Flexible mechanism modeling method integrating kinematics, statics and dynamics analysis methods

Abstract

The present invention discloses a flexible mechanism modeling method integrating kinematics and static and dynamic analysis methods, including the establishment of rigid body (with or without extension) units and flexible unit extension transfer matrices, the establishment of the overall transfer matrix / equation of the flexible mechanism, and the post-processing of the transfer matrix / equation results, which provides a new idea for solving the motion and mechanical characteristics of the flexible mechanism. When solving the characteristics of the flexible mechanism, the present invention establishes the kinematics, statics, and dynamic analysis equations of the mechanism in the same formula, which greatly simplifies the analysis process of the flexible mechanism, and is simpler and more intuitive than the overall flexibility matrix method, kinematic principle, and Lagrangian method. In addition, the mathematical model established by the present invention is not only applicable to the situation where the mechanism unit is subjected to force, but also applicable to the situation under the direct extension and deformation of the rigid body (such as the direct displacement change of the moving pair), and has strong versatility.

Description

Technical Field

[0001] The present invention relates to a flexible mechanism modeling method integrating kinematics and static and dynamic analysis methods, and specifically to a modeling method based on a transfer matrix method, integrating flexible mechanism kinematic analysis, static force-position relationship solution, and dynamic mode and frequency response analysis. Background Art

[0002] With the continuous deepening of research on flexible mechanisms, their applications in aerospace, automation, precision machinery and other fields have become increasingly widespread, especially in precision instruments and precision electronics. Compared with traditional rigid mechanisms, flexible mechanisms work on the principle of elastic deformation of materials. As a highly sensitive transmission mechanism with small size, no friction loss, smooth and gapless motion, it can undoubtedly play a vital role in precision instruments.

[0003] Static and dynamic analysis is the basis of flexible mechanism design and dynamic performance analysis. The quality of modeling directly affects the quality of the mechanism model. An accurate mathematical model can truly reflect the actual working conditions of the model, provide a theoretical basis for the design of the mechanism, and thus improve the quality and efficiency of the product. Therefore, it is of great significance to analyze the static and dynamic characteristics of flexible mechanisms. However, in existing research, there are still the following problems when modeling flexible mechanisms:

[0004] 1. There are two main situations for kinematic / static analysis: a. Analyze the situation where the applied driving form is force input deformation using the flexibility matrix method; b. Analyze the situation where the direct displacement change is analyzed using the kinematic geometric displacement relationship. The problem with this is that the method is not universal, and both methods have limitations. When the input method is displacement or the relationship between the input force and the displacement is unknown, the overall flexibility matrix method is no longer applicable. When the pseudo-rigid body method is used for modeling, there may be situations where the rotation center of the flexible hinge is offset or there are errors in establishing the coordinates of the flexible hinge center.

[0005] 2. The Lagrangian method is mainly used for dynamic analysis. In the process of solving the problem, it is necessary to establish the differential equation of mechanism dynamic motion, that is, a set of equations consisting of n mutually coupled second-order ordinary differential equations. The calculation process is complicated and cumbersome, and the established mathematical model is not intuitive and clear;

[0006] 3. Most importantly, the kinematic / static and dynamic characteristics of the flexible mechanism are analyzed separately without a unified expression, which leads to an increase in the amount of calculation and a decrease in efficiency.

[0007] Therefore, it is particularly important to have a universal method that integrates statics and dynamics analysis. Based on this, the present invention proposes a flexible mechanism modeling method that integrates statics and dynamics analysis, providing a new idea and method for analyzing the characteristics of flexible mechanisms. Summary of the invention

[0008] In view of the problems existing in the above-mentioned prior art, the object of the present invention is to provide a flexible mechanism modeling method integrating kinematics and static and dynamic analysis methods, so that a mathematical model of the flexible mechanism characteristics can be easily and accurately established.

[0009] To achieve the above objectives, the present invention adopts the following technical solution: a flexible mechanism modeling method integrating kinematics and static dynamics analysis methods, which is a flexible mechanism modeling method integrating kinematics and static dynamics analysis methods based on transfer matrix method, the method comprises the following steps, see Figure 1 :

[0010] Step 1: Establish the extended state vector Z of the input and output ends of each feature unit in the physical coordinate system i and Z o The state vector includes displacement and force vectors. The extended term also considers the situation of flexible mechanism unit force or rigid unit expansion and contraction deformation. In the output state vector result, the sub-terms are all nonlinear functions about ω, that is, Z o =f(ω)Z i , therefore, it provides a basis for mechanism kinematics and static / dynamic modeling, which is undoubtedly simpler and more convenient than using the Lagrangian method to solve differential equations.

[0011] Step 2: Construct the transfer matrix of each characteristic unit, including the extended transfer matrix of the flexible unit and the extended transfer matrix of the rigid unit. This step constructs the analysis matrix / equation of the subunits of the mechanism, taking into account the various parameters of the unit under the kinematic, static, and dynamic modeling of the mechanism, including frequency ω, length expansion and contraction change value ΔL, and excitation force F.

[0012] Step 3: Based on the relationship between the characteristic units, the sub-transfer matrices are spliced ​​into an overall transfer matrix / equation. Through this step, a transfer equation is used to establish the connection between the sub-units and the overall mechanism, which plays a role in overall modeling of the mechanism. Figure 2 .

[0013] Step 4: Post-process the results of the total transfer equation. By selecting appropriate values ​​of ω, ΔL, and F parameters, different mathematical model expressions are constructed to provide intuitive expressions for modeling the characteristics of the mechanism (mechanism kinematics, statics / dynamics).

[0014] In the above technical solution, for the establishment of the extended state vector Z of the input end of each characteristic unit in the physical coordinate system in step 1 i and the extended state vector Z at the output o They are:

[0015]

[0016] Among them, X, Y, Θ z ,M z , Fx, Fy are x, y, θ in the physical coordinate system z ,m z ,fx,fy are the state vectors in the corresponding modal coordinates. ΔL is the rigid body unit expansion added to the state vector in the multi-body system transfer matrix analysis method.

[0017] For the transfer matrices of each characteristic unit constructed in step 2, they are:

[0018] a) Establishment of rigid body transfer matrix

[0019] When establishing the transfer matrix for the rigid body unit, it is divided into two parts according to the force or expansion situation: one part is the basis transfer matrix under the force F, and the other part is the displacement transfer matrix containing the rigid body expansion ΔL. Figure 3 Taking the plane formula as an example, according to the geometric parameters and mechanical equilibrium relationship, a rigid body extension transfer matrix with an initial length of L, a stretch amount of ΔL, and a force magnitude of f is established: U = U 1 +U 2 :

[0020] ① Basis transfer matrix U under force F 1

[0021]

[0022] Among them, f 41 =m z +f x (yb y )-f y (b x -x), f x With f x A rigid body is subjected to a simple harmonic excitation force F with a frequency of ω at a certain point (x, y) = [f x cos(ωt),f y cos(ωt)] T and moment m z Effect. i is the inertia matrix relative to the input point I-(0,0), and the coordinates of the output point O (b 1 ,b 2), the coordinates of the center of mass C (c 1 ,c 2 ).

[0023] ② Displacement transfer matrix U including rigid body expansion ΔL 2

[0024]

[0025] For a rigid body with input at one end and output at the other end and length L, it is a special rigid body, that is, ΔL = 0. According to the transfer matrix method of multi-body system, its transfer matrix is ​​the base transfer matrix U under force F: 1 ;

[0026] For a rigid body with multiple inputs and one output, the same as above, the expansion ΔL = 0, and the rows and columns corresponding to the corresponding displacements are added on its basis, and the transfer matrix is:

[0027]

[0028] Among them, L i1o - is the coordinate of the output point relative to the first input point;

[0029] L i2o - is the coordinate of the output point relative to the second input point;

[0030] L i1c - is the coordinate of the centroid relative to the second input point;

[0031] L co - is the coordinates of the output point relative to the centroid;

[0032] I 2×2 is a 2×2 identity matrix, N 1 ×N 2 Zero matrix.

[0033] b) For the transfer matrix of the flexible unit, it is expanded based on the transfer matrix method of the multi-body system as follows:

[0034]

[0035] Among them, f g =[0 0 0 f 41 f x f y ] T ; U 铰 is the transfer matrix of the hinge in the transfer matrix method of multibody systems.

[0036] For step 3, the transfer matrices of the scattered units are spliced ​​into an overall transfer equation according to the unit connection relationship. Figure 2The relationship between the units described in the above is used to establish the transfer equation of the mechanism in the global coordinate system:

[0037] U all Z all =0

[0038] The result is post-processed in step 4. When ω = 0, the equation U established in step 3 all Z all =0, it becomes a function expression about ΔL and F:

[0039] f(ΔL,F)=0

[0040] This mathematical model describes the kinematic and static characteristics of the flexible mechanism. When ω≠0, the equation U established in step 3 all Z all =0 is the function expression about ω, ΔL, and F:

[0041] f(ω,ΔL,F)=0

[0042] This mathematical model describes the dynamic characteristics of the flexible mechanism.

[0043] Compared with the prior art, the present invention has the following advantages:

[0044] 1. When solving the characteristics of flexible mechanisms, the kinematics, statics, and dynamics analysis of the mechanism are established in the same equation, which simplifies the analysis process of the flexible mechanism and has higher computational efficiency than analyzing them one by one.

[0045] 2. The equation established in the present invention is applicable to situations where the mechanism unit is subjected to force or the rigid body is stretched and deformed, and has strong versatility regardless of whether it is a fully compliant mechanism or a rigid-flexible hybrid.

[0046] 3. During the analysis process, different modeling analyses can be selected according to different post-processing conditions. The analysis process is simple and intuitive. BRIEF DESCRIPTION OF THE DRAWINGS

[0047] Figure 1 The modeling process of the flexible mechanism modeling method integrating kinematics and static and dynamic analysis methods of the present invention is as follows;

[0048] Figure 2 It is composed of the total transfer matrix / equation in the mathematical model established by the flexible mechanism modeling method integrating kinematics and static and dynamic analysis methods of the present invention;

[0049] Figure 3 Schematic diagram of a displacement rigid body in a flexible mechanism modeling method integrating kinematics and static and dynamic analysis methods of the present invention, wherein I is an input point, O is an output point, L is the length of the displacement rigid body, and F is the excitation force received;

[0050] Figure 4 It is a schematic diagram of an application model of a flexible mechanism modeling method that integrates kinematics and static and dynamic analysis methods of the present invention, wherein 1, 5, 7, and 11 are rigid body units with one-end input and one-end output, 2, 4, 8, and 10 are flexible units, 6 is a rigid body unit with two-end input and one-end output, and 3 and 9 are retractable rigid body units. DETAILED DESCRIPTION

[0051] The present invention is further described in detail below with reference to the accompanying drawings and specific examples.

[0052] The present invention provides a flexible mechanism modeling method integrating kinematics and static dynamics analysis methods based on the transfer matrix method. The method comprises the following steps: Figure 1 :

[0053] Step 1: Establish the extended state vector Z of the input and output ends of each feature unit in the physical coordinate system i and Z o ;

[0054] Step 2, construct the transfer matrix of each characteristic unit, including the extended transfer matrix of the flexible hinge and the transfer matrix of the displacement rigid body;

[0055] Step 3, according to the relationship between each characteristic unit, the sub-transfer matrix is ​​spliced ​​into an overall transfer equation;

[0056] Step 4, post-processing the results of the total transfer equation;

[0057] In the above technical solution, for the establishment of the extended state vector Z of the input end of each characteristic unit in the physical coordinate system in step 1 i and the extended state vector Z at the output o They are:

[0058]

[0059] On the basis of the state vector in the multi-body system transfer matrix analysis method, the rigid body unit expansion ΔL is added. This can clearly show the displacement changes of the branch, which facilitates the construction of the unit transfer matrix / equation in step 2.

[0060] For the extended transfer matrix of each characteristic unit constructed in step 2, they are:

[0061] 2.1 Establishment of rigid body transfer matrix

[0062] The establishment of the rigid body unit transfer matrix is ​​divided into two parts according to the force or expansion situation: one part is the basis transfer matrix under the force F, and the other part is the displacement transfer matrix containing the rigid body expansion ΔL.

[0063] See also Figure 3 , taking the plane formula as an example, establish a rigid body transfer matrix with an initial length of L and a change of ΔL. I is the input point, O is the output point, and a coordinate system is established with point I as the origin. From the geometric relationship, it can be seen that the rotation angle of point O around the Z axis is the same as the rotation angle of point I, and the displacement of point O can be represented by the displacement of point I and the angular displacement of point I around the Z axis. Considering that the rotation angle around the Z axis is relatively small, and |θ Z,I |<<1, we get the relationship:

[0064]

[0065] According to the force balance formula:

[0066]

[0067] Among them, (bx, by) is the coordinate of the output point O relative to point I, Add its frequency term and write the above formula in matrix form:

[0068] Z O =UZ I

[0069] The transfer matrix of the rigid body is: U = U 1 +U 2 .

[0070] 2.2 Basis transfer matrix U under force F 1

[0071]

[0072] Among them, f 41 =m z +f x (yb y )-f y (b x -x), f x With f x The rigid body receives a simple harmonic excitation force F with a frequency of ω at a certain point (x, y) = [f x cos(ωt),f y cos(ωt)] T and moment m z Effect, J i is the inertia matrix relative to the input point I-(0,0), and the coordinates of the output point O (b 1 ,b 2 ), the coordinates of the center of mass C (c 1 ,c 2 ).

[0073] 2.3 Displacement transfer matrix U including rigid body expansion ΔL 2

[0074]

[0075] For a rigid body with input at one end and output at the other end and length L, it is a special rigid body, that is, ΔL = 0. According to the transfer matrix method of multi-body system, its transfer matrix is ​​the base transfer matrix U under force F: 1 ;

[0076] For a rigid body with multiple inputs and one output, the same as above, the expansion ΔL = 0, and the rows and columns corresponding to the corresponding displacements are added on its basis, and the transfer matrix is:

[0077]

[0078] Among them, L i1o - is the coordinate of the output point relative to the first input point;

[0079] L i2o - is the coordinate of the output point relative to the second input point;

[0080] L i1c - is the coordinate of the centroid relative to the second input point;

[0081] L co - is the coordinate of the output point relative to the centroid;

[0082] I 2×2 is a 2×2 identity matrix, N 1 ×N 2 Zero matrix.

[0083] 2.4 For the transfer matrix of the flexible unit, it is expanded based on the transfer matrix method of the multi-body system as follows:

[0084]

[0085] Among them, f g =[0 0 0 f 41 f x f y ] T ; U 铰 is the transfer matrix of the hinge in the transfer matrix method of multibody systems.

[0086] Step 3: According to the unit connection relationship, the transfer matrix of the scattered units is spliced ​​into an overall transfer equation. Figure 4 The relationship between the units described in the above is used to establish the transfer equation of the mechanism in the global coordinate system:

[0087] U allZ all =0

[0088] Among them, Z all The state vector is Z all =[Z 1.O Z 10.O Z 6.O ], the overall transfer equation is:

[0089]

[0090] in:

[0091]

[0092]

[0093] The coordinates (ax,ay) are the coordinates of the second input point relative to the first input point.

[0094] In step 4, the result is post-processed: when ω = 0, the equation U established in step 3 all Z all =0, it becomes a function expression about ΔL and F:

[0095] f(ΔL,F)=0

[0096] This mathematical model describes the kinematic and static characteristics of the flexible mechanism.

[0097] a1 When the rigid body unit is in the telescopic deformation situation F=0,ΔL≠0, in this case the above expression becomes f(ΔL)=0, which can be used to solve the situation in which the moving pair in the flexible mechanism is driven by motor displacement in actual situations.

[0098] in accordance with Figure 4 The model constraints in (the lower end of the model is completely fixed) are used to establish boundary conditions, and we get:

[0099]

[0100] Substituting this into the overall transfer matrix, removing Z all and U all 1, 2, 3, 8, 9, 10, 18, 19, 20 rows and columns, and delete U all , and then move the column corresponding to the displacement to the right side of the equation and multiply it by the corresponding change to get the nonhomogeneous equation:

[0101]

[0102] Where LX = ΔL 1 U all (:,7)+ΔL 2 Uall (:,14)+(ΔL 1 +ΔL 2 ) all (:,21), solving the above equation yields the state vector equation Thus, the end displacement change XYΘ value is solved:

[0103]

[0104] b1 When the rigid body unit is in the force deformation situation F≠0, ΔL=0, in this case, the above expression becomes f(F)=0, which can be used to solve the situation where the moving pair in the flexible mechanism is flexible deformation in actual situations. In this analysis, the expansion amount ΔL in the state vector involved needs to be set to 1.

[0105] in accordance with Figure 4 The model constraints in (the lower end of the model is completely fixed) are used to establish boundary conditions, and we get:

[0106]

[0107] Substituting this into the overall transfer matrix, removing Z all and U all 1, 2, 3, 8, 9, 10, 18, 19, 20 rows and columns, and delete U all Then move the column corresponding to the force to the right side of the equation and multiply it by the corresponding change to get the nonhomogeneous equation:

[0108]

[0109] Where f = U all (:,7)+U all (:,14)+2×U all (:,21), solving the above equation yields the state vector equation The end displacement change can be calculated:

[0110]

[0111] When ω≠0, the equation U established in step 3 all Z all =0 is the function expression about ω, ΔL, and F:

[0112] f(ω,ΔL,F)=0

[0113] This mathematical model describes the dynamic characteristics of the flexible mechanism.

[0114] a2 When the rigid body unit is in the expansion and contraction deformation situation F=0,ΔL≠0, according to Figure 4The model constraints in (the lower end of the model is completely fixed) are used to establish boundary conditions, and we get:

[0115]

[0116] Substituting this into the overall transfer matrix, removing Z all and U all 1, 2, 3, 8, 9, 10, 18, 19, 20 rows and columns, and delete U all , and then move the column corresponding to the displacement to the right side of the equation and multiply it by the corresponding change to get the nonhomogeneous equation:

[0117]

[0118] Where LX = ΔL 1 U all (:,7)+ΔL 2 U all (:,14)+(ΔL 1 +ΔL 2 ) all (:,21), the total transfer matrix after reorganization When the determinant is zero, the mode of the model can be found.

[0119] b2 When the rigid body unit is in the force deformation state F≠0, ΔL=0, set ΔL to 1. Figure 4 The model constraints in (the lower end of the model is completely fixed) are used to establish boundary conditions, and we get:

[0120]

[0121] Substituting this into the overall transfer matrix, removing Z all and U all 1, 2, 3, 8, 9, 10, 18, 19, 20 rows and columns, and delete U all Then move the column corresponding to the force to the right side of the equation and multiply it by the corresponding change to get the nonhomogeneous equation:

[0122]

[0123] The reorganized total transfer matrix By setting the determinant to zero, the dynamic characteristics (frequency response) of the model can be obtained.

Claims

1. A flexible mechanism modeling method integrating kinematics and static and dynamic analysis methods, characterized in that: The method comprises the following steps: Step 1: Establish the extended state vector of each feature unit at the input end in the physical coordinate system and the extended state vector at the output , the expanded state vector of the input and the extended state vector at the output Including displacement and force vectors, the extended term also takes into account the flexible mechanism unit force or rigid unit expansion and contraction deformation; in the output end of the extended state vector results, the sub-items are about A nonlinear function, that is , thus providing a basis for mechanism kinematics, static / dynamic modeling; Step 2: Construct the transfer matrix of each characteristic unit, including the extended transfer matrix of the flexible unit and the extended transfer matrix of the rigid unit; construct the analysis matrix / equation of the sub-units of the mechanism, taking into account the various parameters of the unit under the kinematic, static and dynamic modeling of the mechanism, including frequency , length expansion change value , and motivation ; Step 3: According to the relationship between each characteristic unit, the sub-transfer matrices are spliced ​​into an overall transfer matrix / equation; a transfer equation is used to establish the connection between the sub-units and the overall mechanism, which plays an overall modeling role for the mechanism; Step 4: Post-process the total transfer equation results by selecting the appropriate , , Parameter values ​​are used to construct different mathematical model expressions, providing intuitive expressions for describing the kinematics of the mechanism and modeling the static / dynamic characteristics of the mechanism.

2. A flexible mechanism modeling method integrating kinematics and static and dynamic analysis methods as claimed in claim 1, characterized in that: The extended state vector of the input end of each characteristic unit in the physical coordinate system is established in step 1 and the extended state vector at the output They are: in, In the physical coordinate system The state vector in the corresponding modal coordinates is In order to increase the rigid body unit expansion and contraction amount based on the state vector in the multi-body system transfer matrix analysis method, the displacement change of the branch can be clearly seen, which facilitates the construction of the unit transfer matrix / equation in step 2.

3. A flexible mechanism modeling method integrating kinematics and static and dynamic analysis methods as claimed in claim 2, characterized in that: The transfer matrices of each characteristic unit constructed in step 2 are: a) Establishment of rigid body transfer matrix When establishing the rigid body unit transfer matrix, it is divided into two parts according to the force or expansion situation: one part contains the force The other part is the basis transfer matrix containing the rigid body expansion For the plane formula, according to the geometric parameters and mechanical equilibrium relationship, the initial length is L and the expansion is , the force magnitude is The rigid body extended transfer matrix is: : ①Including force The basis transfer matrix under in, , and For quality A rigid body at a point The frequency is Simple harmonic excitation and torque effect, Relative to the input point The inertia matrix of Coordinates , centroid Coordinates ; ②Including rigid body expansion The displacement transfer matrix For a rigid body with input at one end and output at the other end and length L, it is a special rigid body, that is, According to the transfer matrix method of multi-body system, the transfer matrix is The basis transfer matrix under ; For a rigid body with multiple inputs and one output, the amount of expansion On this basis, the rows and columns corresponding to the corresponding displacements are added, and the transfer matrix is: in, - is the coordinate of the output point relative to the first input point; - is the coordinate of the output point relative to the second input point; - is the coordinate of the centroid relative to the second input point; - is the coordinates of the output point relative to the centroid; for The identity matrix, for Zero matrix; b) For the transfer matrix of the flexible unit, it is expanded based on the transfer matrix method of the multi-body system as follows: in, ; is the transfer matrix of the hinge in the transfer matrix method of multibody systems.

4. A flexible mechanism modeling method integrating kinematics and static and dynamic analysis methods as claimed in claim 3, characterized in that: For step 3, the sub-unit transfer matrices are spliced ​​into an overall transfer matrix / equation according to the unit connection relationship. For the relationship between the plane model units, the transfer equation of the mechanism in the global coordinate system is constructed: in, The state vector of , the overall transfer equation is: in: coordinate are the coordinates of the second input point relative to the first input point.

5. A flexible mechanism modeling method integrating kinematics and static and dynamic analysis methods as claimed in claim 4, characterized in that: In step 4, the results are post-processed; when When , the equation established in step 3 becomes about , The function expression of This mathematical model describes the kinematic and static characteristics of the flexible mechanism: a1 When the rigid body unit is in the state of expansion and contraction , in which case the above expression becomes , which can be used to solve the situation in which the moving pair in the flexible mechanism is driven by the motor displacement in actual situations; According to the model constraint condition that the lower end of the model is completely fixed, the boundary conditions are established, and we get: Substituting this into the overall transfer matrix, delete and delete the 1st, 2nd, 3rd, 8th, 9th, 10th, 18th, 19th, and 20th rows and columns. Then move the column corresponding to the displacement to the right side of the equation and multiply it by the corresponding change to get the nonhomogeneous equation: in, , L is the initial length, solving the above equation to obtain the state vector equation , and thus solve the end displacement change value: b1 When the rigid body unit is subjected to force and deformation In this case, the above expression becomes , which can be used to solve the situation in which the moving pair in the flexible mechanism is flexible in actual situations. In this analysis, the expansion and contraction amount in the state vector involved needs to be Set to 1; According to the model constraint condition that the lower end of the model is completely fixed, the boundary conditions are established, and we get: Substituting this into the overall transfer matrix, delete and delete the 1st, 2nd, 3rd, 8th, 9th, 10th, 18th, 19th, and 20th rows and columns. Then move the column corresponding to the force to the right side of the equation and multiply it by the corresponding change to get the nonhomogeneous equation: in, , solving the above equations yields the state vector equation , the end displacement change can be calculated: when When , the equation established in step 3 About , , The function expression of This mathematical model describes the dynamic characteristics of the flexible mechanism: a2 When the rigid body unit is in the state of expansion and contraction , based on the model constraint condition that the lower end of the model is completely fixed, the boundary conditions are established, and we get: Substituting this into the overall transfer matrix, delete and delete the 1st, 2nd, 3rd, 8th, 9th, 10th, 18th, 19th, and 20th rows and columns. Then move the column corresponding to the displacement to the right side of the equation and multiply it by the corresponding change to get the nonhomogeneous equation: in, , the total transfer matrix after reorganization is When the determinant is zero, the mode of the model can be found; b2 When the rigid body unit is subjected to force deformation When Set to 1, and establish boundary conditions based on the model constraint condition that the lower end of the model is completely fixed, and we get: Substituting this into the overall transfer matrix, delete and delete the 1st, 2nd, 3rd, 8th, 9th, 10th, 18th, 19th, and 20th rows and columns. Then move the column corresponding to the force to the right side of the equation and multiply it by the corresponding change to get the nonhomogeneous equation: The reorganized total transfer matrix By setting the determinant to zero, the dynamic characteristics of the model can be found.

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