CN106383963A - Branch identification method for spherical six-bar linkage mechanism - Google Patents

Abstract

The present invention discloses a branch identification method for a spherical six-bar linkage mechanism. The method comprises: carrying out analysis on two rings interacted through the spherical six-bar linkage mechanism, and according to a dead point and a branch point of the interaction to the mechanism, determining branches of the entire mechanism. According to the method disclosed by the present invention, by using of the input and output curve of the mechanism and the input and output joint rotation space, branch(loop) identification is realized, and the identification is more accurate, intuitive and efficient; the method provided by the present invention facilitates computer programming simulation, is helpful for mechanical design, and has a high use value; and the method provided by the present invention can be embedded in a plurality of mechanical design business software, and has a good social value and an economic value.

Description

A kind of linear-elsatic buckling method of sphere six bar mechanism

Technical field

The invention belongs to Machine Design manufacturing technology field, it is related to a kind of linear-elsatic buckling side to sphere six bar mechanism Method.

Background technology

For sphere six bar mechanism structure, sphere six bar mechanism is that a kind of of spacing connecting rod mechanism simplifies shape Formula, the extended line of the wherein centrage of the rotary joint of sphere six bar mechanism meet at spherical space a bit.Up to the present, There are a lot of scholars that sphere six bar mechanism is studied.Mccarthy et al. develops on the basis of Burmester theory A kind of computer software is analyzed to sphere double leval jib.Zhang Ji, Lin Guangchun et al. are based on Groebner base method to adjustable ball The position of face 3-freedom parallel mechanism has carried out analyzing and has been made that dynamic simulation.Sun Yang, Yang Suixian et al. will Stephenson sphere six bar mechanism regards sphere quadric chain and the combination of two bar groups as, to Stephenson six-bar linkage The Grashof's criterion of mechanism is probed into.Sancisi, N et al. propose two degrees of freedom Sphere Measurement Model verification method to people Class ankle joint has carried out kinematics analyses.These researchs are propped up to studying sphere six bar mechanism herein and providing certain theory Hold, however, appeal method is all under specific circumstances spherical linkage mechanisms to be analyzed, object of study is relatively single, propose The general feasibility of method relatively inadequate.

Content of the invention

In order to solve above-mentioned technical problem, the present invention proposes a kind of branch of sphere six bar mechanism of simple possible Method of discrimination, will be combined with two loops of input, output, be inputted, the dependent equation of output angle, thus to sphere six The branch of linkage is differentiated.

The technical solution adopted in the present invention is：A kind of sphere six bar mechanism linear-elsatic buckling method it is characterised in that：Logical Two rings crossing the interaction of sphere six bar mechanism are analyzed, the dead point to mechanism being interacted according to it and branch Point, and judge the branch of whole mechanism.

Preferably, implementing of the present invention comprises the following steps：

Step 1：The foundation of Spherical Ring equation；

Define the coordinate system of each bar, set up spherical mechanism mathematical model；The joint rotary shaft defining each rod member is Z axis, X Rotate rule with Y-axis by the right hand to be determined；The bar that these are connected two coordinate systems, is named as border rotation；Along joint Z Axle rotates, and is named as central rotation；Then the Spherical Ring equation of standard is expressed as follows：

Z₁S₁Z₂S₂…Z_k-1S_k-1Z_kS_k=I； (1)

Wherein, k=1,2 ... 7, i=1,2 ... k, Z_iRepresent central rotation, S_iFor border rotation, Z_iAnd S_iBe all 3 × 3 spin matrixs, I is an eigenmatrix；

Anglec of rotation θ according to joint_i, Z_iCan be expressed as follows：

$Z_i=\left[\begin{array}{ccc}{\mathrm{cos\θ}}_i& -{\mathrm{sin\θ}}_i& 0\\ {\mathrm{sin\θ}}_i& {\mathrm{cos\θ}}_i& 0\\ 0& 0& 1\end{array}\right]---\left(2\right)$

Represent that the radian of rod member is long using the corresponding central angle of rod member, border rotates S_iFor：

$S_i=\left[\begin{array}{ccc}1& 0& 0\\ 0& {\mathrm{cos\α}}_i& -{\mathrm{sin\α}}_i\\ 0& {\mathrm{sin\α}}_i& {\mathrm{cos\α}}_i\end{array}\right]---\left(3\right)$

Define z=[0 0 1]^T, because z^TZ_i=z^TAnd Z_iZ=z, equation (1) has,

$z^T{S}_1{Z}_2\mathrm{...}{Z}_{k-1}{S}_{k-1}z=z^T{S}_k^{T}z---\left(4\right)$

Step 2：Obtain sphere quadric chain input-output equation；

According to the construction featuress of sphere six bar mechanism, set up cartesian coordinate system, O point is the center of sphere, X-axis Perpendicular to plane (O, α₁₂), Z axis are exactly axle OB₀, Y-axis is in plane (O, α₁₂) in；In such coordinate system, sphere six bar mechanism Comprise sphere quadric chain A₀ABB₀With a sphere five-bar mechanism A₀ADCC₀B₀；

First to quadric chain A₀ABB₀It is analyzed, give input joint A₀It is respectively input and output joint with A, Ring establishing equation is as follows：

Z₂S₂Z₁S₁Z₄S₄Z₃S₃=I (5)

Equation (5) makes (6) into：

$z^T{S}_2{Z}_1{S}_1{Z}_4{S}_{4}z=z^T{S}_3^{T}z---\left(6\right)$

Equation (6) expands into：

sinα₁₂sinθ₁sin(θ₄+β)sinα₄₃-cosα₁₂sinα₁₄cos(θ₄+β)sinα₄₃

-sinα₁₂cosθ₁sinα₁₄cosα₄₃-sinα₁₂cosθ₁cosα₁₄cos(θ₄+β)sinα₄₃

+cosα₁₄cosα₁₂cosα₄₃-cosα₃₂=0 (7)

Input and output θ is only contained in equation (6) and equation (7)₁And θ₄.

According to half-angle formulas, make x₄=tan (θ₄/ 2), then Bring in equation (7), then equation (7) is represented by：

$a_1{x}_4^2+b_1{x}_4+c_1=0---\left(8\right)$

Wherein：

a₁=sin α₁₂cosθ₁cosα₁₂sinα₄₃cosβ+cosα₁₂sinα₁₄sinα₄₃cosβ

-sinα₁₂sinθ₁sinα₄₃sinβ-sinα₁₂cosθ₁sinα₁₄cosα₄₃

+cosα₁₄cosα₁₂cosα₄₃-cosα₃₂(9.1)

b₁=2sin α₁₂cosθ₁cosα₁₄sinα₄₃sinβ+2sinα₁₂sinθ₁sinα₄₃cosβ

+2cosα₁₂sinα₁₄sinα₄₃sinβ (9.2)

c₁=sin α₁₂sinθ₁sinα₄₃sinβ-sinα₁₂cosθ₁sinα₁₄cosα₄₃

+cosα₁₄cosα₁₂cosα₄₃-sinα₁₂cosθ₁cosα₁₄sinα₄₃

-cosα₁₂sinα₁₄sinα₄₃cosβ-cosα₃₂(9.3)

Here a1, b1, c1 are containing θ₁Undetermined coefficient；

Work as a₁When ≠ 0, the discriminant of equation (8) must is fulfilled for：

${\mathrm{\Δ}}_1=b_1^2-4a_1{c}_1\≥0---\left(10\right)$

Work as △₁When=0, represent that quadric chain is in dead-centre position；

According to equation (10), by input angle θ₁Obtain output angle θ₄：

$x_{4\[1\]}=\frac{-b_1-\sqrt{{\mathrm{\Δ}}_1}}{2a_1},{\mathrm{\θ}}_4=2\arctan x_{4\[1\]}---\left(11.1\right)$

$x_{4\[2\]}=\frac{-b_1+\sqrt{{\mathrm{\Δ}}_1}}{2a_1},{\mathrm{\θ}}_4=2\arctan x_{4\[2\]}---\left(11.2\right)$

Step 3：The joint revolution space of sphere five-bar mechanism；

Sphere six bar mechanism also comprises a sphere five-bar mechanism simultaneously, for sphere five-bar mechanism A₀ADCC₀B₀, the input-output curve of five-bar mechanism can be obtained：

Z₅S₅Z₂S₂Z₁S₁Z₄₁S₄₁Z₇S₇Z₆S₆=I (12)

Equation (12) is rewritten as：

$z^T{S}_5{Z}_2{S}_2{Z}_1{S}_1{Z}_{41}{S}_{41}{Z}_7{S}_{7}z=z^T{S}_6^{T}z---\left(13\right)$

Equation (13) comprises three variable θ₁、θ₄And θ₇；

Using half-angle formulas, make x₇=tan (θ₇/ 2), Equation (13) is rewritten as：

$a_2({\mathrm{\θ}}_1,{\mathrm{\θ}}_4)x_7^2+b_2({\mathrm{\θ}}_1,{\mathrm{\θ}}_4)x_7+c_2({\mathrm{\θ}}_1,{\mathrm{\θ}}_4)=0---\left(14\right)$

Variable θ in equation (14) to be made₇There is solution, the discriminant in equation (14) should meet following condition：

△₂=f (θ₁,θ₄)≥0 (15)

Inequality (15) illustrates rotary joint θ in sphere six bar mechanism₁And θ₄The maximum magnitude of motion, i.e. θ₁And θ₄ Joint revolution space；Work as △₂When=0, i.e. the sideline of joint revolution space, is also the position that the singular point of mechanism occurs；

Step 4：Bifurcation Analysis；

Whether branch point is contained according to mechanism's linear-elsatic buckling in figure, the branch point identification in figure of sphere six bar mechanism whether Bifurcation Analysis containing branch point sphere six bar mechanism are segmented into the following two kinds form：

Class1：Sphere six bar mechanism linear-elsatic buckling in figure no branch point exists, and the joint revolving property of this kind of mechanism is only Acted on by sphere quadric chain；Sphere quadric chain A₀ABB₀Joint revolution space JRS-L completely in sphere five connecting rod Mechanism A₀ADCC₀B₀Joint revolution space JRS-R in；The Bifurcation Analysis of this kind of sphere six bar mechanism can be summarized as following step Suddenly:

Branch's point analysiss：Verify whether containing branch point by joint equation (7) and equation (15), wherein equation (15) is Meet the situation of △=0；

Joint revolution space：By drawing the joint rotation of sphere quadric chain input-output curve and five-bar mechanism Space, obtains its common portion, verifies whether it is to comprise whole quadric chain curve；

Branch curve is analyzed：The boundary curve of analysis effectively public joint revolution space, be respectively designated as Ai, Bi, Ci and Di, i=1,2..., Ai and Bi are JRS-L sideline, Ci and Di is JRS-R sideline；

Which branch Bifurcation Analysis, belonged in by mechanism's place branch curve decision mechanism；

Type 2：Sphere six bar mechanism linear-elsatic buckling in figure is subject to the presence of branch point, the joint revolving property of this kind of mechanism Sphere quadric chain and the collective effect of five-bar mechanism；Branch's situation of sphere six bar mechanism is subject to sphere quadric chain A₀ABB₀With sphere five-bar mechanism A₀ADCC₀B₀Collective effect, sphere six bar mechanism quadric chain chain A₀ABB₀Joint Revolution space JRS-L is intersected with JRS-R and is separated by, and forms different branches, is branched a cut-off between each branch, and this kind of sphere six is even The Bifurcation Analysis of linkage also can be summarized as following steps：

(1) branch's point analysiss：By joint equation (7) and equation (15), the concrete condition of branch point, wherein equation can be obtained (15) it is the situation meeting △=0；

(2) joint revolution space：By drawing the joint revolution space of sphere quadric chain curve and five-bar mechanism, Obtain its common portion；

(3) branch curve analysis：The boundary curve of analysis effectively public joint revolution space, is respectively designated as Ai, Bi, Ci And Di, i=1,2..., Ai and Bi are JRS-L sideline, Ci and Di is JRS-R sideline, branch point is numbered i simultaneously；

(4) which branch Bifurcation Analysis, belonged in by mechanism's place branch curve decision mechanism.

The beneficial effect of patent of the present invention：

1) present invention proposes a kind of sphere six bar mechanism branch and carries out the new method of automatic identification, and the method utilizes machine The input-output curve of structure and input and output joint revolution space realize branch (loop) is identified, identification is more accurate, directly perceived, high Effect；

2) method that the present invention provides is easy to computer programming emulation, contributes to Machine Design, has very high use valency Value；

3) method that the present invention provides can be embedded in various Machine Design class business softwares, has good social value And economic worth.

Brief description

Fig. 1：The flow chart of the embodiment of the present invention；

Fig. 2：The cartesian coordinate system schematic diagram of the sphere six bar mechanism of the embodiment of the present invention；

Fig. 3：The sphere six bar mechanism linear-elsatic buckling schematic diagram without branch point of the embodiment of the present invention；

Fig. 4：The sphere six bar mechanism linear-elsatic buckling schematic diagram containing branch point of the embodiment of the present invention.

Specific embodiment

Understand for the ease of those of ordinary skill in the art and implement the present invention, below in conjunction with the accompanying drawings and embodiment is to this Bright be described in further detail it will be appreciated that described herein enforcement example be merely to illustrate and explain the present invention, not For limiting the present invention.

Ask for an interview Fig. 1, a kind of linear-elsatic buckling method of sphere six-bar linkage that the present invention provides, will be defeated for the input of sphere six-bar linkage Going out curve and input and output joint revolution space, obtaining dead point and the branch point of sphere six bar mechanism, thus differentiating sphere Effective branch of six bar mechanism.

Implementing of the present invention comprises the following steps：

Step 1：The foundation of Spherical Ring equation；

Define the coordinate system of each bar, set up spherical mechanism mathematical model；The joint rotary shaft defining each rod member is Z axis, X Rotate rule with Y-axis by the right hand to be determined；The bar that these are connected two coordinate systems, is named as border rotation；Along joint Z Axle rotates, and is named as central rotation；Then the Spherical Ring equation of standard is expressed as follows：

Z₁S₁Z₂S₂…Z_k-1S_k-1Z_kS_k=I； (1)

Wherein, k=1,2 ... 7, i=1,2 ... k, Z_iRepresent central rotation, S_iFor border rotation, Z_iAnd S_iBe all 3 × 3 spin matrixs, I is an eigenmatrix；

Anglec of rotation θ according to joint_i, Z_iCan be expressed as follows：

$Z_i=\left[\begin{array}{ccc}{\mathrm{cos\θ}}_i& -{\mathrm{sin\θ}}_i& 0\\ {\mathrm{sin\θ}}_i& {\mathrm{cos\θ}}_i& 0\\ 0& 0& 1\end{array}\right]---\left(2\right)$

Represent that the radian of rod member is long using the corresponding central angle of rod member, border rotates S_iFor：

$S_i=\left[\begin{array}{ccc}1& 0& 0\\ 0& {\mathrm{cos\α}}_i& -{\mathrm{sin\α}}_i\\ 0& \sin L& {\mathrm{cos\α}}_i\end{array}\right]---\left(3\right)$

Define z=[0 0 1]^T, because z^TZ_i=z^TAnd Z_iZ=z, equation (1) has,

$z^T{S}_1{Z}_2\mathrm{...}{Z}_{k-1}{S}_{k-1}z=z^T{S}_k^{T}z---\left(4\right)$

Step 2：Obtain sphere quadric chain input-output equation；

According to the construction featuress of sphere six bar mechanism, set up cartesian coordinate system, O point is the center of sphere, X-axis Perpendicular to plane (O, α₁₂), Z axis are exactly axle OB₀, Y-axis is in plane (O, α₁₂) in；In such coordinate system, sphere six bar mechanism Comprise sphere quadric chain A₀ABB₀With a sphere five-bar mechanism A₀ADCC₀B₀；

First to quadric chain A₀ABB₀It is analyzed, give input joint A₀It is respectively input and output joint with A, Ring establishing equation is as follows：

Z₂S₂Z₁S₁Z₄S₄Z₃S₃=I (5)

Equation (5) makes (6) into：

$z^T{S}_2{Z}_1{S}_1{Z}_4{S}_{4}z=z^T{S}_3^{T}z---\left(6\right)$

Equation (6) expands into：

sinα₁₂sinθ₁sin(θ₄+β)sinα₄₃-cosα₁₂sinα₁₄cos(θ₄+β)sinα₄₃

-sinα₁₂cosθ₁sinα₁₄cosα₄₃-sinα₁₂cosθ₁cosα₁₄cos(θ₄+β)sinα₄₃

+cosα₁₄cosα₁₂cosα₄₃-cosα₃₂=0 (7)

Input and output θ is only contained in equation (6) and equation (7)₁And θ₄.

According to half-angle formulas, make x₄=tan (θ₄/ 2), then Bring in equation (7), then equation (7) is represented by：

$a_1{x}_4^2+b_1{x}_4+c_1=0---\left(8\right)$

Wherein：

a₁=sin α₁₂cosθ₁cosα₁₂sinα₄₃cosβ+cosα₁₂sinα₁₄sinα₄₃cosβ

-sinα₁₂sinθ₁sinα₄₃sinβ-sinα₁₂cosθ₁sinα₁₄cosα₄₃

+cosα₁₄cosα₁₂cosα₄₃-cosα₃₂(9.1)

b₁=2sin α₁₂cosθ₁cosα₁₄sinα₄₃sinβ+2sinα₁₂sinθ₁sinα₄₃cosβ

+2cosα₁₂sinα₁₄sinα₄₃sinβ (9.2)

c₁=sin α₁₂sinθ₁sinα₄₃sinβ-sinα₁₂cosθ₁sinα₁₄cosα₄₃

+cosα₁₄cosα₁₂cosα₄₃-sinα₁₂cosθ₁cosα₁₄sinα₄₃

-cosα₁₂sinα₁₄sinα₄₃cosβ-cosα₃₂(9.3)

Here a₁、b₁、c₁It is containing θ₁Undetermined coefficient；

Work as a₁When ≠ 0, the discriminant of equation (8) must is fulfilled for：

${\mathrm{\Δ}}_1=b_1^2-4a_1{c}_1\≥0---\left(10\right)$

Work as △₁When=0, represent that quadric chain is in dead-centre position；

According to equation (10), by input angle θ₁Obtain output angle θ₄：

$x_{4\[1\]}=\frac{-b_1-\sqrt{{\mathrm{\Δ}}_1}}{2a_1},{\mathrm{\θ}}_4=2\arctan x_{4\[1\]}---\left(11.1\right)$

$x_{4\[2\]}=\frac{-b_1+\sqrt{{\mathrm{\Δ}}_1}}{2a_1},{\mathrm{\θ}}_4=2\arctan x_{4\[2\]}---\left(11.2\right)$

Step 3：The joint revolution space of sphere five-bar mechanism；

Sphere six bar mechanism also comprises a sphere five-bar mechanism simultaneously, for sphere five-bar mechanism A₀ADCC₀B₀, the input-output curve of five-bar mechanism can be obtained：

Z₅S₅Z₂S₂Z₁S₁Z₄₁S₄₁Z₇S₇Z₆S₆=I (12)

Equation (12) is rewritten as：

$z^T{S}_5{Z}_2{S}_2{Z}_1{S}_1{Z}_{41}{S}_{41}{Z}_7{S}_{7}z=z^T{S}_6^{T}z---\left(13\right)$

Equation (13) comprises three variable θ₁、θ₄And θ₇；

Using half-angle formulas, make x₇=tan (θ₇/ 2), Equation (13) is rewritten as：

$a_2({\mathrm{\θ}}_1,{\mathrm{\θ}}_4)x_7^2+b_2({\mathrm{\θ}}_1,{\mathrm{\θ}}_4)x_7+c_2({\mathrm{\θ}}_1,{\mathrm{\θ}}_4)=0---\left(14\right)$

Variable θ in equation (14) to be made₇There is solution, the discriminant in equation (14) should meet following condition：

△₂=f (θ₁,θ₄)≥0 (15)

Inequality (15) illustrates rotary joint θ in sphere six bar mechanism₁And θ₄The maximum magnitude of motion, i.e. θ₁And θ₄ Joint revolution space；Work as △₂When=0, i.e. the sideline of joint revolution space, is also the position that the singular point of mechanism occurs；

Step 4：Bifurcation Analysis；

Whether branch point is contained according to mechanism's linear-elsatic buckling in figure, the branch point identification in figure of sphere six bar mechanism whether Bifurcation Analysis containing branch point sphere six bar mechanism are segmented into the following two kinds form：

Class1：Sphere six bar mechanism linear-elsatic buckling in figure no branch point exists, and the joint revolving property of this kind of mechanism is only Acted on by sphere quadric chain；Sphere quadric chain A₀ABB₀Joint revolution space JRS-L completely in sphere five connecting rod Mechanism A₀ADCC₀B₀Joint revolution space JRS-R in；The Bifurcation Analysis of this kind of sphere six bar mechanism can be summarized as following step Suddenly:

Branch's point analysiss：Verify whether containing branch point by joint equation (7) and equation (15), wherein equation (15) is Meet the situation of △=0；

Joint revolution space：By drawing the joint rotation of sphere quadric chain input-output curve and five-bar mechanism Space, obtains its common portion, verifies whether it is to comprise whole quadric chain curve；

Branch curve is analyzed：The boundary curve of analysis effectively public joint revolution space, be respectively designated as Ai, Bi, Ci and Di, i=1,2..., Ai and Bi are JRS-L sideline, Ci and Di is JRS-R sideline；

Which branch Bifurcation Analysis, belonged in by mechanism's place branch curve decision mechanism；

Type 2：Sphere six bar mechanism linear-elsatic buckling in figure is subject to the presence of branch point, the joint revolving property of this kind of mechanism Sphere quadric chain and the collective effect of five-bar mechanism；Branch's situation of sphere six bar mechanism is subject to sphere quadric chain A₀ABB₀With sphere five-bar mechanism A₀ADCC₀B₀Collective effect, sphere six bar mechanism quadric chain chain A₀ABB₀Joint Revolution space JRS-L is intersected with JRS-R and is separated by, and forms different branches, is branched a cut-off between each branch, and this kind of sphere six is even The Bifurcation Analysis of linkage also can be summarized as following steps：

(1) branch's point analysiss：By joint equation (7) and equation (15), the concrete condition of branch point, wherein equation can be obtained (15) it is the situation meeting △=0；

(2) joint revolution space：By drawing the joint revolution space of sphere quadric chain curve and five-bar mechanism, Obtain its common portion；

(3) branch curve analysis：The boundary curve of analysis effectively public joint revolution space, is respectively designated as Ai, Bi, Ci And Di, i=1,2..., Ai and Bi are JRS-L sideline, Ci and Di is JRS-R sideline, branch point is numbered i simultaneously；

(4) which branch Bifurcation Analysis, belonged in by mechanism's place branch curve decision mechanism.

Below by way of being embodied as the present invention is further elaborated；

Example 1：The component parameter providing sphere six bar mechanism is as follows：α₁₂=148 °, α₁₄=65 °, α₃₄=65 °, α₂₃= 79 °, β=103 °, α₂₅=135 °, α₄₇=57 °, α₆₇=65 °, α₅₆=85 °.

According to parameter draw sphere six bar mechanism linear-elsatic buckling figure as shown in figure 3, according to above-mentioned steps analyze such as Under：

(1) branch's point analysiss：By simultaneous equation (7) and equation (15) (△₂=0) (verify), such no branch point；

(2) joint revolution space determines：By drawing the joint of sphere parallel motion curve and five-bar mechanism Revolution space, obtains its common portion, i.e. the cross curve of in figure curve and dash area；

(3) branch curve analysis：According to equation (10) (△₁=0) the dead-centre position 1- of sphere six bar mechanism can be obtained (17 °, 268.2 °) and 2- (341.8 °, 239.3 °)；

(4) Bifurcation Analysis：The branch of such mechanism be sphere double leval jib ring Liang Ge branch, respectively branch 1-2 and Branch 2-1.

Example 2：The component parameter of given sphere six bar mechanism is as follows：α₁₂=135 °, α₁₄=85 °, α₃₄=70 °, α₂₃= 85 °, β=103 °, α₂₅=85 °, α₄₇=57 °, α₆₇=75 °, α₅₆=85 °.

According to parameter draw sphere six bar mechanism linear-elsatic buckling figure as shown in figure 4, according to above-mentioned steps analyze such as Under：

(1) branch's point analysiss：By joint equation (7) and (15) (△₂=0), try to achieve branch point 1- (17.0 °, 333.1°)、2-(30.9°,347.6°)、3-(152.7°,181.5°)、4-(177.3°,158.8°)；

(2) joint revolution space determines, by drawing the curve movement of sphere quadric chain and the pass of five-bar mechanism Section revolution space, obtains its common portion, i.e. the cross curve of in figure curve and dash area；

(3) branch curve：Branch curve A1 and B1 is in joint revolution space

(4) Bifurcation Analysis：Sphere six bar mechanism is in sphere quadric chain A₀ABB₀With sphere five-bar mechanism A₀ADCC₀B₀The interaction of joint revolution space mechanism is divided into Liang Ge branch, be respectively comprise branch point 1- (17.0 °, 333.1 °) and 2- (30.9 °, 347.6 °) branch 2-1 and comprise branch point 3- (152.7 °, 181.5 °) and 4- (177.3 °, 158.8 °) branch 4-3.

It should be appreciated that the part that this specification does not elaborate belongs to prior art.

It should be appreciated that the above-mentioned description for preferred embodiment is more detailed, can not therefore be considered to this The restriction of invention patent protection scope, those of ordinary skill in the art, under the enlightenment of the present invention, is weighing without departing from the present invention Profit requires under protected ambit, can also make replacement or deform, each fall within protection scope of the present invention, this Bright scope is claimed should be defined by claims.

Claims (2)

1. a kind of sphere six bar mechanism linear-elsatic buckling method it is characterised in that：Interaction by sphere six bar mechanism Two rings be analyzed, the dead point to mechanism being interacted according to it and branch point, and judge the branch of whole mechanism.

2. sphere six bar mechanism linear-elsatic buckling method according to claim 1 is it is characterised in that comprise the following steps：

Step 1：The foundation of Spherical Ring equation；

Define the coordinate system of each bar, set up spherical mechanism mathematical model；The joint rotary shaft defining each rod member is Z axis, X and Y Axle rotates rule by the right hand and is determined；The bar that these are connected two coordinate systems, is named as border rotation；Along joint Z axis Rotation, is named as central rotation；Then the Spherical Ring equation of standard is expressed as follows：

Z₁S₁Z₂S₂…Z_k-1S_k-1Z_kS_k=I； (1)

Wherein, k=1,2 ... 7, i=1,2 ... k, Z_iRepresent central rotation, S_iFor border rotation, Z_iAnd S_iIt is all 3 × 3 rotations Matrix, I is an eigenmatrix；

Anglec of rotation θ according to joint_i, Z_iCan be expressed as follows：

$Z_i=\left[\begin{array}{ccc}{\mathrm{cos\θ}}_i& -{\mathrm{sin\θ}}_i& 0\\ {\mathrm{sin\θ}}_i& {\mathrm{cos\θ}}_i& 0\\ 0& 0& 1\end{array}\right]---\left(2\right)$

Represent that the radian of rod member is long using the corresponding central angle of rod member, border rotates S_iFor：

$S_i=\left[\begin{array}{ccc}1& 0& 0\\ 0& {\mathrm{cos\α}}_i& -{\mathrm{sin\α}}_i\\ 0& {\mathrm{sin\α}}_i& {\mathrm{cos\α}}_i\end{array}\right]---\left(3\right)$

Define z=[0 0 1]^T, because z^TZ_i=z^TAnd Z_iZ=z, equation (1) has,

$z^T{S}_1{Z}_2...Z_{k-1}{S}_{k-1}z=z^T{S}_k^{T}z---\left(4\right)$

Step 2：Obtain sphere quadric chain input-output equation；

According to the construction featuress of sphere six bar mechanism, set up cartesian coordinate system, O point is the center of sphere, X-axis is vertical In plane (O, α₁₂), Z axis are exactly axle OB₀, Y-axis is in plane (O, α₁₂) in；In such coordinate system, sphere six bar mechanism comprises Sphere quadric chain A₀ABB₀With a sphere five-bar mechanism A₀ADCC₀B₀；

First to quadric chain A₀ABB₀It is analyzed, give input joint A₀It is respectively input and output joint, ring side with A Cheng Jianli is as follows：

Z₂S₂Z₁S₁Z₄S₄Z₃S₃=I (5)

Equation (5) makes (6) into：

$z^T{S}_2{Z}_1{S}_1{Z}_4{S}_{4}z=z^T{S}_3^{T}z---\left(6\right)$

Equation (6) expands into：

$\begin{array}{c}{\mathrm{sin\α}}_{12}{\mathrm{sin\θ}}_1\sin ({\mathrm{\θ}}_4+\mathrm{\β}){\mathrm{sin\α}}_{43}-{\mathrm{cos\α}}_{12}{\mathrm{sin\α}}_{14}\cos ({\mathrm{\θ}}_4+\mathrm{\β}){\mathrm{sin\α}}_{43}\\ -{\mathrm{sin\α}}_{12}{\mathrm{cos\θ}}_1{\mathrm{sin\α}}_{14}{\mathrm{cos\α}}_{43}-{\mathrm{sin\α}}_{12}{\mathrm{cos\θ}}_1{\mathrm{cos\α}}_{14}\cos ({\mathrm{\θ}}_4+\mathrm{\β}){\mathrm{sin\α}}_{43}\\ +{\mathrm{cos\α}}_{14}{\mathrm{cos\α}}_{12}{\mathrm{cos\α}}_{43}-{\mathrm{cos\α}}_{32}=0\end{array}---\left(7\right)$

Input and output θ is only contained in equation (6) and equation (7)₁And θ₄；

According to half-angle formulas, make x₄=tan (θ₄/ 2), then Band Enter in equation (7), then equation (7) is represented by：

$a_1{x}_4^2+b_1{x}_4+c_1=0---\left(8\right)$

Wherein：

$\begin{array}{c}{a}_1={\mathrm{sin\α}}_{12}{\mathrm{cos\θ}}_1{\mathrm{cos\α}}_{12}{\mathrm{sin\α}}_{43}\cos \mathrm{\β}+{\mathrm{cos\α}}_{12}{\mathrm{sin\α}}_{14}{\mathrm{sin\α}}_{43}\cos \mathrm{\β}\\ -{\mathrm{sin\α}}_{12}{\mathrm{sin\θ}}_1{\mathrm{sin\α}}_{43}\sin \mathrm{\β}-{\mathrm{sin\α}}_{12}{\mathrm{cos\θ}}_1{\mathrm{sin\α}}_{14}{\mathrm{cos\α}}_{43}\\ +{\mathrm{cos\α}}_{14}{\mathrm{cos\α}}_{12}{\mathrm{cos\α}}_{43}-{\mathrm{cos\α}}_{32}\end{array}---\left(9.1\right)$

$\begin{array}{c}{b}_1=2{\mathrm{sin\α}}_{12}{\mathrm{cos\θ}}_1{\mathrm{cos\α}}_{14}{\mathrm{sin\α}}_{43}\sin \mathrm{\β}+2{\mathrm{sin\α}}_{12}{\mathrm{sin\θ}}_1{\mathrm{sin\α}}_{43}\cos \mathrm{\β}\\ +2{\mathrm{cos\α}}_{12}{\mathrm{sin\α}}_{14}{\mathrm{sin\α}}_{43}\sin \mathrm{\β}\end{array}---\left(9.2\right)$

$\begin{array}{c}{c}_1={\mathrm{sin\α}}_{12}{\mathrm{sin\θ}}_1{\mathrm{sin\α}}_{43}\sin \mathrm{\β}-{\mathrm{sin\α}}_{12}{\mathrm{cos\θ}}_1{\mathrm{sin\α}}_{14}{\mathrm{cos\α}}_{43}\\ +{\mathrm{cos\α}}_{14}{\mathrm{cos\α}}_{12}{\mathrm{cos\α}}_{43}-{\mathrm{sin\α}}_{12}{\mathrm{cos\θ}}_1{\mathrm{cos\α}}_{14}{\mathrm{sin\α}}_{43}\\ -{\mathrm{cos\α}}_{12}{\mathrm{sin\α}}_{14}{\mathrm{sin\α}}_{43}\cos \mathrm{\β}-{\mathrm{cos\α}}_{32}\end{array}---\left(9.3\right)$

Here a₁、b₁、c₁It is containing θ₁Undetermined coefficient；

Work as a₁When ≠ 0, the discriminant of equation (8) must is fulfilled for：

${\mathrm{\Δ}}_1=b_1^2-4a_1{c}_1\≥0---\left(10\right)$

Work as △₁When=0, represent that quadric chain is in dead-centre position；

According to equation (10), by input angle θ₁Obtain output angle θ₄：

$x_{4\[1\]}=\frac{-b_1-\sqrt{{\mathrm{\Δ}}_1}}{2a_1},{\mathrm{\θ}}_4=2{\mathrm{arctanx}}_{4\[1\]}---\left(11.1\right)$

$x_{4\[2\]}=\frac{-b_1+\sqrt{{\mathrm{\Δ}}_1}}{2a_1},{\mathrm{\θ}}_4=2{\mathrm{arctanx}}_{4\[2\]}---\left(11.2\right)$

Step 3：The joint revolution space of sphere five-bar mechanism；

Sphere six bar mechanism also comprises a sphere five-bar mechanism simultaneously, for sphere five-bar mechanism A₀ADCC₀B₀, can Obtain the input-output curve of five-bar mechanism：

Z₅S₅Z₂S₂Z₁S₁Z₄₁S₄₁Z₇S₇Z₆S₆=I (12)

Equation (12) is rewritten as：

$z^T{S}_5{Z}_2{S}_2{Z}_1{S}_1{Z}_{41}{S}_{41}{Z}_7{S}_{7}z=z^T{S}_6^{T}z---\left(13\right)$

Equation (13) comprises three variable θ₁、θ₄And θ₇；

Using half-angle formulas, make x₇=tan (θ₇/ 2), Equation (13) it is rewritten as：

$a_2({\mathrm{\θ}}_1,{\mathrm{\θ}}_4)x_7^2+b_2({\mathrm{\θ}}_1,{\mathrm{\θ}}_4)x_7+c_2({\mathrm{\θ}}_1,{\mathrm{\θ}}_4)=0---\left(14\right)$

Variable θ in equation (14) to be made₇There is solution, the discriminant in equation (14) should meet following condition：

△₂=f (θ₁,θ₄)≥0 (15)

Inequality (15) illustrates rotary joint θ in sphere six bar mechanism₁And θ₄The maximum magnitude of motion, i.e. θ₁And θ₄Joint Revolution space；Work as △₂When=0, i.e. the sideline of joint revolution space, is also the position that the singular point of mechanism occurs；

Step 4：Bifurcation Analysis；

Whether branch point is contained according to mechanism's linear-elsatic buckling in figure, whether the branch point identification in figure of sphere six bar mechanism contains The Bifurcation Analysis of branch point sphere six bar mechanism are segmented into the following two kinds form：

Class1：Sphere six bar mechanism linear-elsatic buckling in figure no branch point exists, and the joint revolving property of this kind of mechanism is only subject to ball The effect of face quadric chain；Sphere quadric chain A₀ABB₀Joint revolution space JRS-L completely in sphere five-bar mechanism A₀ADCC₀B₀Joint revolution space JRS-R in；The Bifurcation Analysis of this kind of sphere six bar mechanism can be summarized as following steps：

(1) branch's point analysiss：Verify whether containing branch point by joint equation (7) and equation (15), wherein equation (15) is Meet the situation of △=0；

(2) joint revolution space：By drawing the joint rotation of sphere quadric chain input-output curve and five-bar mechanism Space, obtains its common portion, verifies whether it is to comprise whole quadric chain curve；

(3) branch curve analysis：The boundary curve of analysis effectively public joint revolution space, is respectively designated as A_i、B_i、C_iAnd D_i, i =1,2..., A_iAnd B_iFor JRS-L sideline, C_iAnd D_iFor JRS-R sideline；

(4) which branch Bifurcation Analysis, belonged in by mechanism's place branch curve decision mechanism；

Type 2：Sphere six bar mechanism linear-elsatic buckling in figure is subject to sphere with the presence of branch point, the joint revolving property of this kind of mechanism Quadric chain and the collective effect of five-bar mechanism；Branch's situation of sphere six bar mechanism is subject to sphere quadric chain A₀ABB₀With sphere five-bar mechanism A₀ADCC₀B₀Collective effect, sphere six bar mechanism quadric chain chain A₀ABB₀Joint Revolution space JRS-L is intersected with JRS-R and is separated by, and forms different branches, is branched a cut-off between each branch, and this kind of sphere six is even The Bifurcation Analysis of linkage also can be summarized as following steps：

(1) branch's point analysiss：By joint equation (7) and equation (15), the concrete condition of branch point, wherein equation (15) can be obtained It is the situation meeting △=0；

(2) joint revolution space：By drawing the joint revolution space of sphere quadric chain curve and five-bar mechanism, obtain Its common portion；

(3) branch curve analysis：The boundary curve of analysis effectively public joint revolution space, is respectively designated as A_i、B_i、C_iAnd D_i, i =1,2..., A_iAnd B_iFor JRS-L sideline, C_iAnd D_iFor JRS-R sideline, branch point is numbered i simultaneously；

(4) which branch Bifurcation Analysis, belonged in by mechanism's place branch curve decision mechanism.