CN103345775A - Method for simulating free torsion deformation of hollow elastic cylinder with round section - Google Patents

Abstract

The invention provides a method for simulating free torsion deformation of a hollow elastic cylinder with a round section. According to the method, ridge lines of the hollow elastic cylinder with the round section serve as longitude lines, section lines serve as latitude lines, and the lateral surface of the hollow elastic cylinder with the round section is divided into a series of block-shaped areas through the adjacent longitude lines and the adjacent latitude lines. The equivalent relationship between a thin-coil spiral spring model and the hollow elastic cylinder with the round section under the action that a torque is applied to the free end of the upper end of the hollow elastic cylinder with the round section is established, the deformation of any key point under the torque action is obtained according the corresponding relationship between a relative torsion angle per unit length and the torque, the deformation of any point in any block-shaped area is calculated through an interpolation algorithm, and therefore the model of the twisted elastic cylinder is drawn in an updating mode. The method for simulating the free torsion deformation of the hollow elastic cylinder with the round section is simple in calculation, and is capable of accurately and rapidly calculating the torsion deformation, thereby realizing real-time deformation simulation of the elastic cylinder.

Description

The analogy method of the hollow elastic cylinder FREE TORSION distortion of a kind of round section

Technical field

The invention belongs to field of medical technology, relate in particular to the analogy method of the hollow elastic cylinder FREE TORSION distortion of a kind of round section.

Background technology

Virtual reality is by people's vision, the sense of hearing, power and sense of touch etc., the people is in the environment that a kind of and real world approach very much, experience, experience and estimate scene in the virtual world by virtual unit, power and sense of touch have its outstanding superiority in virtual reality, power and sense of touch make reality environment become true, be unique not only can the accepting the surrounding environment input but also can greatly strengthen the effect of visual expression to the perceptual channel of surrounding environment output of user.Virtual instrument device intelligence degree height, processing power is strong, and reusability is strong, and system cost is low, and is workable, makes it at aspects such as virtual medical treatment, tele-medicine, virtual operations wide prospect be arranged.The power of virtual elastic cylinder and tactilely-perceptible simulation are of paramount importance links in fictitious force and the tactile sense reproduction technology.

Current, many medical diagnosiss all can not guarantee it accurately and security fully, carry out man-machine interaction by flexible fictitious force and tactilely-perceptible equipment, carry out medical simulation, especially to enteron aisle, some column type soft tissues such as blood vessel reverse medical care precesses such as cutting, obtain the result of virtual medical care precess by the information feedback of virtual unit, can help to increase substantially the success ratio of medical operating.Therefore, the torsional deflection virtual analog Study on Technology of elastic cylinder is more significant.

Summary of the invention

The objective of the invention is to overcome the deficiencies in the prior art, the analogy method of the hollow elastic cylinder FREE TORSION distortion of a kind of round section is provided.

For achieving the above object, the technical solution adopted in the present invention is:

The analogy method of the hollow elastic cylinder FREE TORSION distortion of a kind of round section comprises the steps:

Step 1: set up rectangular coordinate system in space, cut apart the hollow elasticity column side face of round section;

The upper end of the hollow elastic cylinder of one round section is set at free end, and the lower end is set at stiff end, is that initial point is set up the XYZ space rectangular coordinate system with the center of circle O of stiff end round section, and wherein, the plane at stiff end place is the XY plane; Choose n bar crest line parallel to each other at the hollow elasticity column side face of round section, and the distance between the adjacent crest line equates that get m and the parallel plane cross section of XY, the cross section comprises the upper and lower side cross section, and the vertical range between the adjacent sections equates;

As key point, the intersection of cross section and described hollow elasticity damaged surface is section line with the intersection point of cross section and crest line; Being warp with the crest line, is parallel with the section line, and adjacent warp, adjacent parallel are divided into boxed area with the hollow elasticity column side face of round section;

Step 2: apply any moment of torsion M at the upper end of the hollow elastic cylinder of round section free end _T, adopt equivalence to dredge circle volute spring method, calculate any key point P _o(x _o, y _o, z _o) displacement variable on X, Y and Z-direction;

If any key point P _o(x _o, y _o, z _o) displacement variable on X, Y and Z-direction is respectively

Then:

${\mathrm{\Δx}}_{p_o}=R\·\cos (\arcsin \frac{y_o}{R}+H_{P_o}\·\mathrm{\θ})-x_o$

$\mathrm{\Δ}{y}_{p_o}=R\·\sin (\arcsin \frac{y_o}{R}+H_{P_o}\·\mathrm{\θ})-y_o$

$\mathrm{\Δ}{z}_{p_o}=\frac{M_T\·R}{{\mathrm{GI}}_F\mathrm{\β}}\sin \mathrm{\α}$

Wherein, R is the hollow elasticity pole section of round section radius,

Be P _o(x _o, y _o, z _o) cross section, place and the fixing vertical range of end face, θ is P _o(x _o, y _o, z _o) angle of twist per unit length on the cross section, place; α is the circular circle volute spring lead angle of dredging; G is the hollow elastic cylinder modulus of shearing of round section, I _FBe the polar moment of inertia of the cylindrical cross-section centre of form, β is the curvature correction factor of the relative round section diameter of the wall thickness of the hollow elastic cylinder of round section;

Step 3: calculate arbitrfary point Q _KAt moment of torsion M _TAct on down, respectively the displacement variable on X-axis, Y-axis, Z-direction;

Set arbitrfary point Q _KCoordinate be

Q _KAt moment of torsion M _TUnder the effect, the displacement variable on X-axis, Y-axis, Z-direction is respectively

Set arbitrfary point Q _KFour key points of place boxed area are respectively A _K, B _K, C _K, D _K, the coordinate of described four key points is respectively

Set

Distance to described four key points is respectively Then described four key points are right respectively Influence coefficient be

The influence coefficient expression formula is as follows:

$\left[\begin{array}{c}{k}_{{\mathrm{AQ}}_K}\\ k_{{\mathrm{BQ}}_K}\\ k_{{\mathrm{CQ}}_K}\\ k_{{\mathrm{DQ}}_K}\end{array}\right]=\left[\begin{array}{c}{d}_{{\mathrm{BQ}}_K}\·d_{{\mathrm{CQ}}_K}\·d_{{\mathrm{DQ}}_K}\\ d_{{\mathrm{AQ}}_K}\·d_{{\mathrm{CQ}}_K}\·d_{{\mathrm{DQ}}_K}\\ d_{{\mathrm{AQ}}_K}\·d_{{\mathrm{BQ}}_K}\·d_{{\mathrm{DQ}}_K}\\ d_{{\mathrm{AQ}}_K}\·d_{{\mathrm{BQ}}_K}\·d_{{\mathrm{CQ}}_K}\end{array}\right]/(d_{{\mathrm{AQ}}_K}\·d_{{\mathrm{BQ}}_K}\·d_{{\mathrm{CQ}}_K}+d_{{\mathrm{BQ}}_K}\·d_{{\mathrm{CQ}}_K}\·d_{{\mathrm{DQ}}_K}+d_{{\mathrm{AQ}}_K}\·d_{{\mathrm{BQ}}_K}\·d_{{\mathrm{DQ}}_K}+d_{{\mathrm{AQ}}_K}\·d_{{\mathrm{CQ}}_K}\·d_{{\mathrm{DQ}}_K})$

And: $k_{{\mathrm{AQ}}_K}+k_{{\mathrm{BQ}}_K}+k_{{\mathrm{CQ}}_K}+k_{{\mathrm{DQ}}_K}=1$

Then

Displacement variable on X-axis, Y-axis, Z-direction respectively

Be expressed as:

$\left[\begin{array}{c}{\mathrm{\Δx}}_{Q_K}\\ {\mathrm{\Δy}}_{Q_K}\\ {\mathrm{\Δz}}_{Q_K}\end{array}\right]=k_{{\mathrm{AQ}}_K}\left[\begin{array}{c}{\mathrm{\Δx}}_{A_K}\\ {\mathrm{\Δy}}_{A_K}\\ {\mathrm{\Δz}}_{A_K}\end{array}\right]+k_{{\mathrm{BQ}}_K}\left[\begin{array}{c}{\mathrm{\Δx}}_{B_K}\\ {\mathrm{\Δy}}_{B_K}\\ {\mathrm{\Δz}}_{B_K}\end{array}\right]+k_{{\mathrm{CQ}}_K}\left[\begin{array}{c}{\mathrm{\Δx}}_{C_K}\\ {\mathrm{\Δy}}_{C_K}\\ {\mathrm{\Δz}}_{C_K}\end{array}\right]+k_{{\mathrm{DQ}}_K}\left[\begin{array}{c}{\mathrm{\Δx}}_{D_K}\\ {\mathrm{\Δy}}_{D_K}\\ {\mathrm{\Δz}}_{D_K}\end{array}\right]$

Wherein, apply any moment of torsion M at the upper end of the hollow elastic cylinder of round section free end _TThe time,

Be respectively key point A _KDisplacement variable on X, Y and Z-direction; Be respectively key point B _KDisplacement variable on X, Y and Z-direction;

Be respectively key point C _KDisplacement variable on X, Y and Z-direction;

Be respectively key point D _KDisplacement variable on X, Y and Z-direction.

In the step 1, described n bar crest line parallel to each other, the span of n is 10 to 360 natural number; Described m and the parallel plane cross section of XY, the span of m is 10 to 360 natural number.

The invention has the beneficial effects as follows: the analogy method that the present invention proposes the hollow elastic cylinder FREE TORSION distortion of a kind of round section, described method is warp with the crest line of the hollow elastic cylinder of round section, be parallel with the section line, by adjacent warp, adjacent parallel the hollow elasticity column side face of round section be divided into a series of boxed area.Apply under the effect of a moment of torsion equivalent relation between the two according to the upper end free end of dredging circle volute spring model and the hollow elastic cylinder of round section; Utilize relative torsional angle and the corresponding relation between the moment of torsion of unit length to obtain the deflection of arbitrary key point under torsional interaction; Adopt interpolation computing method to calculate the interior deflection of any arbitrarily of any boxed area, thereby upgrade the elastic cylinder model of drawing after reversing.The inventive method is calculated simple, can calculate torsional deflection quickly and accurately, realizes the real-time deformation emulation to elastic cylinder.

Description of drawings

Fig. 1 is the analogy method process flow diagram of the hollow elastic cylinder FREE TORSION distortion of a kind of round section of the present invention.

Fig. 2 is the circular circle volute spring model of dredging.

Fig. 3 is that distribution and the interpolation distortion of the interior arbitrfary point of boxed area of any key point of the hollow elastic cylinder of round section calculated.

Fig. 4 is the distortion under the torsional interaction of upper end of key point arbitrarily on the hollow elasticity column side face of round section.

Fig. 5 is the hollow elastic cylinder torsional deflection of round section instance graph, and Fig. 5 (a) is that the hollow elastic cylinder of round section is not subjected to the moment of torsion synoptic diagram, and Fig. 5 (b) is that the hollow elastic cylinder of round section upper end is subjected to the synoptic diagram under the torsional interaction.

Embodiment

Be elaborated below in conjunction with the analogy method of flow process shown in the accompanying drawing to the hollow elastic cylinder FREE TORSION distortion of a kind of round section of the present invention's proposition:

The analogy method of the hollow elastic cylinder FREE TORSION of a kind of round section/distortion, its concrete steps are as follows:

Step 1: set up rectangular coordinate system in space, cut apart the hollow elasticity column side face of round section;

Be provided with the hollow elastic cylinder of a round section, the upper end is free end, the lower end is stiff end, the surface level at stiff end place is the XY plane, be the Z axle with the axis perpendicular to the lower end round section diameter of stiff end, set up the XYZ space rectangular coordinate system, and the initial point of rectangular coordinate system in space is the centre of form O of stiff end, choose n bar crest line parallel to each other at the hollow elasticity column side face of round section, and the distance between the adjacent crest line equates that wherein the span of n is 10～360, gets m and the parallel plane cross section of XY, the cross section comprises the upper and lower side cross section, and the value of section line m is 10～360;

With the intersection point of each cross section and each bar crest line as key point, intersection with cross section and crest line is section line, is warp with the crest line, is parallel with the section line, adjacent warp, adjacent parallel are divided into a series of boxed area with the hollow elasticity column side face of round section, and the boxed area number is:

s=(m-1)·n，m=10,11,12…360，n=10,11,12…360 (1)

Wherein s is the boxed area number that the hollow elasticity column side face of round section is divided into, and m is the cross section number of intercepting, and n is the rib number of lines that the side is evenly divided;

Step 2: by four key points of any K piece boxed area Can obtain arbitrfary point in this zone by interpolation computing method

At moment of torsion M _TUnder the effect, the displacement variable on X-axis, Y-axis, Z-direction is respectively respectively:

Four key points supposing K piece boxed area are respectively

If should in the zone be more arbitrarily

Suppose

Distance to four key points is respectively Then four key points are right Influence coefficient Can be expressed as:

$\left[\begin{array}{c}{k}_{{\mathrm{AQ}}_K}\\ k_{{\mathrm{BQ}}_K}\\ k_{{\mathrm{CQ}}_K}\\ k_{{\mathrm{DQ}}_K}\end{array}\right]=\left[\begin{array}{c}{d}_{{\mathrm{BQ}}_K}\·d_{{\mathrm{CQ}}_K}\·d_{{\mathrm{DQ}}_K}\\ d_{{\mathrm{AQ}}_K}\·d_{{\mathrm{CQ}}_K}\·d_{{\mathrm{DQ}}_K}\\ d_{{\mathrm{AQ}}_K}\·d_{{\mathrm{BQ}}_K}\·d_{{\mathrm{DQ}}_K}\\ d_{{\mathrm{AQ}}_K}\·d_{{\mathrm{BQ}}_K}\·d_{{\mathrm{CQ}}_K}\end{array}\right]/(d_{{\mathrm{AQ}}_K}\·d_{{\mathrm{BQ}}_K}\·d_{{\mathrm{CQ}}_K}+d_{{\mathrm{BQ}}_K}\·d_{{\mathrm{CQ}}_K}\·d_{{\mathrm{DQ}}_K}+d_{{\mathrm{AQ}}_K}\·d_{{\mathrm{BQ}}_K}\·d_{{\mathrm{DQ}}_K}+d_{{\mathrm{AQ}}_K}\·d_{{\mathrm{CQ}}_K}\·d_{{\mathrm{DQ}}_K})---\left(2\right)$

Then

Displacement variable on X-axis, Y-axis, Z-direction respectively

Can be expressed as:

$\left[\begin{array}{c}{\mathrm{\Δx}}_{Q_K}\\ {\mathrm{\Δy}}_{Q_K}\\ {\mathrm{\Δz}}_{Q_K}\end{array}\right]=k_{{\mathrm{AQ}}_K}\left[\begin{array}{c}{\mathrm{\Δx}}_{A_K}\\ {\mathrm{\Δy}}_{A_K}\\ {\mathrm{\Δz}}_{A_K}\end{array}\right]+k_{{\mathrm{BQ}}_K}\left[\begin{array}{c}{\mathrm{\Δx}}_{B_K}\\ {\mathrm{\Δy}}_{B_K}\\ {\mathrm{\Δz}}_{B_K}\end{array}\right]+k_{{\mathrm{CQ}}_K}\left[\begin{array}{c}{\mathrm{\Δx}}_{C_K}\\ {\mathrm{\Δy}}_{C_K}\\ {\mathrm{\Δz}}_{C_K}\end{array}\right]+k_{{\mathrm{DQ}}_K}\left[\begin{array}{c}{\mathrm{\Δx}}_{D_K}\\ {\mathrm{\Δy}}_{D_K}\\ {\mathrm{\Δz}}_{D_K}\end{array}\right]---\left(3\right)$

And $k_{{\mathrm{AQ}}_K}+k_{{\mathrm{BQ}}_K}+k_{{\mathrm{CQ}}_K}+k_{{\mathrm{DQ}}_K}=1---\left(4\right)$

Wherein, meter side's algorithm of four key points of any K piece boxed area can be drawn by step 3;

Step 3: dredge circle volute spring type cylinder models by circle and can find the solution any key point P _o(x _o, y _o, z _o), apply any moment of torsion M at the upper end free end _TEffect under, the displacement variable on X-axis, Y-axis, Z-direction respectively:

Adopt the circular circle volute spring model of dredging to come equivalent simulation, apply a moment of torsion M at the upper end of the hollow elastic cylinder of round section free end _TSituation under, in the boxed area that the hollow elastic cylinder of round section is divided, the displacement variable of any key point;

The upper end free end of dredging circle volute spring model and the hollow elastic cylinder of round section according to circle applies a moment of torsion M _TEffect under, equivalent relation between the two;

Apply a moment of torsion M at the upper end of the hollow elastic cylinder of round section free end _T, then being equivalent to dredge circle volute spring model upper end free end in circle and applying under the effect of a pressure F, the circular circle volute spring of dredging will produce moment of torsion M _T', the upper end free end of dredging circle volute spring model and the hollow elastic cylinder of round section according to circle applies a moment of torsion M _TEffect under, equivalent relation between the two;

M _T′=M _T (5)

Can be by finding the solution M _TUnder ' the effect, the circular circle volute spring angle of twist per unit length θ ' in the horizontal direction that dredges, the upper end free end that can try to achieve the hollow elastic cylinder of round section applies a moment of torsion M _TUnder the effect, the hollow elastic cylinder of round section is angle of twist per unit length θ in the horizontal direction;

θ′=θ (6)

$\mathrm{\θ}=\frac{{\mathrm{FR}\cos }^2\mathrm{\α}}{{2\mathrm{GI}}_F\mathrm{\β}}---\left(7\right)$

Wherein, F is dredging circle volute spring model upper end free end applied pressure for being equivalent to; R is the circular circle volute spring section radius of dredging, and also is the hollow elasticity pole section of round section radius; α is the circular circle volute spring lead angle of dredging; G is the circular circle volute spring modulus of shearing of dredging, and also is the hollow elastic cylinder modulus of shearing of round section; I _FBe cylindrical cross-section centre of form polar moment of inertia, and satisfy:

$I_F=\frac{{\mathrm{\πd}}^4}{32}---\left(8\right)$

Wherein, d is for dredging circle volute spring filament diameter;

β is that the relative round section diameter of the wall thickness of the hollow elastic cylinder of round section is when being not very little, namely enclose compare curvature correction factor β when being not very little of volute spring diameter of section 2R with circular dredging dredging circle volute spring model medi-spring filament diameter d, and β can be expressed as:

$\mathrm{\β}=1+\frac{3{\left(\frac{d}{D}\right)}^2}{16[1-{\left(\frac{d}{D}\right)}^2]}---\left(9\right)$

Wherein, d is the circular circle volute spring filament diameter of dredging, and D is the circular footpath in the circle volute spring of dredging;

According to equivalent relation between the two, the upper end free end of the hollow elastic cylinder of round section applies a moment of torsion M _TAnd circle is dredged the relation that exists between the circle volute spring model upper end free end applied pressure F and can be expressed as:

M _T=FRcosα (10)

Wherein, R is the circular circle volute spring section radius of dredging, and also is the hollow elasticity pole section of round section radius; α is the circular circle volute spring lead angle of dredging;

The upper end free end of the hollow elastic cylinder of round section applies a moment of torsion M _TAfter, any key point P _o(x _o, y _o, z _o) new position P of arrival _e(x _e, y _e, z _e), proportional by the vertical range that torsion angle and this round section of any round section to the lower end is stiff end, obtain from fixing end face vertical range be

The P at place _o(x _o, y _o, z _o) horizontal torsion angle, cross section, place is Wherein, θ is the hollow elastic cylinder of round section angle of twist per unit length in the horizontal direction,

Then apply a moment of torsion M at the upper end of the hollow elastic cylinder of round section free end _TAfter the effect, any key point P _o(x _o, y _o, z _o) along the displacement variable on X-axis, the Y direction:

Can be expressed as respectively:

${\mathrm{\Δx}}_{p_o}=x_e-x_o=R\·\cos (\∂+H_{P_o}\·\mathrm{\θ})-x_o---\left(11\right)$

${\mathrm{\Δy}}_{p_o}=y_e-y_o=R\·\sin (\∂+H_{P_o}\·\mathrm{\θ})-y_o---\left(12\right)$

Wherein, R is the hollow elasticity pole section of round section radius, also is the circular circle volute spring section radius of dredging,

Be P _o(x _o, y _o, z _o) angle between cross section, place centre of form O ' institute's line section and the plane, XOZ place;

$\∂=\mathrm{src}\sin \frac{y_o}{R}---\left(13\right)$

Wherein, y _oBe any key point P _o(x _o, y _o, z _o) at the coordinate of Y-axis, R also is the circular circle volute spring section radius of dredging for dredging circle volute spring section radius;

Any key point P on the hollow elasticity column side face of round section _o(x _o, y _o, z _o) at moment of torsion M _TEffect down perpendicular displacement is dredged the equivalent relation between the two of circle volute spring model and the hollow elastic cylinder of round section also in continuous variation according to circle, and then the hollow elastic cylinder of round section lower end is fixedly the time, and the upper end applies torsional moment M _TUnder the effect, any key point P _o(x _o, y _o, z _o) along the displacement variable on the Z-direction

For:

${\mathrm{\Δz}}_{p_o}=\mathrm{\φ}\·R=\frac{{\mathrm{FR}}^2}{{2\mathrm{GI}}_F\mathrm{\β}}\sin 2\mathrm{\α}---\left(14\right)$

Wherein φ is any key point of the hollow elasticity column side face of round section vertical direction angle changing under twisting action, R also is thin circle volute spring section radius for the circular circle volute spring section radius of dredging, F is upper end applied pressure in the volute spring model, α is the circular circle volute spring lead angle of dredging, G is the circular circle volute spring modulus of shearing of dredging, also be the hollow elastic cylinder modulus of shearing of round section, I _FBe cylindrical cross-section centre of form polar moment of inertia, β is that the wall thickness relative cross-section diameter of the hollow elastic cylinder of round section is not very little, namely at compare with spring diameter of section 2R curvature correction factor β when being not very little of spring model medi-spring filament diameter d;

Step 4 can obtain any some coordinate in three dimensions in each key point and each boxed area according to step 2,3, upgrades the elastic cylinder model of drawing after reversing.

Do corresponding detailed description of drawings according to this invention:

Fig. 1 is the analogy method process flow diagram of the hollow elastic cylinder FREE TORSION distortion of a kind of round section of the present invention.Whole working of an invention procedure is showed with process flow diagram, and namely detailed process of the present invention is divided into: the distance of choosing on the hollow elasticity column side face of round section between n bar crest line and the adjacent crest line equates; Choosing m equates with distance between the parallel plane cross section of XY and the adjacent sections; The hollow elasticity column side face of round section is divided into a series of boxed area; The hollow elastic cylinder of round section lower end is fixed, and the upper end applies moment of torsion M _TAdopt and dredge the displacement variable that circle volute spring equivalent model comes any key point of Equivalent Calculation; By key point interpolation calculation in any K piece boxed area should the zone in the arbitrfary point displacement variable; Upgrade the elastic cylinder model of drawing after reversing;

Fig. 2 is that the model of equivalence among the present invention is round screw thread spring model synoptic diagram, wherein, α is the circular circle volute spring lead angle of dredging, R also is the hollow elasticity pole section of round section radius for the circular circle volute spring section radius of dredging, d is the circular circle volute spring filament diameter of dredging, D is the circular footpath in the circle volute spring of dredging, the torsional deflection that the circular hollow elastic cylinder produces under the effect of upper end moment of torsion among the present invention is subjected to corresponding pressure equivalence with the circular circle volute spring model of dredging in the upper end, change plan and show the concrete manifestation spring model, and to the thin circle of circle volute spring model lead angle α, the circular circle volute spring section radius R that dredges, the circular circle volute spring filament diameter d that dredges, footpath D has done concrete marking explanation in the circular thin circle volute spring;

Fig. 3 is any K piece of the hollow elastic cylinder of round section boxed area distribution situation, and four key points of this boxed area are respectively: A _K, B _K, C _K, D _K, the arbitrfary point is Q in this boxed area _K, can show four key points to arbitrfary point Q by synoptic diagram _KThe influence power magnitude relationship, thereby according to this interpolation thought, calculate this arbitrfary point Q _KThe displacement variable on X-axis, Y-axis, Z-direction under torsional interaction;

Fig. 4 is any key point P on the hollow elasticity column side face of round section _o(x _o, y _o, z _o) distortion synoptic diagram under the torsional interaction of upper end, this any key point P _o(x _o, y _o, z _o) vertical range of end section is under the hollow elastic cylinder of distance circle tee section

At the hollow elastic cylinder of round section upper end moment of torsion M _TThe horizontal torsion angle that effect produces down is

Fig. 5 is the hollow elastic cylinder torsional deflection of round section instance graph, Fig. 5 (a) spatial model figure is the hollow elasticity type cylinder models of round section figure before torsional interaction, the initial point of following end section rectangular coordinate system in space is that the centre of form of stiff end is O, and Fig. 5 (b) is for obtaining in the key point of other boxed area and the corresponding boxed area the hollow elastic cylinder of arbitrfary point round section upper end at moment of torsion M according to the inventive method _TDisplacement variable after the effect obtains the hollow elastic cylinder distortion of round section back model synoptic diagram thereby upgrade.

Claims (2)

1. the analogy method of the hollow elastic cylinder FREE TORSION distortion of round section is characterized in that, comprises the steps:

Step 1: set up rectangular coordinate system in space, cut apart the hollow elasticity column side face of round section;

The upper end of the hollow elastic cylinder of one round section is set at free end, and the lower end is set at stiff end, is that initial point is set up the XYZ space rectangular coordinate system with the center of circle O of stiff end round section, and wherein, the plane at stiff end place is the XY plane; Choose n bar crest line parallel to each other at the hollow elasticity column side face of round section, and the distance between the adjacent crest line equates that get m and the parallel plane cross section of XY, the cross section comprises the upper and lower side cross section, and the vertical range between the adjacent sections equates;

As key point, the intersection of cross section and described hollow elasticity damaged surface is section line with the intersection point of cross section and crest line; Being warp with the crest line, is parallel with the section line, and adjacent warp, adjacent parallel are divided into boxed area with the hollow elasticity column side face of round section;

Step 2: apply any moment of torsion M at the upper end of the hollow elastic cylinder of round section free end _T, adopt equivalence to dredge circle volute spring method, calculate any key point P _o(x _o, y _o, z _o) displacement variable on X, Y and Z-direction;

If any key point P _o(x _o, y _o, z _o) displacement variable on X, Y and Z-direction is respectively

Then:

${\mathrm{\Δx}}_{p_o}=R\·\cos (\arcsin \frac{y_o}{R}+H_{P_o}\·\mathrm{\θ})-x_o$

${\mathrm{\Δy}}_{p_o}=R\·\sin (\arcsin \frac{y_o}{R}+H_{P_o}\·\mathrm{\θ})-y_o$

${\mathrm{\Δz}}_{p_o}=\frac{M_T\·R}{{\mathrm{GI}}_F\mathrm{\β}}\sin \mathrm{\α}$

Wherein, R is the hollow elasticity pole section of round section radius,

Be P _o(x _o, y _o, z _o) cross section, place and the fixing vertical range of end face, θ is P _o(x _o, y _o, z _o) angle of twist per unit length on the cross section, place; α is for dredging circle volute spring lead angle; G is the hollow elastic cylinder modulus of shearing of round section, I _FBe the polar moment of inertia of the cylindrical cross-section centre of form, β is the curvature correction factor of the relative round section diameter of the wall thickness of the hollow elastic cylinder of round section;

Step 3: calculate arbitrfary point Q in any K piece boxed area _KAt moment of torsion M _TAct on down, respectively the displacement variable on X-axis, Y-axis, Z-direction;

Set arbitrfary point Q _KCoordinate be

Q _KAt moment of torsion M _TUnder the effect, the displacement variable on X-axis, Y-axis, Z-direction is respectively

Set arbitrfary point Q _KFour key points of place boxed area are respectively A _K, B _K, C _K, D _K, the coordinate of described four key points is respectively

Set

Distance to described four key points is respectively

Then described four key points are right respectively

Influence coefficient be

The influence coefficient expression formula is as follows:

$\left[\begin{array}{c}{k}_{{\mathrm{AQ}}_K}\\ k_{{\mathrm{BQ}}_K}\\ k_{{\mathrm{CQ}}_K}\\ k_{{\mathrm{DQ}}_K}\end{array}\right]=\begin{array}{c}\\ \end{array}\left[\begin{array}{c}{d}_{{\mathrm{BQ}}_K}\·d_{{\mathrm{CQ}}_K}\·d_{{\mathrm{DQ}}_K}\\ d_{{\mathrm{AQ}}_K}\·d_{{\mathrm{CQ}}_K}\·d_{{\mathrm{DQ}}_K}\\ d_{{\mathrm{AQ}}_K}\·d_{{\mathrm{BQ}}_K}\·d_{{\mathrm{DQ}}_K}\\ d_{{\mathrm{AQ}}_K}\·d_{{\mathrm{BQ}}_K}\·d_{{\mathrm{CQ}}_K}\end{array}\right]/(d_{{\mathrm{AQ}}_K}\·d_{{\mathrm{BQ}}_K}\·d_{{\mathrm{CQ}}_K}+d_{{\mathrm{BQ}}_K}\·d_{{\mathrm{CQ}}_K}\·d_{{\mathrm{DQ}}_K}+d_{{\mathrm{AQ}}_K}\·d_{{\mathrm{BQ}}_K}\·d_{{\mathrm{DQ}}_K}+d_{{\mathrm{AQ}}_K}\·d_{{\mathrm{CQ}}_K}\·d_{{\mathrm{DQ}}_K})$

And: $k_{{\mathrm{AQ}}_K}+k_{{\mathrm{BQ}}_K}+k_{{\mathrm{CQ}}_K}+k_{{\mathrm{DQ}}_K}=1$

Then

Displacement variable on X-axis, Y-axis, Z-direction respectively Be expressed as:

$\left[\begin{array}{c}{\mathrm{\Δx}}_{Q_K}\\ {\mathrm{\Δy}}_{Q_K}\\ {\mathrm{\Δz}}_{Q_K}\end{array}\right]=k_{{\mathrm{AQ}}_K}\left[\begin{array}{c}{\mathrm{\Δx}}_{A_K}\\ {\mathrm{\Δy}}_{A_K}\\ {\mathrm{\Δz}}_{A_K}\end{array}\right]+k_{{\mathrm{BQ}}_K}\left[\begin{array}{c}{\mathrm{\Δx}}_{B_K}\\ {\mathrm{\Δy}}_{B_K}\\ {\mathrm{\Δz}}_{B_K}\end{array}\right]+k_{{\mathrm{CQ}}_K}\left[\begin{array}{c}{\mathrm{\Δx}}_{C_K}\\ {\mathrm{\Δy}}_{C_K}\\ {\mathrm{\Δz}}_{C_K}\end{array}\right]+k_{{\mathrm{DQ}}_K}\left[\begin{array}{c}{\mathrm{\Δx}}_{D_K}\\ {\mathrm{\Δy}}_{D_K}\\ {\mathrm{\Δz}}_{D_K}\end{array}\right]$

Wherein, apply any moment of torsion M at the upper end of the hollow elastic cylinder of round section free end _TThe time, Be respectively key point A _KDisplacement variable on X, Y and Z-direction;

Be respectively key point B _KDisplacement variable on X, Y and Z-direction;

Be respectively key point C _KDisplacement variable on X, Y and Z-direction;

Be respectively key point D _KDisplacement variable on X, Y and Z-direction.

2. the analogy method of the hollow elastic cylinder FREE TORSION distortion of a kind of round section according to claim 1 is characterized in that, in the step 1, and described n bar crest line parallel to each other, the span of n is 10 to 360 natural number; Described m and the parallel plane cross section of XY, the span of m is 10 to 360 natural number.

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