CN101804465A - Multi-scale wavelet antivibration design method of high-speed main shaft of machine tool - Google Patents

Abstract

The invention relates to a multi-scale wavelet antivibration design method of a high-speed main shaft of a machine tool, which comprises the steps of: 1. by using interval cubic Hermite spline wavelets as the multi-scale interpolation basis, establishing a multi-scale wavelet numerical value solving model influenced by comprehensive gyroscopic couple, transverse shear deformation, main shaft rotary inertia, material delay and viscous damping, bearing cross rigidity and cross damping; and 2. carrying out the dynamic characteristic analysis on the high-speed main shaft of the machine tool by using the computation model established in step 1 to obtain a rotation speed range during smooth running and designing the running speed of the main shaft of the machine tool. The experimental result indicates that the invention can greatly enhance the precision of the analysis on the critical eddy speed and the half-speed eddy speed of the emulated high-speed main shaft of the computer, and avoids the resonance region and the unstable region for carrying out the antivibration design of the main shaft rotation speed of the high-speed machine tool according to the acquired dynamic characteristic data , thereby ensuring the machining quality. The invention has the advantages of high design efficiency and low cost.

Description

The multi-scale wavelet antivibration design method of high-speed main shaft of machine tool

Technical field

The invention belongs to machine tool structure dynamic analysis and dynamic design field, be specifically related to a kind of multi-scale wavelet antivibration design method of high-speed main shaft of machine tool.

Background technology

At a high speed being the eternal theme that Machine Tool design is made, and the lathe critical component---dynamic properties such as the modal parameter of main shaft, stability are the indexs of its most critical.At present, the machining center speed of mainshaft is generally (20000-32000) r/min, and the speed of mainshaft of gear machine is also brought up to (9000-12000) r/min, and the superhigh speed grinding machining center reaches 150 especially, 000r/min.This manufactures and designs the new challenge of continuous proposition to lathe manufacturing enterprise and scientific research institution, the an urgent demand people are by multidisciplinary intersection, the Applied Digital technology is quantitatively described and analyzes design process, realize the preview of critical component function by digital to analog simulation, understanding is grasped the essential laws of high-speed main spindle parts dynamic property, thereby instruct the optimum of relevant structural parameters and structural shape configuration to choose, reduce mismachining tolerance, improve high-speed main spindle high accuracy, high efficiency, high reliability working ability.

At present, mainly be the machine dynamic characteristics data that adopt experimental technique to obtain, efficient is low, cost is high, low precision, causes the machine tooling error big.But the research of the high-speed main spindle dynamic property being carried out high accuracy numerical simulation and then minimizing mismachining tolerance still is in the starting stage.The key factor that restricts its development is to lack high efficiency, high-precision method for solving, finds the solution to realize the kinetic model high-fidelity.Therefore, can to find the solution the totally digitilized model of high-speed main shaft of machine tool dynamic characteristic efficiently, accurately be a key technology the most basic in the Machine Tool design, the most essential, that have the greatest impact to structure.

It is the numerical analysis method that a kind of newly-developed gets up that WAVELET NUMERICAL is found the solution, utilize the characteristic of wavelet multiresolution, can obtain to be used for the multiple basic function of structural analysis, at the required precision of finding the solution problem, adopt different basic functions, yet to how realizing high efficiency multi-scale wavelet numerical solution, carry out the high-speed main shaft of machine tool dynamic analysis, obtain the range of speeds of even running, thereby reduce vibration, guarantee that the machine tooling quality yet there are no report.

Summary of the invention

The object of the present invention is to provide a kind of multi-scale wavelet antivibration design method of high-speed main shaft of machine tool.Structure can be found the solution the multi-scale wavelet numerical solution model of high-speed main shaft of machine tool dynamic characteristic efficiently, accurately, carry out the high-speed main shaft of machine tool dynamic analysis, obtain the range of speeds of even running, design machine tool chief axis running speed, thereby the minimizing spindle vibration guarantees the machine tooling quality.

Technical scheme of the present invention comprises the steps:

The first step, adopt interval three Hermite spline wavelets (English Hermite Cubic SplineWavelet on the Interval, be called for short HCSWI) as multiple dimensioned interpolation base, set up the intersection rigidity of having taken all factors into consideration gyroscopic couple, transverse shear deformation, main shaft rotary inertia, material sluggishness and viscous damping, bearing and intersect the multi-scale wavelet numerical solution model that damping influences;

Second step, adopt the computation model of being constructed in the first step, carry out the high-speed main shaft of machine tool dynamic analysis, obtain the range of speeds of even running, design machine tool chief axis running speed, thus reduce spindle vibration, guarantee the machine tooling quality.

The described first step adopts interval three Hermite spline wavelets as multiple dimensioned interpolation base, the intersection rigidity that gyroscopic couple, transverse shear deformation, main shaft rotary inertia, material sluggishness and viscous damping, bearing have been taken all factors into consideration in foundation with intersect the multi-scale wavelet numerical solution model of damping influence, may further comprise the steps:

I, set up interval three Hermite spline wavelets numerical solution models of single scale of high-speed main shaft of machine tool dynamic analysis,

Adopt interval three Hermite spline wavelets bases, metric space V ₁In scaling function φ _{1, k}For

$\left\{\begin{array}{c}{\mathrm{\φ}}_{\mathrm{1,1}}\left(x\right):\sqrt{\frac{5}{24}{\mathrm{\φ}}_1(2x-1)}\\ {\mathrm{\φ}}_{\mathrm{1,2}}\left(x\right):=\sqrt{\frac{15}{4}{\mathrm{\φ}}_2\left(2x\right)}\\ {\mathrm{\φ}}_{\mathrm{1,3}}\left(x\right):=\sqrt{\frac{15}{8}{\mathrm{\φ}}_2(2x-1)}\\ {\mathrm{\φ}}_{\mathrm{1,4}}\left(x\right):=\sqrt{\frac{15}{4}{\mathrm{\φ}}_2(2x-2)}\end{array}\right.$

Wavelet space W _j(j=1,2 ...) in wavelet function ψ _{J, k}For

The Hermite spline wavelets has following characteristic:

$<{\mathrm{\&phi;}}_{1,k}^{\&prime;}{\mathrm{\&psi;}}_{j,k}^{\&prime;}>={\&Integral;}_0^1{\mathrm{\&phi;}}_{1,k}^{\&prime;}{\mathrm{\&psi;}}_{j,k}^{\&prime;}\mathrm{dx}=0$ For j and k arbitrarily and

$<{\mathrm{\&psi;}}_{{\mathrm{<figure-callout\; id="1"\; label="j"\; filenames="HSA00000048392500011.png,HSA00000048392500012.png"\; state="\{\{state\}\}">j</figure-callout>}}_1,k}^{\&prime;},{\mathrm{\&psi;}}_{{\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HSA00000048392500011.png,HSA00000048392500012.png"\; state="\{\{state\}\}">j</figure-callout>}}_2,k}^{\&prime;}>={\&Integral;}_0^1{\mathrm{\&psi;}}_{{\mathrm{<figure-callout\; id="1"\; label="j"\; filenames="HSA00000048392500011.png,HSA00000048392500012.png"\; state="\{\{state\}\}">j</figure-callout>}}_1,k}^{\&prime;}{\mathrm{\&psi;}}_{{\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HSA00000048392500011.png,HSA00000048392500012.png"\; state="\{\{state\}\}">j</figure-callout>}}_2,k}^{\&prime;}\mathrm{dx}=0$ For j arbitrarily ₁≠ j ₂And k

With V _jIn wavelet basis be expressed as:

According to Ruili-timoshenko beam theory, the negligible axial displacement, by the Hamilton variation principle, and the intersection rigidity of considering the sluggish and viscous damping of material, bearing with intersect damping and influence the computation model of acquisition high-speed main shaft of machine tool dynamic analysis:

$\stackrel{\&OverBar;}{M}\stackrel{\&CenterDot;\&CenterDot;}{u}+\stackrel{\&OverBar;}{G}\stackrel{\&CenterDot;}{u}+\stackrel{\&OverBar;}{K}u=F$

In the following formula,

,

,

Represent mass of system matrix, gyro and damping matrix and stiffness matrix respectively with u, the core that the utilization computation model is found the solution is to ask for as lower integral:

$\left\{\begin{array}{c}{\mathrm{\&Gamma;}}^{\mathrm{1,0}}={\&Integral;}_0^1\frac{d{\mathrm{\&Phi;}}_j^T}{\mathrm{d\&xi;}}{\mathrm{\&Phi;}}_j\mathrm{d\&xi;}\\ {\mathrm{\&Gamma;}}^{\mathrm{1,1}}={\&Integral;}_0^1\frac{d{\mathrm{\&Phi;}}_j^T}{\mathrm{d\&xi;}}\frac{d{\mathrm{\&Phi;}}_j}{\mathrm{d\&xi;}}\mathrm{d\&xi;}\\ {\mathrm{\&Gamma;}}^{\mathrm{0,0}}={\&Integral;}_0^1{\mathrm{\&Phi;}}_j^T{\mathrm{\&Phi;}}_j\mathrm{d\&xi;}\end{array}\right.$

II, utilization small echo lifting scheme are constructed high efficiency multiple dimensioned solving equation, are the multi-scale wavelet numerical solution model of high-speed main shaft of machine tool dynamic analysis,

The metric space that is obtained by interval three intrinsic direct sum exploded relationship of Hermite wavelet basis promotes relation.V ₁Be initial gauges space, W _L-1(l=1,2, j-1) be wavelet space.V ₁In scaling function Φ ₁As initially approaching interpolating function, and the wavelet function in the different wavelet space constantly adds to advance in the interpolating function as add-ins, the space nest relation as shown in the formula:

V _j＝V ₁+W ₁+W ₂+…+W _j-1

Realize that the multiple dimensioned key of finding the solution of high efficiency is to realize the decoupling zero of integration item.

For the integration item

The multi-scale product sub matrix is:

Submatrix in the following formula can adopt following formula to calculate:

$A_{x,y}={\&Integral;}_0^1\frac{d{x}^T}{\mathrm{d\&xi;}}\frac{\mathrm{dy}}{\mathrm{d\&xi;}}\mathrm{d\&xi;},(x,y={\mathrm{\&phiv;}}_1,{\mathrm{\&psi;}}_1,...,{\mathrm{\&psi;}}_{j-1})$

Because of the characteristic of interval three Hermite spline wavelets bases, the off diagonal element integration in the following formula is zero.So have

From following formula as can be known, when carrying out multiple dimensioned finding the solution, integration item Г ^1,1Decoupling zero because of diagonal matrix is zero battle array, transfer to calculating A _{X, y}, (x=y=ψ ₁..., ψ _J-1).This can greatly improve multiple dimensioned numerical solution efficient.

And for the integration item

The multi-scale product sub matrix is:

The calculating formula of the submatrix in this formula is:

Because of interval three intrinsic lifting characteristics of Hermite spline wavelets base, when rising to l+1 from yardstick l and carry out Multi-Scale Calculation, integration item Г ^1,0Decoupling zero submatrix only

Need to calculate, and other submatrixs do not need to calculate once more.

For the integration item

The multi-scale product sub matrix is:

The calculating formula of the submatrix in this formula is:

Because of interval three intrinsic lifting characteristics of Hermite spline wavelets base, when rising to l+1 from yardstick l and carry out Multi-Scale Calculation, integration item Г ^0,0Decoupling zero submatrix only

Need to calculate, and other submatrixs do not need to calculate once more.This can improve computational efficiency equally.

Like this, construct multiple dimensioned solving equation, be the multi-scale wavelet numerical solution model of high-speed main shaft of machine tool dynamic analysis.When from low yardstick when high yardstick promotes, the numerical solution equation that can keep low yardstick is constant, only adds each submatrix on the diagonal, greatly improves and finds the solution efficient and solving precision.

In described second step, may further comprise the steps:

The computation model of being constructed in i, the employing first step carries out critical eddy velocity of high-speed main shaft of machine tool and half-speed vortex velocity analysis, obtains the dynamic Characteristic Data in the actual main shaft running;

The dynamic Characteristic Data that ii, foundation obtain is carried out resonance analyzing, avoids resonance region, obtains the range of speeds of even running, the high-speed machine tool main shaft assurance machine tooling quality of design antivibration operation.

The present invention has the following significant advantage that general employing experimental technique carries out the design of lathe antivibration that is different from:

The intersection rigidity that the multi-scale wavelet numerical solution computation model of the high-speed main shaft of machine tool dynamic analysis of 1) being set up has been taken all factors into consideration gyroscopic couple, transverse shear deformation, main shaft rotary inertia, material sluggishness and viscous damping, bearing with intersect the influence of damping, have a high accuracy that can reflect critical eddy velocity of actual high-speed main spindle and half-speed vortex speed;

2) adopt the inventive method carry out high-speed main spindle simulation analysis of computer, according to the dynamic Characteristic Data that obtains, avoid resonance region and unstable region, obtain the range of speeds of even running, design machine tool chief axis running speed, thereby reduce the vibration of high-speed main shaft of machine tool in actual process, guarantee the machine tooling quality;

3) required precision is not only satisfied in the simulation analysis of computer of the inventive method design, and efficient height, cost are low, and alternative experimental technique carries out the design of high-speed main shaft of machine tool antivibration.

Description of drawings

Fig. 1 is interval three Hermite spline wavelets metric space V that the multi-scale wavelet antivibration design method embodiment of this high-speed main shaft of machine tool adopts ₁In scaling function;

Fig. 2 is a small echo space W among Fig. 1 ₁In wavelet basis;

Fig. 3 is the multi-scale wavelet antivibration design method embodiment main shaft model sketch of this high-speed main shaft of machine tool;

Fig. 4 is three Hermite spline wavelets interpolation Quito, interval yardstick lifting scheme of the multi-scale wavelet antivibration design method embodiment of this high-speed main shaft of machine tool;

The computation model that Fig. 5 is constructed for the multi-scale wavelet antivibration design method embodiment of this high-speed main shaft of machine tool obtains has material hysteretic damping coefficient η _H=0.0002 o'clock high-speed main shaft of machine tool eddy velocity figure.

The specific embodiment

Below in conjunction with accompanying drawing embodiments of the invention are described in further detail:

The first step, adopt interval three Hermite spline wavelets as multiple dimensioned interpolation base, the intersection rigidity that gyroscopic couple, transverse shear deformation, main shaft rotary inertia, material sluggishness and viscous damping, bearing have been taken all factors into consideration in foundation with intersect the multi-scale wavelet numerical solution model of damping influence, may further comprise the steps:

I, set up interval three Hermite spline wavelets numerical solution models of single scale of high-speed main shaft of machine tool dynamic analysis.

Classical wavelet function is to be defined on the whole real number axis R or the square integrable real number space L of one-period function ²(R) the complete base on when finding the solution boundary value problem, numerical oscillation can occur on the border.And interval wavelet can overcome this problem.The present invention adopts interval three Hermite spline wavelets bases as multiple dimensioned interpolation base, and utilization small echo lifting scheme is constructed multiple dimensioned solving equation.

L ²(R) the linear square integrable real number space on the expression R.Definition L ²(R) inner product in:

<u，v>：＝∫ _Ru(x)v(x)dx，u，v∈L ²(R).

If＜u, v 〉=0, then u and v quadrature.L ²The norm of the function f (R) is:

If φ ₁And φ ₂For being supported on three Hermite battens on the interval [0,1], that is:

${\mathrm{\&phi;}}_1\left(x\right):=\left\{\begin{array}{cc}{(x+1)}^2(1-2x),& x\&Element;[-\mathrm{1,0}]\\ {(1-x)}^2(1+2x),& x\&Element;\left[\mathrm{0,1}\right]\\ 0,& x\&NotElement;[-\mathrm{1,1}]\end{array}\right.---\left(1\right)$

${\mathrm{\&phi;}}_2\left(x\right):=\left\{\begin{array}{cc}{(x+1)}^{2}x& x\&Element;[-\mathrm{1,0}]\\ {(1-x)}^{2}x& x\&Element;\left[\mathrm{0,1}\right]\\ 0& x\&NotElement;[-\mathrm{1,1}]\end{array}\right.---\left(2\right)$

Obviously, φ ₁And φ ₂Belong to C ¹(R).Being supported on interval [1,1] goes up by φ ₁And φ ₂The corresponding small echo ψ that generates ₁And ψ ₂For

$\left\{\begin{array}{c}{\mathrm{\&psi;}}_1\left(x\right)=-2{\mathrm{\&phi;}}_1(2x+1)+4{\mathrm{\&phi;}}_1\left(2x\right)-2{\mathrm{\&phi;}}_1(2x-1)-21{\mathrm{\&phi;}}_2(2x+1)+21{\mathrm{\&phi;}}_2(2x-1)\\ {\mathrm{\&psi;}}_2\left(x\right)={\mathrm{\&phi;}}_1(2x+1)-{\mathrm{\&phi;}}_1(2x-1)+9{\mathrm{\&phi;}}_1(2x+1)+12{\mathrm{\&phi;}}_2\left(2x\right)+9{\mathrm{\&phi;}}_2(2x-1)\end{array}---\left(3\right)\right.$

Ding Yi scaling function and small echo satisfy following condition like this

$<{\mathrm{\&psi;}}_1^{\&prime;},{\mathrm{\&phi;}}_m^{\&prime;}(\&CenterDot;-j)>=<{\mathrm{\&psi;}}_2^{\&prime;},{\mathrm{\&phi;}}_m^{\&prime;}(\&CenterDot;-j)>=0,m=\mathrm{1,2},\&ForAll;j\&Element;Z$

The translation of wavelet function generates wavelet space, ψ ₁Be symmetric function ψ ₂Be antisymmetric function.

Can further generate H by above-mentioned wavelet function ₀ ¹Small echo in (0,1) space.H ₀ ¹Wavelet decomposition relation in (0,1):

$H_0^1\left(\mathrm{0,1}\right)={\mathrm{<figure-callout\; id="1"\; label="V"\; filenames="HSA00000048392500011.png,HSA00000048392500012.png"\; state="\{\{state\}\}">V</figure-callout>}}_1+{\mathrm{<figure-callout\; id="1"\; label="W"\; filenames="HSA00000048392500011.png,HSA00000048392500012.png"\; state="\{\{state\}\}">W</figure-callout>}}_1+{\mathrm{<figure-callout\; id="2"\; label="W"\; filenames="HSA00000048392500011.png,HSA00000048392500012.png"\; state="\{\{state\}\}">W</figure-callout>}}_2+...---\left(4\right)$

V in the formula (4) ₁Be metric space, and W _j(j=1,2 ...) be the wavelet function under the different scale.

Metric space V ₁In scaling function φ _{1, k}For

$\left\{\begin{array}{c}{\mathrm{\&phi;}}_{\mathrm{1,1}}\left(x\right):\sqrt{\frac{5}{24}{\mathrm{\&phi;}}_1(2x-1)}\\ {\mathrm{\&phi;}}_{\mathrm{1,2}}\left(x\right):=\sqrt{\frac{15}{4}{\mathrm{\&phi;}}_2\left(2x\right)}\\ {\mathrm{\&phi;}}_{\mathrm{1,3}}\left(x\right):=\sqrt{\frac{15}{8}{\mathrm{\&phi;}}_2(2x-1)}\\ {\mathrm{\&phi;}}_{\mathrm{1,4}}\left(x\right):=\sqrt{\frac{15}{4}{\mathrm{\&phi;}}_2(2x-2)}\end{array}\right.---\left(5\right)$

Wavelet space W _j(j=1,2 ...) in wavelet function ψ _{J, k}For

All scaling function φ on interval [0,1] _{1, k}With wavelet function ψ _{1, k}As depicted in figs. 1 and 2.Interval three Hermite spline wavelets have following characteristic:

$<{\mathrm{\&phi;}}_{1,k^{,}}^{\&prime;}{\mathrm{\&psi;}}_{j,k}^{\&prime;}>={\&Integral;}_0^1{\mathrm{\&phi;}}_{1,k}^{\&prime;}{\mathrm{\&psi;}}_{j,k}^{\&prime;}\mathrm{dx}=0$ For j and k (7) arbitrarily

And

$<{\mathrm{\&psi;}}_{{\mathrm{<figure-callout\; id="1"\; label="j"\; filenames="HSA00000048392500011.png,HSA00000048392500012.png"\; state="\{\{state\}\}">j</figure-callout>}}_1,k}^{\&prime;},{\mathrm{\&psi;}}_{{\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HSA00000048392500011.png,HSA00000048392500012.png"\; state="\{\{state\}\}">j</figure-callout>}}_2,k}^{\&prime;}>={\&Integral;}_0^1{\mathrm{\&psi;}}_{{\mathrm{<figure-callout\; id="1"\; label="j"\; filenames="HSA00000048392500011.png,HSA00000048392500012.png"\; state="\{\{state\}\}">j</figure-callout>}}_1,k}^{\&prime;}{\mathrm{\&psi;}}_{{\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HSA00000048392500011.png,HSA00000048392500012.png"\; state="\{\{state\}\}">j</figure-callout>}}_2,k}^{\&prime;}\mathrm{dx}=0$ For j arbitrarily ₁≠ j ₂And k (8)

V _jIn wavelet basis be

In the formula (9),

Expression V ₁In scaling function, ψ _s(s=1,2 ... j-1) expression W _sIn wavelet function, and

This routine high-speed main shaft of machine tool dynamic analysis model as shown in Figure 3, the main shaft diameter is 0.1016m, long 1.27m, elastic modulus E=2.068 * 10 ¹¹Pa, density of material ρ=7833kg/m, Poisson's ratio μ=0.3.The main shaft two end supports is on the elastic damping bearing, and concrete parameter is: K _Wv=K _Vw=-2.917 * 10 ⁶N/m, C _Wv=C _Vw=0, K _Ww=K _Vv=1.7513 * 10 ⁷N/m, C _Ww=C _Vv=1.7513 * 10 ³Ns/m.

According to Ruili-timoshenko beam theory, the negligible axial displacement, main shaft potential energy U is:

$U=\frac{1}{2}{\&Integral;}_0^l\mathrm{EI}{[\left(\frac{d{\mathrm{\&theta;}}_z}{\mathrm{dx}}\right)}^2+{\left(\frac{d{\mathrm{\&theta;}}_y}{\mathrm{dx}}\right)}^2]\mathrm{dx}+\frac{1}{2}{\&Integral;}_0^l\frac{\mathrm{GA}}{k}[{(\frac{\mathrm{dw}}{\mathrm{dx}}+{\mathrm{\&theta;}}_z)}^2+{(\frac{\mathrm{dv}}{\mathrm{dx}}-{\mathrm{\&theta;}}_y)}^2]\mathrm{dx}---\left(10\right)$

E represents Young's modulus in the formula (10), w (ξ, t) and v (ξ t) represents lateral displacement, and l represents main axis length, θ _z(x, t) and θ _y(G represents modulus of shearing for x, the t) rotation displacement that caused by bending of expression, and I represents cross sectional moment of inertia, and A represents area of section, and k represents to shear correction factor (k=(7+12 μ+4 μ ²)/6 (1+ μ) ², μ represents Poisson's Ratio).

Main shaft kinetic energy T is:

$T=\frac{1}{2}{\&Integral;}_0^1\mathrm{\&rho;A}[{\left(\frac{\&PartialD;w}{\&PartialD;t}\right)}^2+{\left(\frac{\&PartialD;v}{\&PartialD;t}\right)}^2]\mathrm{dx}+\frac{1}{2}{\&Integral;}_0^1\mathrm{\&rho;I}[{\left(\frac{{\&PartialD;\mathrm{\&theta;}}_z}{\&PartialD;t}\right)}^2+{\left(\frac{\&PartialD;{\mathrm{\&theta;}}_y}{\&PartialD;t}\right)}^2]\mathrm{dx}-$

(11)

$\mathrm{\&Omega;}{\&Integral;}_0^1{J}_x\mathrm{\&rho;}\frac{{\&PartialD;\mathrm{\&theta;}}_y}{\&PartialD;t}{\mathrm{\&theta;}}_z\mathrm{dx}+\frac{{\mathrm{\&Omega;}}^2}{2}{\&Integral;}_0^1{J}_x\mathrm{\&rho;dx}$

ρ represents spindle material density in the formula (11), and Ω represents rotating speed (rad/s), J _xExpression main shaft second polar moment of area.

W (ξ, t), v (ξ, t), θ _z(ξ, t) and θ _y(ξ t) can be with interval three Hermite spline wavelets base interpolation representation:

$\left\{\begin{array}{c}w(\mathrm{\&xi;},t)={\mathrm{\&Phi;}}_{j}a\\ v(\mathrm{\&xi;},t)={\mathrm{\&Phi;}}_{j}b\\ {\mathrm{\&theta;}}_z(\mathrm{\&xi;},t)={\mathrm{\&Phi;}}_{j}c\\ {\mathrm{\&theta;}}_y(\mathrm{\&xi;},t)={\mathrm{\&Phi;}}_{j}d\end{array}---\left(12\right)\right.$

Small echo interpolation coefficient vector a in the formula (12), b, c, d is:

$\left\{\begin{array}{c}a={\{a_1{a}_2...w_{2^{j+1}}\}}^T\\ b={\{b_1{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="HSA00000048392500011.png,HSA00000048392500012.png"\; state="\{\{state\}\}">b</figure-callout>}}_2...b_{2^{j+1}}\}}^T\\ c={\{c_1{\mathrm{<figure-callout\; id="2"\; label="c"\; filenames="HSA00000048392500011.png,HSA00000048392500012.png"\; state="\{\{state\}\}">c</figure-callout>}}_2...c_{2^{j+1}}\}}^T\\ d={\{d_1{\mathrm{<figure-callout\; id="2"\; label="d"\; filenames="HSA00000048392500011.png,HSA00000048392500012.png"\; state="\{\{state\}\}">d</figure-callout>}}_2...d_{2^{j+1}}\}}^T\end{array}\right.---\left(13\right)$

With formula (12) difference substitution formula (10) and formula (11), obtain:

$\left\{\begin{array}{c}U=\frac{1}{2}{a}^T{K}^{1}a-\frac{1}{2}{a}^T{K}^{2}c-\frac{1}{2}{c}^T{K}^{3}a+\frac{1}{2}{c}^T{K}^{4}c+\frac{1}{2}{c}^T{K}^{5}c\\ +\frac{1}{2}{b}^T{K}^{1}b+\frac{1}{2}{b}^T{K}^{2}d+\frac{1}{2}{d}^T{K}^{3}b+\frac{1}{2}{d}^T{K}^{4}d+\frac{1}{2}{d}^T{K}^{5}d\\ T=\frac{1}{2}{\left(\frac{\&PartialD;a}{\&PartialD;t}\right)}^T{M}^b\left(\frac{\&PartialD;a}{\&PartialD;t}\right)+\frac{1}{2}{\left(\frac{\&PartialD;b}{\&PartialD;t}\right)}^T{M}^b\left(\frac{\&PartialD;b}{\&PartialD;t}\right)+\frac{1}{2}{\left(\frac{\&PartialD;c}{\&PartialD;t}\right)}^T{M}^r\left(\frac{\&PartialD;c}{\&PartialD;t}\right)+\frac{1}{2}{\left(\frac{\&PartialD;d}{\&PartialD;t}\right)}^T{M}^r\left(\frac{\&PartialD;d}{\&PartialD;t}\right)\\ -{\left(\frac{\&PartialD;d}{\&PartialD;t}\right)}^T\mathrm{Gc}+\frac{1}{2}{J}_{x}l{\mathrm{\&Omega;}}^2\end{array}\right.---\left(14\right)$

The rigidity submatrix is respectively in the formula (14):

${\mathrm{<figure-callout\; id="1"\; label="K"\; filenames="HSA00000048392500011.png,HSA00000048392500012.png"\; state="\{\{state\}\}">K</figure-callout>}}^1=\frac{\mathrm{GA}}{\mathrm{kl}}{\mathrm{\&Gamma;}}^{\mathrm{1,1}}---\left(15\right)$

${\mathrm{<figure-callout\; id="2"\; label="K"\; filenames="HSA00000048392500011.png,HSA00000048392500012.png"\; state="\{\{state\}\}">K</figure-callout>}}^2=-\frac{\mathrm{GA}}{k}{\mathrm{\&Gamma;}}^{\mathrm{1,0}}---\left(16\right)$

K ³＝(K ²) ^T (17)

${\mathrm{<figure-callout\; id="4"\; label="K"\; filenames="HSA00000048392500011.png,HSA00000048392500012.png"\; state="\{\{state\}\}">K</figure-callout>}}^4=\frac{\mathrm{EI}}{l}{\mathrm{\&Gamma;}}^{\mathrm{1,1}}---\left(18\right)$

$K^5=\frac{\mathrm{GAl}}{k}{\mathrm{\&Gamma;}}^{\mathrm{0,0}}---\left(19\right)$

The bending quality matrix is:

M ^b＝ρAlГ ^0，0 (20)

The rotary inertia mass matrix is:

M ^r＝ρIlГ ^0，0 (21)

Gyro submatrix G is:

G＝ΩρJ _xlГ ^0，0 (22)

More than various in the integration item be:

$\left\{\begin{array}{c}{\mathrm{\&Gamma;}}^{\mathrm{1,0}}={\&Integral;}_0^1\frac{d{\mathrm{\&Phi;}}_j^T}{\mathrm{d\&xi;}}{\mathrm{\&Phi;}}_j\mathrm{d\&xi;}\\ {\mathrm{\&Gamma;}}^{\mathrm{1,1}}={\&Integral;}_0^1\frac{d{\mathrm{\&Phi;}}_j^T}{\mathrm{d\&xi;}}\frac{d{\mathrm{\&Phi;}}_j}{\mathrm{d\&xi;}}\mathrm{d\&xi;}\\ {\mathrm{\&Gamma;}}^{\mathrm{0,0}}={\&Integral;}_0^1{\mathrm{\&Phi;}}_j^T{\mathrm{\&Phi;}}_j\mathrm{d\&xi;}\end{array}\right.---\left(23\right)$

Use the Hamilton variation principle in Lagrangian L=U-T, can obtain the multi-scale wavelet numerical solution equation of high-speed main shaft of machine tool dynamic analysis, that is:

$M\left[\begin{array}{c}\frac{{\&PartialD;}^{2}a}{{\&PartialD;\mathrm{<figure-callout\; id="2"\; label="t"\; filenames="HSA00000048392500011.png,HSA00000048392500012.png"\; state="\{\{state\}\}">t</figure-callout>}}^2}\\ \frac{{\&PartialD;}^{2}b}{{\&PartialD;\mathrm{<figure-callout\; id="2"\; label="t"\; filenames="HSA00000048392500011.png,HSA00000048392500012.png"\; state="\{\{state\}\}">t</figure-callout>}}^2}\\ \frac{{\&PartialD;}^{2}c}{{\&PartialD;\mathrm{<figure-callout\; id="2"\; label="t"\; filenames="HSA00000048392500011.png,HSA00000048392500012.png"\; state="\{\{state\}\}">t</figure-callout>}}^2}\\ \frac{{\&PartialD;}^{2}d}{{\&PartialD;\mathrm{<figure-callout\; id="2"\; label="t"\; filenames="HSA00000048392500011.png,HSA00000048392500012.png"\; state="\{\{state\}\}">t</figure-callout>}}^2}\end{array}\right]+g\left[\begin{array}{c}\frac{\&PartialD;a}{\&PartialD;t}\\ \frac{\&PartialD;b}{\&PartialD;t}\\ \frac{\&PartialD;c}{\&PartialD;t}\\ \frac{\&PartialD;d}{\&PartialD;t}\end{array}\right]+K\left[\begin{array}{c}a\\ b\\ c\\ d\end{array}\right]=F---\left(24\right)$

In the formula (24), F is a force vector, M, and g and K represent quality, gyro and stiffness matrix respectively, respectively by following various providing:

$M=\left[\begin{array}{cccc}{M}^b& & & \\ & M^b& & \\ & & M^r& \\ & & & M^r\end{array}\right]---\left(25\right)$

$g=\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& G\\ 0& 0& -G& 0\end{array}\right]---\left(26\right)$

$K=\left[\begin{array}{cccc}{\mathrm{<figure-callout\; id="1"\; label="K"\; filenames="HSA00000048392500011.png,HSA00000048392500012.png"\; state="\{\{state\}\}">K</figure-callout>}}^1& 0& -{\mathrm{<figure-callout\; id="2"\; label="K"\; filenames="HSA00000048392500011.png,HSA00000048392500012.png"\; state="\{\{state\}\}">K</figure-callout>}}^2& 0\\ 0& {\mathrm{<figure-callout\; id="1"\; label="K"\; filenames="HSA00000048392500011.png,HSA00000048392500012.png"\; state="\{\{state\}\}">K</figure-callout>}}^1& 0& K^2\\ {-\mathrm{<figure-callout\; id="3"\; label="K"\; filenames="HSA00000048392500011.png,HSA00000048392500012.png"\; state="\{\{state\}\}">K</figure-callout>}}^3& 0& {\mathrm{<figure-callout\; id="4"\; label="K"\; filenames="HSA00000048392500011.png,HSA00000048392500012.png"\; state="\{\{state\}\}">K</figure-callout>}}^4+K^5& 0\\ 0& {\mathrm{<figure-callout\; id="3"\; label="K"\; filenames="HSA00000048392500011.png,HSA00000048392500012.png"\; state="\{\{state\}\}">K</figure-callout>}}^3& 0& {\mathrm{<figure-callout\; id="4"\; label="K"\; filenames="HSA00000048392500011.png,HSA00000048392500012.png"\; state="\{\{state\}\}">K</figure-callout>}}^4+K^5\end{array}\right]---\left(27\right)$

For finding the solution high-speed main spindle nature eddy velocity and unstability threshold value, ignore exciting force, then system equation can be expressed as:

$M\frac{{\&PartialD;}^{2}u}{{\&PartialD;\mathrm{<figure-callout\; id="2"\; label="t"\; filenames="HSA00000048392500011.png,HSA00000048392500012.png"\; state="\{\{state\}\}">t</figure-callout>}}^2}+G\frac{\&PartialD;u}{\&PartialD;t}+\mathrm{Ku}=0---\left(28\right)$

The medium and small wave system number vector of formula (28) u={a ^Tb ^Tc ^Td ^T} ^T

The material internal damping has material impact to the high speed rotor.Consider material sluggishness and viscous damping, the system vibration equation can be expressed as

$M\frac{{\&PartialD;}^{2}u}{{\&PartialD;\mathrm{<figure-callout\; id="2"\; label="t"\; filenames="HSA00000048392500011.png,HSA00000048392500012.png"\; state="\{\{state\}\}">t</figure-callout>}}^2}+({\mathrm{\&eta;}}_{\mathrm{<figure-callout\; id="1"\; label="V\; K\; c"\; filenames="HSA00000048392500011.png,HSA00000048392500012.png"\; state="\{\{state\}\}">V</figure-callout>}}{K}_c{1}_{}+G)\frac{\&PartialD;u}{\&PartialD;t}+({\mathrm{\&eta;}}_{a}K+{\mathrm{\&eta;}}_b{K}_{c2})u=F---\left(29\right)$

In the formula (29), η _HAnd η _VExpression material sluggishness and viscous damping coefficient, and

$\left\{\begin{array}{c}\mathrm{\&eta;}=\left[\begin{array}{cc}{\mathrm{\&eta;}}_a& {\mathrm{\&eta;}}_b\\ -{\mathrm{\&eta;}}_b& {\mathrm{\&eta;}}_a\end{array}\right]\\ {\mathrm{\&eta;}}_a=\frac{1+{\mathrm{\&eta;}}_{\mathrm{<figure-callout\; id="1"\; label="H"\; filenames="HSA00000048392500011.png,HSA00000048392500012.png"\; state="\{\{state\}\}">H</figure-callout>}}}{\sqrt{1+{\mathrm{\&eta;}}_{\mathrm{<figure-callout\; id="2"\; label="H"\; filenames="HSA00000048392500011.png,HSA00000048392500012.png"\; state="\{\{state\}\}">H</figure-callout>}}^2}}\\ {\mathrm{\&eta;}}_b=\frac{{\mathrm{\&eta;}}_{\mathrm{<figure-callout\; id="1"\; label="H"\; filenames="HSA00000048392500011.png,HSA00000048392500012.png"\; state="\{\{state\}\}">H</figure-callout>}}}{\sqrt{1+{\mathrm{\&eta;}}_{\mathrm{<figure-callout\; id="2"\; label="H"\; filenames="HSA00000048392500011.png,HSA00000048392500012.png"\; state="\{\{state\}\}">H</figure-callout>}}^2}}+{\mathrm{\&Omega;\&eta;}}_V\end{array}\right.---\left(30\right)$

And K _C1And K _C2Be respectively:

$\left\{\begin{array}{c}{\mathrm{<figure-callout\; id="1"\; label="K\; c"\; filenames="HSA00000048392500011.png,HSA00000048392500012.png"\; state="\{\{state\}\}">K</figure-callout>}}_c={\stackrel{\&OverBar;}{R}}_1\mathrm{<figure-callout\; id="2"\; label="K\; K\; c"\; filenames="HSA00000048392500011.png,HSA00000048392500012.png"\; state="\{\{state\}\}">K</figure-callout>}& \\ K_c{2}_{}={\stackrel{\&OverBar;}{R}}_{2}K\end{array}---\left(31\right)\right.$

In the formula (31)

With

Find the solution with following formula:

${\stackrel{\&OverBar;}{R}}_1={\left[\begin{array}{cccccc}{R}_1& & & & & \\ & {\mathrm{<figure-callout\; id="1"\; label="R"\; filenames="HSA00000048392500011.png,HSA00000048392500012.png"\; state="\{\{state\}\}">R</figure-callout>}}_1& & & & \\ & & .& & & \\ & & & .& & \\ & & & & .& \\ & & & & & {\mathrm{<figure-callout\; id="1"\; label="R"\; filenames="HSA00000048392500011.png,HSA00000048392500012.png"\; state="\{\{state\}\}">R</figure-callout>}}_1\end{array}\right]}_{2^{j+3}\&times;2^{j+3}}$ With ${\stackrel{\&OverBar;}{R}}_2={\left[\begin{array}{cccccc}{R}_2& & & & & \\ & {\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="HSA00000048392500011.png,HSA00000048392500012.png"\; state="\{\{state\}\}">R</figure-callout>}}_2& & & & \\ & & .& & & \\ & & & .& & \\ & & & & .& \\ & & & & & {\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="HSA00000048392500011.png,HSA00000048392500012.png"\; state="\{\{state\}\}">R</figure-callout>}}_2\end{array}\right]}_{2^{j+3}\&times;2^{j+3}}---\left(32\right)$

And

${\mathrm{<figure-callout\; id="1"\; label="R"\; filenames="HSA00000048392500011.png,HSA00000048392500012.png"\; state="\{\{state\}\}">R</figure-callout>}}_1=\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& -1\end{array}\right]$ With ${\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="HSA00000048392500011.png,HSA00000048392500012.png"\; state="\{\{state\}\}">R</figure-callout>}}_2=\left[\begin{array}{cccc}0& 1& 0& 0\\ -1& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& -1& 0\end{array}\right]---\left(33\right)$

Adopt the linear bearing model, as shown in Figure 3.Its governing equation is:

${\stackrel{\&OverBar;}{C}}^b\frac{{\&PartialD;u}_b}{\&PartialD;t}+{\stackrel{\&OverBar;}{K}}^b{u}_b={\stackrel{\&OverBar;}{F}}^b---\left(34\right)$

In the formula (34),

Be bearing vector, u _b ^eBe the bearing free degree vector, bearing damp and stiffness matrix With

Be expressed as

$\left\{\begin{array}{c}{\stackrel{\&OverBar;}{C}}^b=\left[\begin{array}{cc}{C}_{\mathrm{ww}}& C_{\mathrm{wv}}\\ C_{\mathrm{vw}}& C_{\mathrm{vv}}\end{array}\right]\\ {\stackrel{\&OverBar;}{K}}^b=\left[\begin{array}{cc}{K}_{\mathrm{ww}}& K_{\mathrm{wv}}\\ K_{\mathrm{vw}}& C_{\mathrm{vv}}\end{array}\right]\end{array}\right.---\left(35\right)$

In the formula (35), C _IjAnd K _IjBe bearing damp and stiffness coefficient.

Consider material internal damping and bearing rigidity and damping influence, system equation becomes:

$\stackrel{\&OverBar;}{M}\stackrel{\&CenterDot;\&CenterDot;}{u}+\stackrel{\&OverBar;}{G}\stackrel{\&CenterDot;}{u}+\stackrel{\&OverBar;}{K}u=0---\left(36\right)$

In the formula (36),

,

,

Represent mass of system matrix, gyro and damping matrix and stiffness matrix respectively with u.

For finding the solution conveniently, formula (36) is rewritten as the single order state vector, that is:

$E\stackrel{\&CenterDot;}{q}+\mathrm{Fq}=0---\left(37\right)$

In the formula (37)

$q=\left[\begin{array}{c}\stackrel{\&CenterDot;}{u}\\ u\end{array}\right]---\left(38\right)$

$E=\left[\begin{array}{cc}0& -\stackrel{\&OverBar;}{M}\\ \stackrel{\&OverBar;}{M}& \stackrel{\&OverBar;}{G}\end{array}\right]---\left(39\right)$

$F=\left[\begin{array}{cc}\stackrel{\&OverBar;}{M}& 0\\ 0& \stackrel{\&OverBar;}{K}\end{array}\right]---\left(40\right)$

The corresponding free running frequency equation of formula (37) is

|Eλ+F|＝0 (41)

Among the multiple characteristic root λ=σ of frequency equation+i ω=σ+i2 π f, ω (rad/s) is intrinsic frequency of vortex motion (eddy velocity), and f (Hz) is a model frequency.σ is an attenuation coefficient, definition logarithmic decrement coefficient δ

$\mathrm{\&delta;}=-\frac{2\mathrm{\&pi;\&sigma;}}{\mathrm{\&omega;}}---\left(42\right)$

δ=0 expression unstability threshold value, when δ＜0, main shaft easily brings out corresponding mode unstability.

II, utilization small echo lifting scheme are constructed high efficiency multiple dimensioned solving equation, are the multi-scale wavelet numerical solution model of high-speed main shaft of machine tool dynamic analysis.

Utilize the multiresolution analysis characteristic of interval three Hermite spline wavelets, obtain the intrinsic direct sum exploded relationship of wavelet basis, that is:

${\mathrm{<figure-callout\; id="1"\; label="V"\; filenames="HSA00000048392500011.png,HSA00000048392500012.png"\; state="\{\{state\}\}">V</figure-callout>}}_1={\mathrm{<figure-callout\; id="1"\; label="V"\; filenames="HSA00000048392500011.png,HSA00000048392500012.png"\; state="\{\{state\}\}">V</figure-callout>}}_1\stackrel{\&CenterDot;}{+}{\mathrm{<figure-callout\; id="1"\; label="W"\; filenames="HSA00000048392500011.png,HSA00000048392500012.png"\; state="\{\{state\}\}">W</figure-callout>}}_1\stackrel{\&CenterDot;}{+}{\mathrm{<figure-callout\; id="2"\; label="W"\; filenames="HSA00000048392500011.png,HSA00000048392500012.png"\; state="\{\{state\}\}">W</figure-callout>}}_2\stackrel{\&CenterDot;}{+}...\stackrel{\&CenterDot;}{+}{W}_{l-1}---\left(43\right)$

Symbol in the formula (43)

The expression direct sum.It is to be noted: wavelet space self can form a complete space, and the unknown field function adopts the wavelet basis in the wavelet space to launch.Yet, only comprising limited small echo item when launching for assurance, metric space must comprise the initial gauges SPACE V when decomposing ₁Fig. 4 provides the metric space that is obtained by interval three Hermite small echo direct sum exploded relationship and promotes relation.V ₁Be initial gauges space, W _L-1(l=1,2, j-1) be wavelet space.V ₁In scaling function Φ ₁As initially approaching interpolating function, and the wavelet function in the different wavelet space constantly adds to advance in the interpolating function as add-ins, the space nest relation as shown in the formula:

$V_j={\mathrm{<figure-callout\; id="1"\; label="V"\; filenames="HSA00000048392500011.png,HSA00000048392500012.png"\; state="\{\{state\}\}">V</figure-callout>}}_1\stackrel{\&CenterDot;}{+}{\mathrm{<figure-callout\; id="1"\; label="W"\; filenames="HSA00000048392500011.png,HSA00000048392500012.png"\; state="\{\{state\}\}">W</figure-callout>}}_1\stackrel{\&CenterDot;}{+}{\mathrm{<figure-callout\; id="2"\; label="W"\; filenames="HSA00000048392500011.png,HSA00000048392500012.png"\; state="\{\{state\}\}">W</figure-callout>}}_2\stackrel{\&CenterDot;}{+}...\stackrel{\&CenterDot;}{+}{W}_{j-1}---\left(44\right)$

Realize that the multiple dimensioned key of finding the solution is the multiple dimensioned nested calculating of the integration item shown in the realization formula (23).For the integration item

The multi-scale product sub matrix is:

Submatrix in the formula (45) adopts following formula to calculate:

Consider formula (7) and formula (8), the off diagonal element integration in the formula (46) is zero.So formula (46) becomes

As seen from formula (38), when carrying out multiple dimensioned finding the solution, integration item Г ^1,1Decoupling zero A only _{X, y}, (x=y=ψ ₁..., ψ _J-1) need to calculate, and diagonal matrix is zero battle array.This can greatly improve multiple dimensioned numerical solution efficient.

And for the integration item

The multi-scale product sub matrix is:

The calculating formula of the submatrix in the formula (48) is:

Because of interval three intrinsic lifting characteristics of Hermite spline wavelets base, when rising to l+1 from yardstick l and carry out Multi-Scale Calculation, integration item Г ^1,0Decoupling zero submatrix only

Need to calculate, and other submatrixs do not need to calculate once more, can improve computational efficiency.

For the integration item ${\mathrm{\&Gamma;}}^{\mathrm{0,0}}={\&Integral;}_0^1{\mathrm{\&Phi;}}_j^T{\mathrm{\&Phi;}}_j\mathrm{d\&xi;}$ The multi-scale product sub matrix is:

The calculating formula of the submatrix in the formula (50) is:

Because of interval three intrinsic lifting characteristics of Hermite spline wavelets base, when rising to l+1 from yardstick l and carry out Multi-Scale Calculation, integration item Г ^0,0Decoupling zero submatrix only

Need to calculate, and other submatrixs do not need to calculate once more, can improve computational efficiency.

Like this, construct multiple dimensioned solving equation, be the multi-scale wavelet numerical solution model of high-speed main shaft of machine tool dynamic analysis.When from low yardstick when high yardstick promotes, the numerical solution equation that can keep low yardstick is constant, only adds each submatrix on the diagonal.Can greatly improve and find the solution efficient and solving precision.

Second step, adopt the computation model of being constructed in the first step, carry out the high-speed main shaft of machine tool dynamic analysis, obtain the range of speeds of even running, thereby reduce vibration, guarantee crudy, may further comprise the steps:

The computation model of being constructed in i, the employing first step carries out critical eddy velocity of high-speed main shaft of machine tool and half-speed vortex velocity analysis, obtains the dynamic Characteristic Data in the actual main shaft running;

Yardstick j=1, interval three the Hermite battens multi-scale wavelet base under 2,3 carries out the main shaft frequency of vortex motion as the interpolation base and stability is found the solution, and the solving equation scale is respectively 16 * 16, and 32 * 32,64 * 64.In order to compare intuitively, the utilization experimental result is benchmark as a comparison.The frequency of vortex motion solving result is as shown in table 1.

Eddy velocity (rad/s) multi-scale wavelet numerical solution model solution result and experimental result are relatively under 4000 rev/mins of conditions of table 1

Note: F represents forward eddy velocity (rad/s)

B represents reverse eddy velocity (rad/s)

E1 represents the 1st experimental result (rad/s)

E2 represents the 2nd experimental result (rad/s)

As shown in Table 1, multi-scale wavelet numerical solution model is when yardstick j=3 (64 * 64), and solving result is compared with E2 with twice experimental result E1, and is very identical, and yardstick j=1, and 2,3 times solving result shows: multiple dimensioned solving model converges on experimental result.

Figure 5 shows that η _H=0.0002 o'clock rotor eddy hodograph, abscissa is represented the speed of mainshaft, ordinate is represented frequency of vortex motion.Can obtain 1,2,3 rank unstability critical speeds by Fig. 5.

For Fig. 5, η _H=0.0002 o'clock, preceding 3 rank forward whirling motion rotating speeds are respectively 4976 rev/mins (ω among Fig. 5=Ω straight line and 1F intersections of complex curve), 10476 rev/mins (ω among Fig. 5=2 Ω straight lines and 2F intersections of complex curve) and 21558 rev/mins (ω among Fig. 5=3 Ω straight lines and 2F intersections of complex curve), and the reverse whirling motion rotating speed in preceding 3 rank is respectively 4969 rev/mins (ω among Fig. 5=Ω straight line and 1B intersections of complex curve), 10432 rev/mins (ω among Fig. 5=2 Ω straight lines and 2B intersections of complex curve) and 21298 rev/mins (ω among Fig. 5=3 Ω straight lines and 3B intersections of complex curve).And the half-speed vortex critical speed is 14780 rev/mins.

The dynamic Characteristic Data that ii, foundation obtain is carried out resonance analyzing, avoids resonance region, obtains the range of speeds of machine tool chief axis even running, designs high-speed machine tool main shaft running speed, guarantees the main shaft antivibration of high speed rotating, guarantees the machine tooling quality.

According to high-speed main shaft of machine tool being carried out frequency of vortex motion and stability analysis, the dynamic Characteristic Data in the actual main shaft running of acquisition.For high-speed main spindle design provides foundation.Resonance region is 0.9～1.1 times of critical speed range zone, and the scope of avoiding resonance region is the main shaft even running range of speeds.The even running scope that the present embodiment main shaft is obtained the well processed precision is:

For main shaft, work as η with hysteretic damping material _H=0.0002 o'clock, preceding 3 rank forward whirling motion resonance regions were respectively: 4478～5473 rev/mins, 9628～11523 rev/mins, 19402～23713 rev/mins.And the reverse whirling motion resonance region in preceding 3 rank is respectively: 4472～5465 rev/mins, 9388～11475 rev/mins, 19168～23427 rev/mins.The half-speed vortex resonance region is: 13302～16258 rev/mins.Comprehensively as can be known: resonate to the influence of crudy for avoiding preceding 3 rank forward whirling motions, oppositely whirling motion, half-speed vortex, the main shaft even running range of speeds is: 0～4472 rev/min, 5473～9388 rev/mins, 11523～13302 rev/mins, 16258～19168 rev/mins, 23713 rev/mins～higher rotating speed.

The embodiment analysis result shows: the inventive method is used for the high-speed main shaft of machine tool simulation analysis can carry out the analysis of critical eddy velocity of high-speed main shaft of machine tool and half-speed vortex speed reliably, according to the dynamic Characteristic Data that obtains, avoid resonance region and unstable region, carry out the virtual antivibration design of high-speed machine tool main shaft, reduce the spindle vibration of high speed rotating, reduce the machine tooling error, guarantee that designed lathe has high crudy.Compare with experimental design simultaneously, design efficiency greatly improves, cost also significantly reduces.

The foregoing description is the specific case that purpose of the present invention, technical scheme and beneficial effect are further described only, and the present invention is defined in this.All any modifications of within scope of disclosure of the present invention, being made, be equal to replacement, improvement etc., all be included within protection scope of the present invention.

Claims (3)

1. the multi-scale wavelet antivibration design method of high-speed main shaft of machine tool is characterized in that,

The first step, adopt interval three Hermite spline wavelets as multiple dimensioned interpolation base, the intersection rigidity of setting up comprehensive gyroscopic couple, transverse shear deformation, main shaft rotary inertia, material sluggishness and viscous damping, bearing with intersect the multi-scale wavelet numerical solution model that damping influences;

Second step, adopt the computation model of being constructed in the first step, carry out the high-speed main shaft of machine tool dynamic analysis, obtain the range of speeds of even running, design machine tool chief axis running speed.

2. the multi-scale wavelet antivibration design method of high-speed main shaft of machine tool according to claim 1 is characterized in that the described first step may further comprise the steps:

I, set up interval three Hermite spline wavelets numerical solution models of single scale of high-speed main shaft of machine tool dynamic analysis,

Adopt interval three Hermite spline wavelets bases, the scaling function φ among the metric space V1 _{1, k}For

$\left\{\begin{array}{c}{\mathrm{\&phi;}}_{\mathrm{1,1}}\left(x\right):=\sqrt{\frac{5}{24}{\mathrm{\&phi;}}_1(2x-1)}\\ {\mathrm{\&phi;}}_{\mathrm{1,2}}\left(x\right):=\sqrt{\frac{15}{4}{\mathrm{\&phi;}}_2\left(2x\right)}\\ {\mathrm{\&phi;}}_{\mathrm{1,3}}\left(x\right):=\sqrt{\frac{15}{8}{\mathrm{\&phi;}}_2(2x-1)}\\ {\mathrm{\&phi;}}_{\mathrm{1,4}}\left(x\right):=\sqrt{\frac{15}{4}{\mathrm{\&phi;}}_2(2x-2)}\end{array}\right.$

Wavelet space W _j(j=1,2 ...) in wavelet function ψ _{J, k}For

The Hermite spline wavelets has following characteristic:

$<{\mathrm{\&phi;}}_{1,k}^{\&prime;},{\mathrm{\&psi;}}_{j,k}^{\&prime;}>={\&Integral;}_0^1{\mathrm{\&phi;}}_{1,k}^{\&prime;}{\mathrm{\&psi;}}_{j,k}^{\&prime;}\mathrm{dx}=0$ For j and k arbitrarily

And

$<{\mathrm{\&psi;}}_{j_1,k}^{\&prime;},{\mathrm{\&psi;}}_{j_2,k}^{\&prime;}>={\&Integral;}_0^1{\mathrm{\&psi;}}_{j_1,k}^{\&prime;}{\mathrm{\&psi;}}_{j_2,k}^{\&prime;}\mathrm{dx}=0$ For j arbitrarily ₁≠ j ₂And k

With V _jIn wavelet basis be expressed as:

According to Ruili-timoshenko beam theory, the negligible axial displacement, by the Hamilton variation principle, and the intersection rigidity of considering the sluggish and viscous damping of material, bearing with intersect damping and influence the computation model of acquisition high-speed main shaft of machine tool dynamic analysis:

In this formula,

Represent mass of system matrix, gyro and damping matrix and stiffness matrix respectively with u, the utilization computation model is asked for as lower integral:

$\left\{\begin{array}{c}{\mathrm{\&Gamma;}}^{\mathrm{1,0}}={\&Integral;}_0^1\frac{d{\mathrm{\&Phi;}}_j^T}{\mathrm{d\&xi;}}{\mathrm{\&Phi;}}_j\mathrm{d\&xi;}\\ {\mathrm{\&Gamma;}}^{\mathrm{1,1}}={\&Integral;}_0^1\frac{d{\mathrm{\&Phi;}}_j^T}{\mathrm{d\&xi;}}\frac{d{\mathrm{\&Phi;}}_j}{\mathrm{d\&xi;}}\mathrm{d\&xi;}\\ {\mathrm{\&Gamma;}}^{\mathrm{0,0}}={\&Integral;}_0^1{\mathrm{\&Phi;}}_j^1{\mathrm{\&Phi;}}_j\mathrm{d\&xi;}\end{array}\right.;$

II, utilization small echo lifting scheme are constructed high efficiency multiple dimensioned solving equation, promptly obtain the multi-scale wavelet numerical solution model of high-speed main shaft of machine tool dynamic analysis,

The metric space that is obtained by interval three intrinsic direct sum exploded relationship of Hermite wavelet basis promotes relation, V ₁Be initial gauges space, W _L-1(l=1,2, j-1) be wavelet space, V ₁In scaling function Φ ₁As initially approaching interpolating function, and the wavelet function in the different wavelet space constantly adds to advance in the interpolating function as add-ins, the space nest relation as shown in the formula:

$V_j=V_1\stackrel{\&CenterDot;}{+}{W}_1\stackrel{\&CenterDot;}{+}{W}_2+\stackrel{\&CenterDot;}{+}...\stackrel{\&CenterDot;}{+}{W}_{j-1};$

Carry out the decoupling zero of integration item for realizing multiple dimensioned the finding the solution of high efficiency;

For the integration item The multi-scale product sub matrix is:

The calculating formula of the submatrix in this formula is:

Because of the characteristic of interval three Hermite spline wavelets bases, Γ ^1,1Off diagonal element integration in the formula is zero, so have

When carrying out multiple dimensioned finding the solution, integration item Γ ^1,1Decoupling zero, because of diagonal matrix is zero battle array, transfer to calculating A _{X, y}, (x=y=ψ ₁..., ψ _J-1);

And for the integration item

The multi-scale product sub matrix is:

The calculating formula of the submatrix in this formula is:

Because of interval three intrinsic lifting characteristics of Hermite spline wavelets base, when rising to l+1 from yardstick l and carry out Multi-Scale Calculation, integration item Γ ^1,0Decoupling zero, only need calculated sub-matrix B _{X, y},

For the integration item

The multi-scale product sub matrix is:

The calculating formula of the submatrix in this formula is:

Because of interval three intrinsic lifting characteristics of Hermite spline wavelets base, when rising to l+1 from yardstick l and carry out Multi-Scale Calculation, integration item Γ ^0,0Decoupling zero, only need calculated sub-matrix C _{X, y},

3. the multi-scale wavelet antivibration design method of high-speed main shaft of machine tool according to claim 1 is characterized in that described second step may further comprise the steps:

The computation model of being constructed in i, the employing first step carries out critical eddy velocity of high-speed main shaft of machine tool and half-speed vortex velocity analysis, obtains the dynamic Characteristic Data in the actual main shaft running;

The dynamic Characteristic Data that ii, foundation obtain is carried out resonance analyzing, obtains the range of speeds of even running, the high-speed machine tool speed of mainshaft of design antivibration operation.

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