Parameter optimization method of flexible constant-force mechanism based on multi-section variable cross-section beam

Abstract

The present application relates to a flexible constant force mechanism parameter optimization method based on a multi-section variable cross-section beam, comprising: establishing a flexible constant force mechanism model based on a one-section variable cross-section beam to obtain a set of structural design parameters of the variable cross-section unit; analyzing the set of structural design parameters of the variable cross-section unit; establishing a flexible constant force mechanism model based on a multi-section variable cross-section beam; establishing a theoretical model of the flexible constant force mechanism based on the multi-section variable cross-section beam; establishing constraint conditions, an optimization objective function and a Cost function to obtain a constraint optimization model for solving the optimal objective; obtaining test indexes to obtain an optimal set of structural design parameters that satisfy the minimum Cost function; and performing statics analysis on the model established by the optimal set of structural design parameters. Through parameterization design and optimization solving of the flexible constant force mechanism based on the multi-section variable cross-section beam, the technical problems that the design freedom of the flexible constant force mechanism in the prior art is limited and the optimization method is difficult to consider the constant force value, the pre-travel interval and the constant force interval are solved.

Description

Technical Field

[0001] This invention belongs to the field of flexible constant force mechanisms, and particularly relates to a parameter optimization method for flexible constant force mechanisms based on multi-segment variable cross-section beams. Background Technology

[0002] In recent years, with the rapid development of advanced manufacturing processes, higher requirements have been placed on the stability of contact forces during processing and operation. Traditional solutions typically employ external force sensors combined with complex control algorithms to achieve closed-loop force control. However, this method requires multiple subsystems, including mechanical, electronic, and software components, resulting in high integration complexity and cost. Furthermore, delays in the control loop and oscillations generated when interacting with unknown environments can affect the stability of the output force. Therefore, scholars have proposed the concept of a constant force mechanism. A constant force mechanism is a type of mechanical device that can maintain a constant output contact force within a certain operating range. Based on its construction materials and deformation methods, constant force mechanisms can be divided into rigid constant force mechanisms and flexible constant force mechanisms.

[0003] Rigid constant force mechanisms employ a designed stiffness torsion spring as a kinematic pair, connecting a rigid panel to achieve constant force output within a specific displacement range. However, this mechanism has a large overall size and requires assembly, and friction exists between the kinematic pairs during movement, limiting its application in miniaturization. Compared to rigid constant force mechanisms, flexible constant force mechanisms can achieve constant force output through the elastic deformation of the material itself. They offer advantages such as no assembly required, simple structure, and integral molding, making them more suitable for the manufacture of miniaturized structures in fields such as biomedical devices, precision instruments, and robotics.

[0004] However, existing design and optimization methods for flexible constant-force mechanisms still have significant limitations: on the one hand, at the structural design level, current research is mostly limited to adjusting basic dimensional parameters such as the length and width of beams with uniform cross-sections, lacking design freedom; on the other hand, at the parameter optimization level, existing optimization objectives usually only focus on maximizing the "constant force interval length," while ignoring the matching between the pre-travel interval length and the "target constant force value." Therefore, how to improve design freedom by introducing multi-segment variable cross-section elements, and on this basis, establish a parameter optimization design method that can take into account both interval optimization and matching the target constant force value, is a key technical problem that urgently needs to be solved in this field. Summary of the Invention

[0005] The present invention addresses the problems existing in the prior art, namely, the technical problem to be solved by the present invention is to provide a method for optimizing the parameters of a flexible constant force mechanism based on a multi-segment variable cross-section beam.

[0006] To achieve the above objectives, the technical solution adopted by this invention is: a parameter optimization method for a flexible constant force mechanism based on a multi-segment variable cross-section beam, comprising the following steps:

[0007] Step S1: Using parametric modeling, establish a flexible constant force mechanism model based on a single-section variable cross-section beam, and define the structural design parameters of the variable cross-section element to obtain the structural design parameter set of the variable cross-section element;

[0008] Step S2: Analyze the structural design parameter set of the variable cross-section element using statistical methods combined with static analysis;

[0009] Step S3: Establish a model of a flexible constant force mechanism based on a multi-segment variable cross-section beam and define the structural design parameters of multiple variable cross-section elements; use the Rayleigh-Ritz method based on the energy method to establish a theoretical model of the flexible constant force mechanism based on a multi-segment variable cross-section beam.

[0010] Step S4: Using the structural design parameter set of the flexible constant force mechanism based on multi-segment variable cross-section beam, establish the constraint conditions, optimization objective function and Cost function to obtain the constraint optimization model that solves the objective optimally;

[0011] Step S5: Using the Bayesian optimization algorithm, experimental indicators are obtained based on the theoretical model of a flexible constant force mechanism based on a multi-segment variable cross-section beam. The structural design parameter set is then globally optimized based on the constraints and the Cost function to obtain the optimal structural design parameter set that satisfies the minimum Cost function.

[0012] Step S6: Perform static analysis on the model established by the optimal structural design parameter set, and compare the calculation results of the simulation and the theoretical model.

[0013] Furthermore, step S1 specifically includes: using a positive and negative stiffness combination method, designing a V-shaped beam structure as the positive stiffness structure of the flexible constant force mechanism, and designing a cosine beam structure with a single-section variable cross-section unit as the negative stiffness structure in the flexible constant force mechanism; connecting the positive stiffness structure and the negative stiffness structure through a guide beam to establish a flexible constant force mechanism based on a single-section variable cross-section beam; the thicknesses are all based on the central axis of the beam and are symmetrically distributed on both the upper and lower sides; the structure is symmetrically distributed from left to right, therefore only the left half needs to be analyzed:

[0014] By establishing the origin at the center of the starting section of the cosine beam... A coordinate system is used to define the structural design parameters for the geometry of variable cross-section elements, including: the starting point of the variable cross-section element. Variable cross-section element length And the thickness ratio of variable cross section element The superscript 1 of the above parameter symbols indicates that the constant force mechanism is a flexible constant force mechanism based on a single-section variable cross-section beam, and the subscript... This represents the first variable cross-section element of a flexible constant force mechanism based on a single-section variable cross-section beam; the starting point of the variable cross-section element. This refers to the x-axis coordinate. The center point of the beam section corresponding to the time; the length of the variable cross section element. This refers to the existence of a point on the beam's central axis that is opposite to the starting point of the variable cross-section element. The distance along the positive x-axis is This point is the endpoint of the variable cross-section element. Satisfying the relationship The thickness of a variable cross-section element remains constant throughout its length range; the thickness ratio of the variable cross-section element is defined as... For the thickness of the variable cross section element Compared to the cosine beam thickness in the constant cross-section flexible constant force mechanism used as a control group The ratio satisfies the relation .

[0015] Furthermore, step S2 specifically includes:

[0016] Step S21: Set the constant force range as test index 1 and the pre-stroke range as test index 2. All test indices in this step are determined by the force-displacement curve data obtained from the static analysis of the flexible constant force mechanism. The static analysis of the flexible constant force mechanism employs the following operations: set the boundary conditions of the rigid blocks at both ends of the negative stiffness structure and the rigid block at the bottom of the positive stiffness structure to be completely fixed; set the boundary conditions of the guide beam connecting the positive and negative stiffness structures to - Displacement in direction ;

[0017] Step S22: Define the constant force interval as the force-displacement curve that meets the error requirements: The maximum interval length, using It means that among them This represents the maximum force value within the constant force range. The minimum force within the constant force interval is obtained by analyzing the length of the constant force interval achievable from the starting point of the data points during the full displacement process, according to the error requirements. The longest interval among all intervals is selected as the constant force interval of test index 1, and the corresponding data point is the starting point of the constant force interval. The pre-stroke interval is defined as the length from the starting point of the full displacement process in the force-displacement curve to the starting point of the constant force interval. express;

[0018] Step S23: Use single-factor analysis to set the starting point of the variable cross-section element. Length of variable cross-section element And the thickness ratio of variable cross section element The level range and number of analyses are determined based on the sensitivity of the experimental indicators to changes in the parameter levels and the optimal situation as evaluation criteria. Factor levels for subsequent orthogonal experimental analysis are selected. Based on the selected factor levels, an orthogonal experimental scheme including interactions between parameters is designed. Range analysis and variance analysis are performed based on the orthogonal experimental results to obtain the p-values ​​corresponding to each structural design parameter and its interaction. The significance level is set at 0.05. By comparing the p-values ​​with the significance level, it is determined whether each parameter and its interaction have a significant impact on the experimental indicators, so as to determine the structural design parameters for subsequent mechanism optimization.

[0019] Furthermore, step S3 specifically includes: similar to the design of a flexible constant force mechanism based on a single-section variable cross-section beam, establishing a system with the center of the starting section of the cosine beam as the origin. A coordinate system is used to define the structural design parameters for the geometry of variable cross-section elements, including: the starting point of the variable cross-section element. Variable cross-section element length And the thickness ratio of variable cross section element The superscript of the above parameter symbols This indicates that the constant force mechanism is based on Flexible constant force mechanism of segmented variable cross-section beam, subscript Indicates based on The first flexible constant force mechanism of segmented variable cross-section beam A variable cross-section element; the starting point of the variable cross-section element. It refers to based on Flexible constant force mechanism of segmented variable cross section beam The starting point of each variable cross-section element is at the x-axis coordinate of The center point of the cosine beam section corresponding to the time; the length of the variable cross-section element. It refers to based on Flexible constant force mechanism of segmented variable cross section beam The length of the variable cross-section element, that is, there exists a point on the beam's central axis that is parallel to the first variable cross-section element. Starting point of each variable cross-section element The distance along the positive x-axis is This point is the first... End point of a variable cross section element Satisfying the relationship The thickness of all variable cross-section elements remains constant within their length range, defined based on... Flexible constant force mechanism of segmented variable cross section beam Thickness ratio of each variable cross-section element The thickness of this variable cross-section element Compared to the cosine beam thickness of the constant cross-section flexible constant force mechanism used as a control group The ratio satisfies the relation .

[0020] Furthermore, the theoretical model of the flexible constant force mechanism based on the multi-segment variable cross-section beam is established using the following method:

[0021] Combined with the above definition of the variable cross-section element starting point Length of variable cross-section element Variable cross-section element thickness ratio and basic structural parameters: total length of beam Construct a piecewise thickness function for a multi-segment variable cross-section beam. Therefore, the piecewise thickness of the multi-segment variable cross-section beam can be obtained using... The representation is defined as follows:

[0022]

[0023] Based on the established Coordinate system, foundation structural parameters, beam height and the total length of the beam The initial shape curve of a multi-segment variable cross-section beam can be defined. ;

[0024]

[0025] The shape curve of the multi-segment variable cross-section beam after applying lateral compression to the guide beam can be represented as a linear superposition of cosine modal functions, and the shape curve of the multi-segment variable cross-section beam after deformation can be defined. :

[0026]

[0027] in Indicates the first The amplitude coefficients of the even-order modes are limited by the constraint of the guide beam, restricting the deformation of the even-order modes. Therefore, the multi-segment variable cross-section beam only experiences symmetrical buckling. , This represents the number of periods of the cosine mode function;

[0028] Using the first kind of line integral and based on the small deformation assumption, we can obtain the expression for the beam length:

[0029]

[0030] Based on this expression for beam length, the initial length of a multi-segment variable cross-section beam can be derived. and the length after deformation :

[0031]

[0032] This allows us to obtain the change in beam length before and after deformation of the multi-segment variable cross-section beam. According to the elasticity theorem, the axial force can be obtained. The calculation formula is:

[0033]

[0034] in, The Young's modulus of the material. The out-of-plane thickness is a fundamental structural parameter. The equivalent length of a multi-segment variable cross-section beam is defined as the equivalent length required to produce the same axial stiffness when the multi-segment variable cross-section beam is equivalent to a beam with the same thickness and uniform cross-section. This is based on the segmented thickness function defined above. And the set of structural design parameters related to the variable cross-section element can be used to obtain... Multi-segment variable cross-section beam with variable cross-section element The formula is:

[0035]

[0036] The formula for the compressive strain energy of a beam is:

[0037]

[0038] Based on axial force Calculation formula and beam length variation The specific expression for the compressive strain energy of the beam during deformation can be obtained as follows:

[0039]

[0040] The formula for the bending strain energy of a beam is:

[0041]

[0042] in, For multi-segment variable cross-section beams in coordinate The moment of inertia of the cross section at time t is expressed as:

[0043]

[0044] Combining the shape curve formulas before and after deformation, the specific expression for the bending strain energy of the beam during deformation can be obtained as follows:

[0045]

[0046] in, The moment of inertia of a cosine beam with uniform cross-section, The term representing curvature variation is formulated as follows:

[0047]

[0048] external force Under the action of the guide beam, the displacement generated can be expressed as:

[0049]

[0050] This allows us to obtain external force. Change in system potential energy caused by work The specific expression is:

[0051]

[0052] Total potential energy of the system The compressive strain energy of the beam Bending strain energy Doing work with external forces The formula for obtaining the total energy of the system is as follows:

[0053]

[0054] Based on the above derivations , and The specific expression for the system's total potential energy is obtained. The specific expression is:

[0055]

[0056] Based on the principle of minimum potential energy, the total potential energy of the system is... By variation, we can obtain:

[0057]

[0058] in, By adjusting the curvature change term about The derivative yields the following expression:

[0059]

[0060] in, The modal selection function, defined to ensure that the first-order variational expression of the system's total potential energy is consistent with the physical constraints, is as follows:

[0061]

[0062] The above first-order variational equation describes the general equilibrium state of the structure during the stress process, and is the basis for obtaining... The basis of the curve; for the flexible structure with buckling instability analyzed, when the axial pressure... When the critical value is reached, the system's stiffness undergoes a sudden change, causing the original equilibrium path to bifurcate; therefore, based on the system's total potential energy... The first-order variational equation is used to develop axial pressure. The critical value is calculated and defined as the critical buckling load. ;

[0063] Critical buckling load It is a core intrinsic parameter characterizing structural stability, essentially solving for the value corresponding to zero tangential stiffness of the system. This is achieved by considering the total potential energy of the system. Performing a second-order variational calculus, we obtain the second-order variational equation for the total potential energy of the system as follows:

[0064]

[0065] in, It is a modal coupling integral operator that reflects the variable cross-section effect, and its expression is:

[0066]

[0067] To achieve numerical solutions, the total potential energy of the system is... The second-order variational equation is transformed into a generalized eigenvalue equation. Based on the first part of the second-order variational equation, an elastic stiffness matrix reflecting the structure's own bending recovery capability is established. :

[0068]

[0069] Among them, the diagonal matrix ,

[0070] Based on the second part of the second-order variational equation, a system reflecting the axial force is established. Geometric stiffness matrix of the system stiffness reduction effect Because this part has Therefore, this matrix is ​​a diagonal matrix, and its diagonal elements are defined as follows:

[0071]

[0072] Therefore, the total tangent stiffness matrix of the system can be obtained as follows: When a structure experiences buckling instability, its tangential stiffness drops to zero. Transforming it into a singular matrix is ​​equivalent to solving the following generalized eigenvalue equations:

[0073]

[0074] in, These are the eigenvectors corresponding to the buckling modes. Through eigenvalue decomposition, eigenvalues ​​corresponding to different mode orders can be obtained. According to the minimum critical load principle in elastic stability theory, structures always buckle and fail at the lowest critical load. Therefore, the third-order mode corresponding to... Critical buckling load for multi-segment variable cross-section beams Therefore, the following relationships can be obtained to establish a theoretical model for a multi-segment variable cross-section beam:

[0075]

[0076] Previous research has shown that a pseudo-rigid body method can be used to establish the theoretical model of a V-shaped beam. In the pseudo-rigid body model, the torsion springs are located near the ends of the beam, and the flexible beam section between the two torsion springs is considered as a pseudo-rigid body rod, with a length... , The characteristic radius coefficient is , and the torsional constants of the four torsion springs are all equal in magnitude. , represented as:

[0077]

[0078] in, This is the stiffness coefficient of the torsion spring. The moment of inertia of the V-beam section, .

[0079] Apply a magnitude of to the guide beam The compression displacement was such that only the portion of the beam near the end of the flexible beam deformed, resulting in a pseudo-rigid body angle. The remaining parts remain unchanged. The initial angle between the V-beam and the guide beam. The relationship between them is:

[0080]

[0081] A theoretical model of the V-beam is established based on the principle of virtual work:

[0082]

[0083] The output constant force value of the flexible constant force mechanism based on the multi-segment variable cross-section beam is:

[0084]

[0085] The nonlinear equations of a multi-segment variable cross-section beam are solved using the Newton-Raphson algorithm to obtain the negative stiffness structure. Based on the curve data, and with the goal of achieving the maximum constant force range, an interval optimization algorithm is used to reverse-engineer the matching parameters corresponding to the normal stiffness structure. Based on the above, a theoretical model of a flexible constant force mechanism based on a multi-segment variable cross-section beam was established.

[0086] Furthermore, step S4 specifically includes:

[0087] Step S41: To achieve the pre-travel range To minimize this, the objective function for optimizing the pre-stroke interval of a flexible constant-force mechanism based on a multi-segment variable cross-section beam is established as follows:

[0088]

[0089] In order to avoid the eigenvectors of the corresponding buckling modes Conflict, Setting Let be the starting position vector of the variable cross-section element. Let the length vector of the variable cross-section element be _____. The thickness ratio vector; This is the actual constant force output value obtained based on the combination of structural parameters. For the target constant force value, The constant force output allowable deviation coefficient can be obtained based on the constant force range error requirements. ; and The constant force range and pre-stroke range are obtained from the structural design parameter set based on the geometry of the variable cross-section element. This represents the lower limit of the constant force range;

[0090] Step S42: Establish the constraint conditions for the flexible constant force mechanism based on a multi-segment variable cross-section beam as follows:

[0091]

[0092] in, , They are respectively The upper and lower limits; , They are respectively The upper and lower limits; , They are respectively The upper and lower limits; This is the minimum safety interval value;

[0093] Step S43: To further describe the pre-stroke interval optimization objective function of the flexible constant force mechanism based on a multi-segment variable cross-section beam, a cost function is constructed as follows:

[0094]

[0095] in, and This is a preset penalty weighting coefficient; For constant force value default terms, the expression is: This represents the actual constant force output value. With target value The allowable range of differences; For the constant force interval default term, the expression is: , indicating the target constant force range With respect to the actual constant force range The difference; the optimization objective of the pre-travel interval is The set of structural design parameters for the geometric shape of the variable cross-section element corresponding to the minimum value;

[0096] Step S44: To achieve the constant force range To maximize the constant force range optimization objective function of the flexible constant force mechanism based on a multi-segment variable cross-section beam, the following equation is established:

[0097]

[0098] in, This represents the upper limit of the pre-travel interval. To further describe the constant force interval optimization objective function of the flexible constant force mechanism based on a multi-segment variable cross-section beam, a Cost function is constructed as follows:

[0099]

[0100] in, The expression for the pre-trip interval violation item is: , indicating the actual pre-travel range With the target pre-travel range The difference; the objective of constant force interval optimization is The set of structural design parameters for the variable cross-section element geometry corresponding to the minimum value.

[0101] Furthermore, step S6 specifically includes: performing static analysis on the three-dimensional model established by the optimal variable cross-section unit structural design parameter set with the optimization objective being the pre-stroke interval, and comparing the force-displacement data of the simulation with the theoretical calculation; performing static analysis on the three-dimensional model established by the optimal variable cross-section unit structural design parameter set with the optimization objective being the constant force interval, and comparing the force-displacement data of the simulation with the theoretical calculation.

[0102] Compared with the prior art, the present invention has the following effects: The present invention solves the technical problems of the limited design freedom of flexible constant force mechanisms in the prior art and the difficulty of the optimization method to take into account the constant force value, pre-stroke range and constant force range by parametric design and optimization solution of flexible constant force mechanism based on multi-segment variable cross-section beam. It has important significance and wide application scenarios in the field of constant force mechanism. Attached Figure Description

[0103] Figure 1This is a schematic diagram of the parameter optimization method according to an embodiment of the present invention;

[0104] Figure 2 A schematic diagram of a model based on a single-section variable cross-section flexible constant force mechanism is provided in this embodiment of the invention.

[0105] Figure 3 A schematic diagram of a multi-segment variable cross-section flexible constant force mechanism is shown in the embodiment of the present invention.

[0106] Figure 4 A schematic diagram of the model established by optimizing the optimal variable cross-section unit structure design parameter set for the pre-stroke interval in this embodiment of the invention;

[0107] Figure 5 This embodiment of the invention presents a schematic diagram comparing the simulation and theoretical data of a model established based on the optimal variable cross-section unit structural design parameter set for the pre-stroke interval.

[0108] Figure 6 A schematic diagram of the model established by optimizing the optimal variable cross-section unit structure design parameter set in the constant force range according to an embodiment of the present invention;

[0109] Figure 7 The present invention embodiment is a schematic diagram comparing the simulation and theoretical data of the model established by optimizing the optimal variable cross-section unit structural design parameter set in the constant force range. Detailed Implementation

[0110] The present invention will now be described in further detail with reference to the accompanying drawings and specific embodiments.

[0111] In the description of this invention, it should be understood that the terms "longitudinal", "lateral", "up", "down", "front", "rear", "left", "right", "vertical", "horizontal", "top", "bottom", "inner", "outer", etc., indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings, and are only for the convenience of describing this invention, and are not intended to indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation, and therefore should not be construed as a limitation of this invention.

[0112] like Figures 1-7 As shown, this invention discloses a parameter optimization method for a flexible constant force mechanism based on a multi-segment variable cross-section beam. Based on a flexible constant force mechanism with a known set of foundation structural parameters, it aims to address the limitations in the design freedom of existing flexible constant force mechanisms and the difficulty in simultaneously considering constant force values, pre-stroke intervals, and constant force intervals in optimization methods. The optimization method specifically includes the following steps:

[0113] Step S1: Using parametric modeling, establish a flexible constant force mechanism model based on a single-section variable cross-section beam, and define the structural design parameters of the variable cross-section element to obtain the structural design parameter set of the variable cross-section element;

[0114] Step S2: The structural design parameter set of the variable cross-section element was analyzed using statistical methods combined with static analysis, which verified the significant influence of the structural design parameter set of the variable cross-section element on the test indicators.

[0115] Step S3: Establish a model of a flexible constant force mechanism based on a multi-segment variable cross-section beam and define the structural design parameters of multiple variable cross-section elements; use the Rayleigh-Ritz method based on the energy method to establish a theoretical model of the flexible constant force mechanism based on a multi-segment variable cross-section beam.

[0116] Step S4: Using the structural design parameter set of the flexible constant force mechanism based on multi-segment variable cross-section beam, establish the constraint conditions, optimization objective function and Cost function to obtain the constraint optimization model that solves the objective optimally;

[0117] Step S5: Using the Bayesian optimization algorithm, experimental indicators are obtained based on the theoretical model of a flexible constant force mechanism based on a multi-segment variable cross-section beam. The structural design parameter set is then globally optimized based on the constraints and the Cost function to obtain the optimal structural design parameter set that satisfies the minimum Cost function.

[0118] Step S6: Perform static analysis on the model established by the optimal structural design parameter set, and compare the calculation results of the simulation and theoretical model to verify the reliability of the optimization method.

[0119] Specifically, this optimization method is applicable to a set of structural parameters of a uniform cross-section flexible constant force mechanism, which is used as a control group.

[0120] Specifically, step S1 includes: using a positive and negative stiffness combination method, designing a V-shaped beam structure as the positive stiffness structure of the flexible constant force mechanism, and designing a cosine beam structure with a single-section variable cross-section unit as the negative stiffness structure of the flexible constant force mechanism; connecting the positive and negative stiffness structures through a guide beam to establish a flexible constant force mechanism based on a single-section variable cross-section beam; the thicknesses are all based on the central axis of the beam and are symmetrically distributed on both the upper and lower sides; the structure is symmetrically distributed from left to right, therefore only the left half needs to be analyzed:

[0121] By establishing the origin at the center of the starting section of the cosine beam... A coordinate system is used to define the structural design parameters for the geometry of variable cross-section elements, including: the starting point of the variable cross-section element. Variable cross-section element length And the thickness ratio of variable cross section element The superscript 1 of the above parameter symbols indicates that the constant force mechanism is a flexible constant force mechanism based on a single-section variable cross-section beam, and the subscript... This represents the first variable cross-section element of a flexible constant force mechanism based on a single-section variable cross-section beam; the starting point of the variable cross-section element. This refers to the x-axis coordinate. The center point of the beam section corresponding to the time; the length of the variable cross section element. This refers to the existence of a point on the beam's central axis that is opposite to the starting point of the variable cross-section element. The distance along the positive x-axis is This point is the endpoint of the variable cross-section element. Satisfying the relationship The thickness of a variable cross-section element remains constant throughout its length range; the thickness ratio of the variable cross-section element is defined as... For the thickness of the variable cross section element Compared to the cosine beam thickness in the constant cross-section flexible constant force mechanism used as a control group The ratio satisfies the relation .

[0122] Step S2 specifically includes:

[0123] Step S21: Set the constant force range as test index 1 and the pre-stroke range as test index 2. All test indices in this step are determined by the force-displacement curve data obtained from the static analysis of the flexible constant force mechanism. The static analysis of the flexible constant force mechanism employs the following operations: set the boundary conditions of the rigid blocks at both ends of the negative stiffness structure and the rigid block at the bottom of the positive stiffness structure to be completely fixed; set the boundary conditions of the guide beam connecting the positive and negative stiffness structures to - Displacement in direction ;

[0124] Step S22: Define the constant force interval as the force-displacement curve that meets the error requirements: The maximum interval length, using It means that among them This represents the maximum force value within the constant force range. The minimum force within the constant force interval is obtained by analyzing the length of the constant force interval achievable from the starting point of the data points during the full displacement process, according to the error requirements. The longest interval among all intervals is selected as the constant force interval of test index 1, and the corresponding data point is the starting point of the constant force interval. The pre-stroke interval is defined as the length from the starting point of the full displacement process in the force-displacement curve to the starting point of the constant force interval. express;

[0125] Step S23: Use single-factor analysis to set the starting point of the variable cross-section element. Length of variable cross-section element And the thickness ratio of variable cross section element The level range and number of analyses are determined based on the sensitivity of the experimental indicators to changes in the parameter levels and the optimal situation as evaluation criteria. Factor levels for subsequent orthogonal experimental analysis are selected. Based on the selected factor levels, an orthogonal experimental scheme including interactions between parameters is designed. Range analysis and variance analysis are performed based on the orthogonal experimental results to obtain the p-values ​​corresponding to each structural design parameter and its interaction. The significance level is set at 0.05. By comparing the p-values ​​with the significance level, it is determined whether each parameter and its interaction have a significant impact on the experimental indicators, so as to determine the structural design parameters for subsequent mechanism optimization.

[0126] Specifically, step S3 includes: similar to the design of a flexible constant force mechanism based on a single-section variable cross-section beam, establishing a system with the center of the starting section of the cosine beam as the origin. A coordinate system is used to define the structural design parameters for the geometry of variable cross-section elements, including: the starting point of the variable cross-section element. Variable cross-section element length And the thickness ratio of variable cross section element The superscript of the above parameter symbols This indicates that the constant force mechanism is based on Flexible constant force mechanism of segmented variable cross-section beam, subscript Indicates based on The first flexible constant force mechanism of segmented variable cross-section beam A variable cross-section element; the starting point of the variable cross-section element. It refers to based on Flexible constant force mechanism of segmented variable cross section beam The starting point of each variable cross-section element is at the x-axis coordinate of The center point of the cosine beam section corresponding to the time; the length of the variable cross-section element. It refers to based on Flexible constant force mechanism of segmented variable cross section beam The length of the variable cross-section element, that is, there exists a point on the beam's central axis that is parallel to the first variable cross-section element. Starting point of each variable cross-section element The distance along the positive x-axis is This point is the first... End point of a variable cross section element Satisfying the relationship The thickness of all variable cross-section elements remains constant within their length range, defined based on... Flexible constant force mechanism of segmented variable cross section beam Thickness ratio of each variable cross-section element The thickness of this variable cross-section element Compared to the cosine beam thickness of the constant cross-section flexible constant force mechanism used as a control group The ratio satisfies the relation .

[0127] Specifically, the theoretical model of the flexible constant force mechanism based on a multi-segment variable cross-section beam is established using the following method:

[0128] Combined with the above definition of the variable cross-section element starting point Length of variable cross-section element Variable cross-section element thickness ratio and basic structural parameters: total length of beam Construct a piecewise thickness function for a multi-segment variable cross-section beam. Therefore, the piecewise thickness of the multi-segment variable cross-section beam can be obtained using... The representation is defined as follows:

[0129]

[0130] Based on the established Coordinate system, foundation structural parameters, beam height and the total length of the beam The initial shape curve of a multi-segment variable cross-section beam can be defined. ;

[0131]

[0132] The shape curve of the multi-segment variable cross-section beam after applying lateral compression to the guide beam can be represented as a linear superposition of cosine modal functions, and the shape curve of the multi-segment variable cross-section beam after deformation can be defined. :

[0133]

[0134] in Indicates the first The amplitude coefficients of the even-order modes are limited by the constraint of the guide beam, restricting the deformation of the even-order modes. Therefore, the multi-segment variable cross-section beam only experiences symmetrical buckling. , This represents the number of periods of the cosine mode function;

[0135] Using the first kind of line integral and based on the small deformation assumption, we can obtain the expression for the beam length:

[0136]

[0137] Based on this expression for beam length, the initial length of a multi-segment variable cross-section beam can be derived. and the length after deformation :

[0138]

[0139] This allows us to obtain the change in beam length before and after deformation of the multi-segment variable cross-section beam. According to the elasticity theorem, the axial force can be obtained. The calculation formula is:

[0140]

[0141] in, The Young's modulus of the material. The out-of-plane thickness is a fundamental structural parameter. The equivalent length of a multi-segment variable cross-section beam is defined as the equivalent length required to produce the same axial stiffness when the multi-segment variable cross-section beam is equivalent to a beam with the same thickness and uniform cross-section. This is based on the segmented thickness function defined above. And the set of structural design parameters related to the variable cross-section element can be used to obtain... Multi-segment variable cross-section beam with variable cross-section element The formula is:

[0142]

[0143] The formula for the compressive strain energy of a beam is:

[0144]

[0145] Based on axial force Calculation formula and beam length variation The specific expression for the compressive strain energy of the beam during deformation can be obtained as follows:

[0146]

[0147] The formula for the bending strain energy of a beam is:

[0148]

[0149] in, For multi-segment variable cross-section beams in coordinate The moment of inertia of the cross section at time t is expressed as:

[0150]

[0151] Combining the shape curve formulas before and after deformation, the specific expression for the bending strain energy of the beam during deformation can be obtained as follows:

[0152]

[0153] in, The moment of inertia of a cosine beam with uniform cross-section, The term representing curvature variation is formulated as follows:

[0154]

[0155] external force Under the action of the guide beam, the displacement generated can be expressed as:

[0156]

[0157] This allows us to obtain external force. Change in system potential energy caused by work The specific expression is:

[0158]

[0159] Total potential energy of the system The compressive strain energy of the beam Bending strain energy Doing work with external forces The formula for obtaining the total energy of the system is as follows:

[0160]

[0161] Based on the above derivations , and The specific expression for the system's total potential energy is obtained. The specific expression is:

[0162]

[0163] Based on the principle of minimum potential energy, the total potential energy of the system is... By variation, we can obtain:

[0164]

[0165] in, By adjusting the curvature change term about The derivative yields the following expression:

[0166]

[0167] in, The modal selection function, defined to ensure that the first-order variational expression of the system's total potential energy is consistent with the physical constraints, is as follows:

[0168]

[0169] The above first-order variational equation describes the general equilibrium state of the structure during the stress process, and is the basis for obtaining... The basis of the curve; for the flexible structure with buckling instability analyzed, when the axial pressure... When the critical value is reached, the system's stiffness undergoes a sudden change, causing the original equilibrium path to bifurcate; therefore, based on the system's total potential energy... The first-order variational equation is used to develop axial pressure. The critical value is calculated and defined as the critical buckling load. ;

[0170] Critical buckling load It is a core intrinsic parameter characterizing structural stability, essentially solving for the value corresponding to zero tangential stiffness of the system. This is achieved by considering the total potential energy of the system. Performing a second-order variational calculus, we obtain the second-order variational equation for the total potential energy of the system as follows:

[0171]

[0172] in, It is a modal coupling integral operator that reflects the variable cross-section effect, and its expression is:

[0173]

[0174] To achieve numerical solutions, the total potential energy of the system is... The second-order variational equation is transformed into a generalized eigenvalue equation. Based on the first part of the second-order variational equation, an elastic stiffness matrix reflecting the structure's own bending recovery capability is established. :

[0175]

[0176] Among them, the diagonal matrix ,

[0177] Based on the second part of the second-order variational equation, a system reflecting the axial force is established. Geometric stiffness matrix of the system stiffness reduction effect Because this part has Therefore, this matrix is ​​a diagonal matrix, and its diagonal elements are defined as follows:

[0178]

[0179] Therefore, the total tangent stiffness matrix of the system can be obtained as follows: When a structure experiences buckling instability, its tangential stiffness drops to zero. Transforming it into a singular matrix is ​​equivalent to solving the following generalized eigenvalue equations:

[0180]

[0181] in, These are the eigenvectors corresponding to the buckling modes. Through eigenvalue decomposition, eigenvalues ​​corresponding to different mode orders can be obtained. According to the minimum critical load principle in elastic stability theory, structures always buckle and fail at the lowest critical load. Therefore, the third-order mode corresponding to... Critical buckling load for multi-segment variable cross-section beams Therefore, the following relationships can be obtained to establish a theoretical model for a multi-segment variable cross-section beam:

[0182]

[0183] Previous research has shown that a pseudo-rigid body method can be used to establish the theoretical model of a V-shaped beam. In the pseudo-rigid body model, the torsion springs are located near the ends of the beam, and the flexible beam section between the two torsion springs is considered as a pseudo-rigid body rod, with a length... , The characteristic radius coefficient is , and the torsional constants of the four torsion springs are all equal in magnitude. , represented as:

[0184]

[0185] in, This is the stiffness coefficient of the torsion spring. The moment of inertia of the V-beam section, .

[0186] Apply a magnitude of to the guide beam The compression displacement was such that only the portion of the beam near the end of the flexible beam deformed, resulting in a pseudo-rigid body angle. The remaining parts remain unchanged. The initial angle between the V-beam and the guide beam. The relationship between them is:

[0187]

[0188] A theoretical model of the V-beam is established based on the principle of virtual work:

[0189]

[0190] The output constant force value of the flexible constant force mechanism based on the multi-segment variable cross-section beam is:

[0191]

[0192] The nonlinear equations of a multi-segment variable cross-section beam are solved using the Newton-Raphson algorithm to obtain the negative stiffness structure. Based on the curve data, and with the goal of achieving the maximum constant force range, an interval optimization algorithm is used to reverse-engineer the matching parameters corresponding to the normal stiffness structure. Based on the above, a theoretical model of a flexible constant force mechanism based on a multi-segment variable cross-section beam was established.

[0193] Specifically, step S4 includes:

[0194] Step S41: To achieve the pre-travel range To minimize this, the objective function for optimizing the pre-stroke interval of a flexible constant-force mechanism based on a multi-segment variable cross-section beam is established as follows:

[0195]

[0196] In order to avoid the eigenvectors of the corresponding buckling modes Conflict, Setting Let be the starting position vector of the variable cross-section element. Let the length vector of the variable cross-section element be _____. The thickness ratio vector; This is the actual constant force output value obtained based on the combination of structural parameters. For the target constant force value, The constant force output allowable deviation coefficient can be obtained based on the constant force range error requirements. ; and The constant force range and pre-stroke range are obtained from the structural design parameter set based on the geometry of the variable cross-section element. This represents the lower limit of the constant force range;

[0197] Step S42: Establish the constraint conditions for the flexible constant force mechanism based on a multi-segment variable cross-section beam as follows:

[0198]

[0199] in, , They are respectively The upper and lower limits; , They are respectively The upper and lower limits; , They are respectively The upper and lower limits; This is the minimum safety interval value;

[0200] Step S43: To further describe the pre-stroke interval optimization objective function of the flexible constant force mechanism based on a multi-segment variable cross-section beam, a cost function is constructed as follows:

[0201]

[0202] in, and This is a preset penalty weighting coefficient; For constant force value default terms, the expression is: This represents the actual constant force output value. With target value The allowable range of differences; For the constant force interval default term, the expression is: , indicating the target constant force range With respect to the actual constant force range The difference; the optimization objective of the pre-travel interval is The set of structural design parameters for the geometric shape of the variable cross-section element corresponding to the minimum value;

[0203] Step S44: To achieve the constant force range To maximize the constant force range optimization objective function of the flexible constant force mechanism based on a multi-segment variable cross-section beam, the following equation is established:

[0204]

[0205] in, This represents the upper limit of the pre-travel interval. To further describe the constant force interval optimization objective function of the flexible constant force mechanism based on a multi-segment variable cross-section beam, a Cost function is constructed as follows:

[0206]

[0207] in, The expression for the pre-trip interval violation item is: , indicating the actual pre-travel range With the target pre-travel range The difference; the objective of constant force interval optimization is The set of structural design parameters for the variable cross-section element geometry corresponding to the minimum value.

[0208] Specifically, step S5 includes: The theoretical calculation of the flexible constant force mechanism based on a multi-segment variable cross-section beam involves high-dimensional nonlinear solutions. Each evaluation of the objective function requires solving a complex set of nonlinear equations using the Newton-Raphson algorithm, resulting in a "black box" characteristic between the structural design parameter set of the variable cross-section unit and the mechanism performance, making gradient calculation impossible. Therefore, a Bayesian optimization algorithm is adopted, introducing a Gaussian process as a surrogate model. This method can effectively fit the high-dimensional nonlinear mechanical behavior and significantly improve sample utilization efficiency, achieving convergence within a fewer number of simulation evaluations. Based on the aforementioned constraints and the Cost function, a global optimization of the structural design parameter set is performed, obtaining the optimal structural design parameter set of the variable cross-section unit that minimizes the Cost function.

[0209] Specifically, step S6 includes: performing static analysis on the three-dimensional model established by the optimal variable cross-section unit structural design parameter set with the optimization objective being the pre-stroke interval, and comparing the force-displacement data calculated by simulation and theory; performing static analysis on the three-dimensional model established by the optimal variable cross-section unit structural design parameter set with the optimization objective being the constant force interval, and comparing the force-displacement data calculated by simulation and theory.

[0210] Example

[0211] Step S1: Using parametric modeling, establish a flexible constant force mechanism model based on a single-section variable cross-section beam, and define the structural design parameters of the variable cross-section element to obtain the structural design parameter set of the variable cross-section element:

[0212] The established flexible constant force mechanism is based on a single-section variable cross-section beam, such as Figure 2 As shown, this includes the design parameter set for variable cross-section element structures: variable cross-section element starting point. Variable cross-section element length And the thickness ratio of variable cross section element In this embodiment, the beam length of the cosine beam of the negative stiffness structure of the uniform cross-section flexible constant force mechanism is set. In-plane thickness ,high The beam length is set as that of a V-shaped beam with positive stiffness. In-plane thickness initial angle The thickness of the structure with positive and negative stiffness is equal outside the plane. The material selected is TPU68D, with a Young's modulus of [missing information]. .

[0213] Step S2: The structural design parameter set was analyzed using statistical methods combined with static analysis to verify the significant impact of the structural design parameter set of the variable cross-section element on the test results.

[0214] The boundary conditions for the guide beam with positive and negative stiffness structural connection are set as follows: Displacement in direction Based on static analysis, the constant force range of a flexible constant force mechanism with a uniform cross-section can be determined. Pre-trip range Output constant force value ;

[0215] The starting point of the variable cross-section element was set using a single-factor analysis method. Variable cross-section element length All values ​​are taken at equal intervals of 2mm within the range; the thickness ratio of the variable cross-section unit is set. Values ​​were taken at equal intervals of 0.3 mm within the interval. The evaluation criteria were based on the sensitivity of the experimental index to changes in the parameter level and the optimal situation, ultimately selecting... , , This will be used for subsequent orthogonal experimental analysis. Based on the selected factor levels, a three-factor, three-level interaction experiment will be designed. Orthogonal experimental design, where factor A is the starting point of the variable cross-section element. Factor B is the length of the variable cross-section element. Factor C is the thickness ratio of the variable cross-section element. ;

[0216] Range and variance analyses were performed based on the orthogonal experimental results to obtain the p-values ​​corresponding to the structural design parameters and their interactions. A significance level of 0.05 was set, and the p-values ​​were compared with the significance level; the results are shown in Table 1. Based on these results, the structural design parameters used for subsequent mechanism optimization were determined.

[0217] Table 1. Range Analysis Results of Structural Design Parameters and Their Interactions to Determine Significance

[0218]

[0219] Step S3: Establish a model of a flexible constant force mechanism based on a multi-segment variable cross-section beam, define the structural design parameters of multiple variable cross-section elements, and establish a theoretical model of the flexible constant force mechanism based on the Rayleigh-Ritz method based on the energy method.

[0220] Similar to the design of flexible constant force mechanisms based on a single-section variable cross-section beam, it includes a set of design parameters for variable cross-section element structures: the starting point of the variable cross-section element. Variable cross-section element length And the thickness ratio of the variable cross-section element. In this embodiment, it is set The subsequent analysis will be conducted using a flexible constant force mechanism based on a two-section variable cross-section beam, the schematic model of which is shown below. Figure 3 As shown;

[0221] The theoretical model of the two-segment variable cross-section beam is as follows:

[0222]

[0223] The theoretical model of the V-shaped beam is as follows:

[0224]

[0225] The output constant force value of the flexible constant force mechanism based on the two-section variable cross-section beam is:

[0226]

[0227] The nonlinear equations of the two-segment variable cross-section beam are solved using the Newton-Raphson algorithm to obtain the negative stiffness structure. Based on the curve data, and with the goal of achieving the maximum constant force range, an interval optimization algorithm is used to reverse-engineer the matching parameters corresponding to the normal stiffness structure. Based on the above, a theoretical model of a flexible constant force mechanism based on a two-section variable cross-section beam was established.

[0228] Step S4: Using the structural design parameter set of a flexible constant force mechanism based on a multi-segment variable cross-section beam, establish the constraints, objective function, and Cost function to obtain the constraint optimization model that achieves the optimal solution to the objective.

[0229] To achieve the pre-travel range To minimize this, the objective function for optimizing the pre-stroke interval of a flexible constant-force mechanism based on a two-segment variable cross-section beam is as follows:

[0230]

[0231] Set according to the constant force range error requirement ;

[0232] The constraint conditions for establishing a flexible constant force mechanism based on a two-segment variable cross-section beam are as follows:

[0233]

[0234] Based on the beam length of the cosine beam, set , , , Set according to printing accuracy and strength requirements. , , .

[0235] To further describe the pre-stroke interval optimization objective function of the flexible constant force mechanism based on a multi-segment variable cross-section beam, a cost function is constructed as follows:

[0236]

[0237] In this embodiment, the following settings are provided: The target of the pre-trip interval optimization is The set of structural design parameters for the geometric shape of the variable cross-section element corresponding to the minimum value;

[0238] To achieve constant force range To maximize the constant force range optimization objective function of the flexible constant force mechanism based on a multi-segment variable cross-section beam, the following equation is established:

[0239]

[0240] To further describe the constant force interval optimization objective function of the flexible constant force mechanism based on a multi-segment variable cross-section beam, a Cost function is constructed for description, as follows:

[0241]

[0242] The objective of constant force interval optimization is The set of structural design parameters for the variable cross-section element geometry corresponding to the minimum value.

[0243] Step S5: Using a Bayesian optimization algorithm, experimental indices are obtained based on the theoretical model of a flexible constant-force mechanism with a multi-segment variable cross-section beam. Then, based on the constraints and the Cost function, a global optimization of the structural design parameter set is performed to obtain the optimal structural design parameter set that minimizes the Cost function.

[0244] Based on the aforementioned constraints and the Cost function, a global optimization of the structural design parameter set is performed to obtain an optimized structural parameter set with the goal of minimizing the pre-travel interval. , , , , , The corresponding matching V-beam thickness This achieves constant force output. The minimum travel interval is Compared to the pre-stroke range of the constant cross-section flexible constant force mechanism used as a control group, This represents an improvement of 14.88%. Based on the aforementioned constraints and the Cost function, a global optimization of the structural design parameter set is performed to obtain an optimized structural parameter set with the goal of achieving the maximum constant force range. , , , The corresponding matching V-beam thickness This achieves constant force output. The maximum value of the constant force range is Compared to the constant force range of the uniform cross-section flexible constant force mechanism used as a control group, It increased by 13.91%.

[0245] Step S6: Perform static analysis on the model established by the optimal structural design parameter set, and compare the calculation results of the simulation and theoretical model to verify the reliability of the optimization method.

[0246] A static analysis was performed on a three-dimensional model established using the optimal variable cross-section element structural design parameter set with the optimization objective being the pre-travel range. This model was named... Models such as Figure 4 As shown, comparing the force-displacement data from simulation and theoretical calculations, the data comparison is as follows: Figure 5 As shown; a static analysis is performed on a three-dimensional model established using the optimal variable cross-section element structural design parameter set with the optimization objective of constant force range, and this model is named... Models such as Figure 6 As shown, comparing the force-displacement data from simulation and theoretical calculations, the data comparison is as follows: Figure 7 As shown. In summary, the static analysis data matches the theoretical data well, thus verifying the feasibility of the designed flexible constant force mechanism based on a multi-segment variable cross-section beam.

[0247] The advantages of this invention are:

[0248] (1) By establishing a flexible constant force mechanism based on a multi-segment variable cross-section beam, this invention increases the degree of freedom in structural design, breaks through the limitation of the mechanical response of the structural parameters of the flexible constant force mechanism with uniform cross-section, and enables the optimization algorithm to achieve fine control of stiffness characteristics in a multi-dimensional design space, thereby significantly broadening the optimization boundary of the mechanism performance and improving the upper limit of the target range.

[0249] (2) The present invention uses the Rayleigh-Ritz method based on the energy method to establish a theoretical model of a flexible constant force mechanism based on a multi-segment variable cross-section beam. This theoretical model can quickly and accurately calculate the force-displacement response, providing an efficient computational basis for global parameter optimization in high-dimensional design space.

[0250] (3) This invention achieves coordinated optimization of constant force output, constant force interval length and pre-stroke interval length by establishing a Cost function with multiple constraints.

[0251] If this invention discloses or relates to components or structural parts that are fixedly connected to each other, then, unless otherwise stated, a fixed connection can be understood as: a fixed connection that can be detached (e.g., using bolts or screws), or a fixed connection that cannot be detached (e.g., riveting, welding). Of course, a fixed connection can also be replaced by an integral structure (e.g., manufactured in one piece using a casting process) (except where it is obviously impossible to use an integral molding process).

[0252] In addition, unless otherwise stated, the terms used in any of the technical solutions disclosed in this invention to indicate positional relationships or shapes include states or shapes that are similar to, close to, or approximate with those states or shapes.

[0253] Any component provided by this invention can be assembled from multiple individual components or can be a single component manufactured by a one-piece molding process.

[0254] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit them; although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications can still be made to the specific implementation of the present invention or equivalent substitutions can be made to some technical features without departing from the spirit of the technical solutions of the present invention, and all such modifications and substitutions should be covered within the scope of the technical solutions claimed in the present invention.

Claims

1. A parameter optimization method for a flexible constant force mechanism based on a multi-segment variable cross-section beam, characterized in that: Includes the following steps: Step S1: Using parametric modeling, establish a flexible constant force mechanism model based on a single-section variable cross-section beam, and define the structural design parameters of the variable cross-section element to obtain the structural design parameter set of the variable cross-section element; Step S2: Analyze the structural design parameter set of the variable cross-section element using statistical methods combined with static analysis; Step S3: Establish a model of a flexible constant force mechanism based on a multi-segment variable cross-section beam and define the structural design parameters of multiple variable cross-section elements; use the Rayleigh-Ritz method based on the energy method to establish a theoretical model of the flexible constant force mechanism based on a multi-segment variable cross-section beam. Step S4: Using the structural design parameter set of the flexible constant force mechanism based on multi-segment variable cross-section beam, establish the constraint conditions, optimization objective function and Cost function to obtain the constraint optimization model that solves the objective optimally; Step S5: Using the Bayesian optimization algorithm, experimental indicators are obtained based on the theoretical model of a flexible constant force mechanism based on a multi-segment variable cross-section beam. The structural design parameter set is then globally optimized based on the constraints and the Cost function to obtain the optimal structural design parameter set that satisfies the minimum Cost function. Step S6: Perform static analysis on the model established by the optimal structural design parameter set, and compare the calculation results of the simulation and the theoretical model.

2. The parameter optimization method for a flexible constant force mechanism based on a multi-segment variable cross-section beam according to claim 1, characterized in that: Step S1 specifically includes: using a positive and negative stiffness combination method, designing a V-shaped beam structure as the positive stiffness structure of the flexible constant force mechanism, and designing a cosine beam structure with a single-section variable cross-section unit as the negative stiffness structure of the flexible constant force mechanism; connecting the positive and negative stiffness structures through a guide beam to establish a flexible constant force mechanism based on a single-section variable cross-section beam; the thicknesses are all based on the central axis of the beam and are symmetrically distributed on both the upper and lower sides; the structure is symmetrically distributed from left to right, therefore only the left half needs to be analyzed: By establishing the origin with the center of the starting section of the cosine beam... A coordinate system is used to define the structural design parameters for the geometry of variable cross-section elements, including: the starting point of the variable cross-section element. Variable cross-section element length And the thickness ratio of variable cross section element The superscript 1 of the above parameter symbols indicates that the constant force mechanism is a flexible constant force mechanism based on a single-section variable cross-section beam, and the subscript... This represents the first variable cross-section element of a flexible constant force mechanism based on a single-section variable cross-section beam; the starting point of the variable cross-section element. This refers to the x-axis coordinate. The center point of the beam section corresponding to the time; the length of the variable cross section element. This refers to the existence of a point on the beam's central axis that is opposite to the starting point of the variable cross-section element. The distance along the positive x-axis is This point is the endpoint of the variable cross-section element. Satisfying the relationship The thickness of a variable cross-section element remains constant throughout its length range; the thickness ratio of the variable cross-section element is defined as... For the thickness of the variable cross section element Compared to the cosine beam thickness in the constant cross-section flexible constant force mechanism used as a control group The ratio satisfies the relation .

3. The parameter optimization method for a flexible constant force mechanism based on a multi-segment variable cross-section beam according to claim 1, characterized in that: Step S2 specifically includes: Step S21: Set the constant force range as test index 1 and the pre-stroke range as test index 2. All test indices in this step are determined by the force-displacement curve data obtained from the static analysis of the flexible constant force mechanism. The static analysis of the flexible constant force mechanism employs the following operations: set the boundary conditions of the rigid blocks at both ends of the negative stiffness structure and the rigid block at the bottom of the positive stiffness structure to be completely fixed; set the boundary conditions of the guide beam connecting the positive and negative stiffness structures to - Displacement in direction ; Step S22: Define the constant force interval as the force-displacement curve that meets the error requirements: The maximum interval length, using It means that, among them This represents the maximum force value within the constant force range. The minimum force within the constant force interval is obtained by analyzing the length of the constant force interval achievable from the starting point of the data points during the full displacement process, according to the error requirements. The longest interval among all intervals is selected as the constant force interval of test index 1, and the corresponding data point is the starting point of the constant force interval. The pre-stroke interval is defined as the length from the starting point of the full displacement process in the force-displacement curve to the starting point of the constant force interval. express; Step S23: Use single-factor analysis to set the starting point of the variable cross-section element. Length of variable cross-section element And the thickness ratio of variable cross section element The level range and number of analyses are determined based on the sensitivity of the experimental indicators to changes in the parameter levels and the optimal situation as evaluation criteria. Factor levels for subsequent orthogonal experimental analysis are selected. Based on the selected factor levels, an orthogonal experimental scheme including interactions between parameters is designed. Range analysis and variance analysis are performed based on the orthogonal experimental results to obtain the p-values ​​corresponding to each structural design parameter and its interaction. The significance level is set at 0.

05. By comparing the p-values ​​with the significance level, it is determined whether each parameter and its interaction have a significant impact on the experimental indicators, so as to determine the structural design parameters for subsequent mechanism optimization.

4. The parameter optimization method for a flexible constant force mechanism based on a multi-segment variable cross-section beam according to claim 1, characterized in that: Step S3 specifically includes: similar to the design of a flexible constant force mechanism based on a single-section variable cross-section beam, the origin is also established with the center of the starting section of the cosine beam as the origin. A coordinate system is used to define the structural design parameters for the geometry of variable cross-section elements, including: the starting point of the variable cross-section element. Variable cross-section element length And the thickness ratio of variable cross section element The superscript of the above parameter symbols This indicates that the constant force mechanism is based on Flexible constant force mechanism of segmented variable cross-section beam, subscript Indicates based on The first flexible constant force mechanism of segmented variable cross-section beam A variable cross-section element; the starting point of the variable cross-section element. It refers to based on Flexible constant force mechanism of segmented variable cross section beam The starting point of each variable cross-section element is at the x-axis coordinate of The center point of the cosine beam section corresponding to the time; the length of the variable cross-section element. It refers to based on Flexible constant force mechanism of segmented variable cross section beam The length of the variable cross-section element, that is, there exists a point on the beam's central axis that is parallel to the first variable cross-section element. Starting point of each variable cross-section element The distance along the positive x-axis is This point is the first... End point of a variable cross section element Satisfying the relationship The thickness of all variable cross-section elements remains constant within their length range, defined based on... Flexible constant force mechanism of segmented variable cross section beam Thickness ratio of each variable cross-section element The thickness of this variable cross-section element Compared to the cosine beam thickness of the constant cross-section flexible constant force mechanism used as a control group The ratio satisfies the relation .

5. The parameter optimization method for a flexible constant force mechanism based on a multi-segment variable cross-section beam according to claim 4, characterized in that: The theoretical model of the flexible constant force mechanism based on a multi-segment variable cross-section beam is established using the following method: Combined with the above definition of the variable cross-section element starting point Length of variable cross-section element Variable cross-section element thickness ratio and foundation structural parameters: total length of beam Construct a piecewise thickness function for a multi-segment variable cross-section beam. Therefore, the piecewise thickness of the multi-segment variable cross-section beam can be obtained using... The representation is defined as follows: Based on the established Coordinate system, foundation structural parameters, beam height and the total length of the beam The initial shape curve of a multi-segment variable cross-section beam can be defined. ; The shape curve of the multi-segment variable cross-section beam after applying lateral compression to the guide beam can be represented as a linear superposition of cosine modal functions, and the shape curve of the multi-segment variable cross-section beam after deformation can be defined. : in Indicates the first The amplitude coefficients of the even-order modes are limited by the constraint of the guide beam, restricting the deformation of the even-order modes. Therefore, the multi-segment variable cross-section beam only experiences symmetrical buckling. ... This represents the number of periods of the cosine mode function; Using the first kind of line integral and based on the small deformation assumption, we can obtain the expression for the beam length: Based on this expression for beam length, the initial length of a multi-segment variable cross-section beam can be derived. and the length after deformation : This allows us to obtain the change in beam length before and after deformation of the multi-segment variable cross-section beam. According to the elasticity theorem, the axial force can be obtained. The calculation formula is: in, The Young's modulus of the material. The out-of-plane thickness is a fundamental structural parameter. The equivalent length of a multi-segment variable cross-section beam is defined as the equivalent length required to produce the same axial stiffness when the multi-segment variable cross-section beam is equivalent to a beam with the same thickness and uniform cross-section. This is based on the segmented thickness function defined above. And the set of structural design parameters related to the variable cross-section element can be used to obtain... Multi-segment variable cross-section beam with variable cross-section element The formula is: The formula for the compressive strain energy of a beam is: Based on axial force Calculation formula and beam length variation The specific expression for the compressive strain energy of the beam during deformation can be obtained as follows: The formula for the bending strain energy of a beam is: in, For multi-segment variable cross-section beams in coordinate The moment of inertia of the cross section at time t is expressed as: Combining the shape curve formulas before and after deformation, the specific expression for the bending strain energy of the beam during deformation can be obtained as follows: in, The moment of inertia of a cosine beam with uniform cross-section, The term representing curvature variation is formulated as follows: external force Under the action of the guide beam, the displacement generated can be expressed as: This allows us to obtain external force. Change in system potential energy caused by work The specific expression is: Total potential energy of the system The compressive strain energy of the beam Bending strain energy Doing work with external forces The formula for obtaining the total energy of the system is as follows: Based on the above derivations , and The specific expression for the system's total potential energy is obtained. The specific expression is: Based on the principle of minimum potential energy, the total potential energy of the system is... By variation, we can obtain: in, By adjusting the curvature change term about The derivative yields the following expression: in, The modal selection function, defined to ensure that the first-order variational expression of the system's total potential energy is consistent with the physical constraints, is as follows: The above first-order variational equation describes the general equilibrium state of the structure during the stress process, and is the basis for obtaining... The basis of the curve; for the flexible structure with buckling instability analyzed, when the axial pressure... When the critical value is reached, the system's stiffness undergoes a sudden change, causing the original equilibrium path to bifurcate; therefore, based on the system's total potential energy... The first-order variational equation is used to develop axial pressure. The critical value is calculated and defined as the critical buckling load. ; Critical buckling load It is a core intrinsic parameter characterizing structural stability, essentially solving for the value corresponding to zero tangential stiffness of the system. This is achieved by considering the total potential energy of the system. Performing a second-order variational calculus, we obtain the second-order variational equation for the total potential energy of the system as follows: in, It is a modal coupling integral operator that reflects the variable cross-section effect, and its expression is: To achieve numerical solutions, the total potential energy of the system is... The second-order variational equation is transformed into a generalized eigenvalue equation. Based on the first part of the second-order variational equation, an elastic stiffness matrix reflecting the structure's own bending recovery capability is established. : Among them, the diagonal matrix , Based on the second part of the second-order variational equation, a system reflecting the axial force is established. Geometric stiffness matrix of the system stiffness reduction effect Because this part has Therefore, this matrix is ​​a diagonal matrix, and its diagonal elements are defined as follows: Therefore, the total tangent stiffness matrix of the system can be obtained as follows: When a structure experiences buckling instability, its tangential stiffness drops to zero. Transforming it into a singular matrix is ​​equivalent to solving the following generalized eigenvalue equations: in, These are the eigenvectors corresponding to the buckling modes. Through eigenvalue decomposition, eigenvalues ​​corresponding to different mode orders can be obtained. According to the minimum critical load principle in elastic stability theory, structures always buckle and fail at the lowest critical load. Therefore, the third-order mode corresponding to this principle is selected. Critical buckling load for multi-segment variable cross-section beams Therefore, the following relationships can be obtained to establish a theoretical model for a multi-segment variable cross-section beam: Previous research has shown that a pseudo-rigid body method can be used to establish the theoretical model of a V-shaped beam. In the pseudo-rigid body model, the torsion springs are located near the ends of the beam, and the flexible beam section between the two torsion springs is considered as a pseudo-rigid body rod, with a length... , The characteristic radius coefficient is , and the torsional constants of the four torsion springs are all equal in magnitude. , is represented as: in, This is the stiffness coefficient of the torsion spring. The moment of inertia of the V-beam section, ; Apply a magnitude of to the guide beam The compression displacement was such that only the portion of the beam near the end of the flexible beam deformed, resulting in a pseudo-rigid body angle. The remaining parts remain unchanged. The initial angle between the V-beam and the guide beam. The relationship between them is: A theoretical model of the V-beam is established based on the principle of virtual work: The output constant force value of the flexible constant force mechanism based on the multi-segment variable cross-section beam is: The nonlinear equations of a multi-segment variable cross-section beam are solved using the Newton-Raphson algorithm to obtain the negative stiffness structure. Based on the curve data, and with the goal of achieving the maximum constant force range, an interval optimization algorithm is used to reverse-engineer the matching parameters corresponding to the normal stiffness structure. Based on the above, a theoretical model of a flexible constant force mechanism based on a multi-segment variable cross-section beam was established.

6. The parameter optimization method for a flexible constant force mechanism based on a multi-segment variable cross-section beam according to claim 1, characterized in that: Step S4 specifically includes: Step S41: To achieve the pre-travel range To minimize this, the objective function for optimizing the pre-stroke interval of a flexible constant-force mechanism based on a multi-segment variable cross-section beam is established as follows: In order to avoid the eigenvectors of the corresponding buckling modes Conflict, Setting Let be the starting position vector of the variable cross-section element. Let be the length vector of the variable cross-section element. A thickness ratio vector; This is the actual constant force output value obtained based on the combination of structural parameters. For the target constant force value, The constant force output allowable deviation coefficient can be obtained based on the constant force range error requirements. ; and The constant force range and pre-stroke range are obtained from the structural design parameter set based on the geometry of the variable cross-section element. This represents the lower limit of the constant force range; Step S42: Establish the constraint conditions for the flexible constant force mechanism based on a multi-segment variable cross-section beam as follows: in, , They are respectively The upper and lower limits; , They are respectively The upper and lower limits; , They are respectively The upper and lower limits; This is the minimum safety interval value; Step S43: To further describe the pre-stroke interval optimization objective function of the flexible constant force mechanism based on a multi-segment variable cross-section beam, a cost function is constructed as follows: in, and This is a preset penalty weighting coefficient; For constant force value default terms, the expression is: This represents the actual constant force output value. With target value The allowable range of differences; For the constant force interval default term, the expression is: , indicating the target constant force range With respect to the actual constant force range The difference; the optimization objective of the pre-travel interval is The set of structural design parameters for the geometric shape of the variable cross-section element corresponding to the minimum value; Step S44: To achieve the constant force range To maximize the constant force range optimization objective function of the flexible constant force mechanism based on a multi-segment variable cross-section beam, the following equation is established: in, The upper limit of the pre-travel interval is given; to further describe the constant force interval optimization objective function of the flexible constant force mechanism based on a multi-segment variable cross-section beam, a Cost function is constructed for description, as shown in the following equation: in, The expression for the pre-trip interval violation item is: , indicating the actual pre-travel range With the target pre-travel range The difference; the objective of constant force interval optimization is The set of structural design parameters for the variable cross-section element geometry corresponding to the minimum value.

7. The parameter optimization method for a flexible constant force mechanism based on a multi-segment variable cross-section beam according to claim 1, characterized in that: Step S6 specifically includes: performing static analysis on the three-dimensional model established by the optimal variable cross-section unit structural design parameter set with the optimization objective being the pre-stroke interval, and comparing the force-displacement data of the simulation with the theoretical calculation; performing static analysis on the three-dimensional model established by the optimal variable cross-section unit structural design parameter set with the optimization objective being the constant force interval, and comparing the force-displacement data of the simulation with the theoretical calculation.

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