CN107315869A - A kind of force deformation behavior analysis method of the submissive diameter changing mechanism of diameter-variable wheel - Google Patents

A kind of force deformation behavior analysis method of the submissive diameter changing mechanism of diameter-variable wheel Download PDF

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CN107315869A
CN107315869A CN201710480676.9A CN201710480676A CN107315869A CN 107315869 A CN107315869 A CN 107315869A CN 201710480676 A CN201710480676 A CN 201710480676A CN 107315869 A CN107315869 A CN 107315869A
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msub
mrow
theta
mfrac
delta
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CN107315869B (en
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高峰
曾文
黄川�
张彬
刘本勇
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Beihang University
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/20—Design optimisation, verification or simulation
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B60—VEHICLES IN GENERAL
    • B60B—VEHICLE WHEELS; CASTORS; AXLES FOR WHEELS OR CASTORS; INCREASING WHEEL ADHESION
    • B60B19/00—Wheels not otherwise provided for or having characteristics specified in one of the subgroups of this group
    • B60B19/02—Wheels not otherwise provided for or having characteristics specified in one of the subgroups of this group convertible, e.g. from road wheel to rail wheel; Wheels specially designed for alternative use on road and rail
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2119/00—Details relating to the type or aim of the analysis or the optimisation
    • G06F2119/06—Power analysis or power optimisation

Abstract

The invention discloses a kind of force deformation behavior analysis method of the submissive diameter changing mechanism of diameter-variable wheel, it is characterised in that mainly using following steps：(1) according to force-bearing situation reasonable assumption during diameter changing mechanism reducing, two degree freedom system oblique mining is set up；(2) calculate each active force in the system, the virtual work that active torque and torsion spring torque are done；(3) according to the principle of virtual work, power and the relation of deformation during the mechanism reducing are obtained.Beneficial effects of the present invention：Relation needed for reducing can accurately being obtained between driving torque and wheel footpath size, it may be determined that torque capacity needed for reducing；The force deformation characteristic obtained using this method can provide theoretical foundation for the control of diameter-variable wheel wheel diameter variation.

Description

A kind of power-deformational behavior analysis method of the submissive diameter changing mechanism of diameter-variable wheel
Technical field
The present invention relates to a kind of power of the submissive diameter changing mechanism of diameter-variable wheel-deformational behavior analysis method, belong to submissive machine The power of structure and deformation relationship analysis field.
Background technology
The submissive diameter changing mechanism is primarily to realize the diameter change of diameter-variable wheel, and current this mechanism is applied to one Plant new type lunar rover diameter-variable wheel and new mobile platform diameter-variable wheels of dwelling more.Due to submissive diameter changing mechanism rely primarily on it is soft Deformation along hinge produces the problems such as motion, Wear, lubrication, sealing, makes portable construction compact.Such as Publication No. CN 101503044 application for a patent for invention discloses a kind of machine liquid linkage reducing wheel carrier suitable for diameter-variable wheel, but this The force analysis of diameter changing mechanism simply is regarded as structure to analyze, and not accounting for reducing process is carried out in wheel walking , a safety coefficient is then rule of thumb added to consider the effect of spring torque, therefore needed for can only substantially estimating reducing Torque capacity.The analysis method of neither one system come the relation needed for obtaining reducing between driving torque and wheel footpath size, from And effective control can not be carried out to diameter changing mechanism wheel diameter variation by power-deformation characteristic during the mechanism reducing, resistance The further application and popularization of this kind of mechanism and diameter-variable wheel are hindered.
The content of the invention
For above-mentioned problem, the present invention proposes a kind of power-deformation row of the submissive diameter changing mechanism of diameter-variable wheel It is a kind of analysis for more accurately and reliably obtaining power-deformation characteristic during submissive diameter changing mechanism reducing for analysis method Method.
The technical solution adopted by the present invention is to carry out according to the following steps successively：
Step 1: according to force-bearing situation reasonable assumption during diameter changing mechanism reducing, setting up two degree freedom system pseudo rigid-body Model；
Assuming that moon wheel leg being capable of holding structure stability (spoke bar keeps isosceles-trapezium-shaped), reducing during reducing Carried out in low speed, it is believed that quasistatic, by wheel hub Dcq1With wheel hub Dcq2Regard two cranks as；, can because caster is arc-shaped feet Landed at it to be approximately considered caster and can realize that its central point is contacted with ground in liftoff angle range all the time, in addition, by Radial distance in caster outer rim midpoint to caster connecting rod midpoint is smaller, might as well ignore, i.e., the concentrated force from ground is acted on Caster connecting rod midpoint；The two degree freedom system is using core wheel O as the origin of coordinates, with wheel hub Dcq1Rotational angle theta1With wheel hub Dcq2Rotational angle theta2For Generalized coordinates, submissive hinge is represented with the torsion spring with equivalent stiffness, sets up submissive diameter changing mechanism two-freedom oblique mining.
Then total virtual work of system can be expressed as：
WhereinTo act on i-th of active force in mechanism,For i-th of active point of force application to the position of the origin of coordinates Put vector,That is virtual displacement,To act on i-th of active torque in mechanism,For angular displacement,I.e. empty angle position Move,For i-th of torsion spring torque,For angular displacement,I.e. empty angular displacement.
For the variable-diameter wheel system with n wheel leg, specifically it is represented by：
Step 2: calculate each active force in the system, the virtual work that active torque and torsion spring torque are done；
Sub-step 1：The Vector Closing ring equation set up in the single wheel leg of diameter changing mechanism, to empty angle position can be tried to achieve after its differential Move δ θ3With δ θ1、δθ2Between relation；
It can be obtained by the Vector Closing ring equation in the single wheel leg of the mechanism：
r2cosθz+r3cosθ3+r4cosθ4-r5cosθ5-r1cosθ1=0 (3)
Wherein, r1For wheel hub Dcq1Radius, r2For wheel hub Dcq2Radius, r3For connection wheel hub Dcq2Spoke pole length, r4For wheel The length of pin connecting rod, r5For connection wheel hub Dcq1Spoke pole length, θ3For connection wheel hub Dcq2Spoke bar and horizontal direction angle, θ4 For caster connecting rod and horizontal direction angle, θ5For connection wheel hub Dcq1Spoke bar and horizontal direction angle.
Formula (3) is differentiated, obtained
Wherein, it can obtain by geometrically symmetric relation：
θ5=θ1+θ2-θ3 (6)
After being differentiated to formula (5) and formula (6), obtain
δθ5=δ θ1+δθ2-δθ3 (8)
It can be obtained by structural symmetry：
r1=r2 (9)
r3=r5 (10)
Bring formula (5)~(10) into formula (4), can obtain：
Formula (5), (6) and (10) brings formula (11) into, order
I.e.
δθ3=δ1δθ1+δ2δθ2 (14)
Sub-step 2：Write the position of active force F and active force to the origin of coordinates as vector；
Wherein, FqThe horizontal applied force of caster outer rim central point, F are faced for groundNFor ground hanging down in face of caster outer rim central point Straight active force.
Wherein position vectorMould be：
Sub-step 3：To position vectorDifferentiate and obtain virtual displacement, and by force vector and virtual displacementDot product is done, is obtained Virtual work；
To formula (17) differential, obtain
By formula (15) and formula (18) dot product, virtual work can obtain：
Sub-step 4：The angle turned under active moment loading is differentiated, and obtains active torque and does virtual work；
Θ1=θ1-θ10 (20)
Θ2=θ2-θ20 (21)
Wherein, θ10For wheel hub Dcq1Corner initial value, θ20For wheel hub Dcq2Corner initial value.
Have to formula (20) and formula (21) differential：
δΘ1=δ θ1 (22)
δΘ2=δ θ2 (23)
Then active torque does virtual work：
M1δΘ1+M2δΘ2 (24)
Sub-step 5：The angle turned under acting on spring torque is differentiated；
ψ1=(θ3-θ30)-(θ2-θ20) (25)
ψ2=(θ4-θ40)-(θ3-θ30) (26)
ψ3=-ψ2 (27)
ψ1=-ψ4 (28)
Bring formula (5) into formula (25)~(28) and differentiate, obtain
δψ1=δ θ3-δθ2 (29)
δψ4=δ θ2-δθ3 (32)
Sub-step 6：Obtain spring torque and do virtual work；
Notice open with contraction process spring torque it is in opposite direction, it is therefore, single when diameter changing mechanism, which is in, to be opened Spring is in and returns to nature process from deformation state in individual wheel leg, has：
T1=k1ψ1=k1[(θ3-θ30)-(θ2-θ20)] (33)
T2=k2ψ2=k2[(θ4-θ40)-(θ3-θ30)] (34)
Wherein, k1For the spring rate of spring 1, k2For the spring rate of spring 2.
When diameter changing mechanism, which is in, to be shunk, spring is in from nature to deformation state process in single wheel leg, is had：
T1=-k1ψ1=-k1[(θ3-θ30)-(θ2-θ20)] (35)
T2=-k2ψ2=-k2[(θ4-θ40)-(θ3-θ30)] (36)
By structural symmetry, it is apparent from：
T4=-T1 (37)
T3=-T2 (38)
Association type (14) and formula (29)~(38), then spring does virtual work and is in single wheel leg：
Assuming that diameter-variable wheel has n wheel leg, it is noted that diameter-variable wheel is not in contact with the ground during reducing (n-1) virtual work that each wheel leg spring is done in individual wheel leg is identical with contacting the virtual work that the single wheel leg spring on ground is done, then for For whole wheel system, total virtual work that spring is done is：
Wherein, θ3It can be obtained by following equation
Sub-step 7：Step 3 Chinese style (19), step 4 Chinese style (24) are added and can obtained with step 6 Chinese style (40) virtual work Total virtual work of whole system；
Step 3: according to the principle of virtual work, obtaining power and the relation of deformation during the mechanism reducing；
Under generalized coordinates, total virtual work of two degree freedom system is represented by following form：
δ W=Q1δq1+Q2δq2=Q1δθ1+Q2δθ2 (42)
Here QiRepresent generalized force, qiGeneralized coordinates is represented, it is each wide when system is in standing balance from the principle of virtual work Yi Libi is zero, i.e.,
Qi=0 (43)
It can obtain：
Advantage of the present invention than prior art：
Currently for the analysis method of the submissive diameter changing mechanism power-deformational behavior of diameter-variable wheel, mainly according to traditional Free body diagram method, rule of thumb adds a safety coefficient to consider the effect of spring torque, can only substantially estimate reducing institute Need torque capacity, it is impossible to the relation needed for accurate acquisition reducing between driving torque and wheel footpath size.And utilize present invention proposition The power of the diameter changing mechanism based on oblique mining and the principle of virtual work-deformational behavior analysis method, it is necessary to for determine power with deformation The equation number of relation will be far smaller than free body diagram method, and it is two-freedom system during reducing to consider diameter changing mechanism System.Relation between driving torque and wheel footpath size needed for this method not only can accurately obtain reducing, determines reducing institute Need torque capacity；And power-the deformation characteristic obtained using this method can also be provided for the control of diameter changing mechanism wheel diameter variation Theoretical foundation, realizes the accurate control of diameter-variable wheel wheel diameter variation.
Brief description of the drawings
Fig. 1 is a kind of power-deformational behavior analysis method operating process of the submissive diameter changing mechanism of diameter-variable wheel of the present invention Figure.
Fig. 2 is submissive diameter changing mechanism two-freedom oblique mining figure of the invention.
Fig. 3 is submissive diameter changing mechanism driving torque M of the invention1With θ1、θ2Between graph of a relation.
Fig. 4 is submissive diameter changing mechanism driving torque M of the invention2With θ1、θ2Between graph of a relation.
Embodiment
For a better understanding of the present invention, it is further detailed with reference to example：
This example is only analyzed diameter-variable wheel contraction process (analysis method for opening process is identical), it is known that had The lunar rover diameter-variable wheel of six wheel legs, its submissive diameter changing mechanism dimensional parameters is：Wheel hub Dcq1Radius r1With wheel hub Dcq2Half Footpath r2Equal is 91.7mm, connection wheel hub Dcq2Spoke pole length r3With being connected wheel hub Dcq1Spoke pole length r5Equal is 124mm, Clipping room of the spoke bar on caster is away from r4For 64mm, wheel hub D when opening limiting conditioncq1Corner initial value θ10For 95 °, wheel hub Dcq2 Corner initial value θ20For 25 °, wheel hub D is connectedcq2Spoke bar and horizontal direction angle initial value θ30Below equation (1) can be passed through Calculate.
The power of the submissive diameter changing mechanism of a kind of diameter-variable wheel of the present invention-deformational behavior analysis method operating process, such as Fig. 1 It is shown, specifically include following steps：
Step 1: according to force-bearing situation reasonable assumption during diameter changing mechanism reducing, setting up two degree freedom system pseudo rigid-body Model；
Assuming that moon wheel leg being capable of holding structure stability (spoke bar keeps isosceles-trapezium-shaped), reducing during reducing Carried out in low speed, it is believed that quasistatic, by wheel hub Dcq1With wheel hub Dcq2Regard two cranks as；, can because caster is arc-shaped feet Landed at it to be approximately considered caster and can realize that its central point is contacted with ground in liftoff angle range all the time, in addition, by Radial distance in caster outer rim midpoint to caster connecting rod midpoint is smaller, might as well ignore, i.e., the concentrated force from ground is acted on Caster connecting rod midpoint；The two degree freedom system is using core wheel O as the origin of coordinates, with wheel hub Dcq1Rotational angle theta1With wheel hub Dcq2Rotational angle theta2For Generalized coordinates, submissive hinge represents with the torsion spring of equivalent stiffness, the submissive diameter changing mechanism two-freedom pseudo rigid-body mould of foundation Type is as shown in Figure 2 (other wheel legs not contacted not shown in figure with ground).
Then total virtual work of system can be expressed as：
WhereinTo act on i-th of active force in mechanism,For i-th of active point of force application to the position of the origin of coordinates Put vector,That is virtual displacement,To act on i-th of active torque in mechanism,For angular displacement,I.e. empty angle position Move,For i-th of torsion spring torque,For angular displacement,I.e. empty angular displacement.
For the variable-diameter wheel system with 6 wheel legs, specifically it is represented by：
Step 2: calculate each active force in the system, the virtual work that active torque and torsion spring torque are done；
Sub-step 1：The Vector Closing ring equation set up in the single wheel leg of diameter changing mechanism, to empty angle position can be tried to achieve after its differential Move δ θ3With δ θ1、δθ2Between relation；
It can be obtained by the Vector Closing ring equation in the single wheel leg of the mechanism：
r2cosθ2+r3cosθ3+r4cosθ4-r5cosθ5-r1cosθ1=0 (49)
As shown in Fig. 2 wherein, r1For wheel hub Dcq1Radius, r2For wheel hub Dcq2Radius, r3For connection wheel hub Dcq2Spoke bar it is long Degree, r4For the length of caster connecting rod, r5For connection wheel hub Dcq1Spoke pole length, θ3For connection wheel hub Dcq2Spoke bar and level side To angle, θ4For caster connecting rod and horizontal direction angle, θ5For connection wheel hub Dcq1Spoke bar and horizontal direction angle.
Formula (4) is differentiated, obtained
Wherein, it can obtain by the geometrically symmetric relation in single wheel leg：
θ5=θ1+θ2-θ3 (52)
After being differentiated to formula (6) and formula (7), obtain
δθ5=δ θ 1+ δ θ2-δθ3 (54)
It can be obtained by the structural symmetry in single wheel leg：
r1=r2 (55)
r3=r5 (56)
Bring formula (6)~(11) into formula (5), can obtain：
Formula (6), (7) and (11) brings formula (12) into, order
I.e.
δθ3=δ1δθ1+δ2δθ2 (60)
Sub-step 2：Write the position of active force F and active force to the origin of coordinates as vector；
As shown in Fig. 2 where it is assumed that ground faces the horizontal applied force F of caster connecting rod central pointqFor 45.7N, ground is in face of wheel The vertical F of pin connecting rod central pointNFor 57N.
As shown in Fig. 2 wherein position vectorMould be：
Sub-step 3：To position vectorDifferentiate and obtain virtual displacement, and by force vector and virtual displacementDot product is done, is obtained Virtual work；
To formula (18) differential, obtain
By formula (16) and formula (19) dot product, virtual work can obtain：
Sub-step 4：The angle turned under active moment loading is differentiated, and obtains active torque and does virtual work；
Θ1=θ1-θ10 (66)
Θ2=θ2-θ20 (67)
Have to formula (21) and formula (22) differential：
δΘ1=δ θ1 (68)
δΘ2=δ θ2 (69)
Then active torque does virtual work：
M1δΘ1+M2δΘ2 (70)
Sub-step 5：The angle turned under acting on spring torque is differentiated；
ψ1=(θ3-θ30)-(θ2-θ20) (71)
ψ2=(θ4-θ40)-(θ3-θ30) (72)
ψ3=-ψ2 (73)
ψ1=-ψ4 (74)
Bring formula (6) into formula (26)~(29) and differentiate, obtain
δψ1=δ θ3-δθ2 (75)
δψ4=δ θ2-δθ3 (78)
Sub-step 6：Obtain spring torque and do virtual work；
Because diameter-variable wheel is made up of six wheel legs, then when the centre symmetry line and Y-axis negative direction angle of single wheel leg At [- 30 °, 30 °] it is interval in when, the wheel leg it land with liftoff angle range, can be with reference to geometrical relationship between spoke bar Obtain θ1And θ2Span：
70 ° of ＜ θ1-θ2157 ° of ＜ (80)
It is (false with reference to the ground connection angle range of single wheel leg and the reducing scope of diameter changing mechanism according to formula (34) and formula (35) If wheel diameter changes in the range of 240~400mm), then can proper diameter changing mechanism from opening when extreme position is started to shrink at Initial angle：
θ10=95 ° (81)
θ20=25 ° (82)
When diameter changing mechanism, which is in, to be shunk, spring is in from nature to deformation state process in single wheel leg, is had：
T1=-k1ψ1=-k1[(θ3-θ30)-(θ2-θ20)] (83)
T2=-k2ψ2=-k2[(θ4-θ40)-(θ3-θ30)] (84)
Wherein, the rigidity k of spring 11For 525Nmm/ °, the rigidity k of spring 22For 428Nmm/ °.
As shown in Fig. 2 by the structural symmetry in single wheel leg, being apparent from：
T4=-T1 (85)
T3=-T2 (86)
Because diameter-variable wheel has 6 wheel legs, it is noted that diameter-variable wheel is during reducing, not in contact with the ground 5 The virtual work that each wheel leg spring is done in individual wheel leg is identical with contacting the virtual work that the single wheel leg spring on ground is done, association type (15) and Formula (30)~(41), then for whole wheel system, total virtual work that spring is done is：
Wherein, θ3It can be obtained by following equation
Sub-step 7：Step 3 Chinese style (20), step 4 Chinese style (25) are added and can obtained with step 6 Chinese style (42) virtual work Total virtual work of whole system；
Step 3: according to the principle of virtual work, obtaining power and the relation of deformation during the mechanism reducing；
Under generalized coordinates, total virtual work of two degree freedom system is represented by following form：
δ W=Q1δq1+Q2δq2=Q1δθ1+Q2δθ2 (89)
Here QiRepresent generalized force, qiGeneralized coordinates is represented, it is each wide when system is in standing balance from the principle of virtual work Yi Libi is zero, i.e.,
Qi=0 (90)
It can obtain：
By known geometric parameter, convolution (1)~(46) can be obtained such as Fig. 3, the power of the diameter changing mechanism shown in Fig. 4 with Deformation relationship.Wherein Fig. 3 is M1(ordinate) and θ1、θ2Between graph of a relation, M1Maximum is 30847Nmm；Wherein Fig. 4 is M2(ordinate) and θ1、θ2Between graph of a relation, M2Maximum is 28963Nmm.

Claims (4)

1. a kind of power-deformational behavior analysis method of the submissive diameter changing mechanism of diameter-variable wheel, it is characterised in that：This method is included such as Lower step：
Step 1: according to force-bearing situation reasonable assumption during diameter changing mechanism reducing, setting up two degree freedom system oblique mining；
Step 2: calculate each active force in the system, the virtual work that active torque and torsion spring torque are done；
Step 3: according to the principle of virtual work, obtaining power and the relation of deformation during the mechanism reducing.
2. a kind of power-deformational behavior analysis method of the submissive diameter changing mechanism of diameter-variable wheel according to claim 1, it is special Levy and be：The step one is specific as follows：
Assuming that during reducing moon wheel leg can holding structure stability, reducing carries out in low speed, it is believed that quasistatic, By wheel hub Dcq1With wheel hub Dcq2Regard two cranks as；Due to caster be arc-shaped feet, can be approximately considered caster it land and from All the time it can realize that its central point is contacted with ground in the angle range on ground, further, since caster outer rim midpoint is into caster connecting rod The radial distance of point is smaller, might as well ignore, i.e., the concentrated force from ground acts on caster connecting rod midpoint；The two degree freedom system Using core wheel O as the origin of coordinates, with wheel hub Dcq1Rotational angle theta1With wheel hub Dcq2Rotational angle theta2For generalized coordinates, submissive hinge is used with equivalent The torsion spring of rigidity is represented, sets up submissive diameter changing mechanism two-freedom oblique mining；
Then total virtual work of system can be expressed as：
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WhereinTo act on i-th of active force in mechanism,Sweared for the position of i-th of active point of force application to the origin of coordinates Amount,That is virtual displacement,To act on i-th of active torque in mechanism,For angular displacement,I.e. empty angular displacement,For I-th of torsion spring torque,For angular displacement,I.e. empty angular displacement；
For the variable-diameter wheel system with n wheel leg, specifically it is represented by：
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3. a kind of power-deformational behavior analysis method of the submissive diameter changing mechanism of diameter-variable wheel according to claim 1, it is special Levy and be：The step 2 specifically includes following sub-step：
Sub-step 1：The Vector Closing ring equation set up in the single wheel leg of diameter changing mechanism, to empty angular displacement δ θ can be tried to achieve after its differential3 With δ θ1、δθ2Between relation；
It can be obtained by the Vector Closing ring equation in the single wheel leg of the mechanism：
r2cosθ2+r3cosθ3+r4cosθ4-r5cosθ5-r1cosθ1=0 (3)
Wherein, r1For wheel hub Dcq1Radius, r2For wheel hub Dcq2Radius, r3For connection wheel hub Dcq2Spoke pole length, r4For caster connecting rod Length, r5For connection wheel hub Dcq1Spoke pole length, θ3For connection wheel hub Dcq2Spoke bar and horizontal direction angle, θ4For caster Connecting rod and horizontal direction angle, θ5For connection wheel hub Dcq1Spoke bar and horizontal direction angle；
Formula (3) is differentiated, obtained
<mrow> <mtable> <mtr> <mtd> <mrow> <mo>-</mo> <msub> <mi>r</mi> <mn>1</mn> </msub> <msub> <mi>sin&amp;theta;</mi> <mn>2</mn> </msub> <msub> <mi>&amp;delta;&amp;theta;</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>r</mi> <mn>3</mn> </msub> <msub> <mi>sin&amp;theta;</mi> <mn>3</mn> </msub> <msub> <mi>&amp;delta;&amp;theta;</mi> <mn>3</mn> </msub> <mo>-</mo> <msub> <mi>r</mi> <mn>4</mn> </msub> <msub> <mi>sin&amp;theta;</mi> <mn>4</mn> </msub> <msub> <mi>&amp;delta;&amp;theta;</mi> <mn>4</mn> </msub> <mo>+</mo> <msub> <mi>r</mi> <mn>5</mn> </msub> <msub> <mi>sin&amp;theta;</mi> <mn>5</mn> </msub> <msub> <mi>&amp;delta;&amp;theta;</mi> <mn>5</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <msub> <mi>r</mi> <mn>1</mn> </msub> <msub> <mi>sin&amp;theta;</mi> <mn>1</mn> </msub> <msub> <mi>&amp;delta;&amp;theta;</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow>
Wherein, it can obtain by geometrically symmetric relation：
<mrow> <msub> <mi>&amp;theta;</mi> <mn>4</mn> </msub> <mo>=</mo> <mfrac> <mi>&amp;pi;</mi> <mn>2</mn> </mfrac> <mo>+</mo> <mfrac> <mrow> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> </mrow> <mn>2</mn> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>
θ5=θ1+θ2-θ3 (6)
After being differentiated to formula (5) and formula (6), obtain
<mrow> <msub> <mi>&amp;delta;&amp;theta;</mi> <mn>4</mn> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi>&amp;delta;&amp;theta;</mi> <mn>1</mn> </msub> </mrow> <mn>2</mn> </mfrac> <mo>+</mo> <mfrac> <mrow> <msub> <mi>&amp;delta;&amp;theta;</mi> <mn>2</mn> </msub> </mrow> <mn>2</mn> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow>
δθ5=δ θ1+δθ2-δθ3 (8)
It can be obtained by structural symmetry：
r1=r2 (9)
r3=r5 (10)
Bring formula (5)~(10) into formula (4), can obtain：
<mrow> <mtable> <mtr> <mtd> <mmultiscripts> <mo>=</mo> <mn>3</mn> </mmultiscripts> </mtd> </mtr> <mtr> <mtd> <mfrac> <mrow> <mo>(</mo> <msub> <mi>r</mi> <mn>5</mn> </msub> <msub> <mi>sin&amp;theta;</mi> <mn>5</mn> </msub> <mo>-</mo> <mfrac> <msub> <mi>r</mi> <mn>4</mn> </msub> <mn>2</mn> </mfrac> <msub> <mi>sin&amp;theta;</mi> <mn>4</mn> </msub> <mo>+</mo> <msub> <mi>r</mi> <mn>1</mn> </msub> <msub> <mi>sin&amp;theta;</mi> <mn>1</mn> </msub> <mo>)</mo> <msub> <mi>&amp;delta;&amp;theta;</mi> <mn>1</mn> </msub> <mo>+</mo> <mo>(</mo> <msub> <mi>r</mi> <mn>5</mn> </msub> <msub> <mi>sin&amp;theta;</mi> <mn>5</mn> </msub> <mo>-</mo> <mfrac> <msub> <mi>r</mi> <mn>4</mn> </msub> <mn>2</mn> </mfrac> <msub> <mi>sin&amp;theta;</mi> <mn>4</mn> </msub> <mo>-</mo> <msub> <mi>r</mi> <mn>2</mn> </msub> <msub> <mi>sin&amp;theta;</mi> <mn>2</mn> </msub> <mo>)</mo> <msub> <mi>&amp;delta;&amp;theta;</mi> <mn>2</mn> </msub> </mrow> <mrow> <msub> <mi>r</mi> <mn>3</mn> </msub> <msub> <mi>sin&amp;theta;</mi> <mn>3</mn> </msub> <mo>+</mo> <msub> <mi>r</mi> <mn>5</mn> </msub> <msub> <mi>sin&amp;theta;</mi> <mn>5</mn> </msub> </mrow> </mfrac> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>11</mn> <mo>)</mo> </mrow> </mrow>
Formula (5), (6) and (10) brings formula (11) into, order
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>&amp;delta;</mi> <mn>1</mn> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi>r</mi> <mn>5</mn> </msub> <msub> <mi>sin&amp;theta;</mi> <mn>5</mn> </msub> <mo>-</mo> <mfrac> <msub> <mi>r</mi> <mn>4</mn> </msub> <mn>2</mn> </mfrac> <msub> <mi>sin&amp;theta;</mi> <mn>4</mn> </msub> <mo>+</mo> <msub> <mi>r</mi> <mn>1</mn> </msub> <msub> <mi>sin&amp;theta;</mi> <mn>1</mn> </msub> </mrow> <mrow> <msub> <mi>r</mi> <mn>3</mn> </msub> <msub> <mi>sin&amp;theta;</mi> <mn>3</mn> </msub> <mo>+</mo> <msub> <mi>r</mi> <mn>5</mn> </msub> <msub> <mi>sin&amp;theta;</mi> <mn>5</mn> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mfrac> <mrow> <msub> <mi>r</mi> <mn>3</mn> </msub> <mi>sin</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mfrac> <msub> <mi>r</mi> <mn>4</mn> </msub> <mn>2</mn> </mfrac> <mi>sin</mi> <mrow> <mo>(</mo> <mfrac> <mi>&amp;pi;</mi> <mn>2</mn> </mfrac> <mo>+</mo> <mfrac> <mrow> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> </mrow> <mn>2</mn> </mfrac> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>r</mi> <mn>1</mn> </msub> <msub> <mi>sin&amp;theta;</mi> <mn>1</mn> </msub> </mrow> <mrow> <msub> <mi>r</mi> <mn>3</mn> </msub> <msub> <mi>sin&amp;theta;</mi> <mn>3</mn> </msub> <mo>+</mo> <msub> <mi>r</mi> <mn>3</mn> </msub> <mi>sin</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>12</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>&amp;delta;</mi> <mn>2</mn> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi>r</mi> <mn>5</mn> </msub> <msub> <mi>sin&amp;theta;</mi> <mn>5</mn> </msub> <mo>-</mo> <mfrac> <msub> <mi>r</mi> <mn>4</mn> </msub> <mn>2</mn> </mfrac> <msub> <mi>sin&amp;theta;</mi> <mn>4</mn> </msub> <mo>-</mo> <msub> <mi>r</mi> <mn>2</mn> </msub> <msub> <mi>sin&amp;theta;</mi> <mn>2</mn> </msub> </mrow> <mrow> <msub> <mi>r</mi> <mn>3</mn> </msub> <msub> <mi>sin&amp;theta;</mi> <mn>3</mn> </msub> <mo>+</mo> <msub> <mi>r</mi> <mn>5</mn> </msub> <msub> <mi>sin&amp;theta;</mi> <mn>5</mn> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mfrac> <mrow> <msub> <mi>r</mi> <mn>3</mn> </msub> <mi>sin</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mfrac> <msub> <mi>r</mi> <mn>4</mn> </msub> <mn>2</mn> </mfrac> <mi>sin</mi> <mrow> <mo>(</mo> <mfrac> <mi>&amp;pi;</mi> <mn>2</mn> </mfrac> <mo>+</mo> <mfrac> <mrow> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> </mrow> <mn>2</mn> </mfrac> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>r</mi> <mn>2</mn> </msub> <msub> <mi>sin&amp;theta;</mi> <mn>2</mn> </msub> </mrow> <mrow> <msub> <mi>r</mi> <mn>3</mn> </msub> <msub> <mi>sin&amp;theta;</mi> <mn>3</mn> </msub> <mo>+</mo> <msub> <mi>r</mi> <mn>3</mn> </msub> <mi>sin</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>13</mn> <mo>)</mo> </mrow> </mrow>
I.e.
δθ3=δ1δθ1+δ2δθ2 (14)
Sub-step 2：Write the position of active force F and active force to the origin of coordinates as vector；
<mrow> <mover> <mi>F</mi> <mo>&amp;RightArrow;</mo> </mover> <mo>=</mo> <msub> <mi>F</mi> <mi>q</mi> </msub> <mi>i</mi> <mo>+</mo> <msub> <mi>F</mi> <mi>N</mi> </msub> <mi>j</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>15</mn> <mo>)</mo> </mrow> </mrow>
Wherein, FqThe horizontal applied force of caster outer rim central point, F are faced for groundNThe vertical work of caster outer rim central point is faced for ground Firmly；
<mrow> <mover> <mi>Z</mi> <mo>&amp;RightArrow;</mo> </mover> <mo>=</mo> <mi>Z</mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> </mrow> <mn>2</mn> </mfrac> <mo>)</mo> </mrow> <mi>i</mi> <mo>-</mo> <mi>Z</mi> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> </mrow> <mn>2</mn> </mfrac> <mo>)</mo> </mrow> <mi>j</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>16</mn> <mo>)</mo> </mrow> </mrow>
Wherein position vectorMould be：
<mrow> <mi>Z</mi> <mo>=</mo> <msub> <mi>r</mi> <mn>2</mn> </msub> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> </mrow> <mn>2</mn> </mfrac> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>r</mi> <mn>3</mn> </msub> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> </mrow> <mn>2</mn> </mfrac> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>17</mn> <mo>)</mo> </mrow> </mrow>
Sub-step 3：To position vectorDifferentiate and obtain virtual displacement, and by force vector and virtual displacementDot product is done, virtual work is obtained；
To formula (17) differential, obtain
<mrow> <mtable> <mtr> <mtd> <mrow> <mi>&amp;delta;</mi> <mover> <mi>Z</mi> <mo>&amp;RightArrow;</mo> </mover> <mo>=</mo> <mo>&amp;lsqb;</mo> <mo>-</mo> <mfrac> <msub> <mi>r</mi> <mn>2</mn> </msub> <mn>2</mn> </mfrac> <msub> <mi>sin&amp;theta;</mi> <mn>1</mn> </msub> <mo>+</mo> <mfrac> <msub> <mi>r</mi> <mn>3</mn> </msub> <mn>2</mn> </mfrac> <mi>sin</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mn>3</mn> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <msub> <mi>&amp;delta;&amp;theta;</mi> <mn>1</mn> </msub> <mi>i</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mo>&amp;lsqb;</mo> <mo>-</mo> <mfrac> <msub> <mi>r</mi> <mn>2</mn> </msub> <mn>2</mn> </mfrac> <msub> <mi>sin&amp;theta;</mi> <mn>2</mn> </msub> <mo>-</mo> <mfrac> <msub> <mi>r</mi> <mn>3</mn> </msub> <mn>2</mn> </mfrac> <mi>sin</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <msub> <mi>&amp;delta;&amp;theta;</mi> <mn>2</mn> </msub> <mi>i</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <msub> <mi>r</mi> <mn>3</mn> </msub> <mi>sin</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> </mrow> <mn>2</mn> </mfrac> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> <mi>cos</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> </mrow> <mn>2</mn> </mfrac> <mo>)</mo> </mrow> <msub> <mi>&amp;delta;&amp;theta;</mi> <mn>3</mn> </msub> <mi>i</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mo>&amp;lsqb;</mo> <mo>-</mo> <mfrac> <msub> <mi>r</mi> <mn>2</mn> </msub> <mn>2</mn> </mfrac> <msub> <mi>cos&amp;theta;</mi> <mn>1</mn> </msub> <mo>-</mo> <mfrac> <msub> <mi>r</mi> <mn>3</mn> </msub> <mn>2</mn> </mfrac> <mi>cos</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mn>3</mn> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <msub> <mi>&amp;delta;&amp;theta;</mi> <mn>1</mn> </msub> <mi>j</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mo>&amp;lsqb;</mo> <mo>-</mo> <mfrac> <msub> <mi>r</mi> <mn>2</mn> </msub> <mn>2</mn> </mfrac> <msub> <mi>cos&amp;theta;</mi> <mn>2</mn> </msub> <mo>-</mo> <mfrac> <msub> <mi>r</mi> <mn>3</mn> </msub> <mn>2</mn> </mfrac> <mi>cos</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <msub> <mi>&amp;delta;&amp;theta;</mi> <mn>2</mn> </msub> <mi>j</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <msub> <mi>r</mi> <mn>3</mn> </msub> <mi>sin</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> </mrow> <mn>2</mn> </mfrac> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> <mi>sin</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> </mrow> <mn>2</mn> </mfrac> <mo>)</mo> </mrow> <msub> <mi>&amp;delta;&amp;theta;</mi> <mn>3</mn> </msub> <mi>i</mi> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>18</mn> <mo>)</mo> </mrow> </mrow>
By formula (15) and formula (18) dot product, virtual work can obtain：
<mrow> <mtable> <mtr> <mtd> <mrow> <mover> <mi>F</mi> <mo>&amp;RightArrow;</mo> </mover> <mo>&amp;CenterDot;</mo> <mi>&amp;delta;</mi> <mover> <mi>Z</mi> <mo>&amp;RightArrow;</mo> </mover> <mo>=</mo> <mo>-</mo> <msub> <mi>F</mi> <mi>q</mi> </msub> <mo>&amp;lsqb;</mo> <mfrac> <msub> <mi>r</mi> <mn>2</mn> </msub> <mn>2</mn> </mfrac> <msub> <mi>sin&amp;theta;</mi> <mn>1</mn> </msub> <mo>-</mo> <mfrac> <msub> <mi>r</mi> <mn>3</mn> </msub> <mn>2</mn> </mfrac> <mi>sin</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mn>3</mn> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <msub> <mi>&amp;delta;&amp;theta;</mi> <mn>1</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <msub> <mi>F</mi> <mi>N</mi> </msub> <mo>&amp;lsqb;</mo> <mfrac> <msub> <mi>r</mi> <mn>2</mn> </msub> <mn>2</mn> </mfrac> <msub> <mi>cos&amp;theta;</mi> <mn>1</mn> </msub> <mo>+</mo> <mfrac> <msub> <mi>r</mi> <mn>3</mn> </msub> <mn>2</mn> </mfrac> <mi>cos</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mn>3</mn> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <msub> <mi>&amp;delta;&amp;theta;</mi> <mn>1</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <msub> <mi>F</mi> <mi>q</mi> </msub> <mo>&amp;lsqb;</mo> <msub> <mi>r</mi> <mn>3</mn> </msub> <mi>sin</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mi>+</mi> <msub> <mi>&amp;theta;</mi> <mi>2</mi> </msub> </mrow> <mi>2</mi> </mfrac> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> <mi>cos</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mi>+</mi> <msub> <mi>&amp;theta;</mi> <mi>2</mi> </msub> </mrow> <mi>2</mi> </mfrac> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <msub> <mi>&amp;delta;</mi> <mn>1</mn> </msub> <msub> <mi>&amp;delta;&amp;theta;</mi> <mn>1</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <msub> <mi>F</mi> <mi>N</mi> </msub> <mo>&amp;lsqb;</mo> <msub> <mi>r</mi> <mn>3</mn> </msub> <mi>sin</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mi>+</mi> <msub> <mi>&amp;theta;</mi> <mi>2</mi> </msub> </mrow> <mi>2</mi> </mfrac> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> <mi>sin</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mi>+</mi> <msub> <mi>&amp;theta;</mi> <mi>2</mi> </msub> </mrow> <mi>2</mi> </mfrac> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <msub> <mi>&amp;delta;</mi> <mn>1</mn> </msub> <msub> <mi>&amp;delta;&amp;theta;</mi> <mn>1</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <msub> <mi>F</mi> <mi>q</mi> </msub> <mo>&amp;lsqb;</mo> <mfrac> <msub> <mi>r</mi> <mn>2</mn> </msub> <mn>2</mn> </mfrac> <msub> <mi>sin&amp;theta;</mi> <mn>2</mn> </msub> <mo>+</mo> <mfrac> <msub> <mi>r</mi> <mn>3</mn> </msub> <mn>2</mn> </mfrac> <mi>sin</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <msub> <mi>&amp;delta;&amp;theta;</mi> <mn>2</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <msub> <mi>F</mi> <mi>q</mi> </msub> <mo>&amp;lsqb;</mo> <msub> <mi>r</mi> <mn>3</mn> </msub> <mi>sin</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mi>+</mi> <msub> <mi>&amp;theta;</mi> <mi>2</mi> </msub> </mrow> <mi>2</mi> </mfrac> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> <mi>cos</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mi>+</mi> <msub> <mi>&amp;theta;</mi> <mi>2</mi> </msub> </mrow> <mi>2</mi> </mfrac> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <msub> <mi>&amp;delta;</mi> <mn>2</mn> </msub> <msub> <mi>&amp;delta;&amp;theta;</mi> <mn>2</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <msub> <mi>F</mi> <mi>N</mi> </msub> <mo>&amp;lsqb;</mo> <mfrac> <msub> <mi>r</mi> <mn>2</mn> </msub> <mn>2</mn> </mfrac> <msub> <mi>cos&amp;theta;</mi> <mn>2</mn> </msub> <mo>+</mo> <mfrac> <msub> <mi>r</mi> <mn>3</mn> </msub> <mn>2</mn> </mfrac> <mi>cos</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <msub> <mi>&amp;delta;&amp;theta;</mi> <mn>2</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <msub> <mi>F</mi> <mi>N</mi> </msub> <mo>&amp;lsqb;</mo> <msub> <mi>r</mi> <mn>3</mn> </msub> <mi>sin</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mi>+</mi> <msub> <mi>&amp;theta;</mi> <mi>2</mi> </msub> </mrow> <mi>2</mi> </mfrac> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> <mi>sin</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mi>+</mi> <msub> <mi>&amp;theta;</mi> <mi>2</mi> </msub> </mrow> <mi>2</mi> </mfrac> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <msub> <mi>&amp;delta;</mi> <mn>2</mn> </msub> <msub> <mi>&amp;delta;&amp;theta;</mi> <mn>2</mn> </msub> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>19</mn> <mo>)</mo> </mrow> </mrow>
Sub-step 4：The angle turned under active moment loading is differentiated, and obtains active torque and does virtual work；
Θ1=θ1-θ10 (20)
Θ2=θ2-θ20 (21)
Wherein, θ10For wheel hub Dcq1Corner initial value, θ20For wheel hub Dcq2Corner initial value；
Have to formula (20) and formula (21) differential：
δΘ1=δ θ1 (22)
δΘ2=δ θ2 (23)
Then active torque does virtual work：
M1δΘ1+M2δΘ2 (24)
Sub-step 5：The angle turned under acting on spring torque is differentiated；
ψ1=(θ3-θ30)-(θ2-θ20) (25)
ψ2=(θ4-θ40)-(θ3-θ30) (26)
ψ3=-ψ2 (27)
ψ1=-ψ4 (28)
Bring formula (5) into formula (25)~(28) and differentiate, obtain
δψ1=δ θ3-δθ2 (29)
<mrow> <msub> <mi>&amp;delta;&amp;psi;</mi> <mn>2</mn> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi>&amp;delta;&amp;theta;</mi> <mn>1</mn> </msub> </mrow> <mn>2</mn> </mfrac> <mo>+</mo> <mfrac> <mrow> <msub> <mi>&amp;delta;&amp;theta;</mi> <mn>2</mn> </msub> </mrow> <mn>2</mn> </mfrac> <mo>-</mo> <msub> <mi>&amp;delta;&amp;theta;</mi> <mn>3</mn> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>30</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <msub> <mi>&amp;delta;&amp;psi;</mi> <mn>3</mn> </msub> <mo>=</mo> <msub> <mi>&amp;delta;&amp;theta;</mi> <mn>3</mn> </msub> <mo>-</mo> <mfrac> <mrow> <msub> <mi>&amp;delta;&amp;theta;</mi> <mn>1</mn> </msub> </mrow> <mn>2</mn> </mfrac> <mo>-</mo> <mfrac> <mrow> <msub> <mi>&amp;delta;&amp;theta;</mi> <mn>2</mn> </msub> </mrow> <mn>2</mn> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>31</mn> <mo>)</mo> </mrow> </mrow>
δψ4=δ θ2-δθ3 (32)
Sub-step 6：Obtain spring torque and do virtual work；
Notice open with contraction process spring torque it is in opposite direction, therefore, when diameter changing mechanism be in open when, single wheel Spring is in and returns to nature process from deformation state in leg, has：
T1=k1ψ1=k1[(θ3-θ30)-(θ2-θ20)] (33)
T2=k2ψ2=k2[(θ4-θ40)-(θ3-θ30)] (34)
Wherein, k1For the spring rate of spring 1, k2For the spring rate of spring 2；
When diameter changing mechanism, which is in, to be shunk, spring is in from nature to deformation state process in single wheel leg, is had：
T1=-k1ψ1=-k1[(θ3-θ30)-(θ2-θ20)] (35)
T2=-k2ψ2=-k2[(θ4-θ40)-(θ3-θ30)] (36)
By structural symmetry, it is apparent from：
T4=-T1 (37)
T3=-T2 (38)
Association type (14) and formula (29)~(38), then spring does virtual work and is in single wheel leg：
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>T</mi> <mn>1</mn> </msub> <msub> <mi>&amp;delta;&amp;psi;</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>T</mi> <mn>2</mn> </msub> <msub> <mi>&amp;delta;&amp;psi;</mi> <mn>2</mn> </msub> <mo>+</mo> <msub> <mi>T</mi> <mn>3</mn> </msub> <msub> <mi>&amp;delta;&amp;psi;</mi> <mn>3</mn> </msub> <mo>+</mo> <msub> <mi>T</mi> <mn>4</mn> </msub> <msub> <mi>&amp;delta;&amp;psi;</mi> <mn>4</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mn>2</mn> <msub> <mi>T</mi> <mn>1</mn> </msub> <msub> <mi>&amp;delta;&amp;psi;</mi> <mn>1</mn> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>T</mi> <mn>2</mn> </msub> <msub> <mi>&amp;delta;&amp;psi;</mi> <mn>2</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mo>&amp;lsqb;</mo> <msub> <mi>T</mi> <mn>2</mn> </msub> <mo>+</mo> <mrow> <mo>(</mo> <mn>2</mn> <msub> <mi>T</mi> <mn>1</mn> </msub> <mo>-</mo> <mn>2</mn> <msub> <mi>T</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <msub> <mi>&amp;delta;</mi> <mn>1</mn> </msub> <mo>&amp;rsqb;</mo> <msub> <mi>&amp;delta;&amp;theta;</mi> <mn>1</mn> </msub> <mo>+</mo> <mo>&amp;lsqb;</mo> <mrow> <mo>(</mo> <mn>2</mn> <msub> <mi>T</mi> <mn>1</mn> </msub> <mo>-</mo> <mn>2</mn> <msub> <mi>T</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <msub> <mi>&amp;delta;</mi> <mn>2</mn> </msub> <mo>+</mo> <mrow> <mo>(</mo> <msub> <mi>T</mi> <mn>2</mn> </msub> <mo>-</mo> <mn>2</mn> <msub> <mi>T</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <msub> <mi>&amp;delta;&amp;theta;</mi> <mn>2</mn> </msub> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>39</mn> <mo>)</mo> </mrow> </mrow>
Assuming that diameter-variable wheel has n wheel leg, it is noted that diameter-variable wheel is during reducing, (n-1) not in contact with the ground The virtual work that each wheel leg spring is done in individual wheel leg is identical with contacting the virtual work that the single wheel leg spring on ground is done, then for whole car For wheel system, total virtual work that spring is done is：
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>nT</mi> <mn>1</mn> </msub> <msub> <mi>&amp;delta;&amp;psi;</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>nT</mi> <mn>2</mn> </msub> <msub> <mi>&amp;delta;&amp;psi;</mi> <mn>2</mn> </msub> <mo>+</mo> <msub> <mi>nT</mi> <mn>3</mn> </msub> <msub> <mi>&amp;delta;&amp;psi;</mi> <mn>3</mn> </msub> <mo>+</mo> <msub> <mi>nT</mi> <mn>4</mn> </msub> <msub> <mi>&amp;delta;&amp;psi;</mi> <mn>4</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mn>2</mn> <msub> <mi>nT</mi> <mn>1</mn> </msub> <msub> <mi>&amp;delta;&amp;psi;</mi> <mn>1</mn> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>nT</mi> <mn>2</mn> </msub> <msub> <mi>&amp;delta;&amp;psi;</mi> <mn>2</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mo>&amp;lsqb;</mo> <msub> <mi>nT</mi> <mn>2</mn> </msub> <mo>+</mo> <mrow> <mo>(</mo> <mn>2</mn> <msub> <mi>nT</mi> <mn>1</mn> </msub> <mo>-</mo> <mn>2</mn> <msub> <mi>nT</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <msub> <mi>&amp;delta;</mi> <mn>1</mn> </msub> <mo>&amp;rsqb;</mo> <msub> <mi>&amp;delta;&amp;theta;</mi> <mn>1</mn> </msub> <mo>+</mo> <mo>&amp;lsqb;</mo> <mrow> <mo>(</mo> <mn>2</mn> <msub> <mi>nT</mi> <mn>1</mn> </msub> <mo>-</mo> <mn>2</mn> <msub> <mi>nT</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <msub> <mi>&amp;delta;</mi> <mn>2</mn> </msub> <mo>+</mo> <mrow> <mo>(</mo> <msub> <mi>nT</mi> <mn>2</mn> </msub> <mo>-</mo> <mn>2</mn> <msub> <mi>nT</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <msub> <mi>&amp;delta;&amp;theta;</mi> <mn>2</mn> </msub> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>40</mn> <mo>)</mo> </mrow> </mrow>
Wherein, θ3It can be obtained by following equation
<mrow> <msub> <mi>r</mi> <mn>2</mn> </msub> <mo>&amp;times;</mo> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> </mrow> <mn>2</mn> </mfrac> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>r</mi> <mn>3</mn> </msub> <mo>&amp;times;</mo> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mn>3</mn> </msub> <mo>-</mo> <mfrac> <mrow> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> </mrow> <mn>2</mn> </mfrac> <mo>)</mo> </mrow> <mo>+</mo> <mfrac> <msub> <mi>r</mi> <mn>4</mn> </msub> <mn>2</mn> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>41</mn> <mo>)</mo> </mrow> </mrow>
Sub-step 7：Step 3 Chinese style (19), step 4 Chinese style (24) are added with step 6 Chinese style (40) virtual work can obtain entirely Total virtual work of system.
4. a kind of power-deformational behavior analysis method of the submissive diameter changing mechanism of diameter-variable wheel according to claim 1, it is special Levy and be：The step 3 is specific as follows：
Under generalized coordinates, total virtual work of two degree freedom system is represented by following form：
δ W=Q1δq1+Q2δq2=Q1δθ1+Q2δθ2 (42)
Here QiRepresent generalized force, qiGeneralized coordinates is represented, from the principle of virtual work, when system is in standing balance, each generalized force Must be zero, i.e.,
Qi=0 (43)
It can obtain：
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>M</mi> <mn>1</mn> </msub> <mo>=</mo> <mo>-</mo> <msub> <mi>nT</mi> <mn>2</mn> </msub> <mo>+</mo> <msub> <mi>F</mi> <mi>q</mi> </msub> <mo>&amp;lsqb;</mo> <mfrac> <msub> <mi>r</mi> <mn>2</mn> </msub> <mn>2</mn> </mfrac> <msub> <mi>sin&amp;theta;</mi> <mn>1</mn> </msub> <mo>-</mo> <mfrac> <msub> <mi>r</mi> <mn>3</mn> </msub> <mn>2</mn> </mfrac> <mi>sin</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mn>3</mn> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <msub> <mi>F</mi> <mi>N</mi> </msub> <mo>&amp;lsqb;</mo> <mfrac> <msub> <mi>r</mi> <mn>2</mn> </msub> <mn>2</mn> </mfrac> <msub> <mi>cos&amp;theta;</mi> <mn>1</mn> </msub> <mo>+</mo> <mfrac> <msub> <mi>r</mi> <mn>3</mn> </msub> <mn>2</mn> </mfrac> <mi>cos</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mn>3</mn> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <msub> <mi>F</mi> <mi>q</mi> </msub> <mo>&amp;lsqb;</mo> <msub> <mi>r</mi> <mn>3</mn> </msub> <mi>sin</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> </mrow> <mn>2</mn> </mfrac> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> <mi>cos</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> </mrow> <mn>2</mn> </mfrac> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <msub> <mi>&amp;delta;</mi> <mn>1</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <msub> <mi>F</mi> <mi>N</mi> </msub> <mo>&amp;lsqb;</mo> <msub> <mi>r</mi> <mn>3</mn> </msub> <mi>sin</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> </mrow> <mn>2</mn> </mfrac> <mo>-</mo> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> <mi>sin</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> </mrow> <mn>2</mn> </mfrac> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <msub> <mi>&amp;delta;</mi> <mn>1</mn> </msub> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <msub> <mi>nT</mi> <mn>1</mn> </msub> <mo>-</mo> <mn>2</mn> <msub> <mi>nT</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <msub> <mi>&amp;delta;</mi> <mn>1</mn> </msub> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>44</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>M</mi> <mn>2</mn> </msub> <mo>=</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <msub> <mi>nT</mi> <mn>1</mn> </msub> <mo>-</mo> <mn>2</mn> <msub> <mi>nT</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <msub> <mi>&amp;delta;</mi> <mn>2</mn> </msub> <mo>-</mo> <mrow> <mo>(</mo> <msub> <mi>nT</mi> <mn>2</mn> </msub> <mo>-</mo> <mn>2</mn> <msub> <mi>nT</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>F</mi> <mi>q</mi> </msub> <mo>&amp;lsqb;</mo> <mfrac> <msub> <mi>r</mi> <mn>2</mn> </msub> <mn>2</mn> </mfrac> <msub> <mi>sin&amp;delta;</mi> <mn>2</mn> </msub> <mo>+</mo> <mfrac> <msub> <mi>r</mi> <mn>3</mn> </msub> <mn>2</mn> </mfrac> <mi>sin</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <msub> <mi>F</mi> <mi>q</mi> </msub> <mo>&amp;lsqb;</mo> <msub> <mi>r</mi> <mn>3</mn> </msub> <mi>sin</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> </mrow> <mn>2</mn> </mfrac> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> <mi>cos</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> </mrow> <mn>2</mn> </mfrac> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <msub> <mi>&amp;delta;</mi> <mn>2</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <msub> <mi>F</mi> <mi>N</mi> </msub> <mo>&amp;lsqb;</mo> <mfrac> <msub> <mi>r</mi> <mn>2</mn> </msub> <mn>2</mn> </mfrac> <msub> <mi>cos&amp;theta;</mi> <mn>2</mn> </msub> <mo>+</mo> <mfrac> <msub> <mi>r</mi> <mn>3</mn> </msub> <mn>2</mn> </mfrac> <mi>cos</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <msub> <mi>F</mi> <mi>N</mi> </msub> <mo>&amp;lsqb;</mo> <msub> <mi>r</mi> <mn>3</mn> </msub> <mi>sin</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> </mrow> <mn>2</mn> </mfrac> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> <mi>sin</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> </mrow> <mn>2</mn> </mfrac> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <msub> <mi>&amp;delta;</mi> <mn>2</mn> </msub> </mrow> </mtd> </mtr> </mtable> <mo>.</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>45</mn> <mo>)</mo> </mrow> </mrow> 5
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Cited By (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN111291499A (en) * 2020-03-04 2020-06-16 岭南师范学院 Gas extraction drilling machine modeling method based on multi-body dynamics
CN111846001A (en) * 2020-07-30 2020-10-30 上海交通大学 Wheel-leg variable-structure robot
CN113968101A (en) * 2021-12-03 2022-01-25 北京航空航天大学 Special-shaped combined spring suitable for variable-diameter wheel

Citations (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN101486295A (en) * 2009-02-11 2009-07-22 北京航空航天大学 Reducing mechanism for diameter variable wheel
CN102310710A (en) * 2011-06-03 2012-01-11 北京航空航天大学 Four-bar linkage wheel carrier suitable for diameter-variable semi-walking wheel

Patent Citations (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN101486295A (en) * 2009-02-11 2009-07-22 北京航空航天大学 Reducing mechanism for diameter variable wheel
CN102310710A (en) * 2011-06-03 2012-01-11 北京航空航天大学 Four-bar linkage wheel carrier suitable for diameter-variable semi-walking wheel

Non-Patent Citations (1)

* Cited by examiner, † Cited by third party
Title
CHEN X, GAO F, WANG Z, ET AL.: "Mechanism principle and dynamics simulation on variable diameter walking wheel", 《2011 SECOND INTERNATIONAL CONFERENCE ON DIGITAL MANUFACTURING & AUTOMATION. IEEE》 *

Cited By (5)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN111291499A (en) * 2020-03-04 2020-06-16 岭南师范学院 Gas extraction drilling machine modeling method based on multi-body dynamics
CN111291499B (en) * 2020-03-04 2023-08-01 岭南师范学院 Gas extraction drilling machine modeling method based on multi-body dynamics
CN111846001A (en) * 2020-07-30 2020-10-30 上海交通大学 Wheel-leg variable-structure robot
CN113968101A (en) * 2021-12-03 2022-01-25 北京航空航天大学 Special-shaped combined spring suitable for variable-diameter wheel
CN113968101B (en) * 2021-12-03 2023-11-10 北京航空航天大学 Special-shaped combined spring suitable for variable diameter wheel

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