting generalized coordinatesWherein->，According to Lagrangian equation
，
Wherein,,for generalized speed, ++>Is a generalized moment;
after operation and arrangement, the multi-body dynamics equation of the matrix flexible hydraulic mechanical arm system is as follows:
，
preferably, in the step S7, a kinetic mathematical model of the flexible hydraulic mechanical arm system is derived by matetica scientific calculation software, and data simulation is performed by MATLAB simulation software.
Compared with the prior art, the invention has the beneficial effects that: (1) The invention provides a multi-basis function combination flexible hydraulic mechanical arm system multi-body dynamics modeling method based on a hypothesis mode method, which is used for respectively and equivalently simplifying the mechanical structure of each arm rod, properly combining the basis functions selected and attached according to the simplified structure to describe the flexural deformation of the flexible arm rod, and improving the reliability and the accuracy of the dynamic modeling of the flexible hydraulic mechanical arm system; (2) The invention adopts the idea of rigid-flexible virtual separation to decompose the flexible arm into a virtual rigid part and a virtual flexible part and respectively process the virtual rigid part and the virtual flexible part, thereby reducing unnecessary coupling calculation and achieving the purposes of simplifying calculation and improving efficiency; (3) The flexible characteristic of the rigid-flexible coupling mechanical arm in the modeling method is only embodied in、/>And +.>In the equal coupling terms, when the dynamics modeling is carried out on different types of planar flexible mechanical arms, the dynamics model can be obtained by modifying the rigid-flexible coupling related terms. The method has the advantages of small calculation amount, clear deduction idea and good portability.
Drawings
FIG. 1 is a system diagram of a flexible mechanical arm of the present invention;
FIG. 2 is an equivalent simplified schematic diagram of an exemplary arm lever of the present invention;
FIG. 3 is a simplified schematic diagram of an exemplary second equivalent arm of the present invention;
FIG. 4 is a three equivalent simplified schematic diagram of an exemplary arm lever of the present invention;
FIG. 5 is a coordinate distribution diagram of the flexible hydraulic mechanical arm system of the present invention;
FIG. 6 is a block diagram of a hydraulic cylinder of the present invention;
FIG. 7 is a simplified position diagram of a hydraulic cylinder of the present invention;
FIG. 8 is a modeling flow for the flexible hydro-mechanical arm system of the present invention.
Detailed Description
The invention provides a multi-body dynamics modeling method of a novel flexible hydraulic mechanical arm system, and in order to make the invention more shallow and understandable, the invention is further described below with reference to the accompanying drawings and the specific embodiments.
The modeling method of the flexible hydraulic mechanical arm system comprises the steps of respectively and equivalently simplifying all arm rods, selecting a basis function by combining a hypothesis mode method and the actual deflection condition of the arm rods, respectively processing a virtual rigid part and a flexible part of the arm frame by adopting a rigid-flexible virtual separation idea, and building a complete multi-body dynamics model by using a Lagrange dynamics modeling method. The method mainly comprises the following steps:
s1, respectively and equivalently simplifying the mechanical structure of each arm rod of the mechanical arm into common beam types such as a cantilever beam, a simply supported beam and the like or the combination of different types of beams;
as shown in fig. 2, simplifying the equivalent of the arm rod into a cantilever beam, particularly, the part of the length before the hinge point of the hydraulic cylinder is compared with the length of the whole arm rod, the flexible deformation in the vertical direction is negligible, and the equivalent is approximate rigid part;
as shown in fig. 3, the equivalent of the arm lever is simplified into a combination between the simply supported beam and the cantilever beam, wherein the part before the hinge point is equivalently simplified into the simply supported beam, and the part after the hinge point is simplified into the cantilever beam;
as shown in fig. 4, the three equivalent effects of the arm are simplified as a cantilever beam.
S2, judging deflection forms and curvature changes under the working state according to the actual structure of each arm lever of the mechanical arm, selecting function types capable of approximately describing deformation of the mechanical arm, and respectively deducing specific basis functions of arm frames with different structures according to the Rayleigh-Retzz method;
according to the change of the curvature and the curvature in practical application, the deflection of the arm increases along with the length of the arm. The geometric boundary conditions that the arm base function needs to satisfy are as follows:
，
for a robotic arm system, the arm-basis function based on the rayleigh-litz method that satisfies the boundary condition is:
，
selecting a complete frontThe function of the order mode superposition is too complex as a basis function for the overall calculation of the system, and the improvement brought by the accuracy is very little, so the first two-order mode superposition with larger overall influence is selected as an arm lever basis function,
，
wherein the method comprises the steps ofThe length of the arm lever is the length in front of a hinge point;
the last arm rod at the tail end of the whole arm support can be simplified into a typical cantilever structure, and only the superposition of the first two-order modes is selected as a basis function, so that the precision requirement can be met, and the calculation efficiency is greatly improved;
the invention is exemplified by a planar series three-link flexible hydraulic mechanical arm system, the basis function set of which is as follows:
。
s3, calculating the deflection of the obtained arm lever to be a stable value in an instantaneous steady state according to the principle of minimum potential energy when the potential energy is minimum and the system is in a steady state; according to the Hamiltonian principle, the Hamiltonian action quantity of the arm lever is calculated, and the characteristic vector is calculated and substituted into the basic function set. The present invention is illustrated with a planar tandem three-bar flexible hydraulic robotic arm system, so in the example, the Hamiltonian effect of arm one can be expressed as:
，
the defined variables are:
，
the hamiltonian is represented in matrix form as:
，
solving the approximate solution of the basis function into a solution parameter problem, i.e. solving. The constraint that each parameter has a non-zero solution is that its coefficient determinant be zero,
，
thus, n eigenvalues and corresponding eigenvectors can be obtained as，And will->Substitution into the set of basis functions facilitates subsequent computation substitutions.
S4, calculating modal coordinates of all positions of the arm lever through the common calculation of the basis function and the modal angle equation, further obtaining actual deflection of the arm lever, and performing superposition calculation on the deflection of each arm lever;
s5, decomposing the flexible arm into a virtual rigid part and a virtual flexible part by adopting the idea of rigid-flexible virtual separation, and processing the virtual rigid part and the virtual flexible part respectively. Deflection change of the plane serial mechanical arm system is mainly embodied in the vertical direction, so that the plane serial mechanical arm system is supposed to be in a follow-up coordinate systemIs->The shaft is a flexible arm->Is a virtual rigid part of->The shaft is a flexible portion. Generalized coordinates of the respective virtual rigid movements>And generalized coordinates of flexible deformations ∈>Analysis is carried out to obtain the virtual rigid arm rotary joint coupling item +.>And a coupling item for reflecting the deformation of the flexible arm and the deformation speed +.>、/>；
Rigid joint coupling term:
，
flexible deformation coupling term:
，
wherein the method comprises the steps ofIs arm lever->Inertia matrix of>And->Is arm lever->And the deformation and the coupling effect of the deformation speed.
S6, deducing and calculating the hydraulic driving force and driving moment of the mechanical arm under the condition of flexible deformation;
and neglecting the influence of the valve core and the external leakage of the hydraulic cylinder on the rigid body, and neglecting the friction loss in the pipeline on the assumption that the oil pressure in the hydraulic cylinder is the same. The equation of motion of the hydraulic cylinder piston is:
，
order theThe hydraulic cylinder driving force equation is:
，
is provided with、/>Are respectively arranged as the hinge points of the hydraulic cylinder and the adjacent two arm rods, < >>，/>. The position vector of the hydraulic cylinder 1 between the arm 1 and the base can be expressed as:
，
obtaining a speed vector and an acceleration vector of the hydraulic cylinder 1 after deriving:
，
similarly, the position vectors of the hydraulic cylinders 2 and 3 are respectively:
，
and similarly, the speed vector and the acceleration vector of the second hydraulic cylinder and the third hydraulic cylinder can be obtained. Substituting the hydraulic cylinder position, speed and acceleration vector equation into a hydraulic cylinder driving force equation to obtain the following components:
，
the flexible hydraulic mechanical arm hydraulic cylinder driving moment is expressed as:
。
s7, deducing a mathematical model of the flexible hydraulic mechanical arm system: according to the Lagrange dynamics modeling method, the description quantities such as the position vector, the speed vector and the like of each arm lever of the hydraulic mechanical arm are integrated into a Lagrange equation, a complete system mathematical model is deduced, and system dynamics modeling and analysis are carried out.
The kinetic energy of the flexible arm support isGravitational potential energy is->The elastic potential energy stored due to deformation is +.>As shown below.
，
Defining a Lagrangian function of the flexible mechanical arm:
，
selecting generalized coordinatesWherein->，/>According to Lagrangian equation
，
Wherein,,for generalized speed, ++>Is a generalized moment;
after operation and arrangement, the multi-body dynamics equation of the matrix flexible hydraulic mechanical arm system is as follows:
。
Claims (7)
1. the novel multi-body dynamics modeling method of the flexible hydraulic mechanical arm system is characterized by comprising the following steps of:
s1, respectively and equivalently simplifying the mechanical structure of each arm rod of the mechanical arm into common beam types such as a cantilever beam, a simply supported beam and the like or the combination of different types of beams;
s2, judging deflection forms and curvature changes under the working state according to the actual structure of each arm lever of the mechanical arm, selecting function types capable of approximately describing deformation of the mechanical arm, and respectively deducing specific basis functions of arm frames with different structures according to the Rayleigh-Retzz method;
s3, calculating the deflection of the obtained arm lever to be a stable value in an instantaneous steady state according to the principle of minimum potential energy when the potential energy is minimum and the system is in a steady state; according to the Hamiltonian principle, calculating the Hamiltonian acting quantity of the arm lever, and calculating to obtain a characteristic vector to be substituted into the basic function set;
s4, calculating modal coordinates of all positions of the arm lever through the common calculation of the basis function and the modal angle equation, further obtaining actual deflection of the arm lever, and performing superposition calculation on the deflection of each arm lever;
s5, decomposing the flexible arm into an imaginary rigid part and an imaginary flexible part by adopting the idea of rigid-flexible virtual separation, and respectively processing the imaginary rigid part and the imaginary flexible part;
s6, deducing and calculating the hydraulic driving force and driving moment of the mechanical arm under the condition of flexible deformation;
s7, deducing a mathematical model of the flexible hydraulic mechanical arm system: according to the Lagrange dynamics modeling method, the description quantities such as the position vector, the speed vector and the like of each arm lever of the hydraulic mechanical arm are integrated into a Lagrange equation, a complete system mathematical model is deduced, and system dynamics modeling and analysis are carried out.
2. The novel flexible hydraulic mechanical arm system multi-body dynamics modeling method according to claim 1, wherein the step S1 specifically includes:
according to the equivalent simplification of the actual structure of the mechanical arm rod, the invention takes a plane series three-connecting-rod flexible hydraulic mechanical arm as an example. The method comprises the steps that equivalent simplification is carried out on an example arm lever I, a part in front of a hinge point of the arm lever I is regarded as a rigid part, and a part behind the hinge point of the arm lever I is equivalently simplified into a cantilever beam; the method comprises the steps that an example arm lever II is equivalently simplified to be a combination of a simple support beam and a cantilever beam, wherein the part in front of a hinge point of the arm lever II is the simple support beam, and the part behind the hinge point is the cantilever beam; and the equivalent simplification is carried out on the example arm lever III, and the equivalent simplification is a cantilever beam.
3. The novel multi-body dynamics modeling method of a flexible hydraulic mechanical arm system according to claim 1, wherein the step S2 specifically includes:
according to the change of the curvature and the curvature in practical application, the deflection of the arm increases along with the length of the arm. The geometric boundary conditions that the arm base function needs to satisfy are as follows:
，
for a robotic arm system, the arm-basis function based on the rayleigh-litz method that satisfies the boundary condition is:
，
selecting a complete frontThe function of the order mode superposition is too complex as a basis function for the overall calculation of the system, and the improvement brought by the accuracy is very little, so the first two-order mode superposition with larger overall influence is selected as an arm lever basis function,
，
wherein the method comprises the steps ofThe length of the arm lever is the length in front of a hinge point;
the last arm rod at the tail end of the whole arm support can be simplified into a typical cantilever structure, and only the superposition of the first two-order modes is selected as a basis function, so that the precision requirement can be met, and the calculation efficiency is greatly improved;
the invention is exemplified by a planar series three-link flexible hydraulic mechanical arm system, the basis function set of which is as follows:
。
4. the novel flexible hydraulic mechanical arm system multi-body dynamics modeling method according to claim 1, wherein the step S3 specifically includes:
the invention is illustrated by a planar series three-link flexible hydraulic mechanical arm system, and the Hamiltonian action amount of an example arm rod I can be expressed as follows:
，
the defined variables are:
，
the hamiltonian is represented in matrix form as:
，
solving the approximate solution of the basis function into a solution parameter problem, i.e. solving. The constraint that each parameter has a non-zero solution is that its coefficient determinant be zero,
，
thus, n eigenvalues and corresponding eigenvectors can be obtained as，/>And will->Substitution into the set of basis functions facilitates subsequent computation substitutions.
5. The novel flexible hydraulic mechanical arm system multi-body dynamics modeling method according to claim 1, wherein the step S5 specifically comprises:
the example arms were subjected to a rigid-flexible virtual decomposition. Deflection change of the plane serial mechanical arm system is mainly embodied in the vertical direction, so that the plane serial mechanical arm system is supposed to be in a follow-up coordinate systemIs->The shaft is a flexible arm->Is a virtual rigid part of->The shaft is a flexible portion. Generalized coordinates of the respective virtual rigid movements>And generalized coordinates of flexible deformations ∈>Analysis is carried out to obtain the virtual rigid arm rotary joint coupling item +.>And a coupling item for reflecting the deformation of the flexible arm and the deformation speed +.>、/>；
Rigid joint coupling term:
，
flexible deformation coupling term:
，
wherein the method comprises the steps ofIs arm lever->Inertia matrix of>And->Is arm lever->And the deformation and the coupling effect of the deformation speed.
6. The novel flexible hydraulic mechanical arm system multi-body dynamics modeling method according to claim 1, wherein the step S6 specifically includes:
neglecting the influence of the valve core and the leakage outside the hydraulic cylinder on the cylinder body, and neglecting the friction loss in the pipeline on the premise that the oil pressure in the hydraulic cylinder is the same. The equation of motion of the hydraulic cylinder piston is:
，
order theThe hydraulic cylinder driving force equation is:
，
the flexible hydraulic mechanical arm hydraulic cylinder driving moment is expressed as:
is provided with、/>Are respectively arranged as the hinge points of the hydraulic cylinder and the adjacent two arm rods, < >>，/>. The position vector of the hydraulic cylinder 1 between the arm 1 and the base can be expressed as:
，
obtaining a speed vector and an acceleration vector of the hydraulic cylinder 1 after deriving:
，
similarly, the position vectors of the hydraulic cylinders 2 and 3 are respectively:
，
and similarly, the speed vector and the acceleration vector of the second hydraulic cylinder and the third hydraulic cylinder can be obtained. Substituting the hydraulic cylinder position, speed and acceleration vector equation into a hydraulic cylinder driving force equation to obtain the following components:
。
7. the novel flexible hydraulic mechanical arm system multi-body dynamics modeling method according to claim 1, wherein the step S7 specifically includes:
the kinetic energy, gravitational potential energy and elastic potential energy of the flexible arm support are solved, and the method is expressed as follows:
，
wherein the kinetic energy isGravitational potential energy is->The elastic potential energy stored by deformation is +.>；
Defining a Lagrangian function of the flexible mechanical arm:
，
selecting generalized coordinatesWherein->，/>According to Lagrangian equation
，
Wherein,,for generalized speed, ++>Is a generalized moment;
after operation and arrangement, the multi-body dynamics equation of the matrix flexible hydraulic mechanical arm system is as follows:
。
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| Publication number | Priority date | Publication date | Assignee | Title |
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| CN117171864A (en) * | 2023-11-02 | 2023-12-05 | 鲁东大学 | Beam structure linear vibration prediction method |
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| CN117171864A (en) * | 2023-11-02 | 2023-12-05 | 鲁东大学 | Beam structure linear vibration prediction method |
| CN117171864B (en) * | 2023-11-02 | 2024-02-20 | 鲁东大学 | Beam structure linear vibration prediction method |
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