CN107315869B - Force-deformation behavior analysis method for compliant diameter-variable mechanism of variable-diameter wheel - Google Patents

Force-deformation behavior analysis method for compliant diameter-variable mechanism of variable-diameter wheel Download PDF

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CN107315869B
CN107315869B CN201710480676.9A CN201710480676A CN107315869B CN 107315869 B CN107315869 B CN 107315869B CN 201710480676 A CN201710480676 A CN 201710480676A CN 107315869 B CN107315869 B CN 107315869B
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wheel
force
diameter
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virtual work
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CN107315869A (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 force-deformation behavior analysis method of a flexible reducing mechanism of a variable-diameter wheel, which is characterized by mainly comprising the following steps of: (1) establishing a pseudo rigid body model of a two-degree-of-freedom system according to reasonable assumption of stress conditions in the diameter changing process of the diameter changing mechanism; (2) calculating the virtual work done by each main power, main power moment and torsion spring torque in the system; (3) and obtaining the relation between force and deformation in the diameter changing process of the mechanism according to the virtual work principle. The invention has the beneficial effects that: the relation between the driving torque required by reducing and the wheel diameter can be accurately obtained, and the maximum torque required by reducing can be determined; the force-deformation characteristic obtained by the method can provide a theoretical basis for controlling the wheel diameter change of the variable-diameter wheel.

Description

Force-deformation behavior analysis method for compliant diameter-variable mechanism of variable-diameter wheel
Technical Field
The invention relates to a force-deformation behavior analysis method for a flexible and diameter-variable wheel mechanism, and belongs to the field of force and deformation relation analysis of the flexible and diameter-variable wheel mechanism.
Background
The flexible reducing mechanism is mainly used for realizing the diameter change of the variable-diameter wheel, and is applied to a novel lunar vehicle variable-diameter wheel and a novel multi-purpose mobile platform variable-diameter wheel at present. The flexible reducing mechanism mainly moves by means of deformation of the flexible hinge, so that the problems of abrasion, lubrication, sealing and the like can be avoided, and the structure is light and compact. For example, the invention patent application with the publication number of CN101503044 discloses a mechanical-hydraulic linkage reducing wheel carrier suitable for a variable-diameter wheel, however, the stress analysis of the reducing mechanism is only analyzed by considering the mechanism as a structure, the reducing process is not considered during the wheel walking, and then a safety factor is added according to experience to consider the action of the spring torque, so that the maximum torque required by reducing can only be roughly estimated. No systematic analysis method is available for obtaining the relationship between the driving torque required by reducing and the wheel diameter, so that the wheel diameter change of the reducing mechanism cannot be effectively controlled through the force-deformation characteristic in the reducing process of the mechanism, and further application and popularization of the mechanism and the variable-diameter wheel are hindered.
Disclosure of Invention
Aiming at the existing problems, the invention provides a force-deformation behavior analysis method for a flexible reducing mechanism of a variable-diameter wheel, which is an analysis method for accurately and reliably acquiring the force-deformation characteristic of the flexible reducing mechanism in the reducing process.
The technical scheme adopted by the invention is that the method comprises the following steps in sequence:
step one, establishing a pseudo rigid body model of a two-degree-of-freedom system according to reasonable assumption of stress conditions in the diameter changing process of a diameter changing mechanism;
assuming that the lunar vehicle wheel legs can keep structural stability (the spoke rods keep isosceles trapezoid shape) in the diameter changing process, the diameter changing is carried out at low speed, which can be regarded as quasi-static, and the wheel hub D is connected with the hub through the connecting rodcq1And a hub Dcq2As two cranks; the caster is an arc-shaped caster, so that the caster can be approximately considered to be always in contact with the ground at the center point within the corner range of the caster landing and leaving the ground, and in addition, the radial distance from the center point of the outer edge of the caster to the center point of the caster connecting rod is small and is not ignored, namely, the concentrated force from the ground acts on the center point of the caster connecting rod; the two-degree-of-freedom system takes the wheel center O as the origin of coordinates and the wheel hub Dcq1Angle of rotation theta1And a hub Dcq2Angle of rotation theta2And the flexible hinge is represented by a torsion spring with equivalent rigidity to establish a two-degree-of-freedom pseudo-rigid body model of the flexible reducing mechanism.
The total virtual work of the system can be expressed as:
Figure BDA0001329261990000021
wherein
Figure BDA0001329261990000022
For the ith primary force acting on the mechanism,
Figure BDA0001329261990000023
is the position vector of the ith main force action point to the coordinate origin,
Figure BDA0001329261990000024
namely the virtual displacement, the virtual displacement is,
Figure BDA0001329261990000025
for the i-th active torque acting on the mechanism,
Figure BDA0001329261990000026
in order to be angularly displaced,
Figure BDA0001329261990000027
namely the virtual angular displacement, the angular displacement is reduced,
Figure BDA0001329261990000028
for the ith torsion spring moment, the torsion spring is,
Figure BDA0001329261990000029
in order to be angularly displaced,
Figure BDA00013292619900000210
i.e. a virtual angular displacement.
For a variable diameter wheel system with n wheel legs, this can be expressed in particular as:
Figure BDA00013292619900000211
step two, calculating the virtual work done by each main power, main power moment and torsion spring torque in the system;
substep 1: establishing a vector closed loop equation in a single wheel leg of the reducing mechanism, differentiating the equation and then obtaining the virtual angular displacement theta3And theta1、θ2The relationship between;
from the vector closed loop equation in a single wheel leg of the mechanism we can derive:
r2cosθz+r3cosθ3+r4cosθ4-r5cosθ5-r1cosθ1＝0 (3)
wherein r is1Is a wheel hub Dcq1Radius r2Is a wheel hub Dcq2Radius r3For connecting the wheel hub Dcq2Length of spoke r4Is the length of the caster link, r5For connecting the wheel hub Dcq1Length of spoke, theta3For connecting the wheel hub Dcq2Angle of spoke to the horizontal, theta4Is the angle theta between the caster connecting rod and the horizontal direction5For connecting the wheel hub Dcq1The spoke rods form an included angle with the horizontal direction.
Differentiating the formula (3) to obtain
Figure BDA0001329261990000031
Wherein, the geometric symmetry relationship can obtain:
Figure BDA0001329261990000032
θ5＝θ1+θ2-θ3(6)
obtaining the differential of the formula (5) and the formula (6)
Figure BDA0001329261990000033
θ5＝θ1+θ2-θ3(8)
From the structural symmetry:
r1＝r2(9)
r3＝r5(10)
by bringing formulae (5) to (10) into formula (4), it is possible to obtain:
Figure BDA0001329261990000034
the formulae (5), (6) and (10) carry into formula (11), to
Figure BDA0001329261990000035
Figure BDA0001329261990000041
Namely, it is
θ3＝1θ1+2θ2(14)
Substep 2: writing the main force F and the position of the main force to the origin of coordinates into a vector;
Figure BDA0001329261990000042
wherein, FqFor horizontal forces acting on the ground at the centre point of the outer edge of the wheel foot, FNIs the vertical acting force of the ground to the center point of the outer edge of the wheel foot.
Figure BDA0001329261990000043
Wherein the position vector
Figure BDA0001329261990000044
The mold (A) is as follows:
Figure BDA0001329261990000045
substep 3: to position vector
Figure BDA0001329261990000046
Differentiating to obtain virtual displacement, and comparing the force vector with the virtual displacement
Figure BDA0001329261990000047
The dot product is made by the method,solving the virtual work;
differentiating the formula (17) to obtain
Figure BDA0001329261990000051
The product of the formula (15) and the formula (18) is dot product, so that the virtual work can be obtained:
Figure BDA0001329261990000052
substep 4: differentiating the rotated angle under the action of the main moment, and solving the virtual work done by the main moment;
Θ1＝θ1-θ10(20)
Θ2＝θ2-θ20(21)
wherein, theta10Is a wheel hub Dcq1Initial value of angle of rotation, θ20Is a wheel hub Dcq2And (5) starting the rotation angle.
The differential between equations (20) and (21) is:
Θ1＝θ1(22)
Θ2＝θ2(23)
the main moment does a virtual work:
M1Θ1+M2Θ2(24)
substep 5: differentiating the rotated angle under the action of the spring torque;
ψ1＝(θ3-θ30)-(θ2-θ20) (25)
ψ2＝(θ4-θ40)-(θ3-θ30) (26)
ψ3＝-ψ2(27)
ψ1＝-ψ4(28)
the formula (5) is taken into the formulas (25) to (28) and differentiated to obtain
ψ1＝θ3-θ2(29)
Figure BDA0001329261990000061
Figure BDA0001329261990000062
ψ4＝θ2-θ3(32)
Substep 6: solving the virtual work done by the spring torque;
note that the spring torque during expansion and contraction is in opposite directions, so that when the reducing mechanism is in the expansion state, the spring in a single wheel leg is in the process of returning from the deformed state to the natural state, and the following steps are included:
T1＝k1ψ1＝k1[(θ3-θ30)-(θ2-θ20)](33)
T2＝k2ψ2＝k2[(θ4-θ40)-(θ3-θ30)](34)
wherein k is1Is the spring rate, k, of the spring 12The spring rate of the spring 2.
When reducing mechanism is in the shrink, spring in single round of leg is in the process from natural state to deformation state, has:
T1＝-k1ψ1＝-k1[(θ3-θ30)-(θ2-θ20)](35)
T2＝-k2ψ2＝-k2[(θ4-θ40)-(θ3-θ30)](36)
from structural symmetry, it is easy to know:
T4＝-T1(37)
T3＝-T2(38)
combining equations (14) and (29) to (38), the virtual work done by the spring in a single wheel leg is:
Figure BDA0001329261990000072
assuming that the variable diameter wheel has n wheel legs, it is noted that the virtual work of each wheel leg spring in (n-1) wheel legs which do not contact the ground is the same as the virtual work of a single wheel leg spring which contacts the ground during the diameter changing process of the variable diameter wheel, the total virtual work of the springs is as follows for the whole wheel system:
Figure BDA0001329261990000073
wherein, theta3Can be obtained by the following formula
Figure BDA0001329261990000071
Substep 7: the total virtual work of the whole system can be obtained by adding the virtual work of the step III (19), the step IV (24) and the step VI (40);
thirdly, obtaining the relation between force and deformation in the diameter changing process of the mechanism according to the virtual work principle;
in generalized coordinates, the total imaginary work of a two degree-of-freedom system can be expressed in the form:
W＝Q1q1+Q2q2＝Q1θ1+Q2θ2(42)
here QiRepresents a generalized force, qiRepresenting generalized coordinates, and the principle of virtual work shows that when the system is in static equilibrium, each generalized force must be zero, i.e. the system is in static equilibrium
Qi＝0 (43)
It is possible to obtain:
Figure BDA0001329261990000081
Figure BDA0001329261990000082
compared with the prior art, the invention has the advantages that:
at present, an analysis method for force-deformation behaviors of a flexible reducing mechanism of a variable-diameter wheel mainly comprises the steps of adding a safety factor according to experience according to a traditional free body graphical method to consider the action of spring torque, only roughly estimating the maximum torque required by reducing, and not accurately acquiring the relation between the driving torque required by reducing and the wheel diameter. The force-deformation behavior analysis method of the reducing mechanism based on the pseudo-rigid body model and the virtual work principle, which is provided by the invention, needs to determine the equation number of the relation between the force and the deformation to be far smaller than that of a free body graphical method, and considers that the reducing mechanism is a two-degree-of-freedom system in the reducing process. The method can accurately acquire the relation between the driving torque required by reducing and the wheel diameter, and determine the maximum torque required by reducing; and the force-deformation characteristic obtained by the method can also provide a theoretical basis for controlling the wheel diameter change of the reducing mechanism, and the accurate control of the wheel diameter change of the variable-diameter wheel is realized.
Drawings
FIG. 1 is an operation flow chart of a force-deformation behavior analysis method of a compliant diameter-variable mechanism of a variable diameter wheel according to the present invention.
FIG. 2 is a diagram of a two-degree-of-freedom pseudo-rigid model of the compliant diameter-changing mechanism of the present invention.
FIG. 3 shows the driving torque M of the compliant reducing mechanism of the present invention1And theta1、θ2The relationship between them.
FIG. 4 shows the driving torque M of the compliant reducing mechanism of the present invention2And theta1、θ2The relationship between them.
Detailed Description
For a better understanding of the present invention, the following is further illustrated with reference to the following examples:
in this example, only the contraction process of the variable-diameter wheel is analyzed (the analysis method of the expansion process is the same), and the size parameters of the compliant and variable-diameter mechanism of the known lunar rover variable-diameter wheel with six wheel legs are as follows: wheel hub Dcq1Radius r1And a hub Dcq2Radius r2Equal to 91.7mm and connected with a hub Dcq2Length r of spoke3And is connected withHub Dcq1Length r of spoke5Equal to 124mm, the mounting distance r of the spoke on the caster4Is 64mm, and the hub D is in an opening limit statecq1Initial value of angle of rotation theta10Is at 95 DEG, a hub Dcq2Initial value of angle of rotation theta20Is 25 degrees and is connected with a hub Dcq2Of the spoke to the horizontal30Can be calculated by the following formula (1).
Figure BDA0001329261990000091
The invention relates to an operation flow of a force-deformation behavior analysis method of a flexible and diameter-variable wheel mechanism, which specifically comprises the following steps as shown in figure 1:
step one, establishing a pseudo rigid body model of a two-degree-of-freedom system according to reasonable assumption of stress conditions in the diameter changing process of a diameter changing mechanism;
assuming that the lunar vehicle wheel legs can keep structural stability (the spoke rods keep isosceles trapezoid shape) in the diameter changing process, the diameter changing is carried out at low speed, which can be regarded as quasi-static, and the wheel hub D is connected with the hub through the connecting rodcq1And a hub Dcq2As two cranks; the caster is an arc-shaped caster, so that the caster can be approximately considered to be always in contact with the ground at the center point within the corner range of the caster landing and leaving the ground, and in addition, the radial distance from the center point of the outer edge of the caster to the center point of the caster connecting rod is small and is not ignored, namely, the concentrated force from the ground acts on the center point of the caster connecting rod; the two-degree-of-freedom system takes the wheel center O as the origin of coordinates and the wheel hub Dcq1Angle of rotation theta1And a hub Dcq2Angle of rotation theta2The compliant hinge is represented by a torsion spring with equivalent rigidity, and the established two-degree-of-freedom pseudo-rigid body model of the compliant variable-diameter mechanism is shown in figure 2 (other wheel legs not in contact with the ground are not shown in the figure).
The total virtual work of the system can be expressed as:
Figure BDA0001329261990000101
wherein
Figure BDA0001329261990000102
For the ith primary force acting on the mechanism,
Figure BDA0001329261990000103
is the position vector of the ith main force action point to the coordinate origin,
Figure BDA0001329261990000104
namely the virtual displacement, the virtual displacement is,
Figure BDA0001329261990000105
for the i-th active torque acting on the mechanism,
Figure BDA0001329261990000106
in order to be angularly displaced,
Figure BDA0001329261990000107
namely the virtual angular displacement, the angular displacement is reduced,
Figure BDA0001329261990000108
for the ith torsion spring moment, the torsion spring is,
Figure BDA0001329261990000109
in order to be angularly displaced,
Figure BDA00013292619900001010
i.e. a virtual angular displacement.
For a variable diameter wheel system with 6 wheel legs, this can be expressed in particular as:
Figure BDA00013292619900001011
step two, calculating the virtual work done by each main power, main power moment and torsion spring torque in the system;
substep 1: establishing a vector closed loop equation in a single wheel leg of the reducing mechanism, differentiating the equation and then obtaining the virtual angular displacement theta3And theta1、θ2The relationship between;
from the vector closed loop equation in a single wheel leg of the mechanism we can derive:
r2cosθ2+r3cosθ3+r4cosθ4-r5cosθ5-r1cosθ1＝0 (49)
as shown in FIG. 2, wherein r1Is a wheel hub Dcq1Radius r2Is a wheel hub Dcq2Radius r3For connecting the wheel hub Dcq2Length of spoke r4Is the length of the caster link, r5For connecting the wheel hub Dcq1Length of spoke, theta3For connecting the wheel hub Dcq2Angle of spoke to the horizontal, theta4Is the angle theta between the caster connecting rod and the horizontal direction5For connecting the wheel hub Dcq1The spoke rods form an included angle with the horizontal direction.
Differentiating the formula (4) to obtain
Figure BDA00013292619900001012
Wherein the geometrical symmetry in a single wheel leg can result in:
Figure BDA00013292619900001013
θ5＝θ1+θ2-θ3(52)
obtaining the differential of the formula (6) and the formula (7)
Figure BDA0001329261990000111
θ5＝θ1+θ2-θ3(54)
From the structural symmetry in a single wheel leg:
r1＝r2(55)
r3＝r5(56)
substituting equations (6) to (11) into equation (5) can yield:
Figure BDA0001329261990000112
the formulae (6), (7) and (11) are brought into the formula (12)
Figure BDA0001329261990000113
Figure BDA0001329261990000114
Namely, it is
θ3＝1θ1+2θ2(60)
Substep 2: writing the main force F and the position of the main force to the origin of coordinates into a vector;
Figure BDA0001329261990000115
as shown in fig. 2, wherein a horizontal force F of the ground to the caster link center point is assumedqAt 45.7N, the vertical acting force F of the ground to the center point of the wheel foot connecting rodNIs 57N.
Figure BDA0001329261990000121
As shown in fig. 2, in which the position vector
Figure BDA0001329261990000122
The mold (A) is as follows:
Figure BDA0001329261990000123
substep 3: to position vector
Figure BDA0001329261990000124
Differentiating to obtain virtual displacement, and comparing the force vector with the virtual displacement
Figure BDA0001329261990000125
Performing dot product to obtain the virtual work;
differentiating the formula (18) to obtain
Figure BDA0001329261990000126
The product of the formula (16) and the formula (19) is dot product, so that the virtual work can be obtained:
Figure BDA0001329261990000131
substep 4: differentiating the rotated angle under the action of the main moment, and solving the virtual work done by the main moment;
Θ1＝θ1-θ10(66)
Θ2＝θ2-θ20(67)
the differential between equations (21) and (22) is:
Θ1＝θ1(68)
Θ2＝θ2(69)
the main moment does a virtual work:
M1Θ1+M2Θ2(70)
substep 5: differentiating the rotated angle under the action of the spring torque;
ψ1＝(θ3-θ30)-(θ2-θ20) (71)
ψ2＝(θ4-θ40)-(θ3-θ30) (72)
ψ3＝-ψ2(73)
ψ1＝-ψ4(74)
the formula (6) is taken into the formulas (26) to (29) and differentiated to obtain
ψ1＝θ3-θ2(75)
Figure BDA0001329261990000141
Figure BDA0001329261990000142
ψ4＝θ2-θ3(78)
Substep 6: solving the virtual work done by the spring torque;
because the diameter-variable wheel consists of six wheel legs, when the central symmetry line of a single wheel leg and the Y-axis negative direction form an included angle of [ -30 degrees, 30 degrees DEG ]]When the wheel leg is in the interval, the angle range of the wheel leg between the landing and the leaving of the ground can be obtained by combining the geometrical relationship between the spokes1And theta2The value range of (A):
Figure BDA0001329261990000143
70°＜θ1-θ2＜157° (80)
according to equations (34) and (35), the initial angle when the diameter-changing mechanism starts to contract from the opening limit position can be obtained by combining the ground contact angle range of the single wheel leg and the diameter-changing range of the diameter-changing mechanism (assuming that the wheel diameter varies in the range of 240 to 400 mm):
θ10＝95° (81)
θ20＝25° (82)
when reducing mechanism is in the shrink, spring in single round of leg is in the process from natural state to deformation state, has:
T1＝-k1ψ1＝-k1[(θ3-θ30)-(θ2-θ20)](83)
T2＝-k2ψ2＝-k2[(θ4-θ40)-(θ3-θ30)](84)
wherein the spring rate k of the spring 11Spring rate k of spring 2 at 525 N.mm/°2428 N.mm/°.
As shown in fig. 2, from the structural symmetry in a single wheel leg, it is easy to know that:
T4＝-T1(85)
T3＝-T2(86)
since the variable-diameter wheel has 6 wheel legs, it is noted that in the diameter-changing process of the variable-diameter wheel, the virtual work done by each wheel leg spring in 5 wheel legs without contact with the ground is the same as the virtual work done by a single wheel leg spring in contact with the ground, and if the combination formula (15) and the formulas (30) to (41) are used, the total virtual work done by the springs for the whole wheel system is as follows:
Figure BDA0001329261990000153
wherein, theta3Can be obtained by the following formula
Figure BDA0001329261990000151
Substep 7: adding the virtual work of the step III (20), the step IV (25) and the step VI (42) to obtain the total virtual work of the whole system;
thirdly, obtaining the relation between force and deformation in the diameter changing process of the mechanism according to the virtual work principle;
in generalized coordinates, the total imaginary work of a two degree-of-freedom system can be expressed in the form:
W＝Q1q1+Q2q2＝Q1θ1+Q2θ2(89)
here QiRepresents a generalized force, qiRepresenting generalized coordinates, and the principle of virtual work shows that when the system is in static equilibrium, each generalized force must be zero, i.e. the system is in static equilibrium
Qi＝0 (90)
It is possible to obtain:
Figure BDA0001329261990000152
Figure BDA0001329261990000161
from the known geometric parameters, the force-deformation relationship of the reducing mechanism shown in fig. 3 and 4 can be obtained by combining equations (1) to (46). Wherein FIG. 3 is M1(ordinate) and theta1、θ2Graph of relationship between, M1The maximum value is 30847N mm; wherein FIG. 4 is M2(ordinate) and theta1、θ2Graph of relationship between, M2The maximum value is 28963N mm.

Claims (1)

1. A force-deformation behavior analysis method of a flexible reducing mechanism of a variable-diameter wheel is characterized by comprising the following steps:
the method comprises the following steps:
step one, establishing a pseudo rigid body model of a two-degree-of-freedom system according to reasonable assumption of stress conditions in the diameter changing process of a diameter changing mechanism;
step two, calculating the virtual work done by each main power, main power moment and torsion spring torque in the system;
thirdly, obtaining the relation between force and deformation in the diameter changing process of the mechanism according to the virtual work principle;
the first step is as follows:
assuming that the lunar wheel legs can maintain structural stability in the diameter changing process, the diameter changing is carried out at low speed, which can be regarded as quasi-static, and the wheel hub D is connected with the lunar wheel legscq1And a hub Dcq2As two cranks; the caster is an arc-shaped caster, so that the caster can be approximately considered to be always in contact with the ground at the center point within the corner range of the caster landing and leaving the ground, and in addition, the radial distance from the center point of the outer edge of the caster to the center point of the caster connecting rod is small and is not ignored, namely, the concentrated force from the ground acts on the center point of the caster connecting rod; the two-degree-of-freedom system takes the wheel center O as the origin of coordinates and the wheel hub Dcq1Angle of rotation theta1And a hub Dcq2Angle of rotation theta2The flexible hinge is represented by a torsion spring with equivalent rigidity, and a two-degree-of-freedom pseudo-rigid body model of the flexible reducing mechanism is established;
the total virtual work of the system can be expressed as:
Figure FDA0002484277090000011
wherein
Figure FDA0002484277090000012
For the ith primary force acting on the mechanism,
Figure FDA0002484277090000013
is the position vector of the ith main force action point to the coordinate origin,
Figure FDA0002484277090000014
namely the virtual displacement, the virtual displacement is,
Figure FDA0002484277090000015
for the i-th active torque acting on the mechanism,
Figure FDA0002484277090000016
in order to be angularly displaced,
Figure FDA0002484277090000017
namely the virtual angular displacement, the angular displacement is reduced,
Figure FDA0002484277090000018
for the ith torsion spring moment, the torsion spring is,
Figure FDA0002484277090000019
in order to be angularly displaced,
Figure FDA00024842770900000110
namely virtual angular displacement;
for a variable diameter wheel system with n wheel legs, this can be expressed in particular as:
Figure FDA00024842770900000111
the second step specifically comprises the following substeps:
substep 1: establishing a vector closed loop equation in a single wheel leg of the reducing mechanism, differentiating the equation and then obtaining the virtual angular displacement theta3And theta1、θ2The relationship between;
from the vector closed loop equation in a single wheel leg of the mechanism we can derive:
r2cosθ2+r3cosθ3+r4cosθ4-r5cosθ5-r1cosθ1＝0 (3)
wherein r is1Is a wheel hub Dcq1Radius r2Is a wheel hub Dcq2Radius r3For connecting the wheel hub Dcq2Length of spoke r4Is the length of the caster link, r5For connecting the wheel hub Dcq1Length of spoke, theta3For connecting the wheel hub Dcq2Angle of spoke to the horizontal, theta4Is the angle theta between the caster connecting rod and the horizontal direction5For connecting the wheel hub Dcq1The included angle between the spoke rods and the horizontal direction is formed;
differentiating the formula (3) to obtain
Figure FDA0002484277090000024
Wherein, the geometric symmetry relationship can obtain:
Figure FDA0002484277090000021
θ5＝θ1+θ2-θ3(6)
obtaining the differential of the formula (5) and the formula (6)
Figure FDA0002484277090000022
θ5＝θ1+θ2-θ3(8)
From the structural symmetry:
r1＝r2(9)
r3＝r5(10)
by bringing formulae (5) to (10) into formula (4), it is possible to obtain:
Figure FDA0002484277090000023
the formulae (5), (6) and (10) carry into formula (11), to
Figure FDA0002484277090000031
Figure FDA0002484277090000032
Namely, it is
θ3＝1θ1+2θ2(14)
Substep 2: writing the main force F and the position of the main force to the origin of coordinates into a vector;
Figure FDA0002484277090000033
wherein, FqFor horizontal forces acting on the ground at the centre point of the outer edge of the wheel foot, FNThe vertical acting force of the ground to the center point of the outer edge of the wheel foot;
Figure FDA0002484277090000034
wherein the position vector
Figure FDA0002484277090000035
The mold (A) is as follows:
Figure FDA0002484277090000036
substep 3: to position vector
Figure FDA0002484277090000037
Differentiating to obtain virtual displacement, and comparing the force vector with the virtual displacement
Figure FDA0002484277090000038
Performing dot product to obtain the virtual work;
differentiating the formula (17) to obtain
Figure FDA0002484277090000041
The product of the formula (15) and the formula (18) is dot product, so that the virtual work can be obtained:
Figure FDA0002484277090000042
substep 4: differentiating the rotated angle under the action of the main moment, and solving the virtual work done by the main moment;
Θ1＝θ1-θ10(20)
Θ2＝θ2-θ20(21)
wherein, theta10Is a wheel hub Dcq1Initial value of angle of rotation, θ20Is a wheel hub Dcq2A turning angle initial value;
the differential between equations (20) and (21) is:
Θ1＝θ1(22)
Θ2＝θ2(23)
the main moment does a virtual work:
M1Θ1+M2Θ2(24)
substep 5: differentiating the rotated angle under the action of the spring torque;
ψ1＝(θ3-θ30)-(θ2-θ20) (25)
ψ2＝(θ4-θ40)-(θ3-θ30) (26)
ψ3＝-ψ2(27)
ψ1＝-ψ4(28)
the formula (5) is taken into the formulas (25) to (28) and differentiated to obtain
ψ1＝θ3-θ2(29)
Figure FDA0002484277090000051
Figure FDA0002484277090000052
ψ4＝θ2-θ3(32)
Substep 6: solving the virtual work done by the spring torque;
note that the spring torque during expansion and contraction is in opposite directions, so that when the reducing mechanism is in the expansion state, the spring in a single wheel leg is in the process of returning from the deformed state to the natural state, and the following steps are included:
T1＝k1ψ1＝k1[(θ3-θ30)-(θ2-θ20)](33)
T2＝k2ψ2＝k2[(θ4-θ40)-(θ3-θ30)](34)
wherein k is1Is the spring rate, k, of the spring 12The spring rate of spring 2;
when reducing mechanism is in the shrink, spring in single round of leg is in the process from natural state to deformation state, has:
T1＝-k1ψ1＝-k1[(θ3-θ30)-(θ2-θ20)](35)
T2＝-k2ψ2＝-k2[(θ4-θ40)-(θ3-θ30)](36)
from structural symmetry, it is easy to know:
T4＝-T1(37)
T3＝-T2(38)
combining equations (14) and (29) to (38), the virtual work done by the spring in a single wheel leg is:
Figure FDA0002484277090000061
assuming that the variable diameter wheel has n wheel legs, it is noted that the virtual work of each wheel leg spring in (n-1) wheel legs which do not contact the ground is the same as the virtual work of a single wheel leg spring which contacts the ground during the diameter changing process of the variable diameter wheel, the total virtual work of the springs is as follows for the whole wheel system:
Figure FDA0002484277090000062
wherein, theta3Can be obtained by the following formula
Figure FDA0002484277090000063
Substep 7: the total virtual work of the whole system can be obtained by adding the virtual work of the step III (19), the step IV (24) and the step VI (40);
the third step is as follows:
in generalized coordinates, the total imaginary work of a two degree-of-freedom system can be expressed in the form:
W＝Q1q1+Q2q2＝Q1θ1+Q2θ2(42)
here QiRepresents a generalized force, qiRepresenting generalized coordinates, and the principle of virtual work shows that when the system is in static equilibrium, each generalized force must be zero, i.e. the system is in static equilibrium
Qi＝0 (43)
It is possible to obtain:
Figure FDA0002484277090000071
Figure FDA0002484277090000072
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