CN107182276B - The high rigidity design method of rope drive system in parallel - Google Patents
The high rigidity design method of rope drive system in parallelInfo
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- CN107182276B CN107182276B CN201110016511.9A CN201110016511A CN107182276B CN 107182276 B CN107182276 B CN 107182276B CN 201110016511 A CN201110016511 A CN 201110016511A CN 107182276 B CN107182276 B CN 107182276B
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- 239000002965 rope Substances 0.000 title claims abstract description 96
- 229910000831 Steel Inorganic materials 0.000 claims abstract description 81
- 239000010959 steel Substances 0.000 claims abstract description 81
- 238000004088 simulation Methods 0.000 claims abstract description 21
- 101710031899 moon Proteins 0.000 claims abstract description 13
- 238000006073 displacement reaction Methods 0.000 claims abstract description 9
- 239000000725 suspension Substances 0.000 claims description 26
- 239000011159 matrix material Substances 0.000 claims description 12
- 230000000875 corresponding Effects 0.000 claims description 11
- CWFOCCVIPCEQCK-UHFFFAOYSA-N Chlorfenapyr Chemical compound BrC1=C(C(F)(F)F)N(COCC)C(C=2C=CC(Cl)=CC=2)=C1C#N CWFOCCVIPCEQCK-UHFFFAOYSA-N 0.000 claims description 3
- 238000004458 analytical method Methods 0.000 claims description 3
- 230000001276 controlling effect Effects 0.000 claims description 3
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- 230000001360 synchronised Effects 0.000 claims description 3
- 238000000034 method Methods 0.000 claims description 2
- 230000005611 electricity Effects 0.000 claims 1
- 230000005484 gravity Effects 0.000 description 2
- 230000001133 acceleration Effects 0.000 description 1
- 238000004422 calculation algorithm Methods 0.000 description 1
- 238000004364 calculation method Methods 0.000 description 1
- 238000001514 detection method Methods 0.000 description 1
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- 238000005516 engineering process Methods 0.000 description 1
- 238000010304 firing Methods 0.000 description 1
- 239000000446 fuel Substances 0.000 description 1
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Abstract
The invention discloses a kind of high rigidity design method of rope drive system in parallel, during progress verification experimental verification, realize in the moon land test stand system built to reach and move in the horizontal direction stable purpose for hanging the disk of detector；Steps of the method are：Based on the moon land frame system built, the relative displacement z ' (t) between detector and disk is measured by the encoder VI in vertical mechanism, the rope real-time change amount Δ z (t) on roller is measured by the encoder III in main hoist system simultaneously, ground controller calculates the real-time height z for obtaining disk0(t)；In moonscape simulation region, for R 6 surface drives of evenly over the circumference setting and disk is connected to by 6 steel wire ropes in radius；Controller is according to the real-time height z of disk0(t), calculate the mobile rigidity and rotational stiffness for obtaining disk.
Description
Technical field
The invention belongs to space flight detection design field, and in particular to a kind of high rigidity of rope drive system in parallel
Design method.
Background technology
At present, in the space program of China, to realize soft landing, detector needs to complete in moon overhead
Several actions, including：Hovering, avoidance, slow decline, landing etc..By firing a rocket, thrust is limited,
Lunar orbiter can not carry multiengined fuel again, and this requires whole landing mission will be very short
Completed in time.Therefore, how lunar orbiter completes in the effective coordination of moonscape overhead and accurately these
Action, just becomes the subject matter for realizing soft landing.
Before soft landing is carried out, it is necessary to " hovering, avoidance, slow decline, landing " to lunar orbiter
Verification experimental verification is carried out etc. key operations.Certainly, these experiments can not be carried out on the moon, it is necessary on earth
Complete.Therefore, it is necessary to set up one on earth to analog prober in the space of lunar surface touchdown
Environmental system.
During the key operations simulation before carrying out soft landing to lunar orbiter, lunar orbiter is easy
Influenceed in horizontal plane direction by external interference, so result in whole simulated space environment and there is level
The external force in direction, it is final that built simulation space it is impossible to meet the gravity environment of the moon, that is, is detected
Acceleration of gravity of the device in the 1/6g of moonscape.Therefore, it is necessary to propose a kind of method to realize in the earth
In the landing mission of upper simulation lunar orbiter, attitude stabilization of the lunar orbiter in horizontal plane direction.
The content of the invention
In view of this, the invention provides a kind of high rigidity design method of rope drive system in parallel, with up to
During verification experimental verification is carried out in the moon land test stand system built, realize and visited for hanging
The disk for surveying device moves in the horizontal direction stable purpose.
The present invention to achieve the above object, is adopted the following technical scheme that：
A kind of high rigidity design method of rope drive system in parallel, this method is based on the moon land built
Test stand system, the system includes ground drive system, oblique pull steel wire rope, Rapid Follow-up Systems, main lifting
System, horizontal servomechanism, vertical steel wire rope, horizontal tower, vertical pylon and moonscape simulation region；
Disk, pulling force regulation motor I and pulling force regulation motor II, the two pulling force are provided with Rapid Follow-up Systems
Encoder VI is mounted in regulation motor；Horizontal servomechanism is arranged on the guide rail of horizontal tower,
The upper surface of horizontal servomechanism sets main hoist system, and 6 vertical steel are drawn from the bottom of main hoist system
Cord is connected to the upper surface of disk, draws 6 oblique pull steel wire rope connection ground from the surrounding side of disk and drives
Dynamic system；Moonscape simulation region is disposed with ground between two vertical pylons.
The main hoist system includes 2 servo-drivers, 2 main lifting motors, 2 encoder III, subtracted
Fast machine and roller；Controller in terrestrial operation passes through each servo-driver of cable connection, each servo
Driver is connected to reductor and roller by a main lifting motor, and 2 main lifting motors are operated in synchronous control
An encoder III is installed on mode processed, each main lifting motor.
The ground drive system includes 6 surface drives, and 6 surface drives are evenly distributed in
Radius is on R circumference, R is 60m.Each surface drive is connected to by 1 oblique pull steel wire rope
The ground steel wire suspension centre of disk surrounding；Each surface drive include servo-driver, ground drive motors,
Reductor and reel, the controller are driven by cable connection servo-driver, servo-driver by ground
Motor connection is moved to reductor and reel；Methods described is comprised the following steps that：
S00, controller obtain the real-time height of disk.
Setting up earth coordinates O-XYZ based on the moonscape simulation region is：With moonscape simulation region
Center is origin of coordinates O, using vertical direction as z-axis, using moonscape simulation region place plane as xoy faces,
On this plane, using horizontal direction as x-axis, using perpendicular to x-axis direction as y-axis；The ground is demarcated to drive
The steel wire rope tie point of dynamic device is Pi, i=1,2 ..., 6, the ground steel wire suspension centre demarcated on disk is Qi, with
Horizontal plane where disk sets up disk coordinate system O '-X0Y0Z0。
Each encoder VI obtains the relative displacement z ' (t) between detector and disk and is sent to control by cable
Device processed；When z ' (t) is not equal to relative displacement ideal value, the controller carries out PID calculating according to z ' (t) -3
Δ z ' (t) is obtained, while controlling the main hoist system by cable, the reel in main hoist system is received or is put
Rope regulated quantity is Δ z ' (t)；Otherwise, the reel in main hoist system is without receiving or putting the regulation of rope amount；Meanwhile,
Each encoder III measures the rope real-time change amount Δ z (t) on roller, and is sent to ground control by cable
Device, controller calculates and obtains disk in earth coordinates, and the real-time coordinates on vertical direction are
H+z ' (t)+Δ z (t), and it is designated as z0(t), wherein, h be detector away from moonscape simulate district center initial height
Degree, then real-time height of the disk away from ground is z0(t).The relative displacement ideal value is 3m.
S01, controller calculate the rigidity of disk according to the real-time height of disk.
Crossing a steel wire rope tie point PiWith corresponding suspension centre QiA rectangular coordinate system is set up in plane
XO " Y, PiFor origin of coordinates O ", the positive direction of X-axis points to suspension centre QiProjection in moonscape simulation region,
Y-axis is perpendicular to the direction of x-axis；In the earth coordinates, when Rapid Follow-up Systems make small translation
And minor rotationAfterwards, the energy variation Δ W of Rapid Follow-up Systems is：
Wherein：
For under coordinate system XO " Y, suspension centre QiPoint to steel wire rope tie point PiUnit vector；ROFor the radius of disk；RiFor steel wire rope tie point PiWith accordingly hanging
Point QiThe distance between floor projection；RDFor steel wire rope tie point PiThe radius of place ground circumference；For
Under disk coordinate system, steel wire rope tie point PiVector,βiFor in circle
Under disk coordinate system, disc centre O ' and suspension centre QiThe vector and X constituted0The positive angle of axle；αiFor big
Under ground coordinate system, origin of coordinates O and steel wire rope tie point PiThe vector and the positive angle of X-axis constituted； Represent the unit vector of vertical direction；For in disk coordinate system
Under, suspension centre QiVector；KtFor the rigidity of vertical steel wire rope,E is the modulus of elasticity of steel wire rope,
A is the net sectional area of steel wire rope,L is the initial length of vertical steel wire rope.
In above process, it is contemplated that vectorThe change in direction, by vectorIt is represented by：
Infinitesimal analysis is carried out to formula (7), then had：
For abbreviation formula (8), two unit vectors are introduced below：
So, formula (8) can be converted into：
When Rapid Follow-up Systems make small translationAnd minor rotationAfterwards, due to the pulling force side of oblique pull steel wire rope
To changing, Rapid Follow-up Systems work done W during this3For：
Wherein, FiFor the pulling force of 6 oblique pull steel wire ropes；liFor steel wire rope tie point PiWith corresponding suspension centre QiBetween
Air line distance；Formula (11) is expressed as matrix form, then had：
Wherein,
Similarly, it can obtain the tension variations work done W of vertical steel wire rope4For：
Wherein,
NiFor the initial tension of 6 vertical steel wire ropes；For under disk coordinate system, x0The unit vector of axle；For
Under disk coordinate system, y0The unit vector of axle.
From formula (1), (12) and (16), the tangent stiffness matrix K of rope in parallelKFor：
Wherein, KKFor 6 × 6 matrix, the main diagonal element of the matrix has been followed successively by 3 movements just from top to bottom
Degree, including X-direction rigidity, Y-direction rigidity, Z-direction rigidity and 3 rotational stiffnesses, including θX、θY、
θz。
Beneficial effect：
The method applied in the present invention, based on built moon land test stand system, by change it is quick with
Height of the disk away from ground in dynamic system, and drive Rapid Follow-up Systems to enter water-filling by ground drive system
Flat motion, using the elevation value and value of thrust of the ground steel wire rope obtained by calculating as algorithm input parameter,
The rigidity value of disk is obtained by computing, most the larger operating mode of disk rigidity is used as progress moon land examination at last
Reference value in checking, so as to reach during moon land verification experimental verification is carried out, realizes that disk exists
The purpose of horizontal direction the moving stability.
Brief description of the drawings
Fig. 1 is the structural representation of test stand system provided by the present invention；
Fig. 2 is control principle drawing I provided by the present invention；
Fig. 3 is the distribution schematic diagram of ground drive system；
Fig. 4 is control principle drawing II provided by the present invention；
Wherein, 1- disks, 2- ground drive systems, 3- oblique pull steel wire ropes, 4- Rapid Follow-up Systems, 5- master carries
The system of liter, the horizontal servomechanisms of 6-, the vertical steel wire ropes of 7-, 8- horizontal towers, the vertical pylons of 9-.
Embodiment
The present invention will now be described in detail with reference to the accompanying drawings and examples.
A kind of high rigidity design method of rope drive system in parallel, this method is comprised the following steps that：
(1) based on the moon land frame system built, the real-time height of disk is obtained
As shown in figure 1, moon land frame system include ground drive system 2, oblique pull steel wire rope 3, it is quick with
Dynamic system 4, main hoist system 5, horizontal servomechanism 6, vertical steel wire rope 7, horizontal tower 8, vertical tower
Frame 9 and moonscape simulation region.Wherein：Disk 1, pulling force regulation motor are provided with Rapid Follow-up Systems 4
Encoder VI is mounted in I and pulling force regulation motor II, two pulling force regulation motors.Horizontal servomechanism 6
On the guide rail for being arranged on horizontal tower 8, main hoist system 5 is set in the upper surface of horizontal servomechanism 6；
The upper surface that 6 vertical steel wire ropes 7 are connected to disk 1 is drawn from the bottom of main hoist system 5, from disk 1
Surrounding side draw 6 oblique pull steel wire ropes 3 connection ground drive systems 2.Between two vertical pylons 9
Ground on be disposed with moonscape simulation region.
Using earth coordinates as reference frame, the earth coordinates are：With the center of moonscape simulation region
For origin of coordinates O, using vertical direction as z-axis, plane is xoy face where using moonscape simulation region, at this
In plane, using horizontal direction as x-axis, using perpendicular to x-axis direction as y-axis.
Main hoist system 5 includes 4 servo-drivers, 4 main lifting motors, 4 encoder III, decelerations
Machine and roller, as shown in Figure 2.Controller in terrestrial operation by each servo-driver of cable connection,
Each servo-driver is connected to reductor and roller, 4 main lifting motor work by a main lifting motor
In synchronous control mode, an encoder III is installed on each main lifting motor.Controller passes through cable control
Each servo-driver is made, the corresponding main lifting motor of each servo driver drives passes through decelerator and rolling
Cylinder carries out the control of vertical direction movement to vertical steel wire rope 7.
Each encoder VI obtains the relative displacement z ' (t) between detector and disk 1 and is sent to by cable
Controller.When z ' (t) is not equal to relative displacement ideal value (3m), controller carries out PID according to z ' (t) -3
Calculating obtains Δ z ' (t), while controlling main hoist system 5 by cable, receives the reel in main hoist system 5
Or it is Δ z ' (t) to put rope regulated quantity；Otherwise, the reel in main hoist system 5 is without receiving or putting the regulation of rope amount.
Each encoder III measures the rope real-time change amount Δ z (t) on roller simultaneously, and is sent to ground by cable
Controller, Δ z ' (t) is comprised in Δ z (t).So, disk 1 is can obtain in geodetic coordinates from ground controller
In system, the real-time coordinates on vertical direction are h+z ' (t)+Δ z (t), and are designated as z0(t).Wherein, h is detector
The elemental height of district center is simulated away from moonscape.Can obtain real-time height of the disk 1 away from ground is z0(t)。
The arrangement of ground drive system 2 is as shown in figure 3, ground drive system 2 includes 6 ground driving dresses
Put, 6 surface drives are evenly distributed on the circumference that radius is 60m, each surface drive
The ground steel wire suspension centre of the surrounding of disk 1 is connected to by 1 oblique pull steel wire rope 3,6 oblique pull steel wire ropes 3 exist
The angle between projection in the circumference is 60 °, is " in parallel " connected mode.Ground is demarcated to drive
The steel wire rope tie point of dynamic device is Pi, i=1,2 ..., 6, on disk 1, corresponding ground steel wire suspension centre is Qi,
Disk coordinate system O '-X are set up with the place horizontal plane of disk 10Y0Z0。
Each surface drive includes servo-driver, ground drive motors, reductor and reel, such as Fig. 6
It is shown.Ground controller is connected by cable connection servo-driver, servo-driver by ground drive motors
Reductor and reel are connected to, the oblique pull steel wire rope 3 is drawn from reel.
(2) controller calculates the rigidity of disk
Crossing the steel wire rope tie point P of a surface driveiWith corresponding suspension centre QiPlane (hang down by the plane
Directly in earth coordinates) in set up rectangular coordinate system XO " Y, PiFor origin of coordinates O ", X-axis is just
Point to suspension centre Q in directioniProjection in moonscape simulation region, Y-axis is perpendicular to the direction of X-axis.Built
In vertical earth coordinates, when Rapid Follow-up Systems 4 make small translationAnd minor rotationAfterwards, quickly with
The energy variation Δ W of dynamic system 4 is：
Wherein：
For under coordinate system XO " Y, suspension centre QiPoint to steel wire rope tie point PiUnit vector, i=1,2 ..., 6；ROFor the radius of disk 1；RiFor steel wire rope tie point PiWith it is corresponding
Suspension centre QiThe distance between floor projection；RDFor steel wire rope tie point PiThe radius of place ground circumference；For
Under disk coordinate system, steel wire rope tie point PiVector,βiFor
Under disk coordinate system, disc centre O ' and suspension centre QiThe vector and X constituted0The positive angle of axle；αiFor
Under earth coordinates, origin of coordinates O and steel wire rope tie point PiThe vector and the positive angle of X-axis constituted； Represent the unit vector of vertical direction；For in disk coordinate system
Under, suspension centre QiVector；KtFor the rigidity of vertical steel wire rope 7,E is the springform of steel wire rope
Amount, E=1.2 × 1011Pa；A is the net sectional area of steel wire rope,D=22mm, L are vertical
The initial length of steel wire rope 7.
When Rapid Follow-up Systems 4 make small translationAnd minor rotationAfterwards, it is contemplated that vectorThe change in direction
Change, by vectorIt is represented by：
Infinitesimal analysis is carried out to formula (7), then had：
For abbreviation formula (8), two unit vectors are introduced below：
So, formula (8) can be converted into：
When Rapid Follow-up Systems 4 make small translationAnd minor rotationAfterwards, due to the drawing of oblique pull steel wire rope 3
Force direction changes, the work done W of Rapid Follow-up Systems 4 during this3It is represented by：
Wherein, FiFor the pulling force of 6 oblique pull steel wire ropes 3；liFor steel wire rope tie point PiWith corresponding suspension centre QiIt
Between air line distance.Formula (11) is expressed as matrix form, then had：
Wherein,
Similarly, the tension variations work done that can obtain vertical steel wire rope 7 is W4：
Wherein,
NiFor the initial tension of 6 vertical steel wire ropes 7；For under disk coordinate system, x0The unit vector of axle；
For under disk coordinate system, y0The unit vector of axle.
From formula (1), (12) and (16), the tangent stiffness matrix K of rope in parallelKFor：
Wherein, KKFor 6 × 6 matrix, the main diagonal element of the matrix has been followed successively by 3 movements just from top to bottom
Degree, including X-direction rigidity, Y-direction rigidity, Z-direction rigidity and 3 rotational stiffnesses, including θX、θY、
θz.This six rigidity are using disk coordinate system as referential.
The real-time height z of disk 1 is adjusted by main hoist system 50(t), carried out according to above-mentioned steps (2) firm
Degree is calculated, it is known that, work as z0(t) when being 30m, torsional rigidity θz(unit：Nm/ points) it is maximum, that is, disk
1 is relatively stable when moving in the horizontal direction.
In order to further determine the optimal value of disk rigidity, in above-mentioned selected z0(t) situation for being 30m
Under, change 6 oblique pull steel wire ropes 3 in the tie point position of the surrounding of disk 1, according to above-mentioned steps (3)~
(4) Rigidity Calculation, is carried out, the experimental data of Tables 1 and 2 is obtained.Table 1 is to work as z0(t) it is 30m, oblique pull
When tie point of the steel wire rope 3 on disk 1 biases 15 °, the torsional rigidity θ of disk 1zValue.Table 2 is to work as z0(t)
For 30m, when tie point of the oblique pull steel wire rope 3 on disk 1 biases 10 °, the torsional rigidity θ of disk 1zValue.
Table 1
Table 2
By observing Tables 1 and 2, work as z0(t) it is 30m, and company of the oblique pull steel wire rope 3 on disk 1
When contact biases 15 °, the torsional rigidity θ of disk 1zValue is larger, so, should be by oblique pull steel wire rope 3 in circle
The tie point biasing tie point of 15 °, i.e., two on disk 1 is symmetrically distributed in：By with the central point O of disk 1
The both sides of 3 120 ° of fan-shaped calibration points, i.e., each tie point and corresponding demarcation are divided into for the circumference in the center of circle
The angle of point is 15 °, so that the torsional rigidity θ of disk 1zValue is larger.
In summary, presently preferred embodiments of the present invention is these are only, the guarantor of the present invention is not intended to limit
Protect scope.Within the spirit and principles of the invention, any modification, equivalent substitution and improvements made etc.,
It should be included in the scope of the protection.
Claims (2)
1. a kind of high rigidity design method of rope drive system in parallel, this method based on the moon built
Fall test stand system, the system includes ground drive system (2), oblique pull steel wire rope (3), quick servo-actuated system
Unite (4), main hoist system (5), horizontal servomechanism (6), vertical steel wire rope (7), horizontal tower (8),
Vertical pylon (9) and moonscape simulation region；Disk (1), drawing are provided with Rapid Follow-up Systems (4)
Encoder VI is mounted in power regulation motor I and pulling force regulation motor II, the two pulling force regulation motors；
Horizontal servomechanism (6) is arranged on the guide rail of horizontal tower (8), in horizontal servomechanism (6)
Upper surface sets main hoist system (5), and 6 vertical steel wire ropes (7) are drawn from the bottom of main hoist system (5)
The upper surface of disk (1) is connected to, 6 oblique pull steel wire ropes (3) are drawn from the surrounding side of disk (1)
Connect ground drive system (2)；Moonscape mould is disposed with ground between two vertical pylons (9)
Intend area；
The main hoist system (5) includes 2 servo-drivers, 2 main lifting motors, 2 encoders
III, reductor and roller；Controller in terrestrial operation is by each servo-driver of cable connection, often
Individual servo-driver is connected to reductor and roller by a main lifting motor, and 2 main lifting motors are operated in
An encoder III is installed on synchronous control mode, each main lifting motor；
The ground drive system (2) includes 6 surface drives, and 6 surface drives are equably
It is distributed on the circumference that radius is R, each surface drive is connected to by 1 oblique pull steel wire rope (3)
The ground steel wire suspension centre of disk (1) surrounding；Each surface drive includes servo-driver, ground and driven
Motor, reductor and reel, the controller are passed through by cable connection servo-driver, servo-driver
Ground drive motors are connected to reductor and reel；Characterized in that, methods described is comprised the following steps that：
S00, controller obtain the real-time height of disk；
Setting up earth coordinates O-XYZ based on the moonscape simulation region is：With moonscape simulation region
Center is origin of coordinates O, using vertical direction as z-axis, using moonscape simulation region place plane as xoy faces,
On this plane, using horizontal direction as x-axis, using perpendicular to x-axis direction as y-axis；The ground is demarcated to drive
The steel wire rope tie point of dynamic device is Pi, i=1,2 ..., 6, the ground steel wire suspension centre demarcated on disk (1) is Qi,
Disk coordinate system O '-X are set up with horizontal plane where disk (1)0Y0Z0；
Each encoder VI obtains the relative displacement z ' (t) between detector and disk (1) and transmitted by cable
To controller；Meanwhile, each encoder III measures the rope real-time change amount Δ z (t) on roller, and passes through electricity
Cable is sent to ground controller, and controller calculates and obtains disk (1) in earth coordinates, on vertical direction
Real-time coordinates be h+z ' (t)+Δ z (t), and be designated as z0(t), wherein, h be detector away from moonscape simulation region
The elemental height at center, then real-time height of the disk (1) away from ground is z0(t)；
S01, controller calculate the rigidity of disk according to the real-time height of disk；
Crossing a steel wire rope tie point PiWith corresponding suspension centre QiA rectangular coordinate system is set up in plane
XO " Y, PiFor origin of coordinates O ", the positive direction of X-axis points to suspension centre QiProjection in moonscape simulation region,
Y-axis is perpendicular to the direction of X-axis；In the earth coordinates, when Rapid Follow-up Systems (4) make small
TranslationAnd minor rotationAfterwards, the energy variation Δ W of Rapid Follow-up Systems (4) is：
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</msub>
<mo>&times;</mo>
<msub>
<mover>
<mi>e</mi>
<mo>&OverBar;</mo>
</mover>
<mn>3</mn>
</msub>
</mrow>
</mtd>
<mtd>
<mrow>
<msub>
<mover>
<mi>P</mi>
<mo>&OverBar;</mo>
</mover>
<mn>4</mn>
</msub>
<mo>&times;</mo>
<msub>
<mover>
<mi>e</mi>
<mo>&OverBar;</mo>
</mover>
<mn>4</mn>
</msub>
</mrow>
</mtd>
<mtd>
<mrow>
<msub>
<mover>
<mi>P</mi>
<mo>&OverBar;</mo>
</mover>
<mn>5</mn>
</msub>
<mo>&times;</mo>
<msub>
<mover>
<mi>e</mi>
<mo>&OverBar;</mo>
</mover>
<mn>5</mn>
</msub>
</mrow>
</mtd>
<mtd>
<mrow>
<msub>
<mover>
<mi>P</mi>
<mo>&OverBar;</mo>
</mover>
<mn>6</mn>
</msub>
<mo>&times;</mo>
<msub>
<mover>
<mi>e</mi>
<mo>&OverBar;</mo>
</mover>
<mn>6</mn>
</msub>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>3</mn>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<msubsup>
<mi>J</mi>
<mn>2</mn>
<mi>T</mi>
</msubsup>
<mo>=</mo>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mover>
<mi>k</mi>
<mo>&OverBar;</mo>
</mover>
</mtd>
<mtd>
<mover>
<mi>k</mi>
<mo>&OverBar;</mo>
</mover>
</mtd>
<mtd>
<mover>
<mi>k</mi>
<mo>&OverBar;</mo>
</mover>
</mtd>
<mtd>
<mover>
<mi>k</mi>
<mo>&OverBar;</mo>
</mover>
</mtd>
<mtd>
<mover>
<mi>k</mi>
<mo>&OverBar;</mo>
</mover>
</mtd>
<mtd>
<mover>
<mi>k</mi>
<mo>&OverBar;</mo>
</mover>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msub>
<mover>
<mi>Q</mi>
<mo>&OverBar;</mo>
</mover>
<mn>1</mn>
</msub>
<mo>&times;</mo>
<mover>
<mi>k</mi>
<mo>&OverBar;</mo>
</mover>
</mrow>
</mtd>
<mtd>
<mrow>
<msub>
<mover>
<mi>Q</mi>
<mo>&OverBar;</mo>
</mover>
<mn>2</mn>
</msub>
<mo>&times;</mo>
<mover>
<mi>k</mi>
<mo>&OverBar;</mo>
</mover>
</mrow>
</mtd>
<mtd>
<mrow>
<msub>
<mover>
<mi>Q</mi>
<mo>&OverBar;</mo>
</mover>
<mn>3</mn>
</msub>
<mo>&times;</mo>
<mover>
<mi>k</mi>
<mo>&OverBar;</mo>
</mover>
</mrow>
</mtd>
<mtd>
<mrow>
<msub>
<mover>
<mi>Q</mi>
<mo>&OverBar;</mo>
</mover>
<mn>4</mn>
</msub>
<mo>&times;</mo>
<mover>
<mi>k</mi>
<mo>&OverBar;</mo>
</mover>
</mrow>
</mtd>
<mtd>
<mrow>
<msub>
<mover>
<mi>Q</mi>
<mo>&OverBar;</mo>
</mover>
<mn>5</mn>
</msub>
<mo>&times;</mo>
<mover>
<mi>k</mi>
<mo>&OverBar;</mo>
</mover>
</mrow>
</mtd>
<mtd>
<mrow>
<msub>
<mover>
<mi>Q</mi>
<mo>&OverBar;</mo>
</mover>
<mn>6</mn>
</msub>
<mo>&times;</mo>
<mover>
<mi>k</mi>
<mo>&OverBar;</mo>
</mover>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>5</mn>
<mo>)</mo>
</mrow>
</mrow>
For under coordinate system XO " Y, suspension centre QiPoint to steel wire rope tie point PiUnit vector；ROFor the radius of disk (1)；RiFor steel wire rope tie point PiWith
Corresponding suspension centre QiThe distance between floor projection；RDFor steel wire rope tie point PiThe radius of place ground circumference；
For under disk coordinate system, steel wire rope tie point PiVector,βiFor
Under disk coordinate system, disc centre O ' and suspension centre QiThe vector and X constituted0The positive angle of axle；αiFor
Under earth coordinates, origin of coordinates O and steel wire rope tie point PiThe vector constituted and the positive folder of X-axis
Angle；Represent the unit vector of vertical direction；To be sat in disk
Under mark system, suspension centre QiVector；KtFor the rigidity of vertical steel wire rope (7),E is steel wire rope
Modulus of elasticity, A is the net sectional area of steel wire rope,L is initial for vertical steel wire rope (7)
Length；
In above process, it is contemplated that vectorThe change in direction, by vectorIt is represented by：
<mrow>
<msub>
<mover>
<mi>e</mi>
<mo>&OverBar;</mo>
</mover>
<mi>i</mi>
</msub>
<mo>=</mo>
<mfrac>
<mrow>
<msub>
<mover>
<mi>Q</mi>
<mo>&OverBar;</mo>
</mover>
<mi>i</mi>
</msub>
<mo>-</mo>
<msub>
<mover>
<mi>P</mi>
<mo>&OverBar;</mo>
</mover>
<mi>i</mi>
</msub>
</mrow>
<mrow>
<mo>|</mo>
<msub>
<mover>
<mi>Q</mi>
<mo>&OverBar;</mo>
</mover>
<mi>i</mi>
</msub>
<mo>-</mo>
<msub>
<mover>
<mi>P</mi>
<mo>&OverBar;</mo>
</mover>
<mi>i</mi>
</msub>
<mo>|</mo>
</mrow>
</mfrac>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>7</mn>
<mo>)</mo>
</mrow>
</mrow>
Infinitesimal analysis is carried out to formula (7), then had：
<mrow>
<mi>&Delta;</mi>
<msub>
<mover>
<mi>e</mi>
<mo>&OverBar;</mo>
</mover>
<mi>i</mi>
</msub>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mfrac>
<mrow>
<msub>
<mover>
<mi>e</mi>
<mo>&OverBar;</mo>
</mover>
<mi>i</mi>
</msub>
<mo>&CenterDot;</mo>
<mi>&Delta;</mi>
<msub>
<mover>
<mi>P</mi>
<mo>&OverBar;</mo>
</mover>
<mi>i</mi>
</msub>
</mrow>
<msub>
<mi>l</mi>
<mi>i</mi>
</msub>
</mfrac>
<mo>)</mo>
</mrow>
<msub>
<mover>
<mi>e</mi>
<mo>&OverBar;</mo>
</mover>
<mi>i</mi>
</msub>
<mo>-</mo>
<mfrac>
<mrow>
<mi>&Delta;</mi>
<msub>
<mover>
<mi>P</mi>
<mo>&OverBar;</mo>
</mover>
<mi>i</mi>
</msub>
</mrow>
<msub>
<mi>l</mi>
<mi>i</mi>
</msub>
</mfrac>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>8</mn>
<mo>)</mo>
</mrow>
</mrow>
For abbreviation formula (8), two unit vectors are introduced below：
<mrow>
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<msub>
<mover>
<mi>e</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mn>1</mn>
<mi>i</mi>
</mrow>
</msub>
<mo>=</mo>
<mfrac>
<mrow>
<mover>
<mi>k</mi>
<mo>&OverBar;</mo>
</mover>
<mo>&times;</mo>
<msub>
<mover>
<mi>e</mi>
<mo>&OverBar;</mo>
</mover>
<mi>i</mi>
</msub>
</mrow>
<mrow>
<mo>|</mo>
<mover>
<mi>k</mi>
<mo>&OverBar;</mo>
</mover>
<mo>&times;</mo>
<msub>
<mover>
<mi>e</mi>
<mo>&OverBar;</mo>
</mover>
<mi>i</mi>
</msub>
<mo>|</mo>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msub>
<mover>
<mi>e</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mn>2</mn>
<mi>i</mi>
</mrow>
</msub>
<mo>=</mo>
<mfrac>
<mrow>
<msub>
<mover>
<mi>e</mi>
<mo>&OverBar;</mo>
</mover>
<mi>i</mi>
</msub>
<mo>&times;</mo>
<msub>
<mover>
<mi>e</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mn>1</mn>
<mi>i</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mo>|</mo>
<msub>
<mover>
<mi>e</mi>
<mo>&OverBar;</mo>
</mover>
<mi>i</mi>
</msub>
<mo>&times;</mo>
<msub>
<mover>
<mi>e</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mn>1</mn>
<mi>i</mi>
</mrow>
</msub>
<mo>|</mo>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>9</mn>
<mo>)</mo>
</mrow>
</mrow>
So, formula (8) can be converted into：
<mrow>
<mi>&Delta;</mi>
<msub>
<mover>
<mi>e</mi>
<mo>&OverBar;</mo>
</mover>
<mi>i</mi>
</msub>
<mo>=</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mfrac>
<mrow>
<msub>
<mover>
<mi>e</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mn>1</mn>
<mi>i</mi>
</mrow>
</msub>
<mo>&CenterDot;</mo>
<mi>&Delta;</mi>
<msub>
<mover>
<mi>P</mi>
<mo>&OverBar;</mo>
</mover>
<mi>i</mi>
</msub>
</mrow>
<msub>
<mi>l</mi>
<mi>i</mi>
</msub>
</mfrac>
<mo>)</mo>
</mrow>
<msub>
<mover>
<mi>e</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mn>1</mn>
<mi>i</mi>
</mrow>
</msub>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mfrac>
<mrow>
<msub>
<mover>
<mi>e</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mn>2</mn>
<mi>i</mi>
</mrow>
</msub>
<mo>&CenterDot;</mo>
<mi>&Delta;</mi>
<msub>
<mover>
<mi>P</mi>
<mo>&OverBar;</mo>
</mover>
<mi>i</mi>
</msub>
</mrow>
<msub>
<mi>l</mi>
<mi>i</mi>
</msub>
</mfrac>
<mo>)</mo>
</mrow>
<msub>
<mover>
<mi>e</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mn>2</mn>
<mi>i</mi>
</mrow>
</msub>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>10</mn>
<mo>)</mo>
</mrow>
</mrow>
When Rapid Follow-up Systems (4) make small translationAnd minor rotationAfterwards, due to oblique pull steel wire rope (3)
Direction of pull change, Rapid Follow-up Systems (4) work done W during this3For：
<mrow>
<msub>
<mi>W</mi>
<mn>3</mn>
</msub>
<mo>=</mo>
<mo>-</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<munderover>
<mi>&Sigma;</mi>
<mrow>
<mi>i</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
<mn>6</mn>
</munderover>
<mfrac>
<msub>
<mi>F</mi>
<mi>i</mi>
</msub>
<msub>
<mi>l</mi>
<mi>i</mi>
</msub>
</mfrac>
<mo>&lsqb;</mo>
<msup>
<mrow>
<mo>(</mo>
<msub>
<mover>
<mi>e</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mn>1</mn>
<mi>i</mi>
</mrow>
</msub>
<mo>&CenterDot;</mo>
<mi>&Delta;</mi>
<msub>
<mover>
<mi>P</mi>
<mo>&OverBar;</mo>
</mover>
<mi>i</mi>
</msub>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
<mo>+</mo>
<msup>
<mrow>
<mo>(</mo>
<msub>
<mover>
<mi>e</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mi>2</mi>
<mi>i</mi>
</mrow>
</msub>
<mo>&CenterDot;</mo>
<mi>&Delta;</mi>
<msub>
<mover>
<mi>P</mi>
<mo>&OverBar;</mo>
</mover>
<mi>i</mi>
</msub>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
<mo>&rsqb;</mo>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>11</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein, FiFor the pulling force of 6 oblique pull steel wire ropes (3)；liFor steel wire rope tie point PiWith corresponding suspension centre Qi
Between air line distance；Formula (11) is expressed as matrix form, then had：
<mrow>
<msub>
<mi>W</mi>
<mn>3</mn>
</msub>
<mo>=</mo>
<mo>-</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mo>(</mo>
<msup>
<mover>
<mi>&Delta;</mi>
<mo>&OverBar;</mo>
</mover>
<mi>T</mi>
</msup>
<msubsup>
<mi>J</mi>
<mn>3</mn>
<mi>T</mi>
</msubsup>
<msub>
<mi>K</mi>
<mi>f</mi>
</msub>
<msub>
<mi>J</mi>
<mn>3</mn>
</msub>
<mover>
<mi>&Delta;</mi>
<mo>&OverBar;</mo>
</mover>
<mo>+</mo>
<msup>
<mover>
<mi>&Delta;</mi>
<mo>&OverBar;</mo>
</mover>
<mi>T</mi>
</msup>
<msubsup>
<mi>J</mi>
<mn>4</mn>
<mi>T</mi>
</msubsup>
<msub>
<mi>K</mi>
<mi>f</mi>
</msub>
<msub>
<mi>J</mi>
<mn>4</mn>
</msub>
<mover>
<mi>&Delta;</mi>
<mo>&OverBar;</mo>
</mover>
<mo>)</mo>
</mrow>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>12</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein,
<mrow>
<msubsup>
<mi>J</mi>
<mn>3</mn>
<mi>T</mi>
</msubsup>
<mo>=</mo>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<msub>
<mover>
<mi>e</mi>
<mo>&OverBar;</mo>
</mover>
<mn>11</mn>
</msub>
</mtd>
<mtd>
<msub>
<mover>
<mi>e</mi>
<mo>&OverBar;</mo>
</mover>
<mn>12</mn>
</msub>
</mtd>
<mtd>
<msub>
<mover>
<mi>e</mi>
<mo>&OverBar;</mo>
</mover>
<mn>13</mn>
</msub>
</mtd>
<mtd>
<msub>
<mover>
<mi>e</mi>
<mo>&OverBar;</mo>
</mover>
<mn>14</mn>
</msub>
</mtd>
<mtd>
<msub>
<mover>
<mi>e</mi>
<mo>&OverBar;</mo>
</mover>
<mn>15</mn>
</msub>
</mtd>
<mtd>
<msub>
<mover>
<mi>e</mi>
<mo>&OverBar;</mo>
</mover>
<mn>16</mn>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msub>
<mover>
<mi>P</mi>
<mo>&OverBar;</mo>
</mover>
<mn>1</mn>
</msub>
<mo>&times;</mo>
<msub>
<mover>
<mi>e</mi>
<mo>&OverBar;</mo>
</mover>
<mn>11</mn>
</msub>
</mrow>
</mtd>
<mtd>
<mrow>
<msub>
<mover>
<mi>P</mi>
<mo>&OverBar;</mo>
</mover>
<mn>2</mn>
</msub>
<mo>&times;</mo>
<msub>
<mover>
<mi>e</mi>
<mo>&OverBar;</mo>
</mover>
<mn>12</mn>
</msub>
</mrow>
</mtd>
<mtd>
<mrow>
<msub>
<mover>
<mi>P</mi>
<mo>&OverBar;</mo>
</mover>
<mn>3</mn>
</msub>
<mo>&times;</mo>
<msub>
<mover>
<mi>e</mi>
<mo>&OverBar;</mo>
</mover>
<mn>13</mn>
</msub>
</mrow>
</mtd>
<mtd>
<mrow>
<msub>
<mover>
<mi>P</mi>
<mo>&OverBar;</mo>
</mover>
<mn>4</mn>
</msub>
<mo>&times;</mo>
<msub>
<mover>
<mi>e</mi>
<mo>&OverBar;</mo>
</mover>
<mn>14</mn>
</msub>
</mrow>
</mtd>
<mtd>
<mrow>
<msub>
<mover>
<mi>P</mi>
<mo>&OverBar;</mo>
</mover>
<mn>5</mn>
</msub>
<mo>&times;</mo>
<msub>
<mover>
<mi>e</mi>
<mo>&OverBar;</mo>
</mover>
<mn>15</mn>
</msub>
</mrow>
</mtd>
<mtd>
<mrow>
<msub>
<mover>
<mi>P</mi>
<mo>&OverBar;</mo>
</mover>
<mn>6</mn>
</msub>
<mo>&times;</mo>
<msub>
<mover>
<mi>e</mi>
<mo>&OverBar;</mo>
</mover>
<mn>16</mn>
</msub>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>14</mn>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<msubsup>
<mi>J</mi>
<mn>4</mn>
<mi>T</mi>
</msubsup>
<mo>=</mo>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<msub>
<mover>
<mi>e</mi>
<mo>&OverBar;</mo>
</mover>
<mn>21</mn>
</msub>
</mtd>
<mtd>
<msub>
<mover>
<mi>e</mi>
<mo>&OverBar;</mo>
</mover>
<mn>22</mn>
</msub>
</mtd>
<mtd>
<msub>
<mover>
<mi>e</mi>
<mo>&OverBar;</mo>
</mover>
<mn>23</mn>
</msub>
</mtd>
<mtd>
<msub>
<mover>
<mi>e</mi>
<mo>&OverBar;</mo>
</mover>
<mn>24</mn>
</msub>
</mtd>
<mtd>
<msub>
<mover>
<mi>e</mi>
<mo>&OverBar;</mo>
</mover>
<mn>25</mn>
</msub>
</mtd>
<mtd>
<msub>
<mover>
<mi>e</mi>
<mo>&OverBar;</mo>
</mover>
<mn>26</mn>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msub>
<mover>
<mi>P</mi>
<mo>&OverBar;</mo>
</mover>
<mn>1</mn>
</msub>
<mo>&times;</mo>
<msub>
<mover>
<mi>e</mi>
<mo>&OverBar;</mo>
</mover>
<mn>21</mn>
</msub>
</mrow>
</mtd>
<mtd>
<mrow>
<msub>
<mover>
<mi>P</mi>
<mo>&OverBar;</mo>
</mover>
<mn>2</mn>
</msub>
<mo>&times;</mo>
<msub>
<mover>
<mi>e</mi>
<mo>&OverBar;</mo>
</mover>
<mn>22</mn>
</msub>
</mrow>
</mtd>
<mtd>
<mrow>
<msub>
<mover>
<mi>P</mi>
<mo>&OverBar;</mo>
</mover>
<mn>3</mn>
</msub>
<mo>&times;</mo>
<msub>
<mover>
<mi>e</mi>
<mo>&OverBar;</mo>
</mover>
<mn>23</mn>
</msub>
</mrow>
</mtd>
<mtd>
<mrow>
<msub>
<mover>
<mi>P</mi>
<mo>&OverBar;</mo>
</mover>
<mn>4</mn>
</msub>
<mo>&times;</mo>
<msub>
<mover>
<mi>e</mi>
<mo>&OverBar;</mo>
</mover>
<mn>24</mn>
</msub>
</mrow>
</mtd>
<mtd>
<mrow>
<msub>
<mover>
<mi>P</mi>
<mo>&OverBar;</mo>
</mover>
<mn>5</mn>
</msub>
<mo>&times;</mo>
<msub>
<mover>
<mi>e</mi>
<mo>&OverBar;</mo>
</mover>
<mn>25</mn>
</msub>
</mrow>
</mtd>
<mtd>
<mrow>
<msub>
<mover>
<mi>P</mi>
<mo>&OverBar;</mo>
</mover>
<mn>6</mn>
</msub>
<mo>&times;</mo>
<msub>
<mover>
<mi>e</mi>
<mo>&OverBar;</mo>
</mover>
<mn>26</mn>
</msub>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>15</mn>
<mo>)</mo>
</mrow>
</mrow>
Similarly, it can obtain the tension variations work done W of vertical steel wire rope (7)4For：
<mrow>
<msub>
<mi>W</mi>
<mn>4</mn>
</msub>
<mo>=</mo>
<mo>-</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mo>(</mo>
<msup>
<mover>
<mi>&Delta;</mi>
<mo>&OverBar;</mo>
</mover>
<mi>T</mi>
</msup>
<msubsup>
<mi>J</mi>
<mn>5</mn>
<mi>T</mi>
</msubsup>
<msub>
<mi>K</mi>
<mi>n</mi>
</msub>
<msub>
<mi>J</mi>
<mn>5</mn>
</msub>
<mover>
<mi>&Delta;</mi>
<mo>&OverBar;</mo>
</mover>
<mo>+</mo>
<msup>
<mover>
<mi>&Delta;</mi>
<mo>&OverBar;</mo>
</mover>
<mi>T</mi>
</msup>
<msubsup>
<mi>J</mi>
<mn>6</mn>
<mi>T</mi>
</msubsup>
<msub>
<mi>K</mi>
<mi>n</mi>
</msub>
<msub>
<mi>J</mi>
<mn>6</mn>
</msub>
<mover>
<mi>&Delta;</mi>
<mo>&OverBar;</mo>
</mover>
<mo>)</mo>
</mrow>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>16</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein,
<mrow>
<msub>
<mi>K</mi>
<mi>n</mi>
</msub>
<mo>=</mo>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mfrac>
<msub>
<mi>N</mi>
<mn>1</mn>
</msub>
<mi>L</mi>
</mfrac>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mfrac>
<msub>
<mi>N</mi>
<mn>2</mn>
</msub>
<mi>L</mi>
</mfrac>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mfrac>
<msub>
<mi>N</mi>
<mn>3</mn>
</msub>
<mi>L</mi>
</mfrac>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mfrac>
<msub>
<mi>N</mi>
<mn>4</mn>
</msub>
<mi>L</mi>
</mfrac>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mfrac>
<msub>
<mi>N</mi>
<mn>5</mn>
</msub>
<mi>L</mi>
</mfrac>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mfrac>
<msub>
<mi>N</mi>
<mn>6</mn>
</msub>
<mi>L</mi>
</mfrac>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>17</mn>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<msubsup>
<mi>J</mi>
<mn>5</mn>
<mi>T</mi>
</msubsup>
<mo>=</mo>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mover>
<mi>i</mi>
<mo>&OverBar;</mo>
</mover>
</mtd>
<mtd>
<mover>
<mi>i</mi>
<mo>&OverBar;</mo>
</mover>
</mtd>
<mtd>
<mover>
<mi>i</mi>
<mo>&OverBar;</mo>
</mover>
</mtd>
<mtd>
<mover>
<mi>i</mi>
<mo>&OverBar;</mo>
</mover>
</mtd>
<mtd>
<mover>
<mi>i</mi>
<mo>&OverBar;</mo>
</mover>
</mtd>
<mtd>
<mover>
<mi>i</mi>
<mo>&OverBar;</mo>
</mover>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msub>
<mover>
<mi>P</mi>
<mo>&OverBar;</mo>
</mover>
<mn>1</mn>
</msub>
<mo>&times;</mo>
<mover>
<mi>i</mi>
<mo>&OverBar;</mo>
</mover>
</mrow>
</mtd>
<mtd>
<mrow>
<msub>
<mover>
<mi>P</mi>
<mo>&OverBar;</mo>
</mover>
<mn>2</mn>
</msub>
<mo>&times;</mo>
<mover>
<mi>i</mi>
<mo>&OverBar;</mo>
</mover>
</mrow>
</mtd>
<mtd>
<mrow>
<msub>
<mover>
<mi>P</mi>
<mo>&OverBar;</mo>
</mover>
<mn>3</mn>
</msub>
<mo>&times;</mo>
<mover>
<mi>i</mi>
<mo>&OverBar;</mo>
</mover>
</mrow>
</mtd>
<mtd>
<mrow>
<msub>
<mover>
<mi>P</mi>
<mo>&OverBar;</mo>
</mover>
<mn>4</mn>
</msub>
<mo>&times;</mo>
<mover>
<mi>i</mi>
<mo>&OverBar;</mo>
</mover>
</mrow>
</mtd>
<mtd>
<mrow>
<msub>
<mover>
<mi>P</mi>
<mo>&OverBar;</mo>
</mover>
<mn>5</mn>
</msub>
<mo>&times;</mo>
<mover>
<mi>i</mi>
<mo>&OverBar;</mo>
</mover>
</mrow>
</mtd>
<mtd>
<mrow>
<msub>
<mover>
<mi>P</mi>
<mo>&OverBar;</mo>
</mover>
<mn>6</mn>
</msub>
<mo>&times;</mo>
<mover>
<mi>i</mi>
<mo>&OverBar;</mo>
</mover>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>18</mn>
<mo>)</mo>
</mrow>
</mrow>
4
<mrow>
<msubsup>
<mi>J</mi>
<mn>6</mn>
<mi>T</mi>
</msubsup>
<mo>=</mo>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mover>
<mi>j</mi>
<mo>&OverBar;</mo>
</mover>
</mtd>
<mtd>
<mover>
<mi>j</mi>
<mo>&OverBar;</mo>
</mover>
</mtd>
<mtd>
<mover>
<mi>j</mi>
<mo>&OverBar;</mo>
</mover>
</mtd>
<mtd>
<mover>
<mi>j</mi>
<mo>&OverBar;</mo>
</mover>
</mtd>
<mtd>
<mover>
<mi>j</mi>
<mo>&OverBar;</mo>
</mover>
</mtd>
<mtd>
<mover>
<mi>j</mi>
<mo>&OverBar;</mo>
</mover>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msub>
<mover>
<mi>P</mi>
<mo>&OverBar;</mo>
</mover>
<mn>1</mn>
</msub>
<mo>&times;</mo>
<mover>
<mi>j</mi>
<mo>&OverBar;</mo>
</mover>
</mrow>
</mtd>
<mtd>
<mrow>
<msub>
<mover>
<mi>P</mi>
<mo>&OverBar;</mo>
</mover>
<mn>2</mn>
</msub>
<mo>&times;</mo>
<mover>
<mi>j</mi>
<mo>&OverBar;</mo>
</mover>
</mrow>
</mtd>
<mtd>
<mrow>
<msub>
<mover>
<mi>P</mi>
<mo>&OverBar;</mo>
</mover>
<mn>3</mn>
</msub>
<mo>&times;</mo>
<mover>
<mi>j</mi>
<mo>&OverBar;</mo>
</mover>
</mrow>
</mtd>
<mtd>
<mrow>
<msub>
<mover>
<mi>P</mi>
<mo>&OverBar;</mo>
</mover>
<mn>4</mn>
</msub>
<mo>&times;</mo>
<mover>
<mi>j</mi>
<mo>&OverBar;</mo>
</mover>
</mrow>
</mtd>
<mtd>
<mrow>
<msub>
<mover>
<mi>P</mi>
<mo>&OverBar;</mo>
</mover>
<mn>5</mn>
</msub>
<mo>&times;</mo>
<mover>
<mi>j</mi>
<mo>&OverBar;</mo>
</mover>
</mrow>
</mtd>
<mtd>
<mrow>
<msub>
<mover>
<mi>P</mi>
<mo>&OverBar;</mo>
</mover>
<mn>6</mn>
</msub>
<mo>&times;</mo>
<mover>
<mi>j</mi>
<mo>&OverBar;</mo>
</mover>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>19</mn>
<mo>)</mo>
</mrow>
</mrow>
NiFor the initial tension of 6 vertical steel wire ropes (7)；For under disk coordinate system, x0The unit vector of axle；For under disk coordinate system, y0The unit vector of axle；
From formula (1), (12) and (16), the tangent stiffness matrix K of rope in parallelKFor：
<mrow>
<msub>
<mi>K</mi>
<mi>K</mi>
</msub>
<mo>=</mo>
<mo>-</mo>
<msubsup>
<mi>J</mi>
<mn>1</mn>
<mi>T</mi>
</msubsup>
<msub>
<mi>J</mi>
<mrow>
<mi>k</mi>
<mn>1</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>K</mi>
<mi>t</mi>
</msub>
<msubsup>
<mi>J</mi>
<mn>2</mn>
<mi>T</mi>
</msubsup>
<msub>
<mi>J</mi>
<mn>2</mn>
</msub>
<mo>+</mo>
<msubsup>
<mi>J</mi>
<mi>3</mi>
<mi>T</mi>
</msubsup>
<msub>
<mi>K</mi>
<mi>f</mi>
</msub>
<msub>
<mi>J</mi>
<mn>3</mn>
</msub>
<mo>+</mo>
<msubsup>
<mi>J</mi>
<mi>4</mi>
<mi>T</mi>
</msubsup>
<msub>
<mi>K</mi>
<mi>f</mi>
</msub>
<msub>
<mi>J</mi>
<mn>4</mn>
</msub>
<mo>+</mo>
<msubsup>
<mi>J</mi>
<mi>5</mi>
<mi>T</mi>
</msubsup>
<msub>
<mi>K</mi>
<mi>n</mi>
</msub>
<msub>
<mi>J</mi>
<mn>5</mn>
</msub>
<mo>+</mo>
<msubsup>
<mi>J</mi>
<mn>6</mn>
<mi>T</mi>
</msubsup>
<msub>
<mi>K</mi>
<mi>n</mi>
</msub>
<msub>
<mi>J</mi>
<mn>6</mn>
</msub>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>20</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein, KKFor 6 × 6 matrix, the main diagonal element of the matrix has been followed successively by 3 movements just from top to bottom
Degree, including X-direction rigidity, Y-direction rigidity, Z-direction rigidity and 3 rotational stiffnesses, including θX、θY、
θz。
2. a kind of high rigidity design method of rope drive system in parallel as claimed in claim 1, its feature
Be, in step S00, when z ' (t) is not equal to relative displacement ideal value, the controller according to z ' (t) -3,
Carry out PID calculating and obtain Δ z ' (t), while controlling the main hoist system (5) by cable, make main lifting system
It is Δ z ' (t) that rope regulated quantity is received or put to reel in system (5)；Otherwise, the reel in main hoist system (5) is not
Received or put the regulation of rope amount；The relative displacement ideal value is 3m；The R is 60m.
Publications (1)
| Publication Number | Publication Date |
|---|---|
| CN107182276B true CN107182276B (en) | 2014-06-18 |
Family
ID=
Cited By (5)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN109431791A (en) * | 2018-10-22 | 2019-03-08 | 珠海市万瑙特健康科技有限公司 | Automatic moxibustion system, control method, device, computer equipment and storage medium |
| CN109724482A (en) * | 2019-01-07 | 2019-05-07 | 哈尔滨工业大学 | A kind of reusable rocket landing Work condition analogue equipment for driving parallel robot based on rope |
| CN112987691A (en) * | 2021-02-25 | 2021-06-18 | 北京空间飞行器总体设计部 | Soft landing closed-loop follow-up control test method for surface of extraterrestrial celestial body |
| CN111947901B (en) * | 2020-09-23 | 2022-11-22 | 北京强度环境研究所 | Novel spring steel cable free boundary simulation system |
| CN117921745B (en) * | 2024-03-25 | 2024-05-24 | 中国科学院长春光学精密机械与物理研究所 | Time-varying stiffness base system for multidirectional motion conversion |
Cited By (7)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN109431791A (en) * | 2018-10-22 | 2019-03-08 | 珠海市万瑙特健康科技有限公司 | Automatic moxibustion system, control method, device, computer equipment and storage medium |
| CN109724482A (en) * | 2019-01-07 | 2019-05-07 | 哈尔滨工业大学 | A kind of reusable rocket landing Work condition analogue equipment for driving parallel robot based on rope |
| CN109724482B (en) * | 2019-01-07 | 2021-03-23 | 哈尔滨工业大学 | Recoverable rocket landing condition simulation equipment based on rope-driven parallel robot |
| CN111947901B (en) * | 2020-09-23 | 2022-11-22 | 北京强度环境研究所 | Novel spring steel cable free boundary simulation system |
| CN112987691A (en) * | 2021-02-25 | 2021-06-18 | 北京空间飞行器总体设计部 | Soft landing closed-loop follow-up control test method for surface of extraterrestrial celestial body |
| CN112987691B (en) * | 2021-02-25 | 2023-02-03 | 北京空间飞行器总体设计部 | Soft landing closed-loop follow-up control test method for surface of extraterrestrial celestial body |
| CN117921745B (en) * | 2024-03-25 | 2024-05-24 | 中国科学院长春光学精密机械与物理研究所 | Time-varying stiffness base system for multidirectional motion conversion |
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