CN106354974A - Calculation method of equivalent stiffness and equivalent damping of rolling bearing - Google Patents

Abstract

The invention discloses a calculation method of equivalent stiffness and equivalent damping of a rolling bearing. The calculation method comprises steps as follows: 1) a contact area among a rolling element, an oil film and a raceway is refined on the basis of an elastic fluid dynamic lubrication theory; 2) the elastic deformation stiffness of contact, the oil film pressure and the oil film thickness in each refined area are obtained by using a pressure distribution bilinear approximation function and a difference method, and the oil film dynamic stiffness and the oil film damping in each refined area are obtained by the aid of a linear disturbance equation; the contact stiffness and the contact damping of contact of the rolling element and the raceway are calculated; 3) according to the calculated contact stiffness and the contact damping of contact of the rolling element and the raceway in step 2), the equivalent stiffness and the equivalent damping of the rolling bearing are calculated. The calculation method has the advantage of high calculation result accuracy, the technical support is provided for design of a bearing rotor system, and the fault rate of the designed bearing rotor system is decreased.

Description

A kind of rolling bearing equivalent stiffness and the computational methods of equivalent damping

Technical field

The present invention relates to the computational methods of a kind of equivalent stiffness of rolling bearing and equivalent damping.

Background technology

Bearing rotor system design needs to consider rigidity and the damping of bearing.Move carrying out rolling bearing such as critical speed etc. The dynamic characteristic parameter of rolling bearing: equivalent stiffness and equivalent damping must be given during mechanical characteristic analysis.Equivalent stiffness with etc. The inaccurate dynamical property analysis directly to rolling bearing rotor-support-foundation system of effect damping bring very big error, cause bearing to turn Subsystem design time error is big, critical speed can be caused close with rated speed, cause the major break down of fatal crass when serious.

Now, when dynamic analyses being carried out to rolling bearing, often using rolling bearing interior contact elastic stiffness as rolling Bearing integral rigidity and do not consider the impact of oil film, have ignored simultaneously damping to the materially affect under systematic steady state, for damping Acquisition with rigidity relies primarily on empirical value, and the equivalent stiffness obtaining is low with the degree of accuracy of equivalent damping.Fang Bing etc. is special to bearing Property when being analyzed using equivalent damping and the equivalent stiffness of experiment measurement bearing.When He Zhixian etc. analyzes for bearing system dynamics Consider rigidity and ignore the impact of damping.The impact simplification damping is processed by the model of gupta.Hagiu proposes one and moves State theoretical analysis model, highlights elastic stiffness and the tactile area entrance that high speed scroll contact dynamically depends on the contact of mechanism hertz The lubricant rigidity at place and damping.When Liu Xiuhai is translated in a fluid using rolling element, suffered damping is simulating suffered by roller Viscous damping, but have ignored damping compression properties.Elsermans and walford think ball bearing rigidity radially, axially and resistance Buddhist nun's experimental measurements are all big than desired value, and this result can be explained by the analysis of outer ring-rolling element-inner ring-axle, But Theoretical Calculation is not perfect.Harsha and kankar proposes the ball bearing nonlinear model based on hertz elastic deformation, and draws Enter experience and damped to analyze its vibration transmission.Chen Bin etc. has carried out Theoretical Calculation to oil-film damping, but lacks strong experiment Checking.

Content of the invention

In order to solve above-mentioned technical problem, the present invention provides a kind of rolling bearing equivalent stiffness and equivalent damping calculation result The rolling bearing equivalent stiffness of high precision and equivalent damping calculation method, it is that the design of bearing rotor system provides technology and props up Hold, reduce the fault rate of the bearing rotor system of design.

The technical scheme that the present invention solves above-mentioned technical problem is: a kind of equivalent stiffness of rolling bearing and equivalent damping Computational methods, comprise the steps: 1) it is based on elastohydrodynamic lubrication theory, by between rolling element, oil film, raceway three Contact area is refined；2) pass through pressure distribution bilinearity approximating function and calculus of finite differences, obtain in each region after refinement The elastic deformation rigidity of contact, oil film pressure and oil film thickness, using linear perturbation equation solution local oil film dynamic stiffness with Oil-film damping；3) it is distributed according to rolling bearing dynamic load, and combine step 2) calculated elastic deformation rigidity, oil film be firm Degree and oil-film damping, calculate equivalent stiffness and the equivalent damping of rolling bearing.

In above-mentioned rolling bearing equivalent stiffness and the computational methods of equivalent damping, step 1) in rolling element, oil film, raceway Between three contact refinement when, do not consider rolling element relative with inside and outside raceway slip, by rolling element, oil film, raceway three it Between contact, according to contact resilient deformation be divided into oil film inlet region, flexible contact areas and three areas of oil film outlet area.

In above-mentioned rolling bearing equivalent stiffness and the computational methods of equivalent damping, step 2) in each region after refinement Oil film dynamic stiffness with oil-film damping method for solving is: simultaneous elastic deformation equation and reynolds equation, and with compound direct Iterative method, obtains the pressure under oil film static state and thickness value, then sets up perturbation equation group using perturbation equation and solve fine motion Amount and single order fine motion amount, try to achieve linear approximation oil film rigidity value and oil-film damping value.

In above-mentioned rolling bearing equivalent stiffness and the computational methods of equivalent damping, step 2) in when solving elastic deformation, Approach pressure distribution using bilinear function on the four nodes element of rectangle and solve elastic deformation.

In the computational methods of above-mentioned rolling bearing equivalent stiffness and equivalent damping, when perturbation equation group solves, utilize Oil film thickness is launched by Taylor series with oil film pressure, takes zero to be equipoise.

In the computational methods of above-mentioned rolling bearing equivalent stiffness and equivalent damping, step 2) in, ignore oil film outlet area Oil film rigidity and oil-film damping, to the oil film rigidity of oil film inlet region and oil-film damping and flexible contact areas rolling element and raceway Elastic contact stiffness, oil film rigidity and oil-film damping calculate respectively, then calculate rolling element and raceway phase on this basis The contact stiffness of contact and contact damping.

Above-mentioned rolling bearing equivalent stiffness and the computational methods of equivalent damping, calculate oil film inlet region oil-film damping and oil It is assumed that following condition during film rigidity:

Because in rolling bearing, rolling element and the shape in raceway contact face are long and narrow ellipses, therefore rotating direction is short by half Axle is little more than major semiaxis, so contact surface shape can be similar to the rectangular contact face of equal major diameter and minor axis, and ignores Oil leak on contact area edge；

Gap between rolling element and raceway is parabola；

In bearing, oil film thickness is more than rolling element, the surface roughness of Internal and external cycle；

The value of inertia force is less than viscous force；

Ignore gravity impact；

In view of the reynolds equation suitability it is believed that viscosity is constant；

Ignore the counter-pressure that in contact process, cavitation produces.

In above-mentioned rolling bearing equivalent stiffness and the computational methods of equivalent damping, calculation bearing radial direction equivalent stiffness with etc. It is considered to there is end-play in rolling bearing between rolling element and raceway and in bearing radial load lower bearing during effect damping Relative displacement is produced between outer snare.

Compared with prior art, the method have the advantages that

(1) present invention has refined rolling element, oil film, the contact between raceway three, is divided into three according to contact resilient deformation Individual area, oil film inlet region, flexible contact areas and oil film outlet area, and consider in rolling bearing and deposit between rolling element and raceway In end-play, under bearing radial load, the relative displacement of the inside and outside snare generation of bearing, so more conform to actual feelings Condition, the result calculating is more accurate；Design for bearing rotor system provides technical support, and the bearing reducing design turns The fault rate of subsystem.

(2) present invention is on the basis of simultaneous elastic deformation equation and reynolds equation, with compound Direct Iterative Method, Obtain the pressure under oil film static state and thickness value, using fine motion amount and the single order fine motion amount of perturbation equation solving equation group, finally Try to achieve linear approximation oil film rigidity value and oil-film damping value, there is the advantages of calculating speed is fast, and computational accuracy is high.

(3) calculating of the present invention does not need rolling bearing is carried out various tests, obtains relevant parameter, but directly permissible Application, has the advantages that simple and practical.

Brief description

Fig. 1 is the elastohydrodynamic behaviour contact model of the simplification of the present invention.

Fig. 2 is the oil film geometry of the rolling bearing of the present invention.

Fig. 3 is the distribution of rolling bearing load and the deformation of the present invention.

Fig. 4 is the equivalent stiffness of single rolling element contact and the equivalent damping of the rolling bearing of the present invention.

Fig. 5 is the position angle with radial direction for the rolling element of rolling bearing of the present inventionWhen between outer ring and rolling element Oil film pressure value.

Fig. 6 is the position angle with radial direction for the rolling element of rolling bearing of the present inventionWhen inner ring contact with rolling element Oil film pressure value.

Fig. 7 is the position angle with radial direction for the rolling element of rolling bearing of the present inventionWhen outer ring contact with rolling element Oil film pressure value.

Fig. 8 is the position angle with radial direction for the rolling element of rolling bearing of the present inventionWhen inner ring contact with rolling element Oil film pressure value.

Fig. 9 is the front quadravalence natural frequency measurement result of the axle-bearing-base systems for experimental verification of the present invention.

Figure 10 is that the axle-bearing-base systems for experimental verification of the present invention are based on the simplification of dyrobes rotor-bearing Analysis result.

Specific embodiment

The present invention is further illustrated below in conjunction with the accompanying drawings.

The present invention comprises the steps:

The first step: set up the ehl contact model of rolling element and raceway.

In the case of not considering that rolling element is relative with inside and outside raceway and sliding, rolling element, oil film, connecing between raceway three Tactile region (as shown in Figure 1), is divided into three areas according to contact resilient deformation: oil film inlet region a, flexible contact areas b and oil film go out Mouth region c.k_ef、c_efOil film rigidity for oil film inlet region a and oil-film damping.k_c、k_f、c_fIt is respectively flexible contact areas b rolling element With the Elastic contact stiffness of raceway, oil film rigidity and oil-film damping.Oil film due to its oil film outlet area c starts to stretch its shadow The power of sound weakens, and compares overall contact damping and stiffness effect is very little, do not consider here the oil film rigidity of oil film outlet area c with Oil-film damping.So main rigidity with damp as k_ef、c_ef、k_c、k_f、c_fThis five parameters, in order to equivalent to rolling bearing firm Degree is calculated with equivalent damping, answers analysis meter to calculate this five parameters first.

(1) resiliency deforming contact's rigidity k_c

Hertz contact theory is to be drawn according to the Static Contact condition of perfectly elastic body, is usually used to as different To the foundation of the secondary elastic deformation of curved face contact and stress field calculation, when rolling element is contacted with raceway groove, the width of its contact surface It is far smaller than the radius of curvature of contact surface, due to contact is considered as a contact, then its contact surface is regarded as ellipse.

By calculating the major axis a of Contact Ellipse and short axle b and to contact deformation δ as follows:

$a=a^{*}{\left(\frac{3q}{2{\mathrm{\σ\ρe}}^{\′}}\right)}^{1/3}---\left(1\right)$

$b=b^{*}{\left(\frac{3q}{2{\mathrm{\σ\ρe}}^{\′}}\right)}^{1/3}---\left(2\right)$

$\mathrm{\δ}={\mathrm{\δ}}^{*}{\left(\frac{3q}{2{\mathrm{\σ\ρe}}^{\′}}\right)}^{2/3}\frac{\mathrm{\σ}\mathrm{\ρ}}{2}---\left(3\right)$

$a^{*}={\left(\frac{2e\left(e\right)}{\mathrm{\π}(1-e^2)}\right)}^{1/3},b^{*}={\left(\frac{2\sqrt{1-e^2}e\left(e\right)}{\mathrm{\π}}\right)}^{1/3},{\mathrm{\δ}}^{*}=\frac{2k\left(e\right)}{\mathrm{\π}}{\left(\frac{(1-e^2)\mathrm{\π}}{2e\left(e\right)}\right)}^{1/3}$

$\frac{1}{e^{\′}}=\frac{1}{2}(\frac{1-v_1^2}{e_1}+\frac{1-v_2^2}{e_2}),\mathrm{\σ}\mathrm{\ρ}=\frac{1}{r_{1x}}+\frac{1}{r_{1y}}+\frac{1}{r_{2x}}+\frac{1}{r_{2y}},$

$\frac{1}{r_x}=\frac{1}{r_{x1}}+\frac{1}{r_{x2}},\frac{1}{r_y}=\frac{1}{r_{y1}}+\frac{1}{r_{y2}}$

$e\left(e\right)={\&integral;}_0^{\mathrm{\π}/2}\sqrt{1-e^2{\sin }^2\mathrm{\θ}}d\mathrm{\θ},k\left(e\right)={\&integral;}_0^{\mathrm{\π}/2}\frac{1}{\sqrt{1-e^2{\sin }^2\mathrm{\θ}}}d\mathrm{\θ},$

K (e)=1.5277+0.6023ln (r_y/r_x), e (e)=1.0003+0.5968 (r_x/r_y)

K=1.0339 (r_y/r_x)^0.6360、

Two face nonreentrant surfaces of sign of curvature regulation contact are that just concave surface is negative.E (e), k (e) be respectively the first kind and Complete elliptic integral of the second kind function, a is the major axis radius of elliptic contact surface, and b is the minor axis radius of elliptic contact surface, and q is to connect The mutual pressure of contacting surface, δ is elasticity maximum variable quantity, and ∑ ρ is curvature with ν is Poisson's ratio, and e is elastic modelling quantity, and e is oval ginseng Number.

The hertz theory elastic deformation of single rolling element can be obtained using (3) formula, (3) formula is reduced to:

δ=gq^2/3(4)

WhereinSingle rolling element with the hertz contact stiffness of inner ring or outer ring is:

$k_c=\frac{dq}{d\mathrm{\δ}}=\frac{3}{2}{g}^{-1.5}{\mathrm{\δ}}^{0.5}---\left(5\right)$

There it can be seen that its rigidity is not a constant, it can change with the change of displacement (or load).

(2) oil film rigidity k of flexible contact areas_fWith oil-film damping c_f

As shown in Fig. 2 the Point contact between two elastomeric objects can be considered as with equivalent principal radius of curvature r_z, r_yWith The contact with rigid plane for the ellipsoid of elasticity body of equivalent elastic modelling quantity e'.There is lubricating oil film between two surfaces, and in contact The actual oil film thickness of heart point o is h_c, under oil film pressure effect, the elastic deformation that contact surface produces is δ (x, y), and oil film is thick The expression formula of degree can be write as:

$h(x,y)=h_c+\frac{x^2}{2r_x}+\frac{y^2}{2r_y}+\mathrm{\δ}(x,y)-\mathrm{\δ}(0,0)---\left(6\right)$

h_c=h₀+ δ (0,0), h₀For rigid body center oil film thickness.

Based on the reynolds equation under isothermy, the parameter of oil film is calculated, the reynolds equation under isothermy Common form (assumes u₂And v₂Do not change with x and y) as follows:

$\begin{array}{c}\frac{\∂}{\∂x}\left(\frac{{\mathrm{\ρh}}^3}{\mathrm{\η}}\frac{\∂p}{\∂x}\right)+\frac{\∂}{\∂y}\left(\frac{{\mathrm{\ρh}}^3}{\mathrm{\η}}\frac{\∂p}{\∂y}\right)=6(u_1+u_2)\\ \×\frac{\∂\left(\mathrm{\ρ}h\right)}{\∂x}+6(v_1+v_2)\frac{\∂\left(\mathrm{\ρ}h\right)}{\∂y}+12\frac{\∂\left(\mathrm{\ρ}h\right)}{\∂t}\end{array}---\left(7\right)$

Simultaneous formula (6), formula (7) are solved, and on the basis of the mutual relation of oil film thickness and oil-film force, acquisition connects The oil film rigidity in tactile area and oil-film damping.The point that some scholars derive in hamrock with dowson contacts minimum oil film thickness Derivation is directly carried out on formula to obtain rigidity value, so exist two problems: 1. oil film rigidity refer to firm under micro deformation Angle value, 2. because the change of local oil film thickness is inconsistent under the influence of elastic deformation, oil film rigidity is unable to immediate derivation and obtains Take, and oil-film damping value cannot direct access.Therefore on the basis of simultaneous elastic deformation equation and reynolds equation, With compound Direct Iterative Method, obtain the pressure under oil film static state and thickness value, using the fine motion of perturbation equation solving equation group Amount and single order fine motion amount, finally try to achieve linear approximation oil film rigidity value and oil-film damping value.

Approach pressure distribution using bilinear function on the four nodes element of rectangle and solve elastic deformation, make in oil film pressure With under, the Normal Displacement summation of two contact surfaces is:

$\mathrm{\δ}(x,y)=\frac{2}{{\mathrm{\πe}}^{\′}}\&integral;{\&integral;}_{\mathrm{\ω}}\frac{p(\mathrm{\ξ},\mathrm{\ζ})d\mathrm{\ξ}d\mathrm{\ζ}}{\sqrt{{(x-\mathrm{\ξ})}^2+{(y-\mathrm{\ζ})}^2}}---\left(8\right)$

ω is domain, and e is unit area, p_ijCenter pressure value for unit area e, λ (x, y) is coefficient value.

Using Dimensionless Form, formula (6) and formula (7) abbreviation are contacted for point and lubricate dimensionless reynolds equation:

$\frac{\∂}{\∂x}\left(\frac{\stackrel{\&overbar;}{\mathrm{\ρ}}{h}^3}{\stackrel{\&overbar;}{\mathrm{\η}}}\frac{\∂p}{\∂x}\right)+\frac{1}{k^2}\frac{\∂}{\∂y}\left(\frac{\stackrel{\&overbar;}{\mathrm{\ρ}}{h}^3}{\stackrel{\&overbar;}{\mathrm{\η}}}\frac{\∂p}{\∂y}\right)=a_1\frac{\∂\left(\stackrel{\&overbar;}{\mathrm{\ρ}}h\right)}{\∂x}---\left(9\right)$

Oil film thickness equation:

$h_{kl}=h_0+a_3{x}_k^2+a_4{y}_l^2+\stackrel{i}{\σ}\stackrel{j}{\σ}{d}_{ij}^{kl}{p}_{ij}---\left(10\right)$

Thickness parameterLoad parameterSpeed parameterMaterial parameter g=α e¹α is viscous Pressure coefficient,For maximum hertz pressure, w is load value here.

$a_1=\frac{12e^1{\mathrm{ur}}_x^3}{a^2{\mathrm{bp}}_h},a_3=\frac{1}{2k},a_4=\frac{{\mathrm{kr}}_x}{2r_y},a_5=\frac{12e^1{r}_x^2}{a^2{p}_h},d_{ij}^{kl}=\frac{c_{ij}^{kl}{r}_x{p}_h}{ab}$

Glue pressure relational expression for dimensionless roelands.

For non-dimensional density equation.

By formula (9) and formula (10) simultaneous, and to substitute local derviation using three dot center's difference schemes, collated to obtain as follows DIFFERENCE EQUATIONS:

$\begin{array}{c}{e}_{kl}^e{p}_{k+1,l}+e_{kl}^n{p}_{k,l+1}+e_{kl}^w{p}_{k-1,l}+e_{kl}^s{p}_{k,l-1}+e_{kl}^o{p}_{kl}\\ -\stackrel{i}{\σ}\stackrel{j}{\σ}\left\{a_1\[{\left(\frac{\∂\stackrel{\&overbar;}{\mathrm{\ρ}}}{\∂x}\right)}_{kl}{d}_{ij}^{kl}+\stackrel{\&overbar;}{{\mathrm{\ρ}}_{kl}}{\mathrm{dx}}_{ij}^{kl}\]\right\}{p}_{ij}=\\ a_1{\left(\frac{\∂\stackrel{\&overbar;}{\mathrm{\ρ}}}{\∂x}\right)}_{kl}(h_0+a_3{x}_k^2+a_4{y}_l^2)+2a_1{a}_3{\stackrel{\&overbar;}{\mathrm{\ρ}}}_{kl}{x}_k\end{array}---\left(11\right)$

Try to achieve static oil film thickness and oil film pressure value, on the basis of perturbation equation, with Taylor series by oil film thickness with Oil film pressure launches, and takes zero to be equipoise, then oil film pressure near quiescent value during fine motion, oil film pressure and oil Film thickness can be represented by following linear relationships formula:

$p=p_0+k\mathrm{\δ}h+c\mathrm{\δ}\stackrel{\·}{h}---\left(12\right)$

H=h₀+δh (13)

In formula:p₀For static balance oil film pressure, h₀For static balance oil film thickness, k is that oil film is approximate Rigidity, c approximately damps for oil film, δ h,It is disturbance parameter, its value very little.The then reynolds side under isothermy Dimensionless Form after journey simplifies is:

$\begin{array}{c}\frac{\∂}{\∂x}\left(\frac{\stackrel{\&overbar;}{\mathrm{\ρ}}{h}^3}{\stackrel{\&overbar;}{\mathrm{\η}}}\right)\frac{\∂p}{\∂x}+\frac{1}{k^2}\frac{\∂}{\∂y}\left(\frac{\stackrel{\&overbar;}{\mathrm{\ρ}}{h}^3}{\stackrel{\&overbar;}{\mathrm{\η}}}\right)\frac{\∂p}{\∂y}\\ +\left(\frac{\stackrel{\&overbar;}{\mathrm{\ρ}}{h}^3}{\stackrel{\&overbar;}{\mathrm{\η}}}\right)\left(\frac{{\∂}^{2}p}{\∂x^2}+\frac{1}{k^2}\frac{{\∂}^{2}p}{\∂y^2}\right)=a_1\frac{\∂\left(\stackrel{\&overbar;}{\mathrm{\ρ}}h\right)}{\∂x}+a_5\stackrel{\&overbar;}{\mathrm{\ρ}}\frac{\∂\left(\stackrel{\&overbar;}{\mathrm{\ρ}}h\right)}{\∂t}\end{array}---\left(14\right)$

h₀With time variableUnrelated, formula (12), formula (13) dimensionless formula are substituted in formula (14), by δ h, Item more than quadratic power and quadratic power omits, and by homogeneous item normalizing, obtains following three equation:

$\frac{\∂}{\∂x}\left(\frac{\stackrel{\&overbar;}{\mathrm{\ρ}}{h_0}^3}{\stackrel{\&overbar;}{\mathrm{\η}}}\frac{\∂p_0}{\∂x}\right)+\frac{1}{k^2}\frac{\∂k}{\∂y}\left(\frac{\stackrel{\&overbar;}{\mathrm{\ρ}}{h_0}^3}{\stackrel{\&overbar;}{\mathrm{\η}}}\frac{\∂p_0}{\∂y}\right)=a_1\frac{\∂\left(\stackrel{\&overbar;}{\mathrm{\ρ}}{h}_0\right)}{\∂x}---\left(15\right)$

$\begin{array}{c}\frac{\∂}{\∂x}\left(\frac{\stackrel{\&overbar;}{\mathrm{\ρ}}{h_0}^3}{\stackrel{\&overbar;}{\mathrm{\η}}}\left(\frac{\∂k}{\∂x}+\frac{3}{h_0}\frac{\∂p_0}{\∂x}\right)\right)\\ +\frac{1}{k^2}\frac{\∂}{\∂y}\left(\frac{\stackrel{\&overbar;}{\mathrm{\ρ}}{h_0}^3}{\stackrel{\&overbar;}{\mathrm{\η}}}\left(\frac{\∂k}{\∂y}+\frac{3}{h_0}\frac{\∂p_0}{\∂y}\right)\right)=a_1\frac{\∂\stackrel{\&overbar;}{\mathrm{\ρ}}}{\∂x}\end{array}---\left(16\right)$

$\frac{\∂}{\∂x}\left(\frac{\stackrel{\&overbar;}{\mathrm{\ρ}}{h_0}^3}{\stackrel{\&overbar;}{\mathrm{\η}}}\frac{\∂c}{\∂x}\right)+\frac{1}{k^2}\frac{\∂}{\∂y}\left(\frac{\stackrel{\&overbar;}{\mathrm{\ρ}}{h_0}^3}{\stackrel{\&overbar;}{\mathrm{\η}}}\frac{\∂c}{\∂y}\right)=\frac{12e^1{r}_x^2}{p_h{a}^2}\stackrel{\&overbar;}{\mathrm{\ρ}}---\left(17\right)$

Using direct combination iteration Method formula (16) and formula (17), draw oil film rigidity k_fWith oil-film damping c_f.

(3) oil film inlet region oil film rigidity k_efWith oil-film damping c_ef

Simplified analysis inlet region oil-film damping and oil film rigidity are it is assumed that following condition:

1. because the shape in rolling element and raceway contact face in rolling bearing is long and narrow ellipse, therefore rotating direction is short Semiaxis b is little more than major semiaxis a, so contact surface shape can be similar to the rectangular contact face of equal major diameter and minor axis, and Ignore the oil leak on contact area edge.

2. the gap between rolling element and raceway is parabolaAnd x >=b, r is y direction here Composite curve radius.

3. in bearing, oil film thickness is more than rolling element, the surface roughness of Internal and external cycle.

4. the value of inertia force is less than viscous force.

5. ignore gravity impact.

6. consider the reynolds equation suitability it is believed that viscosity is constant.

7. ignore the counter-pressure that in contact process, cavitation produces.

According to above-mentioned it is assumed that being reduced to reynolds equation (7):

$\frac{\∂}{\∂x}\left(\frac{{\mathrm{\ρh}}^3}{\mathrm{\η}}\frac{\∂p}{\∂x}\right)=6u_s\frac{\∂\left(\mathrm{\ρ}h\right)}{\∂x}+12u_z---\left(18\right)$

Wherein u_s=u₁+u₂, u_zFor normal direction extrusion speed.η and ρ is definite value.To the x integration in formula (18), and consider Sommerfeld condition and half sommerfeld condition can obtain:

$q_e=\frac{2\mathrm{\η}url}{h_0}+\frac{3}{\sqrt{2}}\frac{{\mathrm{\πv\ηr}}^{1.5}l}{h_0^{1.5}}---\left(19\right)$

η is film viscosity, u rolling element peripheral speed, l Contact Ellipse region major axis, v oil film admission velocity.So can obtain The oil-film damping of inlet region:

$c_{ef}=\frac{{\mathrm{dq}}_e}{dv}=\frac{3}{\sqrt{2}}\frac{{\mathrm{\π\ηr}}^{1.5}l}{h_0^{1.5}}---\left(20\right)$

By numerical simulation, find that the deformation of oil film inlet region (includes the elastic deformation of contact surface and the thickness change of oil film Change) deformation of resilience in comparison contact area is very little, therefore, load is micro- become in the case of, rigidity k of oil film inlet region_ef It is negligible.By calculated c_ef、c_f、k_c、k_fBring formula (19) into, the final rolling element that obtains of formula (20) is connected with raceway Tactile contact stiffness k and contact damping c:

$k=\frac{k_c{k}_f}{k_c+k_f}---\left(21\right)$

C=c_ef+c_f(22)

Second step: calculate rolling bearing equivalent damping and equivalent stiffness.

The contact stiffness k being contacted with raceway by calculated rolling element and contact damping c, calculation bearing radial direction etc. Effect rigidity k_rcWith equivalent damping c_re.Consider there is end-play between rolling element and raceway in rolling bearing, radially carry in bearing Lotus f_rUnder, the relative displacement that the inside and outside snare of bearing produces is δ_r, as shown in Figure 3.

In equilibrium conditions, the radial load of inner ring is necessarily equal to the vertical component sum of rolling element load:

HereFor the position angle of single rolling element and radial direction, After noting Integral TransformationIt is continuous, α is correction factor.

$q_{max}=\frac{f_r}{{\mathrm{zj}}_r\left(\mathrm{\ϵ}\right)}---\left(24\right)$

Simultaneous formula (24), formula (25) formula can obtain the load distribution of rolling element 2 and raceway, thus obtaining single rolling element 2 with inner ring 1, the ct clamping of outer ring 3, as shown in figure 4, obtain single rolling element 2 contacting with inner ring 1, outer ring 3 Contact stiffness and contact damping:

$k_e=\frac{k_1{k}_2}{(k_1+k_2)}---\left(26\right)$

$c_e=\frac{c_1{c}_2}{c_1+c_2}---\left(27\right)$

The contact stiffness that contacted each rolling element 2 with inner ring 1, outer ring 3 and contact damping simultaneous, try to achieve rolling bearing Radial direction equivalent stiffness k_rcWith equivalent damping c_re:

In the same manner, herePosition angle for single rolling element and radial direction.

Experimental verification

Using b＆k test system, on the basis of parts of bearings mode, measured using hammering method, obtain its system Rigid body natural frequency, tries to achieve bearing equivalent damping using formula.

In view of bearing light weight itself, small volume, belong to Appendage mode test, be therefore unfavorable for the arrangement of sensor, And transmission function calculates the factors such as difficulty, bearing outer ring is fixed on bearing spider, the axle of certain mass is arranged on bearing Inner ring is simultaneously fastened by tight lock screw, and such axle-bearing-pedestal becomes system.

Using single-point pick-up method, in one section of placement sensor of axle, axle is hammer point with the white point of institute's labelling on bearing block. The transmission function that measurement is obtained obtains bearing arrangement part rigid body natural frequency by model analyses.This method of testing is in axle Hold and carry out after working 10 minutes so that 600rpm is continuous, the natural frequency obtained by measurement part system is as shown in Figure 9.

From Fig. 9 it can be seen that: former rank natural frequency value ω of Appendage mode_nAnd dampingratioζ, take here 2 rank Modal frequencies and damping ratio (part rigid body mode) substitute into c_b=2 ζ m ω_nMiddle calculating, the damping calculating is 483.2, this and our Theoretical Calculation obtained value c_re=484.0255 differences 0.17%.

It is that rigidity is verified, soft mode is analyzed by dyrobes rotor-bearing and draws up under duplex bearing-mono- rotating shaft one Rank critical speed 57.97hz, as shown in Figure 10, centre adds the mass of 0.82kg

Here countershaft has carried out equivalent, acquisition equivalent mass m=1.64kg respectively, equivalent bearing rigidity k_s= 219450, equivalent damping c_k=121.3705.

$k=\frac{2k_b{k}_s}{2k_b+k_s}---\left(30\right)$

$c=\frac{c_{re}{c}_s}{c_{re}+c_s}---\left(31\right)$

$2\mathrm{\π}f=\sqrt{1-{\left(\frac{c}{2\sqrt{mk}}\right)}^2}\·\sqrt{\frac{k}{m}}---\left(32\right)$

Simultaneous formula (30), (31), (32), trying to achieve bearing radial direction dynamic rate is k_b=5.4743 × 10⁷, this is managed with us By value of calculation k_rc=5.3067 × 10⁷Deviation be 3.06%.

Claims (8)

1. a kind of rolling bearing equivalent stiffness and the computational methods of equivalent damping, comprise the steps: 1) moved based on elastic fluid Hydrodynamic lubrication is theoretical, and the contact area between rolling element, oil film, raceway three is refined；2) pass through pressure distribution bilinearity Approximating function and calculus of finite differences, obtain the elastic deformation rigidity of contact in each region after refinement, oil film pressure and oil film thick Degree, using the oil film dynamic stiffness in each region after the refinement of linear perturbation equation solution and oil-film damping；And then calculate rolling element with Contact stiffness and contact damping that raceway contacts；3) in step 2) contact with raceway contacts of calculated rolling element is firm On the basis of degree and contact damping, calculate equivalent stiffness and the equivalent damping of rolling bearing.

2. the computational methods of rolling bearing equivalent stiffness according to claim 1 and equivalent damping, rolling element in step 1), When contact between oil film, raceway three refines, do not consider that rolling element is relative with inside and outside raceway and slide, by rolling element, oil film, rolling Contact between road three, is divided into oil film inlet region, flexible contact areas and three areas of oil film outlet area according to contact resilient deformation.

3. the computational methods of rolling bearing equivalent stiffness according to claim 1 and equivalent damping, step 2) in after refinement Oil film dynamic stiffness in each region and oil-film damping method for solving are: simultaneous elastic deformation equation and reynolds equation, and transport With being combined Direct Iterative Method, obtain the pressure under oil film static state and thickness value, then set up perturbation equation group using perturbation equation And solve fine motion amount and single order fine motion amount, try to achieve linear approximation oil film rigidity value and oil-film damping value.

4. the computational methods of rolling bearing equivalent stiffness according to claim 1 and equivalent damping, step 2) middle solution bullet Property deformation when, on the four nodes element of rectangle using bilinear function approach pressure distribution solve elastic deformation.

5. rolling bearing equivalent stiffness according to claim 3 and the computational methods of equivalent damping, ask in perturbation equation group Oil film thickness is launched with oil film pressure using Taylor series, takes zero to be equipoise by Xie Shi.

6. the computational methods of rolling bearing equivalent stiffness according to claim 2 and equivalent damping, step 2) in, ignore oil The oil film rigidity of film outlet area and oil-film damping, roll to the oil film rigidity and oil-film damping and flexible contact areas of oil film inlet region The Elastic contact stiffness of body and raceway, oil film rigidity and oil-film damping calculate respectively, then calculate rolling element on this basis The contact stiffness contacting with raceway and contact damping.

7. rolling bearing equivalent stiffness according to claim 6 and the computational methods of equivalent damping, calculate oil film inlet region It is assumed that following condition when oil-film damping and oil film rigidity:

Because in rolling bearing, rolling element and the shape in raceway contact face are long and narrow ellipses, the semi-minor axis of therefore rotating direction is remote Less than major semiaxis, so contact surface shape can be similar to the rectangular contact face of equal major diameter and minor axis, and ignore contact Oil leak in edges of regions；

Gap between rolling element and raceway is parabola；

In bearing, oil film thickness is more than rolling element, the surface roughness of Internal and external cycle；

The value of inertia force is less than viscous force；

Ignore gravity impact；

In view of the reynolds equation suitability it is believed that viscosity is constant；

Ignore the counter-pressure that in contact process, cavitation produces.

8. the computational methods of rolling bearing equivalent stiffness according to claim 1 and equivalent damping, calculation bearing radial direction etc. It is considered to there is end-play in rolling bearing between rolling element and raceway and in bearing radial load when effect rigidity and equivalent damping Relative displacement is produced between the inside and outside snare of lower bearing.