Mathematical modeling method and system applying mechanics

Abstract

The invention provides a mathematical modeling method and system applying mechanics. The method comprises the steps that a symmetric piecewise linear model of an auxiliary spring system is obtained based on a motion equation; obtaining an integral interval of a corrected phase according to the corrected symmetric linear model, and obtaining a stability discriminant of a steady solution; adjusting the symmetric piecewise linear model according to the clear bifurcation point; two adjacent initial state vectors in the adjusted symmetric piecewise linear model are extracted, motion state tracks of the two initial state vectors are obtained, and a chaotic boundary is established according to the boundary of the motion state tracks; and adjusting the chaos boundary based on the periodic excitation force to obtain a regular motion state, and further correcting the symmetric piecewise linear model according to the regular motion state to obtain a mathematical model of the auxiliary spring system. According to the method, the instability of the symmetric piecewise linear model caused by the vibration diversity of the mechanical structure can be avoided, and the symmetric piecewise linear model is further corrected according to the regular motion state so as to improve the stability of the piecewise linear system.

Description

Technical Field

[0001] The present invention relates to the technical field of mathematical modeling, and in particular to a mathematical modeling method and system for applied mechanics. Background Art

[0002] With the development of industry, humans have created more and more mechanical systems. Most mechanical systems will cause structural imbalance due to improper design or unqualified processing, which will further cause vibration problems: for example, the imbalance in diesel engines can cause ground waves strong enough to generate urban noise, and the structural imbalance of some locomotives can cause the wheels to deviate from the track by as much as 1mm when running at high speed. In all these cases, the structures or parts that are subjected to vibration will cause material fatigue failure due to the alternating stress caused by vibration. In addition, vibration will increase the wear of mechanical parts, such as bearings and gears. Vibration can also loosen fasteners in machinery, such as nuts. When cutting metal, vibration will cause chatter, resulting in poor surface processing quality.

[0003] Despite the negative side of vibration, vibration is often effectively used in daily life and production practice. In fact, the application of vibration equipment has grown rapidly in recent years. For example, vibration conveyors, vibration distributors, vibration compactors, washing machines, electric toothbrushes, small electric drills used by dentists, clocks and other equipment are all examples of equipment that use vibration.

[0004] Due to wear or human design, many mechanical systems have internal gaps, which can be simplified into piecewise linear systems in theoretical analysis. Piecewise smooth systems containing gaps, constraints, friction, hysteresis and other factors in vibration problems are a very important part in the field of vibration engineering.

[0005] Among the existing technologies, piecewise linear systems have a wide range of application scenarios in the engineering field. However, the randomness and diversity of wear in existing vibrations, such as amplitude instability, lead to unsatisfactory performance of piecewise linear models, which in turn makes the stability of piecewise linear systems poor. Summary of the invention

[0006] Based on this, the purpose of the present invention is to provide a mathematical modeling method and system for applied mechanics to at least solve the deficiencies in the above-mentioned prior art.

[0007] In a first aspect, the present invention provides a mathematical modeling method for applied mechanics, the method comprising: Acquire a mechanical model of the auxiliary spring system in the mechanical model, acquire a motion equation of the auxiliary spring system according to the mechanical model through numerical simulation, and acquire a symmetric piecewise linear model of the auxiliary spring system based on the motion equation; Modifying the symmetric piecewise linear model, obtaining an integral interval of a modified phase according to the modified symmetric linear model, and obtaining a steady-state solution of the symmetric piecewise linear model based on the integral interval and an averaging method, and obtaining a stability discriminant of the steady-state solution; Extracting a bifurcation diagram that changes with the excitation frequency in the modified symmetric piecewise linear model, modifying the bifurcation diagram based on the stability discriminant to obtain a clear bifurcation point, and adjusting the symmetric piecewise linear model according to the clear bifurcation point; Extracting two adjacent initial state vectors in the adjusted symmetric piecewise linear model, obtaining motion state trajectories of the two initial state vectors, and establishing a chaotic boundary according to the boundary of the motion state trajectory; The chaotic boundary is adjusted based on the periodic exciting force to obtain a regular motion state, and the symmetrical piecewise linear model is further corrected according to the regular motion state to obtain a mathematical model of the secondary spring system.

[0008] Compared with the prior art, the beneficial effects of the present invention are: by obtaining the steady-state solution of the symmetric piecewise linear model based on the integral interval and the averaging method, and obtaining the stability discriminant of the steady-state solution, and correcting the bifurcation diagram through the stability discriminant, and adjusting the symmetric piecewise linear model by clarifying the bifurcation point, it is possible to avoid the instability of the symmetric piecewise linear model caused by the diversity of mechanical structure vibration, and establish a chaotic boundary through the boundary of the motion state trajectory, adjust the chaotic boundary through periodic excitation force, and further correct the symmetric piecewise linear model according to the regular motion state, so as to improve the stability of the piecewise linear system.

[0009] Furthermore, the step of obtaining the mechanical model of the auxiliary spring system in the mechanical model and obtaining the motion equation of the auxiliary spring system according to the mechanical model through numerical simulation includes: Obtaining the mass, stiffness and damping coefficient of the secondary spring system in the mechanical model; generating a mechanical model according to the mass, the stiffness, the damping coefficient and the numerical simulation; Numerical simulation is performed using MATLAB to obtain the motion equation of the secondary spring system.

[0010] Furthermore, the expression of the symmetric piecewise linear model is: ; In the formula, represents the mass of the secondary spring system, represents the displacement of the secondary spring system, represents the damping coefficient of the secondary spring system, represents the stiffness of the secondary spring system, represents the excitation frequency, Indicates the cycle.

[0011] Furthermore, the step of correcting the symmetric piecewise linear model, obtaining an integral interval of a corrected phase according to the corrected symmetric linear model, and obtaining a steady-state solution of the symmetric piecewise linear model based on the integral interval and an averaging method, and obtaining a stability discriminant of the steady-state solution includes: The phase interval of the symmetrical piecewise linear model is corrected by using an averaging method to correct the symmetrical piecewise linear model; According to the modified separation condition of the symmetric piecewise linear model, an integral interval of the modified phase is obtained; Based on the averaging method and the integral interval, a steady-state solution of the symmetric piecewise linear model is obtained, and a stability discriminant is obtained according to the parameters of the auxiliary spring system and the steady-state solution.

[0012] Furthermore, the step of extracting the bifurcation diagram of the modified symmetric piecewise linear model that changes with the excitation frequency includes: Extracting the Poincare section after movement of one point in the modified symmetric piecewise linear model through numerical simulation; Based on the Poincare section, a bifurcation diagram of the modified symmetric piecewise linear model that changes with the excitation frequency is drawn.

[0013] Furthermore, the step of extracting two adjacent initial state vectors in the adjusted symmetric piecewise linear model, obtaining the motion state trajectories of the two initial state vectors, and establishing a chaotic boundary according to the boundary of the motion state trajectory includes: Extracting two similar but different initial state vectors from the adjusted symmetric piecewise linear model, and solving the state trajectories of the two initial state vectors based on the distance between the two initial state vectors and the initial point; The motion state of the symmetric piecewise linear model is acquired according to the state trajectory, and a chaos boundary is established based on the state trajectory and the motion state.

[0014] Furthermore, the step of adjusting the chaotic boundary based on the periodic exciting force to obtain a regular motion state, and further correcting the symmetric piecewise linear model according to the regular motion state includes: Obtaining the chaos factor of the chaotic world through a periodic exciting force method and the periodic motion of the secondary spring system; Selecting a preset amplitude and a preset disturbance frequency to adjust the chaotic factor to obtain a regular motion state; The boundary of the chaotic world is controlled based on the regular motion state to further modify the symmetric piecewise linear model.

[0015] In a second aspect, the present invention provides a mathematical modeling system for applied mechanics, the system comprising: An acquisition module, used for acquiring a mechanical model of a secondary spring system in a mechanical model, acquiring a motion equation of the secondary spring system according to the mechanical model through numerical simulation, and acquiring a symmetric piecewise linear model of the secondary spring system based on the motion equation; A correction module, used for correcting the symmetric piecewise linear model, obtaining an integral interval of a corrected phase according to the corrected symmetric linear model, and obtaining a steady-state solution of the symmetric piecewise linear model based on the integral interval and an average method, and obtaining a stability discriminant of the steady-state solution; A first extraction module is used to extract a bifurcation diagram that changes with the excitation frequency in the modified symmetric piecewise linear model, modify the bifurcation diagram based on the stability discriminant to obtain a clear bifurcation point, and adjust the symmetric piecewise linear model according to the clear bifurcation point; A second extraction module is used to extract two adjacent initial state vectors in the adjusted symmetric piecewise linear model, obtain the motion state trajectory of the two initial state vectors, and establish a chaotic boundary according to the boundary of the motion state trajectory; The adjustment module is used to adjust the chaotic boundary based on the periodic exciting force to obtain a regular motion state, and further correct the symmetrical piecewise linear model according to the regular motion state to obtain a mathematical model of the secondary spring system.

[0016] In a third aspect, the present invention provides a readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the above-mentioned mathematical modeling method of applied mechanics.

[0017] In a fourth aspect, the present invention provides a computer device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor implements the above-mentioned mathematical modeling method of applied mechanics when executing the computer program. BRIEF DESCRIPTION OF THE DRAWINGS

[0018] Figure 1 A flowchart of a mathematical modeling method for applied mechanics in a first embodiment of the present invention; Figure 2 It is a structural block diagram of a mathematical modeling system for applied mechanics in a second embodiment of the present invention; Figure 3 It is a structural block diagram of a computer device in the third embodiment of the present invention.

[0019] Description of main component symbols: 11. Acquisition module; 12. Correction module; 13. First extraction module; 14. Second extraction module; 15. Adjustment module; 10. Memory; 20. Processor; 30. Computer program.

[0020] The following specific implementation manner will further illustrate the present invention in conjunction with the above-mentioned drawings. DETAILED DESCRIPTION

[0021] In order to facilitate the understanding of the present invention, the present invention will be described more fully below with reference to the relevant drawings. Several embodiments of the present invention are given in the drawings. However, the present invention can be implemented in many different forms and is not limited to the embodiments described herein. On the contrary, the purpose of providing these embodiments is to make the disclosure of the present invention more thorough and comprehensive.

[0022] It should be noted that when an element is referred to as being "fixed to" another element, it may be directly on the other element or there may be a central element. When an element is considered to be "connected to" another element, it may be directly connected to the other element or there may be a central element at the same time. The terms "vertical", "horizontal", "left", "right" and similar expressions used herein are for illustrative purposes only.

[0023] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as those commonly understood by those skilled in the art to which the present invention belongs. The terms used herein in the specification of the present invention are only for the purpose of describing specific embodiments and are not intended to limit the present invention. The term "and / or" used herein includes any and all combinations of one or more of the related listed items.

[0024] See also Figure 1 , which shows a mathematical modeling method of applied mechanics in a first embodiment of the present invention, the method comprises steps S1 to S5: S1, obtaining a mechanical model of the auxiliary spring system in the mechanical model, obtaining a motion equation of the auxiliary spring system according to the mechanical model through numerical simulation, and obtaining a symmetric piecewise linear model of the auxiliary spring system based on the motion equation; Specifically, the step S1 includes steps S11 to S13: S11, obtaining the mass, stiffness and damping coefficient of the secondary spring system in the mechanical model; S12, generating a mechanical model according to the mass, the stiffness, the damping coefficient and the numerical simulation; S13, performing numerical simulation using MATLAB to obtain the motion equation of the secondary spring system; It can be understood that by obtaining the mass, stiffness and damping coefficient of the auxiliary spring system in the mechanical system, a mechanical model of the auxiliary spring system is generated by a numerical simulation method, and numerical simulation is performed using MATLAB based on the mechanical model, thereby obtaining the motion equation of the auxiliary spring system; It is worth noting that after the motion equation of the auxiliary spring system is obtained, the motion equation is simulated by the mechanical model in the classical model, so that a symmetric piecewise linear model of the auxiliary spring system can be obtained. In this embodiment, the expression of the symmetric piecewise linear model is: ; In the formula, represents the mass of the secondary spring system, represents the displacement of the secondary spring system, represents the damping coefficient of the secondary spring system, represents the stiffness of the secondary spring system, represents the excitation frequency, Indicates the cycle.

[0025] S2, correcting the symmetric piecewise linear model, obtaining an integral interval of a corrected phase according to the corrected symmetric linear model, and obtaining a steady-state solution of the symmetric piecewise linear model based on the integral interval and an averaging method, and obtaining a stability discriminant of the steady-state solution; Specifically, the step S2 includes steps S21 to S23: S21, correcting the phase interval of the symmetric piecewise linear model by using an averaging method to correct the symmetric piecewise linear model; It can be understood that the averaging method is a method for processing piecewise linear models. The phase of the symmetric piecewise linear model is integrated by the averaging method. The upper and lower limits of the phase also correspond to the parameters of the auxiliary spring system, so that the symmetric piecewise linear model will not have abnormal parameters, thereby improving the stability of the symmetric piecewise linear model.

[0026] S22, obtaining an integral interval of a modified phase according to the modified separation condition of the symmetric piecewise linear model; It can be understood that after the symmetric piecewise linear model is corrected by adopting the averaging method, the corrected phase interval is obtained according to the separation condition of the symmetric piecewise linear model. In this embodiment, the separation condition refers to the position of the separation point when the symmetric piecewise linear model is corrected.

[0027] S23, obtaining a steady-state solution of the symmetrical piecewise linear model based on an average method and the integral interval, and obtaining a stability discriminant according to the parameters of the auxiliary spring system and the steady-state solution; It can be understood that the steady-state solution of the symmetric piecewise linear model is solved by the averaging method and the modified integration interval. In the specific implementation, the maximum value in the integration interval is selected as the amplitude, so that the numerical solution of the amplitude-frequency response can be obtained. This numerical solution is the steady-state solution, and the stability discriminant is obtained by the degree of consistency between the steady-state solution and the amplitude.

[0028] S3, extracting a bifurcation diagram of the modified symmetric piecewise linear model that changes with the excitation frequency, modifying the bifurcation diagram based on the stability discriminant to obtain a clear bifurcation point, and adjusting the symmetric piecewise linear model according to the clear bifurcation point; Specifically, the step S3 includes steps S31 to S32: S31, extracting the Poincare section after movement of one point in the modified symmetric piecewise linear model through numerical simulation; S32, drawing a modified bifurcation diagram of the symmetric piecewise linear model that changes with the excitation frequency based on the Poincare section; It can be understood that the dynamic model of the auxiliary spring system is analyzed by numerical simulation, and one of the points is selected to move to the end to take the Poincare section, and then several points on the Poincare section are taken to draw a bifurcation diagram that changes with the excitation frequency.

[0029] It is worth noting that after the bifurcation diagram is obtained, the bifurcation diagram is corrected by the stability discriminant, so that a clear bifurcation point can be obtained according to the correction process, and then the symmetric piecewise linear model is adjusted according to the bifurcation point. The adjusted symmetric piecewise linear model has stable periodic motion.

[0030] S4, extracting two adjacent initial state vectors in the adjusted symmetric piecewise linear model, obtaining motion state trajectories of the two initial state vectors, and establishing a chaotic boundary according to the boundary of the motion state trajectory; Specifically, the step S4 includes step S41 to step S42: S41, extracting two similar but different initial state vectors from the adjusted symmetric piecewise linear model, and solving the state trajectories of the two initial state vectors based on the distance between the two initial state vectors and the initial point; It can be understood that in the adjusted symmetric piecewise linear model, two similar but different initial state vectors are taken, the initial state vector includes displacement and velocity, is a 6-dimensional vector, and the state trajectory of the two initial state vectors is solved according to the distance between the two initial state vectors, and the state trajectory is corrected.

[0031] S42, acquiring the motion state of the symmetric piecewise linear model according to the state trajectory, and establishing a chaotic boundary based on the state trajectory and the motion state; It can be understood that the motion state of the symmetric piecewise linear model can be obtained through the state trajectory, and then the motion boundary can be obtained according to the motion state, so that the chaos boundary can be established according to the state trajectory and the motion state, that is, bounded motion is used as a chaos boundary.

[0032] S5, adjusting the chaotic boundary based on the periodic exciting force to obtain a regular motion state, and further correcting the symmetrical piecewise linear model according to the regular motion state to obtain a mathematical model of the secondary spring system; Specifically, the step S5 includes steps S51 to S53: S51, obtaining a chaos factor of the chaos world through a periodic exciting force method and a periodic motion of the secondary spring system; S52, selecting a preset amplitude and a preset disturbance frequency to adjust the chaos factor to obtain a regular motion state; It should be explained that the internal information of the chaotic factor is unstable, so the chaotic factor of the auxiliary spring system is obtained through periodic motion and periodic excitation force method, and then the chaotic factor is adjusted by the preset amplitude and preset disturbance frequency corresponding to the chaotic factor to achieve the purpose of controlling the chaotic factor, and then a regular motion state can be obtained; S53, controlling the boundary of the chaotic world based on the regular motion state to further correct the symmetric piecewise linear model; It can be understood that after controlling the chaotic factors in the chaotic world, that is, controlling the boundary of the chaotic world through regular motion states, the state of the chaotic world can be controlled, thereby achieving the purpose of correcting the symmetric piecewise linear model, so that the mathematical model of the secondary spring system can be stably corrected after it is established.

[0033] In summary, the mathematical modeling method of applied mechanics in the above-mentioned embodiment of the present invention obtains the steady-state solution of the symmetric piecewise linear model based on the integral interval and the averaging method, and obtains the stability discriminant of the steady-state solution, and corrects the bifurcation diagram through the stability discriminant, and adjusts the symmetric piecewise linear model by clarifying the bifurcation point, so as to avoid the instability of the symmetric piecewise linear model caused by the diversity of mechanical structure vibration, and establishes a chaotic boundary through the boundary of the motion state trajectory, adjusts the chaotic boundary through periodic excitation force, and further corrects the symmetric piecewise linear model according to the regular motion state to improve the stability of the piecewise linear system.

[0034] Embodiment 2 See also Figure 2 , shown is a mathematical modeling system of applied mechanics in a second embodiment of the present invention, the system comprising: An acquisition module 11 is used to acquire a mechanical model of a secondary spring system in a mechanical model, acquire a motion equation of the secondary spring system according to the mechanical model through numerical simulation, and acquire a symmetric piecewise linear model of the secondary spring system based on the motion equation; The expression of the symmetric piecewise linear model is: ; In the formula, represents the mass of the secondary spring system, represents the displacement of the secondary spring system, represents the damping coefficient of the secondary spring system, represents the stiffness of the secondary spring system, represents the excitation frequency, Indicates a cycle; A correction module 12 is used to correct the symmetric piecewise linear model, obtain an integral interval of a corrected phase according to the corrected symmetric linear model, and obtain a steady-state solution of the symmetric piecewise linear model based on the integral interval and an average method, and obtain a stability discriminant of the steady-state solution; A first extraction module 13 is used to extract a bifurcation diagram that changes with the excitation frequency in the modified symmetric piecewise linear model, modify the bifurcation diagram based on the stability discriminant to obtain a clear bifurcation point, and adjust the symmetric piecewise linear model according to the clear bifurcation point; A second extraction module 14 is used to extract two adjacent initial state vectors in the adjusted symmetric piecewise linear model, obtain the motion state trajectory of the two initial state vectors, and establish a chaotic boundary according to the boundary of the motion state trajectory; The adjustment module 15 is used to adjust the chaotic boundary based on the periodic exciting force to obtain a regular motion state, and further correct the symmetrical piecewise linear model according to the regular motion state to obtain a mathematical model of the secondary spring system.

[0035] In some optional embodiments, the acquisition module 11 includes: A first acquisition unit, used for acquiring the mass, stiffness and damping coefficient of the auxiliary spring system in the mechanical model; A simulation unit, used for generating a mechanical model according to the mass, the stiffness, the damping coefficient and the numerical simulation; The simulation unit is used for performing numerical simulation using MATLAB to obtain the motion equation of the secondary spring system.

[0036] In some optional embodiments, the correction module 12 includes: A correction unit, used for correcting the phase interval of the symmetric piecewise linear model by adopting an averaging method to correct the symmetric piecewise linear model; A second acquisition unit, configured to acquire an integral interval of a modified phase according to the modified separation condition of the symmetric piecewise linear model; The third acquisition unit is used to obtain a steady-state solution of the symmetric piecewise linear model based on an averaging method and the integral interval, and to obtain a stability discriminant according to the parameters of the auxiliary spring system and the steady-state solution.

[0037] In some optional embodiments, the first extraction module 13 includes: An extraction and correction unit, used for extracting the modified Poincare section of one point in the symmetric piecewise linear model through numerical simulation; A drawing unit is used to draw a bifurcation diagram of the modified symmetric piecewise linear model that changes with the excitation frequency based on the Poincare section.

[0038] In some optional embodiments, the second extraction module 14 includes: A solving unit, used for extracting two similar but different initial state vectors from the adjusted symmetric piecewise linear model, and solving the state trajectory of the two initial state vectors based on the distance between the two initial state vectors and the initial point; An establishing unit is used to obtain the motion state of the symmetric piecewise linear model according to the state trajectory, and to establish a chaotic boundary based on the state trajectory and the motion state.

[0039] In some optional embodiments, the adjustment module 15 includes: A fourth acquisition unit, used for acquiring a chaotic factor of the chaotic world through a periodic exciting force method and a periodic motion of the secondary spring system; A selection unit, used for selecting a preset amplitude and a preset disturbance frequency to adjust the chaotic factor to obtain a regular motion state; A control unit is used to control the boundary of the chaotic world based on the regular motion state to further correct the symmetric piecewise linear model.

[0040] The functions or operation steps implemented when the above modules and units are executed are generally the same as those in the above method embodiments, and will not be repeated here.

[0041] The mathematical modeling system of applied mechanics provided in the embodiment of the present invention has the same implementation principle and technical effects as those of the aforementioned method embodiment. For the sake of brief description, for matters not mentioned in the system embodiment, reference may be made to the corresponding contents in the aforementioned method embodiment.

[0042] Embodiment 3 See also Figure 3, shown is a computer device in the third embodiment of the present invention, including a memory 10, a processor 20, and a computer program 30 stored in the memory 10 and executable on the processor 20, and when the processor 20 executes the computer program 30, the above-mentioned mathematical modeling method of applied mechanics is implemented.

[0043] The memory 10 includes at least one type of readable storage medium, and the readable storage medium includes a flash memory, a hard disk, a multimedia card, a card-type memory (for example, an SD or DX memory, etc.), a magnetic memory, a magnetic disk, an optical disk, etc. In some embodiments, the memory 10 may be an internal storage unit of a computer device, such as a hard disk of the computer device. In other embodiments, the memory 10 may also be an external storage device, such as a plug-in hard disk, a smart memory card (Smart MediaCard, SMC), a secure digital (Secure Digital, SD) card, a flash card, etc. Further, the memory 10 may also include both an internal storage unit of a computer device and an external storage device. The memory 10 may be used not only to store application software and various types of data installed in the computer device, but also to temporarily store data that has been output or is to be output.

[0044] Among them, in some embodiments, the processor 20 can be an electronic control unit (Electronic Control Unit, abbreviated as ECU, also known as a vehicle computer), a central processing unit (Central Processing Unit, CPU), a controller, a microcontroller, a microprocessor or other data processing chip, used to run the program code stored in the memory 10 or process data, such as executing access restriction programs, etc.

[0045] It should be pointed out that Figure 3 The structure shown does not constitute a limitation on the computer device. In other embodiments, the computer device may include fewer or more components than shown in the figure, or combine certain components, or arrange the components differently.

[0046] An embodiment of the present invention further provides a readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the mathematical modeling method of applied mechanics as described above.

[0047] Those skilled in the art will appreciate that the logic and / or steps represented in the flowchart or otherwise described herein, for example, may be considered as an ordered list of executable instructions for implementing logical functions, and may be specifically implemented in any computer-readable medium for use by an instruction execution system, device or apparatus (such as a computer-based system, a system including a processor, or other system that can fetch instructions from an instruction execution system, device or apparatus and execute instructions), or in conjunction with such instruction execution systems, devices or apparatuses. For purposes of this specification, "computer-readable medium" may be any device that can contain, store, communicate, propagate or transmit a program for use by an instruction execution system, device or apparatus, or in conjunction with such instruction execution systems, devices or apparatuses.

[0048] More specific examples of computer-readable media (a non-exhaustive list) include the following: an electrical connection with one or more wires (electronic device), a portable computer disk case (magnetic device), a random access memory (RAM), a read-only memory (ROM), an erasable and programmable read-only memory (EPROM or flash memory), an optical fiber device, and a portable compact disk read-only memory (CDROM). In addition, the computer-readable medium may even be a paper or other suitable medium on which the program is printed, since the program may be obtained electronically, for example, by optically scanning the paper or other medium, followed by editing, deciphering or, if necessary, processing in another suitable manner, and then stored in a computer memory.

[0049] It should be understood that the various parts of the present invention can be implemented by hardware, software, firmware or a combination thereof. In the above-mentioned embodiments, multiple steps or methods can be implemented by software or firmware stored in a memory and executed by a suitable instruction execution system. For example, if implemented by hardware, as in another embodiment, it can be implemented by any one of the following technologies known in the art or a combination thereof: a discrete logic circuit having a logic gate circuit for implementing a logic function for a data signal, a dedicated integrated circuit having a suitable combination of logic gate circuits, a programmable gate array (PGA), a field programmable gate array (FPGA), etc.

[0050] In the description of this specification, the description with reference to the terms "one embodiment", "some embodiments", "examples", "specific examples", or "some examples" means that the specific features, structures, materials or characteristics described in conjunction with the embodiment or example are included in at least one embodiment or example of the present invention. In this specification, the schematic representation of the above terms does not necessarily refer to the same embodiment or example. Moreover, the specific features, structures, materials or characteristics described may be combined in any one or more embodiments or examples in a suitable manner.

[0051] The above-mentioned embodiments only express several implementation methods of the present invention, and the description thereof is relatively specific and detailed, but it cannot be understood as limiting the scope of the patent of the present invention. It should be pointed out that, for ordinary technicians in this field, several variations and improvements can be made without departing from the concept of the present invention, which all belong to the protection scope of the present invention. Therefore, the protection scope of the patent of the present invention shall be subject to the attached claims.

Claims

1. A mathematical modeling method for applied mechanics, characterized in that: The method comprises: Acquire a mechanical model of the auxiliary spring system in the mechanical model, acquire a motion equation of the auxiliary spring system according to the mechanical model through numerical simulation, and acquire a symmetric piecewise linear model of the auxiliary spring system based on the motion equation; Modifying the symmetric piecewise linear model, obtaining an integral interval of a modified phase according to the modified symmetric linear model, and obtaining a steady-state solution of the symmetric piecewise linear model based on the integral interval and an averaging method, and obtaining a stability discriminant of the steady-state solution; Extracting a bifurcation diagram that changes with the excitation frequency in the modified symmetric piecewise linear model, modifying the bifurcation diagram based on the stability discriminant to obtain a clear bifurcation point, and adjusting the symmetric piecewise linear model according to the clear bifurcation point; Extracting two adjacent initial state vectors in the adjusted symmetric piecewise linear model, obtaining motion state trajectories of the two initial state vectors, and establishing a chaotic boundary according to the boundary of the motion state trajectory; The chaotic boundary is adjusted based on the periodic exciting force to obtain a regular motion state, and the symmetrical piecewise linear model is further corrected according to the regular motion state to obtain a mathematical model of the secondary spring system.

2. The mathematical modeling method of applied mechanics according to claim 1, characterized in that: The step of obtaining a mechanical model of the auxiliary spring system in the mechanical model and obtaining the motion equation of the auxiliary spring system according to the mechanical model through numerical simulation comprises: Obtaining the mass, stiffness and damping coefficient of the secondary spring system in the mechanical model; generating a mechanical model according to the mass, the stiffness, the damping coefficient and the numerical simulation; Numerical simulation is performed using MATLAB to obtain the motion equation of the secondary spring system.

3. The mathematical modeling method of applied mechanics according to claim 1, characterized in that: The expression of the symmetric piecewise linear model is: ； In the formula, represents the mass of the secondary spring system, represents the displacement of the secondary spring system, represents the damping coefficient of the secondary spring system, represents the stiffness of the secondary spring system, represents the excitation frequency, Indicates the cycle.

4. The mathematical modeling method of applied mechanics according to claim 1, characterized in that: The step of correcting the symmetric piecewise linear model, obtaining an integral interval of a corrected phase according to the corrected symmetric linear model, and obtaining a steady-state solution of the symmetric piecewise linear model based on the integral interval and an averaging method, and obtaining a stability discriminant of the steady-state solution includes: The phase interval of the symmetrical piecewise linear model is corrected by using an averaging method to correct the symmetrical piecewise linear model; According to the modified separation condition of the symmetric piecewise linear model, an integral interval of the modified phase is obtained; Based on the averaging method and the integral interval, a steady-state solution of the symmetric piecewise linear model is obtained, and a stability discriminant is obtained according to the parameters of the auxiliary spring system and the steady-state solution.

5. The mathematical modeling method of applied mechanics according to claim 1, characterized in that: The step of extracting the modified bifurcation diagram of the symmetric piecewise linear model that changes with the excitation frequency comprises: Extracting the Poincare section after movement of one point in the modified symmetric piecewise linear model through numerical simulation; Based on the Poincare section, a bifurcation diagram of the modified symmetric piecewise linear model that changes with the excitation frequency is drawn.

6. The mathematical modeling method of applied mechanics according to claim 1, characterized in that: The steps of extracting two adjacent initial state vectors in the adjusted symmetric piecewise linear model, obtaining the motion state trajectories of the two initial state vectors, and establishing a chaotic boundary according to the boundary of the motion state trajectory include: Extracting two similar but different initial state vectors from the adjusted symmetric piecewise linear model, and solving the state trajectories of the two initial state vectors based on the distance between the two initial state vectors and the initial point; The motion state of the symmetric piecewise linear model is obtained according to the state trajectory, and a chaos boundary is established based on the state trajectory and the motion state.

7. The mathematical modeling method of applied mechanics according to claim 1, characterized in that: The step of adjusting the chaotic boundary based on the periodic exciting force to obtain a regular motion state, and further correcting the symmetrical piecewise linear model according to the regular motion state comprises: Obtaining the chaos factor of the chaotic world through a periodic exciting force method and the periodic motion of the secondary spring system; Selecting a preset amplitude and a preset disturbance frequency to adjust the chaotic factor to obtain a regular motion state; The boundary of the chaotic world is controlled based on the regular motion state to further modify the symmetric piecewise linear model.

8. A mathematical modeling system for applied mechanics, characterized in that: The system comprises: An acquisition module, used for acquiring a mechanical model of a secondary spring system in a mechanical model, acquiring a motion equation of the secondary spring system according to the mechanical model through numerical simulation, and acquiring a symmetric piecewise linear model of the secondary spring system based on the motion equation; A correction module, used for correcting the symmetric piecewise linear model, obtaining an integral interval of a corrected phase according to the corrected symmetric linear model, and obtaining a steady-state solution of the symmetric piecewise linear model based on the integral interval and an average method, and obtaining a stability discriminant of the steady-state solution; A first extraction module is used to extract a bifurcation diagram that changes with the excitation frequency in the modified symmetric piecewise linear model, modify the bifurcation diagram based on the stability discriminant to obtain a clear bifurcation point, and adjust the symmetric piecewise linear model according to the clear bifurcation point; A second extraction module is used to extract two adjacent initial state vectors in the adjusted symmetric piecewise linear model, obtain the motion state trajectory of the two initial state vectors, and establish a chaotic boundary according to the boundary of the motion state trajectory; The adjustment module is used to adjust the chaotic boundary based on the periodic exciting force to obtain a regular motion state, and further correct the symmetrical piecewise linear model according to the regular motion state to obtain a mathematical model of the secondary spring system.

9. A readable storage medium having a computer program stored thereon, characterized in that: When the program is executed by a processor, the mathematical modeling method of applied mechanics as described in any one of claims 1 to 7 is implemented.

10. A computer device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that: When the processor executes the computer program, the mathematical modeling method of applied mechanics as described in any one of claims 1 to 7 is implemented.

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