CN115045904A - Double-shaft flexible guide mechanism for nano positioning platform and rigidity modeling method thereof - Google Patents

Double-shaft flexible guide mechanism for nano positioning platform and rigidity modeling method thereof Download PDF

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CN115045904A
CN115045904A CN202110250767.XA CN202110250767A CN115045904A CN 115045904 A CN115045904 A CN 115045904A CN 202110250767 A CN202110250767 A CN 202110250767A CN 115045904 A CN115045904 A CN 115045904A
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flexible
flexible hinge
hinge
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牛禹涵
陈星有
王凯杰
宋佳奇
周志文
朱紫辉
朱志伟
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Nanjing University of Science and Technology
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    • F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F16—ENGINEERING ELEMENTS AND UNITS; GENERAL MEASURES FOR PRODUCING AND MAINTAINING EFFECTIVE FUNCTIONING OF MACHINES OR INSTALLATIONS; THERMAL INSULATION IN GENERAL
    • F16C—SHAFTS; FLEXIBLE SHAFTS; ELEMENTS OR CRANKSHAFT MECHANISMS; ROTARY BODIES OTHER THAN GEARING ELEMENTS; BEARINGS
    • F16C11/00—Pivots; Pivotal connections
    • F16C11/04—Pivotal connections
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    • F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F16—ENGINEERING ELEMENTS AND UNITS; GENERAL MEASURES FOR PRODUCING AND MAINTAINING EFFECTIVE FUNCTIONING OF MACHINES OR INSTALLATIONS; THERMAL INSULATION IN GENERAL
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    • F16M11/00—Stands or trestles as supports for apparatus or articles placed thereon ; Stands for scientific apparatus such as gravitational force meters
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    • F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F16—ENGINEERING ELEMENTS AND UNITS; GENERAL MEASURES FOR PRODUCING AND MAINTAINING EFFECTIVE FUNCTIONING OF MACHINES OR INSTALLATIONS; THERMAL INSULATION IN GENERAL
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Abstract

The invention discloses a double-shaft flexible guide mechanism for a nanometer positioning platform and a rigidity modeling method thereof. Including workstation base and fixed frame, the workstation base is set in fixed frame's middle part, and the centre has 8 to be the same sinusoidal flexible hinge of symmetric distribution's structure and is connected with fixed frame. The rigidity modeling method comprises the following steps: and deducing the compliance of the sine-shaped flexible hinge unit by utilizing a second card-type theorem, further obtaining a rigidity matrix of the flexible hinge by adopting a matrix-based compliance modeling method, and finally obtaining the total rigidity of the mechanism by using the same method. The invention discloses a double-shaft flexible guide mechanism for a nanometer positioning platform by utilizing the characteristic that a sinusoidal flexible hinge can simultaneously generate elastic extension and bending deformation, which can realize the decoupling motion of two shafts, has larger motion stroke, reduces the overall size and motion quality of the platform and improves the working bandwidth of the positioning platform.

Description

Double-shaft flexible guide mechanism for nano positioning platform and rigidity modeling method thereof
Technical Field
The invention belongs to the technical field of precision manufacturing, and particularly relates to a double-shaft flexible guide mechanism for a nanometer positioning platform and a rigidity modeling method thereof.
Background
The nano positioning platform is a motion, positioning and operation platform with nano-level precision, is widely applied to the fields of nano measurement, nano processing, biological manufacturing and the like, is one of core modules of nano manufacturing equipment, directly influences the development level of the technology in the field of microscopic science, and has extremely important practical research significance. The guide mechanism is a main body component of the nanometer positioning platform, and the compactness, decoupling property, motion resolution, motion part quality and the like of the structure directly determine the performance of the nanometer positioning platform. The most common structure in a guide mechanism is a flexible hinge, which uses elastic deformation of a material to achieve deformation of a structure to output displacement. The device has the advantages of no mechanical friction, no gap, high motion sensitivity, compact structure, small volume, light weight and the like, and is widely applied to various nano positioning platforms.
For the flexible guide mechanism with two degrees of freedom, the mainstream design method at present is to combine the flexible hinges with one-way degrees of freedom in a series-parallel connection mode, and the developed flexible hinges with one-way degrees of freedom have a right-angle section type, an elliptical section type, a transition fillet section type, a perfect circle section type and the like, however, the flexible guide mechanism formed by combining the flexible hinges inevitably increases the structural size and the motion quality of the platform. In order to improve the working performance, another method is to directly adopt a flexible hinge with two degrees of freedom to be connected in parallel, but the existing flexible hinge with two degrees of freedom, such as an arc type and a corrugated spring type, all face common problems as follows: the amount of elastic deformation that can be produced per unit length of hinge is limited. Therefore, how to break through this limit is one of the key technologies for studying flexible guide mechanism.
Disclosure of Invention
The invention aims to provide a double-shaft flexible guide mechanism for a nanometer positioning platform and a rigidity modeling method thereof.
The technical solution for realizing the purpose of the invention is as follows:
a double-shaft flexible guide mechanism for a nanometer positioning platform comprises a fixed frame, a plurality of flexible hinges and a workbench base,
the workbench base is located in the middle of the fixed frame, at least one flexible hinge is connected between the four outer sides of the workbench base and the inner side of the fixed frame respectively, and the flexible hinges are sine-shaped flexible hinges.
Further, still include drive module, bearing structure and workstation, fixed frame's bottom is connected with bearing structure, the top and the workstation of workstation base are connected.
Further, fixed frame is last to include a plurality of frame mounting holes, including a plurality of workstation mounting holes on the workstation base, fixed frame's bottom is passed through the bolt and is connected with bearing structure, the top of workstation base is passed through the bolt and is connected with the workstation.
The bottom surface of the workbench base is an armature mounting surface, and the armature is connected with the armature mounting surface in an adhesive mode.
Further, the device also comprises a sensor mounting seat, wherein the sensor mounting seat is mounted at the top of the workbench base, and a sensor mounting hole is formed in the sensor mounting seat.
Furthermore, the free end of the flexible hinge is connected with the workbench base, and the fixed end of the flexible hinge is connected with the fixed frame.
Furthermore, the number of the flexible hinges is eight, the eight flexible hinges are divided into four groups and symmetrically arranged on the four outer side surfaces of the workbench base, and each group of flexible hinges is perpendicular to the adjacent group of flexible hinges.
The rigidity modeling method of the biaxial flexible guide mechanism for the nanometer positioning platform comprises the following steps of:
the method comprises the following steps: determining a functional expression of a sinusoid formed by the center line of the single sinusoidal flexible hinge:
a rectangular coordinate system o is established at the central point of the fixed end of the sine flexible hinge 0 -x 0 y 0 z 0 Wherein A, p x And n is used to represent the amplitude of the sinusoid, the length of one period, and the number of periods of one sinusoidal flexible hinge, respectively, the functional expression of the sinusoid formed by the center line of the sinusoidal flexible hinge is:
Figure BDA0002965960850000021
step two: determining the normal stress and the shear stress of any position of a single-period sine curve:
selecting a periodic sine flexible hinge x ∈ [0, p ] x ]Its free end load is F ═ F x ,f y ,f z ,m x ,m y ,m z ] T Wherein f is x ,f y ,f z Each representing a free end x of a periodic flexible hinge 0 ,y 0 ,z 0 Force m received in the axial direction x ,m y ,m z Each representing a free end x of a periodic flexible hinge 0 ,y 0 ,z 0 The free end is deformed into u ═ u by the torque applied in the axial direction x ,u y ,u z ,θ x ,θ y ,θ z ] T Wherein u is x ,u y ,u z Each representing a free end x of a periodic flexible hinge 0 ,y 0 ,z 0 Free end displacement in axial direction, theta x ,θ y ,θ z Each representing a free end x of a periodic flexible hinge 0 ,y 0 ,z 0 The rotation angle in the axial direction is in any x 0 Bending moment M at location b Comprises the following steps:
M b ＝[M xy ,M xz ,M yz ]
wherein,
Figure BDA0002965960850000031
wherein M is xy ，M xz ，M yz Each is represented at an arbitrary x 0 At position x 0 y 0 ，x 0 z 0 ，y 0 z 0 Bending moment received in plane, at arbitrary x 0 The plane normal force N and shear force S at the positions are expressed as follows:
Figure BDA0002965960850000032
wherein,
Figure BDA0002965960850000033
step three: determining elastic strain energy of a single cycle of a flexible hinge:
according to the second theorem of clamping, the elastic strain energy U due to deformation is expressed as
Figure BDA0002965960850000034
Wherein
Figure BDA0002965960850000035
Is the second moment of inertia, A, of the sinusoidal flexible hinge cross section c Where w represents the width of the flexural hinge, t represents the thickness of the flexural hinge, and E represents the materialThe Young modulus of the material is as follows,
Figure BDA0002965960850000036
is the corresponding shear modulus, mu represents the Poisson's ratio of the material, alpha t Represents the shear coefficient of a rectangular cross section, L being the actual length of a single period flexible hinge in the direction of the sinusoid;
step four: determining the deformation amount of the flexible hinge in a single period:
the space flexibility expression of the sine-shaped flexible hinge obtained according to the second theorem of the clamping type is as follows
Figure BDA0002965960850000037
Wherein
Figure BDA0002965960850000041
The relationship between single cycle compliant hinge deflection and end load is therefore: u ═ C s F；
Step five: determining a spatial compliance matrix for a single sinusoidal compliant hinge:
the single sine flexible hinge is formed by connecting n periodic flexible hinge units in series, and a space flexibility matrix of the random first sine flexible hinge is obtained according to a space position conversion method of the flexibility matrix
Figure BDA0002965960850000042
Wherein T is i Is from the coordinate system o i -x i y i z i To o c1 -x c1 y c1 z 0c1 Position conversion matrix of o c1 -x c1 y c1 z 0c1 Point o for the free end coordinate system of the flexible hinge c1 Global origin of coordinates representing a single sinusoidal flexible hinge, at the free end, point o i Representing the ith cycle of the flexible hingeOrigin of part coordinate, T i Is shown as
Figure BDA0002965960850000043
Wherein R is i (α i ) Is from the coordinate system o i -x i y i z i To o c1 -x c1 y c1 z 0c1 Of the rotation matrix S i (r i ) Is r i Has a rotation angle alpha around the z-axis i ，R i (α i ) Is defined as
Figure BDA0002965960850000044
r i ＝[x i ,y i ,z i ]Is in a coordinate system o c1 -x c1 y c1 z 0c1 Point o of (1) i Position of (S), S i (r i ) Is given by
Figure BDA0002965960850000045
Since the single-period flexible hinge is in a serial relationship in the single sinusoidal flexible hinge, the position of the ith period flexible hinge is in the coordinate system o c1 -x c1 y c1 z 0c1 In is represented by r i ＝[-ip x ,0,0]And a rotation angle alpha around the z-axis i ＝0；
Step six: determining the rigidity of the whole double-shaft flexible guide mechanism:
stiffness at the center of the mechanism is expressed as
Figure BDA0002965960850000046
Wherein T is j Is at an angle around the z-axis of
Figure BDA0002965960850000047
The position transformation matrix of (2), i.e. the free end coordinate system o of the flexible hinge involved in the calculation cj -x cj y cj z cj A position conversion matrix to o-xyz, j represents the number of groups of flexible hinges, j is 1,2,3,4, o-xyz is a coordinate system at the center position of the base of the workbench,
in modeling the compliance in the x and y directions, considering the table base as a rigid body, the stiffness in the x and y directions at the center of the mechanism is represented as k in consideration of the characteristics of the biaxial flexible guide mechanism symmetrical about the x and y axes x ＝k y ＝K(1,1)＝K(2,2)。
Compared with the prior art, the invention has the following remarkable advantages:
the invention designs the sine-shaped flexible hinge, improves the motion stroke and reduces the motion quality under the same size, and simultaneously the hinge has better double-shaft motion guiding capability, and the hinge is used for constructing the two-shaft nanometer positioning platform, so that the working performance of the flexible hinge can be fully exerted, the structural size and the motion quality of the platform are greatly reduced, the working performance of the platform is improved, and meanwhile, a corresponding rigidity analysis model is established, so that the invention can provide a guiding function at the design stage, simplify the design process and improve the efficiency.
Drawings
FIG. 1 is a schematic diagram of a part of the structure of a biaxial flexible guide mechanism for a nanometer positioning platform.
FIG. 2 is a rear view of the biaxial flexible guide mechanism for the nanopositioning platform of the present invention.
FIG. 3 is a front view of the sinusoidal flexible hinge of the biaxial flexible guide mechanism for the nanopositioning platform of the present invention.
FIG. 4 is a schematic structural diagram of a sinusoidal flexible hinge of a biaxial flexible guide mechanism for a nano-positioning platform according to the present invention.
FIG. 5 is a schematic structural diagram of a biaxial flexible guiding mechanism for a nanopositioning platform according to the present invention.
In the figure, 1, a fixed frame, 2, a flexible hinge, 3, a workbench base, 4, a frame mounting hole, 5, a workbench mounting hole, 6, an armature mounting surface, 7, a driving module, 8, a supporting structure, 9, a workbench, 10, a sensor mounting seat and 11, a sensor mounting hole are arranged.
Detailed Description
The present invention is described in further detail below with reference to the attached drawing figures.
The structure of the double-shaft flexible guide mechanism for the nanometer positioning platform is shown in figures 1 and 5, and comprises a fixed frame 1, wherein the middle part of the frame 1 is a workbench base 3, and the middle parts of the fixed frame 1 and the workbench base are connected by 8 same sine-shaped flexible hinges which are symmetrically distributed. The fixing frame 1 is provided with a frame mounting hole 4 for mounting and positioning. The table base 3 has a table mounting hole 5 for connecting to a table 9.
The working principle of the invention is as follows: referring to fig. 5, the fixed frame 1 is first bolted to the corresponding support structure 8 through the frame mounting holes 4, then the table 9 is mounted on the table base 3 through the table mounting holes 5, then the armature is connected to the armature mounting surface 6 by gluing, and finally the sensor mounting seat 10 is bolted to the fixed frame 1, and the sensor mounting seat 10 has sensor mounting holes 11. Thus, when the driving module 7 (specifically, the normal force electromagnetic driving module, which is used for generating maxwell electromagnetic force to generate an effect on the armature) acts on the armature with a driving force, the armature will drive the table base 3 to move together, that is, the free end of the sine-shaped flexible hinge will be influenced by the acting force of the driving module 7 to generate elastic deformation. Different from the traditional double-shaft mechanism, the free end and the fixed end of the sine-shaped flexible hinge are respectively connected to the workbench base 3 and the fixed frame 1, when the sine-shaped flexible hinge is driven, axial extension and/or compression and plane bending can occur simultaneously, 8 identical flexible hinges 2 are used for forming two groups of mutually perpendicular parallelogram structures, good two-shaft decoupling motion can be achieved, and unwanted additional motion is restrained. Therefore, the working performance of the sine-shaped flexible hinge can be fully utilized, and the problem that the size structure and the movement quality of a traditional flexible mechanism are overlarge due to the influence of single degree of freedom of a common flexible hinge is solved.
The rigidity is an important parameter of the flexible guide mechanism, and when the driving force is constant, the size of the flexible guide mechanism directly determines the working stroke of the positioning platform. In order to provide a reference model for the optimal design of the flexible guide mechanism, the invention establishes a flexibility matrix of the sine-shaped flexible hinge.
The flexible mechanism rigidity modeling method comprises the following steps:
the method comprises the following steps: determining the functional expression of the sinusoidal curve formed by the central line of the single sinusoidal flexible hinge 2:
a rectangular coordinate system o is established at the center point of the left fixed end of the sine flexible hinge in the manner shown in FIG. 3 0 -x 0 y 0 z 0 Wherein A, p x And n are used to represent the amplitude of the sinusoid, the length of one cycle, and the number of cycles of one sinusoidal hinge, respectively. The functional expression of the sine curve formed by the center line of the sine flexible hinge is
Figure BDA0002965960850000061
Step two: determining the normal stress and the shear stress at any position of a single period:
selecting a periodic sine flexible hinge x ∈ [0, p ] x ]Its free end load is F ═ F x ,f y ,f z ,m x ,m y ,m z ] T Wherein f is x ,f y ,f z Each representing a free end x of a periodic flexible hinge 0 ,y 0 ,z 0 Force m received in the axial direction x ,m y ,m z Each representing a free end x of a periodic flexible hinge 0 ,y 0 ,z 0 The free end is deformed into u ═ u by the torque applied in the axial direction x ,u y ,u z ,θ x ,θ y ,θ z ] T Wherein u is x ,u y ,u z Each representing a periodic flexible hinge free end x 0 ,y 0 ,z 0 Free end displacement in axial direction, theta x ,θ y ,θ z Each representing a free end x of a periodic flexible hinge 0 ,y 0 ,z 0 The rotation angle in the axial direction is in any x 0 Bending moment M at location b Comprises the following steps:
M b ＝[M xy ,M xz ,M yz ]
wherein
Figure BDA0002965960850000071
Wherein M is xy ，M xz ，M yz Respectively denote x at an arbitrary x position 0 y 0 ，x 0 z 0 ，y 0 z 0 Bending moment received in plane, at arbitrary x 0 The plane normal force N and shear force S at the positions are expressed as follows:
Figure BDA0002965960850000072
wherein
Figure BDA0002965960850000073
Step three: determining elastic strain energy of a single cycle of a flexible hinge:
according to the second theorem of clamping, the elastic strain energy U due to deformation is expressed as
Figure BDA0002965960850000074
Wherein
Figure BDA0002965960850000075
Is the second moment of inertia of the sinusoidal flexural hinge cross section, and A c Wt is the cross-sectional area of the sinusoidal flexural hinge, where w represents the width of the flexural hinge as shown in fig. 4 and t represents the thickness of the flexural hinge as shown in fig. 3. In addition, E represents the Young's modulus of the material,
Figure BDA0002965960850000076
is the corresponding shear modulus, mu represents the Poisson's ratio of the material, alpha t Representing the shear coefficient of a rectangular cross section, L is the actual length of a single period flexible hinge in the direction of the sinusoid.
Step four: determining the deformation amount of the flexible hinge in a single period:
the space flexibility expression of the sine flexible hinge obtained according to the second theorem of the clamping type is as follows
Figure BDA0002965960850000077
Wherein
Figure BDA0002965960850000081
The relationship between single cycle hinge deflection and end load is therefore: u ═ C s F
Step five: determining spatial compliance matrix for a single sinusoidal compliant hinge
The single sine flexible hinge can be regarded as n (taking 3 here) periodic flexible hinge units which are connected in series, and the space flexibility matrix of the first random sine flexible hinge can be obtained according to the space position conversion method of the flexibility matrix
Figure BDA0002965960850000082
Wherein T is i Is from the coordinate system o i -x i y i z i To o c1 -x c1 y c1 z 0c1 Position conversion matrix of o c1 -x c1 y c1 z 0c1 Point o being the free end coordinate system of the flexible hinge c1 Global origin of coordinates representing a single sinusoidal flexible hinge, at the free end, point o i Local origin of coordinates representing the ith flexible hinge, shown as a single cycle in FIG. 3The geometric positional relationship between the sinusoidal compliance hinge and the single sinusoidal compliance hinge is determined and is represented by
Figure BDA0002965960850000083
Wherein R is i (α i ) Is from the coordinate system o i -x i y i z i To o c1 -x c1 y c1 z 0c1 Rotation matrix of S i (r i ) Is r of i Of an angle of rotation alpha around the z-axis i ，R i (α i ) Is defined as
Figure BDA0002965960850000084
r i ＝[x i ,y i ,z i ]Is in a coordinate system o c1 -x c1 y c1 z 0c1 Point o of (1) i Position of (S) i (r i ) Is given by
Figure BDA0002965960850000085
Since the flexible single-period flexible hinges are in series relation in the single sinusoidal flexible hinge, the position of the ith period flexible hinge in the coordinate system o can be obtained from fig. 3 c1 -x c1 y c1 z 0c1 Can be represented as r i ＝[-ip x ,0,0]And a rotation angle alpha around the z-axis i ＝0。
Step six: determining stiffness of an entire dual-axis flexible guide mechanism
The double-shaft flexible guide mechanism is formed by connecting eight parallel sine-shaped flexible hinges in parallel, and the flexible guide mechanisms are symmetrical about x and y axes, so that when the rigidity of the center of the mechanism is calculated according to a space position conversion method of a flexibility matrix, the rigidity of the center of the mechanism can be calculated by only half of the hinge, and then the final rigidity is twice of that of the center of the mechanism, so that the rigidity at the center of the mechanism can be expressed by
Figure BDA0002965960850000091
At this time T therein j Is at an angle around the z-axis of
Figure BDA0002965960850000092
I.e. the free end coordinate system o of the flexible hinge shown in fig. 2, which participates in the calculation cj -x cj y cj z cj (j represents the number of sets of flexible hinges, j is 1,2,3,4) to o-xyz (coordinate system at the center of the table base).
Although the table base, rigidly connected to the flexible hinge, has some deformation during movement, this has little effect on the stiffness of the planar movement. Thus, when modeling the x and y direction compliance, it can be considered a rigid body. Finally, considering the symmetric properties of the biaxial flexible guide mechanism about the x and y axes, the stiffness in the x and y directions at the center of the mechanism can be expressed as k x ＝k y ＝K(1,1)＝K(2,2)。
The foregoing illustrates and describes the principles, general features, and advantages of the present invention. It will be understood by those skilled in the art that the present invention is not limited to the embodiments described above, which are described in the specification and illustrated only to illustrate the principle of the present invention, but that various changes and modifications may be made therein without departing from the spirit and scope of the present invention, which fall within the scope of the invention as claimed. The scope of the invention is defined by the appended claims and equivalents thereof.

Claims (8)

1. A double-shaft flexible guide mechanism for a nanometer positioning platform comprises a fixed frame (1), a plurality of flexible hinges (2) and a workbench base (3), and is characterized in that,
the workbench base (3) is located in the middle of the fixed frame (1), at least one flexible hinge (2) is connected between the four outer side faces of the workbench base (3) and the inner side face of the fixed frame (1) respectively, and the flexible hinge (2) is a sine-shaped flexible hinge.
2. The biaxial flexible guide mechanism for the nanopositioning platform of claim 1 further comprising a drive module (7), a support structure (8) and a table (9), the bottom of the fixed frame (1) being connected to the support structure (8), the top of the table base (3) being connected to the table (9).
3. The biaxial flexible guide mechanism for the nanopositioning platform of claim 2, wherein the fixed frame (1) comprises a plurality of frame mounting holes (4), the table base (3) comprises a plurality of table mounting holes (5), the bottom of the fixed frame (1) is connected to the support structure (8) by bolts, and the top of the table base (3) is connected to the table (9) by bolts.
4. The biaxial flexible guide mechanism for the nanopositioning platform of claim 2 further comprising an armature, wherein the bottom surface of the table base (3) is an armature mounting surface (6), and the armature is connected with the armature mounting surface (6) by gluing.
5. The biaxial flexible guide mechanism for the nanometer positioning platform as set forth in claim 2, further comprising a sensor mounting seat (10), wherein the sensor mounting seat (10) is mounted on the top of the worktable base (3), and a sensor mounting hole (11) is provided on the sensor mounting seat (10).
6. The biaxial flexible guide mechanism for the nanopositioning platform of any one of claims 1 to 5, wherein the free end of the flexible hinge (2) is connected to the workbench base (3) and the fixed end of the flexible hinge (2) is connected to the fixed frame (1).
7. The biaxial flexible guide mechanism for the nano positioning platform according to claim 6, characterized in that the number of the flexible hinges (2) is eight, the eight flexible hinges (2) are divided into four groups and symmetrically arranged on four outer sides of the workbench base (3), and each group of flexible hinges (2) is perpendicular to the adjacent group of flexible hinges (2).
8. The stiffness modeling method for the biaxial flexible guiding mechanism of the nanopositioning platform of claim 7, comprising the steps of:
the method comprises the following steps: determining a functional expression of a sinusoid formed by the centre line of the single sinusoidal flexible hinge (2):
a rectangular coordinate system o is established at the central point of the fixed end of the sine flexible hinge 0 -x 0 y 0 z 0 Therein A, p x And n is used to represent the amplitude of the sinusoid, the length of one period, and the number of periods of one sinusoidal flexible hinge, respectively, the functional expression of the sinusoid formed by the center line of the sinusoidal flexible hinge is:
Figure FDA0002965960840000021
step two: determining the normal stress and the shear stress of any position of a single-period sine curve:
selecting a periodic sine flexible hinge x ∈ [0, p ] x ]Its free end load is F ═ F x ，f y ，f z ，m x ，m y ，m z ] T Wherein f is x ，f y ，f z Each representing a free end x of a periodic flexible hinge 0 ，y 0 ，z 0 Force applied in the axial direction, m x ，m y ，m z Each representing a free end x of a periodic flexible hinge 0 ，y 0 ，z 0 The free end is deformed into u ═ u by the torque applied in the axial direction x ，u y ，u z ，θ x ，θ y ，θ z ] T Wherein u is x ，u y ，u z Each representing a free end x of a periodic flexible hinge 0 ，y 0 ，z 0 Free end displacement in axial direction, theta x ，θ y ，θ z Each representing a periodic flexible hinge free end x 0 ，y 0 ，z 0 The rotation angle in the axial direction is in any x 0 Bending moment M at location b Comprises the following steps:
M b ＝[M xy ，M xz ，M yz ]
wherein,
Figure FDA0002965960840000022
wherein M is xy ，M xz ，M yz Each is represented at an arbitrary x 0 At position x 0 y 0 ，x 0 z 0 ，y 0 z 0 Bending moment received in plane, at arbitrary x 0 The plane normal force N and shear force S at the positions are expressed as follows:
Figure FDA0002965960840000023
wherein,
Figure FDA0002965960840000024
step three: determining elastic strain energy of the flexible hinge in a single cycle:
according to the second theorem of clamping, the elastic strain energy U due to deformation is expressed as
Figure FDA0002965960840000025
Wherein
Figure FDA0002965960840000026
Is the second moment of inertia, A, of the sinusoidal flexible hinge cross-section c Where w represents the width of the flexural hinge, t represents the thickness of the flexural hinge, E represents the young's modulus of the material,
Figure FDA0002965960840000027
is the corresponding shear modulus, mu represents the Poisson's ratio of the material, alpha s Represents the shear coefficient of a rectangular cross section, L being the actual length of a single period flexible hinge in the direction of the sinusoid;
step four: determining the deformation amount of the flexible hinge in a single period:
the space flexibility expression of the sine-shaped flexible hinge obtained according to the second theorem of the clamping type is as follows
Figure FDA0002965960840000031
Wherein
Figure FDA0002965960840000032
The relationship between single cycle compliant hinge deflection and end load is therefore: u ═ C s F；
Step five: determining a spatial compliance matrix for a single sinusoidal compliant hinge:
the single sine flexible hinge is formed by connecting n periodic flexible hinge units in series, and a space flexibility matrix of the random first sine flexible hinge is obtained according to a space position conversion method of the flexibility matrix
Figure FDA0002965960840000033
Wherein T is i Is from the coordinate system o i -x i y i z i To o c1 -x c1 y c1 z 0c1 Position conversion matrix of o c1 -x c1 y c1 z 0c1 Point o for the free end coordinate system of the flexible hinge c1 Global origin of coordinates representing a single sinusoidal flexible hinge, at the free end, point o i Local origin of coordinates, T, representing the ith periodic flexure hinge i Is shown as
Figure FDA0002965960840000034
Wherein R is i (α i ) Is from the coordinate system o i -x i y i z i To o c1 -x c1 y c1 z 0c1 Of the rotation matrix S i (r i ) Is r of i Has a rotation angle alpha around the z-axis i ，R i (α i ) Is defined as
Figure FDA0002965960840000035
r i ＝[x i ，y i ，z i ]Is in a coordinate system o c1 -x c1 y c1 z 0c1 Point o of (1) i Position of (S), S i (r i ) Is given by
Figure FDA0002965960840000036
Since the single periodic flexure hinges are in series relationship in the single sinusoidal flexure hinge, the ith periodic flexure hinge is located in the coordinate system o c1 -x c1 y c1 z 0c1 In is represented by r i ＝[-ip x ，0，0]And a rotation angle alpha around the z-axis i ＝0；
Step six: determining the rigidity of the whole double-shaft flexible guide mechanism:
stiffness at the center of the mechanism is expressed as
Figure FDA0002965960840000041
Wherein T is j Is at an angle around the z-axis of
Figure FDA0002965960840000042
The position transformation matrix of (1), i.e. the free end coordinate system o of the flexible hinge involved in the calculation cj -x cj y cj z cj A position conversion matrix to o-xyz, j represents the number of sets of flexible hinges, j is 1,2,3,4, o-xyz is a coordinate system at the center of the table base,
in modeling the compliance in the x and y directions, considering the table base as a rigid body, the stiffness in the x and y directions at the center of the mechanism is represented as k in consideration of the characteristics of the biaxial flexible guide mechanism symmetrical about the x and y axes x ＝k y ＝K(1，1)＝K(2，2)。
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