CN104933257A - Analysis method of wire system distortion rule - Google Patents

Abstract

The invention discloses an analysis method of a wire system distortion rule. The analysis method specifically comprises the following steps: establishing a wire model; on the basis of a [pi] theorem, exporting a dimensionless item of a kinetic characteristic parameter in the wire model; according to the obtained dimensionless item, establishing a similarity relationship between a wire prototype and the wire model; on the basis of a Buckingham theory, establishing a similarity equation about [pi] after the dimensionless item is functional; taking the obtained similarity relationship as a basis for judging whether the wire model is subjected to distortion or not, importing a distortion coefficient into the obtained similarity equation to obtain the wire model distortion rule for the wire mode subjected to the distortion; and according to the obtained distortion rule, after a model similar to the kinetic performance of the wire prototype is constructed, analyzing the kinetic performance of the model to obtain the relevant performance of the prototype.

Description

A kind of analytical approach of cable system Distortion Law

Technical field

The invention belongs to distortion theory analysis techniques field, relate to a kind of analytical approach of cable system Distortion Law.

Background technology

In the past in decades, many researchers to derive the physical characteristics of prototype by using model like little but dynamical phase, and result of study display is done like this and greatly can be reduced financial cost.Such as automobile industry, there is many scale models now to complete the prediction of the car prortotype such as characteristic such as rectilinearity, handling, ride quality.Therefore, in Practical Project, in order to analyze the power performance of large scale prototype, needing to build the model similar to prototype dynamic performance, being judged the correlated performance of prototype by the dynamic performance of analytical model.But find in reality test, there is distortion between model and prototype, only obtain distortion model and distortion factor just go to predict prototype performance by model, in order to obtain Wire driven robot parallel institution (Wire Driven Parallel Manipulator, be called for short WDPM) dynamic performance, need to adopt contracting to analyze than modeling method, but when carrying out contracting than modeling, first Water demand goes out the Distortion Law of cable system, just draws the dynamics of rope prototype by the dynamics of rope model.

Summary of the invention

The object of this invention is to provide a kind of analytical approach of cable system Distortion Law, according to the Distortion Law drawn, after building the model similar to rope prototype dynamic performance, by the dynamic performance of analytical model, thus the correlated performance of prototype can be known by inference.

The technical solution adopted in the present invention is, a kind of analytical approach of cable system Distortion Law, specifically comprises the following steps:

Step 1, sets up rope model;

Step 2, derives the dimensionless item of described rope model medium power characteristic parameter based on Buckingham theorem;

Step 3, the dimensionless item obtained according to step 2 sets up the similarity relation between rope prototype and described rope model;

Step 4, theoretical based on Buckingham, about the Similarity equations of π after the dimensionless item minuend that establishment step 2 obtains;

Step 5, the similarity relation obtained using step 3, as judging whether described rope model the foundation distorted occurs, to the rope model that distortion occurs, introduces by the Similarity equations obtained step 4 Distortion Law that distortion factor draws described rope model.

Feature of the present invention is also,

Wherein the detailed process of step 1 is:

Step 1.1, chooses seven rope driven Parallel Kinematic Manipulator as analytic target in WDPM model, and use Lagrangian method to obtain the kinetics equation of described seven rope driven Parallel Kinematic Manipulator under fixing pose, kinetics equation is as follows:

$M\mathrm{\δ}\stackrel{\·\·}{x}+C(\mathrm{\δ}x,\mathrm{\δ}\stackrel{\·}{x})\mathrm{\δ}\stackrel{\·}{x}+K\mathrm{\δ}x={\mathrm{\δF}}_e---\left(3\right);$

Wherein, $M=\[\begin{array}{cc}{m}_{c}I& -m_c{r}_w\×\\ m_c{r}_w\×& m_{c}I\end{array}\],C(\mathrm{\δ}x,\mathrm{\δ}\stackrel{\·}{x})=\[\begin{array}{cc}0& m_c(\mathrm{\ω}\×r_w)\×\\ m_c(\mathrm{\ω}\×r_w)\×& \left(I\mathrm{\ω}\right)\×\end{array}\],$ $J^T=\left[\begin{array}{ccc}{u}_1& ...& u_6\\ r_1\×u_1& ...& r_6\×u_6\end{array}\right],$ represent diagonal matrix, cornerwise numerical value is k entirely ₀, it is the elastic stiffness (N/m) of every root drag-line; δ F _efor cutting force;

Step 1.2, makes the δ F in described kinetics equation _ebe zero, obtain the free vibration differential equation of described seven rope driven Parallel Kinematic Manipulator, the described differential equation is the rope model of structure, and the differential equation obtained is as follows:

$M\mathrm{\δ}\stackrel{\·\·}{x}+C(\mathrm{\δ}x,\mathrm{\δ}\stackrel{\·}{x})\mathrm{\δ}\stackrel{\·}{x}+K\mathrm{\δ}x=0---\left(5\right).$

Wherein the concrete derivation of the dimensionless item of the rope model medium power characteristic parameter of step 2 is as follows:

Based on statics research object:

For the element stiffness equation in described rope model, dimensionless item is derived as follows:

Wherein, P, M represent concentrated force and concentrated moment respectively, use represent linear displacement and angle displacement, l, A, E, I represent length, area, elastic section modulus, inertia torque respectively;

When putting on described rope model for gravity as external force, dimensionless item is derived as follows:

${\mathrm{\π}}_5=\frac{P}{\mathrm{\ρ}Al}---\left(7\right);$

For cable wire material, rope strain and Suo Yingli meet following formula:

$\mathrm{\ϵ}=\frac{P}{EA}+\frac{M}{EW\mathrm{\ϵ}}---\left(8\right);$

Wherein, ε represents that rope strains, and EW represents bending stiffness;

σ＝Eε (9)；

Wherein, σ represents Suo Yingli;

Dimensionless item is derived as follows:

$\left\{\begin{array}{c}{\mathrm{\π}}_6\frac{P}{EA\mathrm{\ϵ}}\\ {\mathrm{\π}}_7\frac{M}{EW\mathrm{\ϵ}}\\ {\mathrm{\π}}_8\frac{\mathrm{\σ}}{E\mathrm{\ϵ}}\end{array}\right.---\left(10\right);$

Based on dynamics research object:

Steps A, ignores the damping in described rope model, and according to model analysis, draw dynamics research model by described rope model, described kinetic model formula is as follows:

[K-ω ²M]{x}＝0 (11)；

Wherein, the rigidity element in stiffness matrix K comprises m is mass matrix, and in mass matrix, matrix element comprises rope density p, rope area A, the long l of rope, and { x} is displacement array, and ω represents angular frequency;

Step B, the kinetic model obtained according to steps A, draws following dimensionless item

${\mathrm{\π}}_9=\frac{EA}{l^2{\mathrm{\ω}}^2\mathrm{\ρ}A}---\left(12\right).$

Wherein the detailed process of step 3 is:

Step 3.1, set up the statics similarity relation between rope prototype and described rope model, detailed process is as follows:

Make λ _l=λ, λ _{Δ l}=λ and λ _p=λ _eλ ²; Wherein, λ _lrepresent the contracting ratio of Suo Changdu, λ _{Δ l}represent the contracting ratio of rope length variations, λ _prepresent the contracting ratio of the change of power between rope model and prototype; The dimensionless item π obtained by step 2 ₁, π ₂, π ₃, π ₄, π ₅, π ₆, π ₇, π ₈show that the contracting ratio of following parameter is:

In formula (13), the contracting ratio of each parameter is the statics similarity relation between rope model and prototype;

Step 3.2, set up the kinematic similarity relation between rope prototype and described rope model, detailed process is as follows:

According to the dimensionless item π obtained in step 2 ₉the statics similarity relationships obtained with step 3.1 draws the contracting ratio λ of angular frequency _ω, bring in formula (13) by formula (12) and draw λ _ω, the contracting ratio λ of angular frequency _ωfor:

${\mathrm{\λ}}_{\mathrm{\ω}}=\sqrt{\mathrm{\λ}}---\left(14\right);$

The contracting ratio λ of angular frequency _ωbe the kinematic similarity sexual intercourse between rope prototype and described rope model.

In step 4 about the Similarity equations of π be wherein:

Wherein, C ' _nrepresent based on π ₁, π ₂, π ₃, π ₄, π ₅, π ₆, π ₇, π ₈matching π ₉function dimensionless constant, a ₁, a ₂, a ₃, a ₄, a ₅, a ₆, a ₇, a ₈undetermined coefficient.

Detailed process in step 5 is:

Step 5.1, judges whether described rope model distorts, and detailed process is as follows:

By the dimensionless item π in described rope model ₁, π ₂, π ₃, π ₄, π ₅, π ₆, π ₇, π ₈middle any one substitutes in the statics similarity relation between rope model and rope prototype obtained in step 3 and verifies, works as π ₁, π ₂, π ₃, π ₄, π ₅, π ₆, π ₇, π ₈when middle any one can not meet the statics similarity relation between rope model and rope prototype that step 3 obtains, then judge to exist between rope model and rope prototype to distort;

, there is the judged result of distortion based on step 5.1, introduce distortion factor δ to what obtain in step 4 about in the Similarity equations of π in step 5.2 _i, draw the Distortion Law of described rope model, the equation of Distortion Law is as follows:

The invention has the beneficial effects as follows, in the present invention with seven rope driven Parallel Kinematic Manipulator in WDPM model for model, dimensionless item in model is quantitatively deduced based on π, the similarity relation between rope prototype and rope model is set up by the dimensionless item obtained, using this similarity relation as rope model, relatively whether there is the basis for estimation of distortion in rope prototype, the Similarity equations about π after dimensionless item minuend is established based on Buckingham (Buckingham) theory in the present invention, namely show that the rule of distortion occurs rope model to introducing distortion factor in Similarity equations, the Analysis of dynamics performance of the Distortion Law drawn to large scale prototype is laid a good foundation.

Accompanying drawing explanation

Fig. 1 is the seven rope driven Parallel Kinematic Manipulator model schematic adopted in the analytical approach of a kind of cable system Distortion Law of the present invention;

Fig. 2 is for verifying the front view of single rope vibration-testing apparatus of distortion similarity rule in the embodiment of the present invention;

For verifying the vertical view of single rope vibration-testing apparatus of distortion similarity rule in Fig. 3 embodiment of the present invention;

Fig. 4 is the structural representation of pedestal A in Fig. 2;

Fig. 5 is the structural representation of pedestal B in Fig. 2;

Fig. 6 is the structural representation of platform in Fig. 2.

In figure, 1. platform, 2. pedestal A, 2-1.L shape support plate, 3. vibration suppression adjusting mechanism; 4. pedestal B, 4-1. ladder-shaped support, 5. pedestal C, 6. pedestal D, 7. rope; 8. scale, 9. horizontal guide pin bushing, 10. block, 11. holddown springs; 12. solid rope slide blocks, 13. over caps, 14. hollow studs, 15. holding screw A; 16. tightening stopper A, 17. holding screw B, 18. tightening stopper B, 19. pedestal E; 20. track A, 21. track B, 22. pulleys, 23. floors.

Embodiment

Below in conjunction with the drawings and specific embodiments, the present invention is described in detail.

The analytical approach of a kind of cable system Distortion Law of the present invention, adopts seven rope driven Parallel Kinematic Manipulator as analytic target, and structure is as shown in Fig. 1 (a), and figure (a) represents seven rope driven Parallel Kinematic Manipulator experiment table schematic diagram, and its global coordinate is O ₀-X ₀y ₀z ₀, moving platform reference point is O ₁, the relative O of barycenter ₁distance be set to r _w=[0,0, c _g] ^t.Local coordinate is moving platform moving coordinate system O ₁-X ₁y ₁z ₁.Moving coordinate system { O ₁reference point O ₁at global coordinate { O ₀position vector athletic posture is represented, i.e. attitude coordinate by roll angle (Roll), the angle of pitch (Pitch) and deflection angle (Yaw) ^o0=(θ _x, θ _y, θ _z) ^t.Point B _i(i=1 ~ 7) are at { O ₁on distribution as shown in Fig. 1 (b), (c), figure (b) represent the distributing position of bitter end at moving coordinate system, figure (c) represent the distributing position of bitter end at fixed coordinate system, set up an office B _iat { O ₁in position vector be then B can be obtained according to coordinate transform _i(i=1 ~ 7) are at { O ₀in expression formula be: $b_i=R_{O1}^{O_0}\·{b_{O_1}}_i+p_{O_0}=r_i+p_{O_0},$ Wherein, B _iwith O ₁link vector be $r_i=R_{O1}^{O_0}\·b_i.$ Regulation A _i{ O relatively ₀displacement vector a _irepresent, then the long l of rope _i=a _i-b _iif, l _ibe that i-th rope is at B _ithe tangential vector of end, make A=[a ₁, a ₂..., a ₇], B=[b ₁, b ₂..., b ₇] then

The analytical approach of a kind of cable system Distortion Law of the present invention, concrete steps are as follows:

Step 1, sets up rope model;

The detailed process of step 1 is:

Step 1.1, chooses seven rope driven Parallel Kinematic Manipulator as analytic target in WDPM model, and use Lagrangian method to obtain the kinetics equation of described seven rope driven Parallel Kinematic Manipulator under fixing pose, kinetics equation is as follows:

$M\mathrm{\δ}\stackrel{\·\·}{x}+C(\mathrm{\δ}x,\mathrm{\δ}\stackrel{\·}{x})\mathrm{\δ}\stackrel{\·}{x}+K\mathrm{\δ}x={\mathrm{\δF}}_e---\left(3\right);$

Wherein, $M=\left[\begin{array}{cc}{m}_{c}I& -m_c{r}_w\×\\ m_c{r}_w\×& m_{c}I\end{array}\right].C(\mathrm{\δ}x,\stackrel{\·}{x})=\left[\begin{array}{cc}0& m_c\left(\mathrm{\ω}\×r_w\right)\×\\ m_c(\mathrm{\ω}\×r_w)\×& \left(I\mathrm{\ω}\right)\×\end{array}\right],$ $J^T=\left[\begin{array}{ccc}{u}_1& ...& u_6\\ r_1\×u_1& ...& r_6\×u_6\end{array}\right],$ represent diagonal matrix, cornerwise numerical value is k entirely ₀, it is the elastic stiffness (N/m) of every root drag-line.δ F _efor cutting force, the equation of cutting force is as follows:

$\left\{\begin{array}{c}{\mathrm{\δF}}_f={\[f_x,f_y,0\]}^T\\ {\mathrm{\δM}}_f=r_f\×{\mathrm{\δF}}_f\end{array}\right.,{\mathrm{\δF}}_e{\[{\mathrm{\δF}}_f,{\mathrm{\δM}}_f\]}^T---\left(4\right);$

Step 1.2, makes the δ F in described kinetics equation _ebe zero, equation (3) becomes the free vibration differential equation of system:

$M\mathrm{\δ}\stackrel{\·\·}{x}+C(\mathrm{\δ}x,\mathrm{\δ}\stackrel{\·}{x})\mathrm{\δ}\stackrel{\·}{x}+K\mathrm{\δ}x=0---\left(5\right);$

Formula (5) is the rope model of structure.

Step 2, derives the dimensionless item of rope model medium power characteristic parameter in step 1 based on Buckingham theorem;

The concrete derivation of the dimensionless item of the rope model medium power characteristic parameter of step 2 is as follows:

Based on statics research object:

The finite element model that the vibration analysis of WDPM is united by space bar realizes, and for element stiffness equation, dimensionless item is derived as follows:

Wherein, P, M represent concentrated force and concentrated moment respectively.With represent linear displacement and angle displacement respectively, length, area, elastic section modulus and inertia torque use l respectively, and A, E and I represent.

When putting on rope model for gravity as external force, dimensionless item is derived as follows:

${\mathrm{\π}}_5=\frac{P}{\mathrm{\ρ}Al}---\left(7\right);$

Wherein, ρ represents Suo Midu;

For cable wire material, strain and stress meets following formula:

$\mathrm{\ϵ}=\frac{P}{EA}+\frac{M}{EW\mathrm{\ϵ}}---\left(8\right);$

Wherein, ε represents that rope strains, and EW represents bending stiffness;

σ＝Eε (9)；

Wherein, σ represents Suo Yingli;

Dimensionless item is derived as follows:

$\left\{\begin{array}{c}{\mathrm{\π}}_6=\frac{P}{EA\mathrm{\ϵ}}\\ {\mathrm{\π}}_7=\frac{M}{EW\mathrm{\ϵ}}\\ {\mathrm{\π}}_8=\frac{\mathrm{\σ}}{E\mathrm{\ϵ}}\end{array}\right.---\left(10\right);$

Based on dynamics research object:

Steps A, the main research freedom vibration of model experiment due to WDPM, therefore, ignore damping, according to model analysis, the rope model (i.e. formula (5)) set up by step 1 draws dynamics research model, and draw the similarity criterion between rope model and rope prototype according to kinetic model, kinetic model formula is as follows:

[K-ω ²M]{x}＝0 (11)；

Wherein, the rigidity element in stiffness matrix K comprises m is mass matrix, and in mass matrix, matrix element comprises rope density p, rope area A, the long l of rope, and { x} is displacement array, and ω represents angular frequency;

Step B, the kinetic model (i.e. formula (11)) obtained according to steps A draws following dimensionless item

${\mathrm{\π}}_9=\frac{EA}{l^2{\mathrm{\ω}}^2\mathrm{\ρ}A}=\frac{EI}{l^4{\mathrm{\ω}}^2\mathrm{\ρ}A}---\left(12\right);$

Step 3, the dimensionless item obtained according to step 2 sets up the similarity relation between rope prototype and rope model.

Step 3.1, seeks the statics similarity relation between model and prototype;

The detailed process of step 3.1 is as follows:

Make λ _l=λ, λ _{Δ l}=λ and λ _p=λ _eλ ², wherein, λ _lrepresent the contracting ratio of Suo Changdu, λ _{Δ l}represent the contracting ratio of rope length variations, λ _prepresent the contracting ratio of the change of power between rope model and prototype; The dimensionless item π obtained by step 2 ₁, π ₂, π ₃, π ₄, π ₅, π ₆, π ₇, π ₈show that the contracting ratio of following parameter is:

In formula (13), the contracting ratio of each parameter is the statics similarity relation between rope model and prototype.

Step 3.2, seeks the kinematic similarity sexual intercourse between model and prototype;

The concrete steps of step 3.2 are:

According to dimensionless item π ₉the contracting ratio λ of angular frequency is drawn with statics similarity relationships _ω, bring in formula (13) by formula (12) and draw λ _ω, the contracting ratio λ of angular frequency _ωfor:

${\mathrm{\λ}}_{\mathrm{\ω}}=\sqrt{\mathrm{\λ}}---\left(14\right);$

The contracting ratio λ of angular frequency _ωbe the kinematic similarity sexual intercourse between rope model and prototype.

For general dynamic model, the ratio of the ratio of inertia and gravity, inertia and external force must be used for the accurate number of dimensionless, namely with in formula, g and a is the acceleration of acceleration of gravity and some applying power.Given λ _a=λ _g=1, be easy to get but, can derive thus ${\mathrm{\λ}}_v=\sqrt{{\mathrm{\λ}}_l}=\sqrt{\mathrm{\λ}},{\mathrm{\λ}}_t=\sqrt{\mathrm{\λ}}=1/{\mathrm{\λ}}_{\mathrm{\ω}}\⇒{\mathrm{\λ}}_{\mathrm{\ω}}=1/\sqrt{\mathrm{\λ}},$ Result does not meet formula (14).For WDPM system, introducing distortion model and distortion factor is clearly needed to go prediction prototype performance.

Step 4, theoretical based on Buckingham (Buckingham), about the Similarity equations of π after the dimensionless item minuend that establishment step 2 obtains, equation is as follows:

${\mathrm{\π}}_9=g({\mathrm{\π}}_1,{\mathrm{\π}}_2,...,{\mathrm{\π}}_8)=C_n^{\′}.\underset{i=1}{\overset{8}{\Π}}\left({\mathrm{\π}}_i^{a_1}\right)---\left(15\right);$

Formula (6), (7), (10), (12) are substituted in formula (15) and obtain

Wherein, C ' _nrepresent based on π ₁, π ₂, π ₃, π ₄, π ₅, π ₆, π ₇, π ₈matching π ₉function dimensionless constant, a _iit is the undetermined coefficient of fitting function.

Step 5, the similarity relation obtained using step 3, as judging whether rope model the foundation distorted occurs, to the rope model that distortion occurs, introduces by the Similarity equations obtained step 4 Distortion Law that distortion factor draws rope model.

Show in step 5 that the detailed process of Distortion Law is:

As the dimensionless item π in rope model ₁, π ₂, π ₃, π ₄, π ₅, π ₆, π ₇, π ₈when middle any one can not meet the similarity relation between rope prototype and rope model that step 3 obtains, then by distortion factor δ _iintroduce in formula (16), draw following Distortion Law:

Step 5.1, judges whether described rope model distorts, and detailed process is as follows:

By the dimensionless item π in described rope model ₁, π ₂, π ₃, π ₄, π ₅, π ₆, π ₇, π ₈middle any one substitutes in the statics similarity relation between rope model and rope prototype obtained in step 3 and verifies, works as π ₁, π ₂, π ₃, π ₄, π ₅, π ₆, π ₇, π ₈when middle any one can not meet the statics similarity relation between rope model and rope prototype that step 3 obtains, then judge to exist between rope model and rope prototype to distort;

Step 5.2, based on the judged result of step 5.1, introduces distortion factor δ to what obtain in step 4 about in the Similarity equations of π _i, draw the Distortion Law of described rope model, the equation of Distortion Law is as follows:

In order to carry out model test, determine that the similarity relationships between the physical quantitys such as physical dimension, interface geometrical property, material behavior, system load, internal force, stress, strain, displacement and boundary condition is very important.

Embodiment, the Distortion Law of single rope is analyzed:

Based on dimensional analysis, if similar for its meeting geometric of single rope, kinematic similitude, dynamic similarity are similar with border, it is not difficult for deriving similarity relationships.When using gravity and elastic force as mainly implementing power, inertia and gravity ratio and inertia and tension force ratio are also referred to as similarity criterion or characteristic, and it can be used respectively with represent.Thus the contracting ratio of following parameter can be obtained, expression is as follows:

$\{\begin{array}{c}{\mathrm{\λ}}_{\mathrm{\ϵ}}=I\\ {\mathrm{\λ}}_{\mathrm{\σ}}={\mathrm{\λ}}_E\\ {\mathrm{\λ}}_t={\mathrm{\λ}}_{\mathrm{\ν}}=\sqrt{{\mathrm{\λ}}_L}\\ {\mathrm{\λ}}_E={\mathrm{\λ}}_{\mathrm{\ρ}}{\mathrm{\λ}}_l\\ {\mathrm{\λ}}_T={\mathrm{\λ}}_{\mathrm{\ρ}}{\mathrm{\λ}}_l^2\\ {\mathrm{\λ}}_{\mathrm{\ω}}=\frac{1}{{\mathrm{\λ}}_l}{\left(\frac{{\mathrm{\λ}}_E}{{\mathrm{\λ}}_{\mathrm{\ρ}}}\right)}^{1/2}\end{array}---\left(18\right);$

Ideally, physical quantity elastic modulus E and density p must keep absolute contracting ratio (namely completely similar between prototype and model).But actual conditions are, λ _l, λ _ρand λ _econtracting ratio is all known, does not meet formula (13) conditional λ _e=λ _ρλ _l.Therefore, there is distortion term in system, needs the prediction realizing field test characteristic based on distortion factor.

Obtain the vibration frequency of single rope: given boundary condition, the natural frequency of elastic cable determines ω ₁.In fact the vibration of rope is exactly ripple at the kinetic energy of rope and the direct propagation phenomenon of potential energy.Kinetic energy originates from intermediate medium---and rope particulate, it represents by rope density p.Potential energy originates from elastic cable deformation-recovery, and it represents with elastic modulus E and Poisson ratio ν.If given boundary condition displacement is fixed, the fundamental frequency omega of rope ₁be exactly the function about the long L of rope, rope section A, elastic modulus E, internal force T and density p, be expressed as follows:

ω ₁＝f(l,A,E,T,ρ) (19)；

Wherein, 6 parameters can represent by 3 basic dimensions, i.e. length l (L), density p (ML ^-3) and axial rigidity EA (MLT ^-2), the expression of 6 parameters is as follows:

$\{\begin{array}{c}\[{\mathrm{\ω}}_1\]=L^{-1}{\left({\mathrm{ML}}^{-3}\right)}^{-1/2}{\left({\mathrm{MLT}}^{-2}\right)}^{1/2}\\ \[A\]=L^2{\left({\mathrm{ML}}^{-3}\right)}^0{\left({\mathrm{MLT}}^{-2}\right)}^0\\ \[l\]=L^1{\left({\mathrm{ML}}^{-3}\right)}^0{\left({\mathrm{MLT}}^{-2}\right)}^0\\ \[E\]=L^{-2}{\left({\mathrm{ML}}^{-3}\right)}^0{\left({\mathrm{MLT}}^{-2}\right)}^1\\ \[T\]=L^0{\left({\mathrm{ML}}^{-3}\right)}^0{\left({\mathrm{MLT}}^{-2}\right)}^1\\ \[\mathrm{\ρ}\]=L^0{\left({\mathrm{ML}}^{-3}\right)}^1{\left({\mathrm{MLT}}^{-2}\right)}^0\end{array}---\left(20\right);$

Formula (20) also available following dimensional matrix represents:

The order (r) of formula (21) is 3, and the number (n) of parameter is 6.Theoretical according to Buckingham, independent dimensionless π item is n-r, and it is all M that its dimension is expressed ⁰l ⁰t ⁰.With ω ₁, A and T is as repeated variable, and dimensional matrix can be tried to achieve by following formula:

Wherein, ξ ₁, ξ ₂and ξ ₃formula (22) repeated variable ω ₁, A and T exponential term, dimension π ₁, π ₂and π ₃expression formula be:

$\{\begin{array}{c}{\mathrm{\π}}_1={\mathrm{\ω}}_1^{{\mathrm{\ξ}}_1}{A}^{{\mathrm{\ξ}}_2}{T}^{{\mathrm{\ξ}}_3}l\\ {\mathrm{\π}}_2={\mathrm{\ω}}_1^{{\mathrm{\ξ}}_1}{A}^{{\mathrm{\ξ}}_2}{T}^{{\mathrm{\ξ}}_3}E\\ {\mathrm{\π}}_3={\mathrm{\ω}}_1^{{\mathrm{\ξ}}_1}{A}^{{\mathrm{\ξ}}_2}{T}^{{\mathrm{\ξ}}_3}\mathrm{\ρ}\end{array}---\left(23\right);$

Because all π items are nondimensional M ⁰l ⁰t ⁰, so application dimension Coordination Theory can balance each π item, thus obtain the expression formula of each π item:

$\{\begin{array}{c}{\mathrm{\π}}_1={\mathrm{\ω}}_1^2{A}^2\mathrm{\ρ}/T\\ {\mathrm{\π}}_2=A^{-1/2}l\\ {\mathrm{\π}}_3=T^{-1}EA\end{array}---\left(24\right);$

Theoretical according to Buckingham (Buckingham), express immeasurable parameter by characteristic variable, expression formula is as follows:

π ₁＝f(π ₂,π ₃) (25)；

(16) are foundation with the formula, by formula (24) and formula (25) simultaneous, draw ω ₁for:

${\mathrm{\ω}}_1=C_n{\left(\frac{A}{l^2}\right)}^{\mathrm{\α}}{\left(\frac{T}{EA}\right)}^{\mathrm{\β}}\left(\frac{\sqrt{EA/\mathrm{\ρ}}}{l^2}\right)=C_n{A}^{\mathrm{\α}}{l}^{-2\mathrm{\α}-2}{T}^{\mathrm{\β}}{\left(EA\right)}^{1/2-\mathrm{\β}}{\mathrm{\ρ}}^{-1/2}---\left(26\right);$

Wherein, C _nbe dimensionless constant, α, β are coefficients to be determined, and it obtains by experiment.Clearly, for model and prototype system, they all meet formula (16), introduce distortion factor when similarity relation between the rope prototype that the contracting ratio step 3 of physical parameter obtains and rope model.Such as given introduce distortion factor δ _afor predicting field test fundamental frequency.According to formula (17), predictive equation is:

${\mathrm{\λ}}_{{\mathrm{\ω}}_1}={\mathrm{\δ}}_A.\left({\mathrm{\λ}}_A^{\mathrm{\α}}{\mathrm{\λ}}_l^{-2\mathrm{\α}-2}{\mathrm{\λ}}_T^{\mathrm{\β}}{\left({\mathrm{\λ}}_{EA}\right)}^{1/2-\mathrm{\β}}{\mathrm{\λ}}_{\mathrm{\ρ}}^{-1/2}\right)---\left(27\right);$

${\mathrm{\ω}}_{1p}={\mathrm{\λ}}_{{\mathrm{\ω}}_1}.{\mathrm{\ω}}_{1m}---\left(28\right);$

Wherein, p represents prototype, and m represents model.

In order to obtain the dynamics of large-scale rope drive system, be necessary to adopt contracting to analyze than modeling method, owing to relating to structure in the many rope driving mechanism distortion laws of similitude, the parameters such as dynamic process, parameter is many, experiment is complicated, and the research parameter of the single rope distortion law of similitude is relatively less, be convenient to experimental design, therefore, the reading that the dynamic (dynamical) experiment porch of the single rope of meter can realize the long change of rope and rope tensility is proposed at this, the placement of vibration testing instrument and system restructural, obtain the general rule of distortion similarity by the model experiment of this simple power system and then establish experiment basis for the kinematic similarity Journal of Sex Research of many ropes drive system.

For verifying single rope vibration-testing apparatus of distortion similarity rule, as shown in Figure 2,3, comprise platform 1, platform 1 is provided with two parallel track A20, track B21, the two ends of track A20 are respectively equipped with pedestal A2, pedestal E19, pedestal A2 is provided with vibration suppression adjusting mechanism 3, pedestal E19 is provided with pulley 22, moveable pedestal B4 is provided with between pedestal A2 and pedestal E19, one end of rope 7 is fixed on pedestal A2, and the other end of rope 7 is successively through the pedestal C5 be fixed on after pedestal B4, pulley 22 for placing counterweight; Track B21 is provided with the pedestal D6 for installing single rope vibration-testing head; Platform 1 between track A20 and track B21 has been horizontally disposed with scale 8.

As shown in Figure 4, pedestal A2 comprises L shape support plate 2-1, and hollow stud 14 level is by through for the vertical edge of L shape support plate 2-1, and one end of hollow stud 14 is provided with solid lock set, and the other end of hollow stud 14 is sleeved in vibration suppression adjusting mechanism 3.

Vibration suppression adjusting mechanism 3 comprises horizontal guide pin bushing 9, one end of horizontal guide pin bushing 9 is fixed on inside the vertical edge of L shape support plate 2-1, the other end of horizontal guide pin bushing 9 is provided with over cap 13, solid rope slide block 12, block 10 is respectively equipped with in horizontal guide pin bushing 9, Gu be provided with holddown spring 11 between rope slide block 12 and block 10, one end of rope is successively through being fixed on solid lock slide 12 after solid lock set, hollow stud 14, block 10, holddown spring 11; The other end of hollow stud 14 to be socketed in horizontal guide pin bushing 9 and to arrange near block 10.

The horizontal edge of L shape support plate 2-1 and vertically between be provided with floor 23.

Gu lock set comprises tightening stopper A16, rope 7 is wrapped in tightening stopper A16, and rope 7 is fixed with tightening stopper A16 by holding screw A15.

Pedestal A2 is positioned at the zero-bit of scale 8, and scale 8 is for measuring rope 7 length between pedestal A2 and pedestal B4, and scale 8 is parallel with track A20.

As shown in Figure 5, pedestal B4 comprises ladder-shaped support 4-1, and the upper base of ladder-shaped support 4-1 is provided with the tightening stopper B18 for wrapping up rope 7, and tightening stopper B18 and rope 7 fix by holding screw B17.

As shown in Figure 6, platform 1 is rectangle, and track A20, track B21 are T-slot, and pedestal A2, pedestal B4, pedestal E19 are bolted on track A20 respectively by T-shaped; Pedestal D6 is bolted on track B21 by T-shaped.

Concrete verification step is as follows:

The first step, determines dimensionless constant C _nwith undetermined coefficient α, β.By designed test platform by adjustment rope length and Suo Li, be combined into and organize measurement data more, the factor alpha in fitting formula (26), β and C _n, wherein, the distance on track A20 between movable base B4 and pedestal A2 realizes rope length and adjusts, and on pedestal C5, the Mass adjust-ment of counterweight realizes rope tensility control, and single rope vibration-testing head of track B21 top base D6 realizes the measurement of rope vibration frequency;

Second step, the introducing of distortion factor, select in two ropes wherein one as prototype rope, all parameter subscripts of this rope are set to p, another root is as model rope, and all parameter subscripts of this rope are set to m, assuming that known the two area exist distortion, meet

${\mathrm{\λ}}_A^{*}=\mathrm{\κ}\·{\mathrm{\λ}}_A---\left(29\right);$

Wherein κ is the dissimilar coefficient of model and prototype, and all the other physical quantity l, T, E, ρ do not exist distortion, and formula (28) is substituted into formula (26) distortion factor

δ _A＝κ ^α+1/2-β(30)；

3rd step, according to the distortion theoretical formula (27) of the law of similitude, the theoretical values of (28) derivation prototype frequency.Known measurement obtains the vibration frequency ω of model rope _1m, the predictive coefficient of formula (30) is substituted in formula (28), thus obtains prototype rope vibration frequency ω _{1p is theoretical};

4th step, the correctness of the checking distortion law of similitude.By the prototype frequencies omega measured _{1p tests}with the prototype frequency theory value ω of the 3rd step _{1p is theoretical}make comparisons, if consistent, illustrate that the formula (27) of the distortion law of similitude and formula (28) are correct, otherwise be incorrect.

Claims (6)

1. an analytical approach for cable system Distortion Law, is characterized in that: specifically comprise the following steps:

Step 1, sets up rope model;

Step 2, derives the dimensionless item of described rope model medium power characteristic parameter based on Buckingham theorem;

Step 3, the dimensionless item obtained according to step 2 sets up the similarity relation between rope prototype and described rope model;

Step 4, theoretical based on Buckingham, about the Similarity equations of π after the dimensionless item minuend that establishment step 2 obtains;

Step 5, the similarity relation obtained using step 3, as judging whether described rope model the foundation distorted occurs, to the rope model that distortion occurs, introduces by the Similarity equations obtained step 4 Distortion Law that distortion factor draws described rope model.

2. the analytical approach of a kind of cable system Distortion Law according to claim 1, is characterized in that: the detailed process of described step 1 is:

Step 1.1, chooses seven rope driven Parallel Kinematic Manipulator as analytic target in WDPM model, and use Lagrangian method to obtain the kinetics equation of described seven rope driven Parallel Kinematic Manipulator under fixing pose, kinetics equation is as follows:

$M\mathrm{\δ}\stackrel{\·\·}{x}+C(\mathrm{\δ}x,\mathrm{\δ}\stackrel{\·}{x})\mathrm{\δ}\stackrel{\·}{x}+K\mathrm{\δ}x={\mathrm{\δF}}_e---\left(3\right);$

Wherein, $M=\left[\begin{array}{cc}{m}_{c}I& -m_c{r}_w\×\\ m_c{r}_w\×& m_{c}I\end{array}\right],C(\mathrm{\δ}x,\mathrm{\δ}\stackrel{\·}{x})=\left[\begin{array}{cc}\mathrm{\θ}& m_c(\mathrm{\ω}\×r_w)\×\\ m_c(\mathrm{\ω}\×r_w)\×& \left(I\mathrm{\ω}\right)\×\end{array}\right],$

represent diagonal matrix, cornerwise numerical value is k entirely ₀, it is the elastic stiffness (N/m) of every root drag-line; δ F _efor cutting force;

Step 1.2, makes the δ F in described kinetics equation _ebe zero, obtain the free vibration differential equation of described seven rope driven Parallel Kinematic Manipulator, the described differential equation is the rope model of structure, and the differential equation obtained is as follows:

$M\mathrm{\δ}\stackrel{\·\·}{x}+C(\mathrm{\δ}x,\mathrm{\δ}\stackrel{\·}{x})\mathrm{\δ}\stackrel{\·}{x}+K\mathrm{\δ}x=0---\left(5\right).$

3. the analytical approach of a kind of cable system Distortion Law according to claim 2, is characterized in that: the concrete derivation of the dimensionless item of the rope model medium power characteristic parameter of described step 2 is as follows:

Based on statics research object:

For the element stiffness equation in described rope model, dimensionless item is derived as follows:

Wherein, P, M represent concentrated force and concentrated moment respectively, use Δ l, represent linear displacement and angle displacement respectively, l, A, E, I represent length, area, elastic section modulus, inertia torque respectively;

When putting on described rope model for gravity as external force, dimensionless item is derived as follows:

${\mathrm{\π}}_5=\frac{P}{\mathrm{\ρ}Al}---\left(7\right);$

For cable wire material, rope strain and Suo Yingli meet following formula:

$\mathrm{\ϵ}=\frac{P}{EA}+\frac{M}{EW\mathrm{\ϵ}}---\left(8\right);$

Wherein, ε represents that rope strains, and EW represents bending stiffness;

σ＝Eε (9)；

Wherein, σ represents Suo Yingli;

Dimensionless item is derived as follows:

$\left\{\begin{array}{c}{\mathrm{\π}}_6=\frac{P}{EA\mathrm{\ϵ}}\\ {\mathrm{\π}}_7=\frac{M}{EW\mathrm{\ϵ}}\\ {\mathrm{\π}}_8=\frac{\mathrm{\σ}}{E\mathrm{\ϵ}}\end{array}\right.---\left(10\right);$

Based on dynamics research object:

Steps A, ignores the damping in described rope model, and according to model analysis, draw dynamics research model by described rope model, described kinetic model formula is as follows:

[K-ω ²M]{x}＝0 (11)；

Wherein, the rigidity element in stiffness matrix K comprises m is mass matrix, and in mass matrix, matrix element comprises rope density p, rope area A, the long l of rope, and { x} is displacement array, and ω represents angular frequency;

Step B, the kinetic model obtained according to steps A, draws following dimensionless item

${\mathrm{\π}}_9=\frac{EA}{l^2{\mathrm{\ω}}^2\mathrm{\ρ}A}---\left(12\right).$

4. the analytical approach of a kind of cable system Distortion Law according to claim 3, is characterized in that:

The detailed process of described step 3 is:

Step 3.1, set up the statics similarity relation between rope prototype and described rope model, detailed process is as follows:

Make λ _l=λ, λ _{Δ l}=λ and λ _p=λ _eλ ²; Wherein, λ _lrepresent the contracting ratio of Suo Changdu, λ _{Δ l}represent the contracting ratio of rope length variations, λ _prepresent the contracting ratio of the change of power between rope model and prototype; The dimensionless item π obtained by step 2 ₁, π ₂, π ₃, π ₄, π ₅, π ₆, π ₇, π ₈show that the contracting ratio of following parameter is:

In formula (13), the contracting ratio of each parameter is the statics similarity relation between rope model and prototype;

Step 3.2, set up the kinematic similarity relation between rope prototype and described rope model, detailed process is as follows:

According to the dimensionless item π obtained in step 2 ₉the statics similarity relationships obtained with step 3.1 draws the contracting ratio λ of angular frequency _ω, bring in formula (13) by formula (12) and draw λ _ω, the contracting ratio λ of angular frequency _ωfor:

${\mathrm{\λ}}_{\mathrm{\ω}}=\sqrt{\mathrm{\λ}}---\left(14\right);$

The contracting ratio λ of described angular frequency _ωbe the kinematic similarity sexual intercourse between rope prototype and described rope model.

5. the analytical approach of a kind of cable system Distortion Law according to claim 4, is characterized in that: in described step 4 about the Similarity equations of π be:

Wherein, C' _nrepresent based on π ₁, π ₂, π ₃, π ₄, π ₅, π ₆, π ₇, π ₈matching π ₉function dimensionless constant, a ₁, a ₂, a ₃, a ₄, a ₅, a ₆, a ₇, a ₈undetermined coefficient.

6. the analytical approach of a kind of cable system Distortion Law according to claim 5, is characterized in that: the detailed process of described step 5 is:

Step 5.1, judges whether described rope model distorts, and detailed process is as follows:

By the dimensionless item π in described rope model ₁, π ₂, π ₃, π ₄, π ₅, π ₆, π ₇, π ₈middle any one substitutes in the statics similarity relation between rope model and rope prototype obtained in step 3 and verifies, works as π ₁, π ₂, π ₃, π ₄, π ₅, π ₆, π ₇, π ₈when middle any one can not meet the statics similarity relation between rope model and rope prototype that step 3 obtains, then judge to exist between rope model and rope prototype to distort;

Step 5.2, the judged result that the existence based on step 5.1 distorts, introduces distortion factor δ to what obtain in step 4 about in the Similarity equations of π _i, draw the Distortion Law of described rope model, the equation of Distortion Law is as follows: