CN103268378A - Globoidal cam motion curve identification method based on multi-order correlation analysis - Google Patents

Abstract

The invention discloses a globoidal cam motion curve identification method based on multi-order correlation analysis. The globoidal cam motion curve identification method includes: (1) identifying a globoidal cam motion curve; (2) performing correlation analysis on obtained globoidal cam motion acquisition data, fitting an actual motion curve, and determining similarity of the globoidal cam actual motion curve and an ideal motion law; and (3) performing multi-order correlation analysis on the obtained globoidal cam motion acquisition data, performing correlation analysis on a first-order derivative, a second-order derivative and a multi-order derivative of a globoidal cam motion law and the ideal motion law, obtaining correlation coefficients of the position, the speed and the accelerated speed of the actual motion curve of the globoidal cam on the globoidal cam and similar morphology categories of the actual motion curve of the globoidal cam, and finally identifying the actual motion curve the globoidal cam. The globoidal cam motion curve identification method based on the multi-order correlation analysis facilitates improvement of position accuracy of a mechanical motion mechanism in the mechanical industry, facilitates improvement of the speed and the accelerated speed of the mechanical motion mechanism, and further improves motion stability.

Description

A kind of globoid cam curve movement discrimination method based on multistage correlation analysis

Technical field

The invention belongs to the design field of the process globoid cam characteristics of motion, relate to a kind of globoid cam curve movement discrimination method based on multistage correlation analysis.

Background technology

At mechanical engineering field, globoid cam is one of vital part of realizing mechanical motion, its curve movement is the basis that guarantees globoid cam kinematic accuracy and stationarity, and therefore, each globoid cam manufacturing enterprise and researcher have carried out a large amount of theoretical researches and test to this.

The identification of globoid cam curve movement mainly is to ask the characteristics of motion of globoid cam is counter, obtains the mathematical expression of the characteristics of motion.But existing data fitting and curvilinear correlation analysis and research all concentrate on once fitting or the correlation analysis aspect of various curves, existing discrimination method adopts least square method directly to carry out curve fitting, the matched curve that obtains of match is all very relevant with multiple curve movement like this, related coefficient can't be distinguished all near 1.Therefore, for fear of impact, in the face of the design of globoid cam, need a kind of discrimination method of new curve movement, thereby make that globoid cam structure motion ground is more steady.In the face of the processing of globoid cam, need a kind of discrimination method of curve movement, analyze the characteristics of motion of existing globoid cam, the kinematic error of the globoid cam that check is made.

Summary of the invention

The object of the present invention is to provide a kind of globoid cam curve movement discrimination method based on multistage correlation analysis.

For achieving the above object, the present invention has adopted following technical scheme:

(1) least square fitting of globoid cam curve movement: to the globoid cam exercise data that collects, adopt least square method to carry out curve fitting, obtain the actual motion curve of globoid cam;

(2) correlation analysis: according to the actual motion curve of globoid cam and desirable globoid cam curve movement, the correlation analysis of carrying out the globoid cam characteristics of motion obtains related coefficient;

(3) finish the identifying of the actual motion curve of globoid cam according to the difference of related coefficient.

Described step (1) comprises following idiographic flow:

1. suppose to comprise n collection point on the actual motion curve of certain globoid cam, then the actual motion curve table with this globoid cam is shown C={y _t=f _c(x _t), t=1,2,3 ..., n; Desirable globoid cam curve movement is expressed as P _k={ y _Kt=f _k(x _t), t=1,2,3 ..., n; K=1,2,3 ..., K, K are desirable globoid cam curve movement kind sum;

2. the polynomial fitting that defines the actual motion curve is:

f _C(x)=α ₀g ₀(x)+α ₁g ₁(x)+α ₂g ₂(x)+...α _mg _m(x) (1)

In the formula (1), g _j(x)=x ^j, j=0,1,2 ..., m; M represents the exponent number of polynomial fitting;

3. by the actual motion curve collection point of globoid cam and the polynomial fitting error of calculation quadratic sum of actual motion curve:

$S_C=\underset{t=1}{\overset{n}{\begin{array}{c}\mathrm{\Σ}\end{array}}}{(y_t-f_c\left(x_t\right))}^2\→\min ---\left(2\right)$

4. ask local derviation then:

$\frac{\∂S_C}{\∂{\mathrm{\α}}_j}=0,j=\mathrm{0,1,2,3},...m---\left(3\right)$

Definition:

$A_{\mathrm{ij}}=\underset{t=1}{\overset{n}{\mathrm{\Σ}}}{x}_t^{i+j},i=\mathrm{0,1,2},...,m;j=\mathrm{0,1},...m;---\left(4\right)$

$b_j=\underset{t=1}{\overset{n}{\mathrm{\Σ}}}{y}_t\×x_t^j,j=\mathrm{0,1,2},...,m---\left(5\right)$

So have:

$\left[\begin{array}{cccc}{A}_{00}& A_{01}& ...& A_{0\mathrm{<figure-callout\; id="10"\; label="m\; A"\; filenames="HDA00003203133600011.png"\; state="\{\{state\}\}">m</figure-callout>}}& \\ A_{}{}_{10}& & ...& A_{1m}\\ ...& ...& ...& ...\\ A_{m0}& {\mathrm{<figure-callout\; id="1"\; label="A\; m"\; filenames="HDA00003203133600011.png,HDA00003203133600012.png"\; state="\{\{state\}\}">A</figure-callout>}}_m& ...& A_{\mathrm{mm}}\end{array}\right]\left[\begin{array}{c}{\mathrm{\&alpha;}}_0\\ {\mathrm{\&alpha;}}_1\\ ...\\ {\mathrm{\&alpha;}}_m\end{array}\right]=\left[\begin{array}{c}{b}_0\\ {\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="HDA00003203133600011.png,HDA00003203133600012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1\\ ...\\ b_m\end{array}\right]---\left(6\right)$

Obtain the factor alpha of polynomial fitting by solution formula (6) _j, with factor alpha _jCorrespondence is brought the actual motion curvilinear equation that polynomial fitting gets globoid cam into.

The correlation analysis of the described globoid cam characteristics of motion comprises the multistage correlation analysis of actual motion curve and the desirable globoid cam curve movement of globoid cam, described multistage correlation analysis comprises actual motion curve and the first order derivative of desirable globoid cam curve movement or the correlation analysis of second derivative of globoid cam, and described related coefficient is the related coefficient of the globoid cam speed corresponding with correlation analysis or the related coefficient of globoid cam acceleration.

The correlation analysis of the described globoid cam characteristics of motion is further comprising the steps of: before multistage correlation analysis, the actual motion curve of globoid cam and desirable globoid cam curve movement are carried out correlation analysis, obtain related coefficient γ _Xy

The actual motion curve of described globoid cam and desirable globoid cam curve movement carry out correlation analysis, comprise following idiographic flow:

Calculate related coefficient γ by actual motion curve collection point and ideal movements curve point _Xy:

${\mathrm{\&gamma;}}_{\mathrm{xy}}=\frac{n\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ci}}{y}_{\mathrm{ki}}-\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ci}}\&CenterDot;\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ki}}}{\sqrt{n\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y_{\mathrm{ci}}}^2-{\left(\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ci}}\right)}^2}\&CenterDot;\sqrt{n\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y_{\mathrm{ki}}}^2-{\left(\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ki}}\right)}^2}}---\left(8\right)$

Wherein, y _CiCorresponding x on the actual motion curve of expression match _iValue, y _KiRepresent that desirable globoid cam curve movement k goes up corresponding x _iValue, n represents the collection point number.

Described desirable globoid cam curve movement comprises cosine curve, sinusoidal curve, five order polynomial curves, revises curve of equal velocity, revises step curve and revises sinusoidal curve, and described related coefficient is the related coefficient of globoid cam acceleration.

Described step (2) comprises following idiographic flow:

Related coefficient γ when the actual motion curve of the desirable globoid cam curve movement of K kind and globoid cam _Xy≈ 1, illustrates that the desirable globoid cam curve movement of the actual motion curve of globoid cam and K kind is very high at the similarity degree of position curve, need carry out first order derivative or second derivative correlation analysis;

Calculate related coefficient by actual motion curve collection point and ideal movements curve point, if the related coefficient γ ' of actual motion rate curve and ideal movements rate curve _XyCan't distinguish the speed attribute of the actual motion curve of globoid cam, then carry out the related coefficient γ of second order differentiate and calculating acceleration " _Xy, up to the come out actual movement rule of globoid cam of identification:

${\mathrm{\&gamma;}}_{\mathrm{xy}}^{\&prime;}=\frac{n\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ci}}^{\&prime;}{y}_{\mathrm{ki}}^{\&prime;}-\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ci}}^{\&prime;}\&CenterDot;\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ki}}^{\&prime;}}{\sqrt{n\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ci}}^{\&prime;2}-{\left(\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ci}}^{\&prime;}\right)}^2}\&CenterDot;\sqrt{n\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ki}}^{\&prime;2}-{\left(\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ki}}^{\&prime;}\right)}^2}}---\left(11\right)$

${\mathrm{\&gamma;}}_{\mathrm{xy}}^{\&prime;\&prime;}=\frac{n\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ci}}^{\&prime;\&prime;}{y}_{\mathrm{ki}}^{\&prime;\&prime;}-\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ci}}^{\&prime;\&prime;}\&CenterDot;\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ki}}^{\&prime;\&prime;}}{\sqrt{n\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ci}}^{\&prime;\&prime;2}-{\left(\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ci}}^{\&prime;\&prime;}\right)}^2}\&CenterDot;\sqrt{n\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ki}}^{\&prime;\&prime;2}-{\left(\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ki}}^{\&prime;\&prime;}\right)}^2}}---\left(12\right)$

Wherein, y ' _KiRepresent corresponding x on the desirable globoid cam curve movement k first order derivative _iValue, y ' _CiCorresponding x on the actual motion curve first order derivative of expression match _iValue, y " _CiCorresponding x on the actual motion curve second derivative of expression match _iValue, y " _KiRepresent corresponding x on the desirable globoid cam curve movement k second derivative _iValue, n represents the collection point number.

The present invention compared with prior art, its advantage is:

1) the present invention provides complete reference solution and control flow clearly for the identification of globoid cam curve movement.

2) the present invention proposes a kind of globoid cam curve movement discrimination method based on multistage correlation analysis, and be applied to the identification of globoid cam curve movement first.The identification of globoid cam actual motion curve comprises: the 1. data acquisition of globoid cam actual motion curve; 2. the correlation analysis of globoid cam actual motion curve data; 3. globoid cam actual motion curve is in first order derivative, second derivative and the more correlation analysis of higher derivative.

3) the present invention not only is conducive to improve the positional precision of mechanical motion mechanism in the mechanical industry, and is conducive to improve speed and the acceleration precision of mechanical motion mechanism, and then improves the stationarity of mechanical motion.

Description of drawings

Fig. 1 revises the sinusoidal motion law curve; Wherein, (a) for revising the displacement curve of sinusoidal motion rule, (b) for revising the rate curve of sinusoidal motion rule, (c) for revising the accelerating curve of sinusoidal motion rule, (d) for revising the acceleration change curve of sinusoidal motion rule;

Fig. 2 is acceleration theoretical curve and matched curve figure; Wherein, solid line is the actual acceleration curve, (a) dotted line is for revising constant speed acceleration theoretical curve in, (b) dotted line is for revising the trapezoidal acceleration theoretical curve in, (c) dotted line is for revising the sinusoidal acceleration theoretical curve in, (d) dotted line is cosine acceleration theoretical curve in, and (e) middle dotted line is five order polynomial acceleration theoretical curves, and (f) middle dotted line is the sinusoidal acceleration theoretical curve;

Fig. 3 is displacement curve identification program FB(flow block).

Embodiment

The present invention will be further described below in conjunction with accompanying drawing:

A kind of globoid cam curve movement discrimination method based on multistage correlation analysis comprises three steps: correlation analysis and the multistage correlation analysis of the least square fitting of globoid cam curve movement, globoid cam actual motion curve data.

The least square fitting of step (1) globoid cam curve movement

In order to identify the actual movement rule with the identification globoid cam, further improve the machining precision of globoid cam, carry out data acquisition by the actual motion curve to globoid cam, on this basis, carry out least square curve fitting, obtain the mathematical expression of the actual movement rule of globoid cam.Therefore, in globoid cam curve movement identification process, after collecting the globoid cam exercise data, adopt least square method to carry out curve fitting, obtain the actual motion curve; Specifically may further comprise the steps:

1. suppose to comprise n collection point on certain globoid cam actual motion curve, then this globoid cam actual motion curve table is shown C={y _t=f _c(x _t) (t=1,2,3 ..., n); Desirable globoid cam curve movement is expressed as P _k={ y _Kt=f _k(x _t) (t=1,2,3 ..., n; K=1,2,3 ..., K), wherein: K is the ideal movements curve kind sum that globoid cam may occur, and P is arranged _k={ P ₁, P ₂... P _kPlant; P _k={ y _Kt=f _k(x _t) t point on the expression k kind ideal movements curve;

2. the polynomial fitting that defines the actual motion curve is:

f _C(x)=α ₀g ₀(x)+α ₁g ₁(x)+α ₂g ₂(x)+...α _mg _m(x) (1)

In the reality, simple in order to calculate, General Definition: g _j(x)=x ^jBe power function commonly used.J=0,1,2 ..., m; M represents the exponent number of polynomial fitting, and exponent number is more high, and precision is more high, for mechanical motion, through commonly used 5～10.

3. by actual motion curve collection point and polynomial fitting error of calculation quadratic sum:

$S_C=\underset{t=1}{\overset{n}{\begin{array}{c}\mathrm{\&Sigma;}\end{array}}}{(y_t-f_c\left(x_t\right))}^2\&RightArrow;\min ---\left(2\right)$

This error sum of squares has represented the departure degree of actual motion curve and polynomial fitting.

4. ask the local derviation of polynomial fitting and solve an equation, obtain polynomial coefficient:

$\frac{\&PartialD;S_C}{\&PartialD;{\mathrm{\&alpha;}}_j}=0,j=\mathrm{0,1,2,3},...m---\left(3\right)$

Wherein, definition:

$A_{\mathrm{ij}}=\underset{t=1}{\overset{n}{\mathrm{\&Sigma;}}}{x}_t^{i+j},i=\mathrm{0,1,2},...,m;j=\mathrm{0,1},...m;---\left(4\right)$

$b_j=\underset{t=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_t\&times;x_t^j,j=\mathrm{0,1,2},...,m---\left(5\right)$

So have:

$\left[\begin{array}{cccc}{A}_{00}& A_{01}& ...& A_{0\mathrm{<figure-callout\; id="10"\; label="m\; A"\; filenames="HDA00003203133600011.png"\; state="\{\{state\}\}">m</figure-callout>}}& \\ A_{}{}_{10}& & ...& A_{1m}\\ ...& ...& ...& ...\\ A_{m0}& A_{\mathrm{<figure-callout\; id="1"\; label="m"\; filenames="HDA00003203133600011.png,HDA00003203133600012.png"\; state="\{\{state\}\}">m</figure-callout>}1}& ...& A_{\mathrm{mm}}\end{array}\right]\left[\begin{array}{c}{\mathrm{\&alpha;}}_0\\ {\mathrm{\&alpha;}}_1\\ ...\\ {\mathrm{\&alpha;}}_m\end{array}\right]=\left[\begin{array}{c}{b}_0\\ {\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="HDA00003203133600011.png,HDA00003203133600012.png"\; state="\{\{state\}\}">b</figure-callout>}}_1\\ ...\\ b_m\end{array}\right]---\left(6\right)$

So, just can obtain the actual motion curvilinear equation of globoid cam by finding the solution this equation, and then can carry out the actual motion curvilinear correlation analysis of globoid cam.

Step (2) correlation analysis

The related coefficient of the actual motion curve by calculating globoid cam and desirable curve movement is carried out the correlation analysis of actual motion curve and the desirable curve movement of globoid cam, specifically comprises:

1. by actual motion curve collection point and ideal movements curve point error of calculation quadratic sum:

$S_k=\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{(y_{\mathrm{ci}}-y_{\mathrm{ki}})}^2,k=\mathrm{1,2,3}...,K---\left(7\right)$

Error sum of squares is one of foundation of distinguishing relevant curve movement in the following formula, and this error sum of squares has represented the departure degree of actual motion curve and k kind ideal movements curve, and use can match with normalized related coefficient.

2. calculate related coefficient by actual motion curve collection point and ideal movements curve point:

${\mathrm{\&gamma;}}_{\mathrm{xy}}=\frac{n\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ci}}{y}_{\mathrm{ki}}-\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ci}}\&CenterDot;\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ki}}}{\sqrt{n\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y_{\mathrm{ci}}}^2-{\left(\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ci}}\right)}^2}\&CenterDot;\sqrt{n\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y_{\mathrm{ki}}}^2-{\left(\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ki}}\right)}^2}}---\left(8\right)$

This related coefficient has represented the similarity degree of actual motion curve and k kind ideal movements curve.Wherein, y _CiCorresponding x on the actual motion curve of expression match _iValue, y _KiExpression ideal movements curve k goes up corresponding x _iValue, n represents the collection point number.

The multistage correlation analysis of step (3)

On the closely related basis of globoid cam actual motion curve and the desirable characteristics of motion, carry out first order derivative, second derivative and the more correlation analysis of higher derivative of the globoid cam characteristics of motion, obtain globoid cam actual motion curve in related coefficient and the similar form classification thereof of globoid cam speed, acceleration etc., finish the identifying of globoid cam actual motion curve, specifically comprise:

1. the related coefficient γ that works as K kind ideal movements curve and actual motion curve _CP≈ 1, illustrates that actual motion curve and K kind ideal movements curve are very high at the similarity degree of position curve, need carry out first order derivative, second derivative and the more correlation analysis of higher derivative.

2. actual motion curve and K kind ideal movements curve are carried out differentiate respectively, calculate actual motion curve and the single order of K kind ideal movements curve and the derivative error sum of squares of second order:

$S_k^{\&prime;}=\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{(y_{\mathrm{ci}}^{\&prime;}-y_{\mathrm{ki}}^{\&prime;})}^2,---\left(9\right)$

$S_k^{\&prime;\&prime;}=\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{(y_{\mathrm{ci}}^{\&prime;\&prime;}-y_{\mathrm{ki}}^{\&prime;\&prime;})}^2,---\left(10\right)$

k=1,2,3…,K；

Error sum of squares is one of foundation of distinguishing relevant curve movement in the following formula, represents the degree that actual curve and ideal curve depart from, and use matches with normalized related coefficient.For mechanical motion mechanism, the error sum of squares of this first order derivative has represented the speed deviations degree of actual motion curve and ideal movements curve, and the error sum of squares of second derivative has represented the acceleration of globoid cam and the acceleration offset degree of ideal movements.

3. calculate related coefficient by actual motion curve collection point and ideal movements curve point, if the related coefficient of actual motion rate curve and ideal movements rate curve can't be distinguished the speed attribute of the actual motion curve of globoid cam, then repeat the second order differentiate and calculate the related coefficient of acceleration, up to the come out actual movement rule of globoid cam of identification:

${\mathrm{\&gamma;}}_{\mathrm{xy}}^{\&prime;}=\frac{n\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ci}}^{\&prime;}{y}_{\mathrm{ki}}^{\&prime;}-\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ci}}^{\&prime;}\&CenterDot;\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ki}}^{\&prime;}}{\sqrt{n\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ci}}^{\&prime;2}-{\left(\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ci}}^{\&prime;}\right)}^2}\&CenterDot;\sqrt{n\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ki}}^{\&prime;2}-{\left(\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ki}}^{\&prime;}\right)}^2}}---\left(11\right)$

${\mathrm{\&gamma;}}_{\mathrm{xy}}^{\&prime;\&prime;}=\frac{n\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ci}}^{\&prime;\&prime;}{y}_{\mathrm{ki}}^{\&prime;\&prime;}-\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ci}}^{\&prime;\&prime;}\&CenterDot;\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ki}}^{\&prime;\&prime;}}{\sqrt{n\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ci}}^{\&prime;\&prime;2}-{\left(\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ci}}^{\&prime;\&prime;}\right)}^2}\&CenterDot;\sqrt{n\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ki}}^{\&prime;\&prime;2}-{\left(\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ki}}^{\&prime;\&prime;}\right)}^2}}---\left(12\right)$

Wherein, y ' _KiCorresponding x on the expression ideal movements curve k first order derivative _iValue, y ' _CiCorresponding x on the actual motion curve first order derivative of expression match _iValue, y " _CiCorresponding x on the actual motion curve second derivative of expression match _iValue, y " _KiCorresponding x on the expression ideal movements curve k second derivative _iValue.

Globoid cam curve movement discrimination method application example based on multistage correlation analysis

Globoid cam curve movement commonly used has: six kinds of versions such as cosine accelerating curve, sinusoidal acceleration curve, five order polynomial curves, correction curve of equal velocity, correction step curve and correction sinusoidal curve.Wherein, be illustrated in figure 1 as correction sinusoidal displacement curve (0 rank), rate curve (1 rank), accelerating curve (2 rank) and acceleration change curve (3 rank).

The identification of these curve movements mainly is to ask the characteristics of motion of globoid cam is counter, obtains the mathematical expression of the characteristics of motion.Therefore, in order to verify correctness of the present invention, carry out the data acquisition of actual arc face cam, obtain the data of one group of globoid cam.

1) least square fitting of globoid cam curve movement

Carry out least square fitting according to above-mentioned steps (1), the actual motion curve polynomial expression that obtains globoid cam is:

p(x)=91.4645×x ¹⁰-448.3554×x ⁹+934.9087×x ⁸-1.078×10 ³×x ⁷

+746.8583×x ⁶-312.223×x ⁵+70.0229×x ⁴-5.7668×x ³+2.156×x ²

-0.1052×x ¹+0.0011

2) correlation analysis

Carry out correlation analysis according to above-mentioned steps (2), the related coefficient that obtains the actual motion curve of globoid cam and ideal movements curve is as shown in table 1:

Six kinds of errors of displacement curve and related coefficient that the operation of table 1 matlab program obtains

Theoretical curve

Cosine

Sinusoidal

Five order polynomials

Revise constant speed

Revise trapezoidal

Revise sinusoidal

Displacement error

2.151×10 -7

8.687×10 -8

8.1775×10 -8

5.9926×10 -6

3.0269×10 -6

4.8481×10 -9

Related coefficient

1

1

1

1

1

1

3) multistage correlation analysis

Shown in the above table 1, because the actual motion curve of globoid cam and the related coefficient of ideal movements curve all equal 1, all strong correlations of this globoid cam curve movement and six kinds of ideal movements curves are described, therefore, need further carry out multistage correlation analysis, carry out multistage correlation analysis according to above-mentioned steps (3), the related coefficient of second derivative that obtains the actual motion curve of globoid cam and ideal movements curve is as shown in table 2:

Six kinds of errors of accelerating curve and related coefficient that the operation of table 2 matlab program obtains

4) effect

Identification effect as shown in Figure 2, the accelerating curve of actual cam and six kinds of results that the theoretical acceleration curve is compared, wherein solid line is the actual acceleration curve, dotted line is various theoretical acceleration curves, as can be seen, the actual acceleration curve only and among Fig. 2 c accelerating curve overlapped, all the other are not overlapping.As seen, when the characteristics of motion of globoid cam is carried out identification, because the displacement law of various cams is extremely similar, and acceleration law has very big difference, and it is more directly perceived with accelerating curve the globoid cam curve movement to be carried out identification, therefore, shows through correlation analysis and figure, actual globoid cam curve is for revising cam curve, and its program flow diagram as shown in Figure 3.

The correction sinusoidal curve is the improvement to cosine curve, it had both overcome the discontinuous shortcoming of cosine curve, kept the less advantage of maximal rate and peak acceleration again, the balance of this curve is fabulous, under the unclear situation of load character, use dangerous least, and have absorbing preferably, can realize easy motion.Revise sinusoidal curve and be applied to high-speed motion heavy duty in addition, have special superiority.

At the globoid cam curve movement, prior art is only carried out the match of position curve movement, and do not consider the curvilinear correlation analysis of higher derivative correspondences such as the speed of curve correspondence and acceleration, mechanical motion mechanism at the curve movement correspondence, do not consider the correlation analysis result of speed continuity and flatness like this, can bring the impact of motion to machinery.The result of above-mentioned application example has proved the accuracy based on the globoid cam curve movement discrimination method of multistage correlation analysis that proposes, and has also proved feasibility and the necessity implemented in the globoid cam process based on the globoid cam curve movement discrimination method of multistage correlation analysis.

Claims (7)

1. globoid cam curve movement discrimination method based on multistage correlation analysis is characterized in that this discrimination method may further comprise the steps:

(1) least square fitting of globoid cam curve movement: to the globoid cam exercise data that collects, adopt least square method to carry out curve fitting, obtain the actual motion curve of globoid cam;

(2) correlation analysis: according to the actual motion curve of globoid cam and desirable globoid cam curve movement, the correlation analysis of carrying out the globoid cam characteristics of motion obtains related coefficient;

(3) finish the identifying of the actual motion curve of globoid cam according to the difference of related coefficient.

2. according to the described a kind of globoid cam curve movement discrimination method based on multistage correlation analysis of claim 1, it is characterized in that: described step (1) comprises following idiographic flow:

1. suppose to comprise n collection point on the actual motion curve of certain globoid cam, then the actual motion curve table with this globoid cam is shown C={y _t=f _c(x _t), t=1,2,3 ..., n; Desirable globoid cam curve movement is expressed as P _k={ y _Kt=f _k(x _t), t=1,2,3 ..., n; K=1,2,3 ..., K, K are desirable globoid cam curve movement kind sum;

2. the polynomial fitting that defines the actual motion curve is:

f _C(x)=α ₀g ₀(x)+α ₁g ₁(x)+α ₂g ₂(x)+...α _mg _m(x) (1)

In the formula (1), g _j(x)=x ^j, j=0,1,2 ..., m; M represents the exponent number of polynomial fitting;

3. by the actual motion curve collection point of globoid cam and the polynomial fitting error of calculation quadratic sum of actual motion curve:

$S_C=\underset{t=1}{\overset{n}{\begin{array}{c}\mathrm{\&Sigma;}\end{array}}}{(y_t-f_c\left(x_t\right))}^2\&RightArrow;\min ---\left(2\right)$

4. ask local derviation then:

$\frac{\&PartialD;S_C}{\&PartialD;{\mathrm{\&alpha;}}_j}=0,j=\mathrm{0,1,2,3},...m---\left(3\right)$

Definition:

$A_{\mathrm{ij}}=\underset{t=1}{\overset{n}{\mathrm{\&Sigma;}}}{x}_t^{i+j},i=\mathrm{0,1,2},...,m;j=\mathrm{0,1},...m;---\left(4\right)$

$b_j=\underset{t=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_t\&times;x_t^j,j=\mathrm{0,1,2},...,m---\left(5\right)$

So have:

$\left[\begin{array}{cccc}{A}_{00}& A_{01}& ...& A_{0m}\\ A_{10}& & ...& A_{1m}\\ ...& ...& ...& ...\\ A_{m0}& A_{m1}& ...& A_{\mathrm{mm}}\end{array}\right]\left[\begin{array}{c}{\mathrm{\&alpha;}}_0\\ {\mathrm{\&alpha;}}_1\\ ...\\ {\mathrm{\&alpha;}}_m\end{array}\right]=\left[\begin{array}{c}{b}_0\\ b_1\\ ...\\ b_m\end{array}\right]---\left(6\right)$

Obtain the factor alpha of polynomial fitting by solution formula (6) _j, with factor alpha _jCorrespondence is brought the actual motion curvilinear equation that polynomial fitting gets globoid cam into.

3. according to the described a kind of globoid cam curve movement discrimination method based on multistage correlation analysis of claim 1, it is characterized in that: the correlation analysis of the described globoid cam characteristics of motion comprises the multistage correlation analysis of actual motion curve and the desirable globoid cam curve movement of globoid cam, described multistage correlation analysis comprises actual motion curve and the first order derivative of desirable globoid cam curve movement or the correlation analysis of second derivative of globoid cam, and described related coefficient is the related coefficient of the globoid cam speed corresponding with correlation analysis or the related coefficient of globoid cam acceleration.

4. according to the described a kind of globoid cam curve movement discrimination method based on multistage correlation analysis of claim 3, it is characterized in that: the correlation analysis of the described globoid cam characteristics of motion is further comprising the steps of: before multistage correlation analysis, the actual motion curve of globoid cam and desirable globoid cam curve movement are carried out correlation analysis, obtain related coefficient γ _Xy

5. according to the described a kind of globoid cam curve movement discrimination method based on multistage correlation analysis of claim 4, it is characterized in that: the actual motion curve of described globoid cam and desirable globoid cam curve movement carry out correlation analysis, comprise following idiographic flow:

Calculate related coefficient γ by actual motion curve collection point and ideal movements curve point _Xy:

${\mathrm{\&gamma;}}_{\mathrm{xy}}=\frac{n\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ci}}{y}_{\mathrm{ki}}-\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ci}}\&CenterDot;\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ki}}}{\sqrt{n\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y_{\mathrm{ci}}}^2-{\left(\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ci}}\right)}^2}\&CenterDot;\sqrt{n\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y_{\mathrm{ki}}}^2-{\left(\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ki}}\right)}^2}}---\left(8\right)$

Wherein, y _CiCorresponding x on the expression actual motion curve _iValue, y _KiRepresent that desirable globoid cam curve movement k goes up corresponding x _iValue, n represents the collection point number.

6. according to the described a kind of globoid cam curve movement discrimination method based on multistage correlation analysis of claim 1, it is characterized in that: described desirable globoid cam curve movement comprises cosine curve, sinusoidal curve, five order polynomial curves, revises curve of equal velocity, revises step curve and revises sinusoidal curve, and described related coefficient is the related coefficient of globoid cam acceleration.

7. according to the described a kind of globoid cam curve movement discrimination method based on multistage correlation analysis of claim 1, it is characterized in that: described step (2) comprises following idiographic flow:

Related coefficient γ when the actual motion curve of the desirable globoid cam curve movement of K kind and globoid cam _Xy≈ 1, illustrates that the desirable globoid cam curve movement of the actual motion curve of globoid cam and K kind is very high at the similarity degree of position curve, need carry out first order derivative or second derivative correlation analysis;

Calculate related coefficient by actual motion curve collection point and ideal movements curve point, if the related coefficient γ ' of actual motion rate curve and ideal movements rate curve _XyCan't distinguish the speed attribute of the actual motion curve of globoid cam, then carry out the related coefficient γ of second order differentiate and calculating acceleration " _Xy, up to the come out actual movement rule of globoid cam of identification:

${\mathrm{\&gamma;}}_{\mathrm{xy}}^{\&prime;}=\frac{n\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ci}}^{\&prime;}{y}_{\mathrm{ki}}^{\&prime;}-\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ci}}^{\&prime;}\&CenterDot;\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ki}}^{\&prime;}}{\sqrt{n\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ci}}^{\&prime;2}-{\left(\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ci}}^{\&prime;}\right)}^2}\&CenterDot;\sqrt{n\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ki}}^{\&prime;2}-{\left(\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ki}}^{\&prime;}\right)}^2}}---\left(11\right)$

${\mathrm{\&gamma;}}_{\mathrm{xy}}^{\&prime;\&prime;}=\frac{n\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ci}}^{\&prime;\&prime;}{y}_{\mathrm{ki}}^{\&prime;\&prime;}-\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ci}}^{\&prime;\&prime;}\&CenterDot;\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ki}}^{\&prime;\&prime;}}{\sqrt{n\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ci}}^{\&prime;\&prime;2}-{\left(\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ci}}^{\&prime;\&prime;}\right)}^2}\&CenterDot;\sqrt{n\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ki}}^{\&prime;\&prime;2}-{\left(\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ki}}^{\&prime;\&prime;}\right)}^2}}---\left(12\right)$

Wherein, y ' _KiRepresent corresponding x on the desirable globoid cam curve movement k first order derivative _iValue, y ' _CiCorresponding x on the expression actual motion curve first order derivative _iValue, y " _CiCorresponding x on the expression actual motion curve second derivative _iValue, y " _KiRepresent corresponding x on the desirable globoid cam curve movement k second derivative _iValue, n represents the collection point number.

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