CN103912315B - Structural dynamics design method of rotor of aerial engine - Google Patents

Abstract

Disclosed is a structural dynamics design method of the rotor of an aerial engine. Based on the characteristics of the working status of a high propulsion ratio aerial engine, the structural dynamics design method of the rotor of the aerial engine presents the concept of thermal modal to adapt to structural dynamics design of the rotor of the aerial engine under variable working conditions. The structural dynamics design method of the rotor of the aerial engine is characterized by optimizing the parameters of the rotor and bearings, enabling the thermal modal of a rotor system to avoid the modal of the rotor under the absolute rigidity of the bearings, meanwhile, giving play to the effects of a damper and enabling the rotor system to meet the requirements of vibration standards in the thermal modal. The structural dynamics design method of the rotor of the aerial engine comprises taking the margin between the modal of the rotor with elastic bearings and the modal of the rotor under the absolute rigidity of the bearings as optimized parameters, meanwhile giving play to the damping effects of the damper to meet the dynamics requirements of the rotor system. The structural dynamics design method of the rotor of the aerial engine changes the traditional design concept of design first and checking calculation later and performs active design of the dynamics characteristics of the rotor to achieve circulating design, thereby improving the working efficiency and having significant engineering practical value.

Description

A kind of design method of aeroengine rotor structural dynamics

Technical field

The present invention relates to aeroengine field, specifically a kind of dynamic (dynamical) method of designed engines rotor structure.

Background technique

Reduce vibration, control gap, and reduction supporting structure load is the key content involved by rotor dynamics design criterion.Along with to the lifting of opportunity of combat operational requirements and the diversity of aerial mission, aeroengine variable working condition feature is further outstanding.Therefore, aeroengine rotor Structural Dynamic Design is extremely important.

The design criterion that existing aeroengine rotor Structural Dynamic Design is continued to use always " ensureing that working speed and critical speed of rotation leave enough nargin ", no matter the rotor of i.e. subcritical or overcritical operation, its working speed must keep enough nargin with critical speed of rotation, and such as 15%.Due in overcritical work, therefore damper will must be arranged to be decreased through vibratory response during critical speed of rotation.Height push away than performance motor during operation, rotor frequently crosses that single order is critical, even second order critical speed of rotation, and cannot ensure the margin requirement between working speed and critical speed of rotation, critical speed of rotation position or neighborhood are even dropped in operation point.In other words, critical speed of rotation may become working speed.Therefore, the design criterion " ensureing that working speed and critical speed of rotation leave enough nargin " no longer meets the requirement of this kind of aeroengine rotor dynamics Design.

Content about the rotor dynamics of rotor-support-foundation system in domestic and international Patents document and paper focuses mostly in the design of certain rotor experiment table and the technique study of rotor dynamic Epidemiological Analysis.Its research background depends on current version and rotor.Not by the whole design phase of rotor dynamics design through rotor-support-foundation system, its expansion is subject to a definite limitation.Be disclose one " dynamics characteristic experimental apparatus of heavy type gas turbine pull rod rotor " in the innovation and creation of CN 101329220B at publication number, the invention provides a kind of experimental setup.This experimental setup comprises rod fastening rotor and supporting vibration test system, is suitable for carrying out rotor dynamics experimental research.This invention is a kind of experimental setup of certain rotor, does not obtain the dynamic (dynamical) design method of rotor structure.Be in the innovation and creation of 201410033310.3, disclose one " aircraft engine high pressure rotor Structural Design " at application number, this invention determines the relation between high pressure rotor model design parameter and rotor oscillation characteristic, thus providing design method and criterion for the dynamics Design of high pressure rotor, the design for engine high pressure rotor has important directive significance.Rotor is thought of as stiffness rotor by this invention, is only applicable to the design of aircraft engine high pressure rotor, also needs the design method improving low pressure rotor further.

L.E.Barrett, E.J.Gunter and P.E.Allaire describes a kind of quick approximation method calculating the damping of multi rotor system optimal support in paper " Optimum bearing and supportdamping for unbalance response and stability of rotating machinery " (ISSN:0022-0825), and this optimal damping makes the minimum stability of rotor-support-foundation system unbalance response near first critical speed the highest.The work of this paper belongs to the analysis of rotor dynamics, and the method for proposition makes rotor dynamics analysis be simplified, and saves time and is convenient to application, but does not propose the concrete grammar step of rotor structure dynamics Design.Jiang Yun-fan, Liao Ming-fu, Wang Siji devises a kind of aeroengine of simulating to the supporting means turning two-spool dynamics simulations device and two kinds of high-pressure shaft according to dynamic similarity principle in paper " The Design of Counter-Rotating Dual Rotor Experimental Apparatus " (ISBN:978-7-5612-3441-9).Carry out double rotor with turning, to rotational kinematics and dynamics experiment, have studied birotary engine dynamic balancing technique.The double rotor tester designed in this paper, according to dynamic similarity principle, is a kind of passive method, and object is the dynamics of research birotary engine, instead of the method for active designs rotor.

Summary of the invention

In order to adapt to highly to push away designing requirement more dynamic (dynamical) than performance aeroengine rotor, the present invention proposes a kind of dynamic (dynamical) method of designed engines rotor structure.

Push away than performance aeroengine for height, the present invention proposes a kind of definition of rotor mode, be hot-die state, it is critical that corresponding critical speed of rotation is defined as heat.Why be defined as hot-die state, one is that rotor frequently crosses critical speed of rotation, and critical speed of rotation even may become working speed because motor run duration; Two is because under working state, and material behavior, cooperation rigidity and coupling stiffness generation significant change, cause critical speed of rotation to change in a big way, is difficult to rotating speed nargin when ensureing design; Three is because under hot-die state, the criterion of rotor structure dynamics Design is, the hot threshold response of rotor be controlled under the limits value allowed, and be no longer the rotating speed nargin deliberately ensureing to expect.

Concrete steps of the present invention are:

The first step: set up rotor dynamics model, obtains the vibratory response of rotor-support-foundation system at critical speed of rotation place and the relation of each design parameter.The relation of the vibratory response of described rotor-support-foundation system at critical speed of rotation place and each design parameter comprises the relation of rotor-support-foundation system threshold response and Structural Parameters of its Rotor and the relation of rotor-support-foundation system threshold response and rotor modal parameter.

The equation matrix expression (1) of rotor dynamics model is:

$M\stackrel{\·\·}{Q}+D\stackrel{\·}{Q}+\mathrm{SQ}=u---\left(1\right)$

In formula (1):

$M=\left[\begin{array}{cccc}m& 0& 0& 0\\ 0& I& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right],$ M is mass matrix, and m is the quality of dish, and I is the diameter rotary inertia of dish;

$D=\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& -j{I}_p\mathrm{\Ω}& 0& 0\\ 0& 0& -d_{b1}& -d_{b2}\\ 0& 0& {\mathrm{ajd}}_{b1}& -(L-a){\mathrm{jd}}_{b2}\end{array}\right],$ D is damping matrix, and j is imaginary unit, I _pfor the pole axis rotary inertia of dish, Ω is rotary speed of rotator, d _b1, d _b2be respectively the damping constant supported before and after rotor-support-foundation system, L is the span of rotor, and a is the distance of rotor front fulcrum to rotor c.g.;

$S=\left[\begin{array}{cccc}{s}_{11}& -{\mathrm{js}}_{11}& -(1-\frac{a}{L})s_{11}-\frac{s_{12}}{L}& -\frac{a}{L}{s}_{11}+\frac{s_{12}}{L}\\ {\mathrm{js}}_{21}& s_{22}& -j(1-\frac{a}{L})s_{21}-j\frac{s_{22}}{L}& -j\frac{a}{L}{s}_{21}+j\frac{s_{22}}{L}\\ s_{11}& -j{s}_{12}& -(1-\frac{a}{L})s_{11}-\frac{s_{12}}{L}-s_{b1}& -\frac{a}{L}{s}_{11}+\frac{s_{12}}{L}-s_{b2}\\ {\mathrm{js}}_{21}& s_{22}& -j(1-\frac{a}{L})s_{21}-j\frac{s_{22}}{L}+{\mathrm{jas}}_{b1}& -j\frac{a}{L}{s}_{21}+j\frac{s_{22}}{L}-(L-a){\mathrm{js}}_{b2}\end{array}\right],$ S is stiffness matrix, s ₁₁, s ₁₂, s ₁₃and s ₁₄be respectively the stiffness coefficient of rotor, s _b1, s _b2be respectively the stiffness coefficient supported before and after rotor-support-foundation system;

$u=\left\{\begin{array}{c}\begin{array}{cc}\mathrm{m\ϵ}{\mathrm{\Ω}}^2& e^{j(\mathrm{\Ωt}+\mathrm{\β})}\end{array}\\ 0\\ 0\\ 0\end{array}\right\},$ U is the inertial centrifugal force that rotor-support-foundation system bears due to imbalance, and ε is the uneven throw of eccentric of rotor; β is phase angle of unbalance;

q is the response of rotor-support-foundation system, and r is the amount of deflection at rotor c.g. place, for the deflection angle at rotor c.g. place, r _b1, r _b2be respectively the amount of deflection of rotor two supportings place.

Rotor-support-foundation system is at the threshold response of supporting place

$\begin{array}{c}{Q}_{2\mathrm{cr}}=-j\frac{1}{m(I-I_p)({\widetilde{\mathrm{\Ω}}}_{\mathrm{crl}}-{\mathrm{\Ω}}_{\mathrm{cri}}^2)({\widetilde{\mathrm{\Ω}}}_{\mathrm{cr}2}^2-{\mathrm{\Ω}}_{\mathrm{cri}}^2)}\left[\mathrm{\Φ}\right]\left[\begin{array}{cc}\frac{{\mathrm{\Ω}}_{\mathrm{cri}}}{2m_{\mathrm{crl}}{D}_{\mathrm{crl}}{\mathrm{\Ω}}_{\mathrm{crl}}}& 0\\ 0& \frac{{\mathrm{\Ω}}_{\mathrm{cri}}}{{2I}_{\mathrm{cr}2}{D}_{\mathrm{cr}2}{\mathrm{\Ω}}_{\mathrm{cr}2}}\end{array}\right]{\left[\mathrm{\Φ}\right]}^T.\\ S_s\left[\begin{array}{cc}{s}_{22}-(I-I_p){\mathrm{\Ω}}_{\mathrm{cri}}^2& -s_{12}\\ -s_{21}& s_{11}-m{\mathrm{\Ω}}_{\mathrm{cri}}^2\end{array}\right]U_1\end{array}---\left(17\right)$

The threshold response at rotor c.g. place

$Q_{\mathrm{lcr}}=S_s^{-1}{S}_{\mathrm{bb}}{Q}_{2\mathrm{cr}}+S_s^{-1}{S}_{\mathrm{bd}}{\stackrel{\·}{Q}}_{2\mathrm{cr}}---\left(18\right)$

Formula (17), formula (18) describe the vibratory response Q of flexibly mounted rotor-support-foundation system at critical speed of rotation place _1cr, Q _2cr, include the modal parameter of rotor-support-foundation system structural dynamic parameter and rotor-support-foundation system.Formula (17), formula (18) describe flexibly mounted rotor-support-foundation system threshold response and rotor critical speed during supporting absolute rigidity relation.

In formula (17) and (18),

Q _1crfor the threshold response amplitude at rotor c.g. place; Q _2crfor the threshold response amplitude at rotor bearing place.

Ω _crifor rotor i-th rank critical speed of rotation.

The modal matrix that [Φ] is rotor-support-foundation system, $D_{\mathrm{crl}}=\frac{d_{\mathrm{crl}}}{{2m}_{\mathrm{crl}}{\mathrm{\Ω}}_{\mathrm{crl}}},D_{\mathrm{cr}2}=\frac{d_{\mathrm{cr}2}}{{2I}_{\mathrm{cr}2}{\mathrm{\Ω}}_{\mathrm{cr}2}}$ Be respectively the first rank and second-order damping ratios, d _cr1, d _cr2be respectively the first rank and the second-order modal damping coefficient of rotor-support-foundation system, m _cr1, I _cr2be respectively the first rank and second-order modal mass, Ω _cr1, Ω _cr2be respectively the first rank and the second-order critical speed of rotation of rotor.

when being respectively supporting rigidity, the critical speed of rotation of rotor-support-foundation system.

$S_s=\left[\begin{array}{cc}{s}_{11}& -j{s}_{12}\\ {\mathrm{js}}_{21}& s_{22}\end{array}\right]$

$S_{\mathrm{bb}}=\left[\begin{array}{cc}(1-\frac{a}{L})s_{11}+\frac{s_{12}}{L}+s_{b1}& \frac{a}{L}{s}_{11}-\frac{s_{12}}{L}+s_{b2}\\ j(1-\frac{a}{L})s_{21}+j\frac{s_{22}}{L}-{\mathrm{jas}}_{b1}& j\frac{a}{L}{s}_{21}-j\frac{s_{22}}{L}+(L-a){\mathrm{js}}_{b2}\end{array}\right]$

$S_S\left\{\begin{array}{ccc}\left[\begin{array}{cc}(1-\frac{a}{L})& \frac{a}{L}\\ \frac{j}{L}& \frac{-j}{L}\end{array}\right]+S_S^{-1}& \left[\begin{array}{cc}1& 1\\ -\mathrm{ja}& j(L-a)\end{array}\right]& \left[\begin{array}{cc}{s}_{b1}& 0\\ 0& s_{b2}\end{array}\right]\end{array}\right\}$

$S_{\mathrm{bd}}=\left[\begin{array}{cc}{d}_{b1}& d_{b2}\\ -{\mathrm{ajd}}_{b1}& (L-a){\mathrm{jd}}_{b2}\end{array}\right]=\left[\begin{array}{cc}1& 1\\ -\mathrm{ja}& j(L-a)\end{array}\right]\left[\begin{array}{cc}{d}_{b1}& 0\\ 0& d_{b2}\end{array}\right]$

U ₁for the amount of unbalance of rotor-support-foundation system.

Ω _crifor the critical speed of rotation of rotor, Q _2crfor rotor is in the threshold response amplitude of supporting place.

Second step: the hot-die state during work of setting rotor-support-foundation system.

According to operating rotational speed range and the performance requirement of motor, the hot-die state of setting rotor-support-foundation system.When setting described hot-die state, must meet the following conditions successively: first step mode critical speed of rotation is positioned at below slow train rotating speed, and allow in slow train position; Second-order mode critical speed of rotation and the 3rd rank mode critical speed of rotation are all positioned at below cruising speed, nargin equal 10%; Critical speed of rotation is not established between cruising speed and maximum (top) speed; When the design parameter of rotor requires to establish critical speed of rotation between the cruising speed and maximum (top) speed of rotor-support-foundation system, then quadravalence mode critical speed of rotation is arranged on maximum (top) speed, nargin 8-10%.

3rd step: the mode of rotor-support-foundation system when determining to support absolute rigidity.

If each support stiffness of rotor-support-foundation system is infinitely great, transfer matrix method or finite element method is adopted to determine the mode of rotor-support-foundation system with this understanding.

4th step: the initial value choosing support stiffness.The initial value of support stiffness is selected according to the rigidity of supporting absolute rigidity rotor-support-foundation system.Each support stiffness of rotor-support-foundation system should be the 20-50% of rigid support rotor rigidity, adopts finite element method or determines the lateral stiffness of rotor at barycenter place by formula (19):

$S=\frac{243\mathrm{EI}}{4L^3}=1.0577\×{10}^{7}N---\left(19\right)$

E is the Young's modulus of material, and I is the cross section moments of inertia of rotating shaft.

Get 50% of s as the initial value of rotor-support-foundation system elastic supporting rigidity:

$S_{b1}=S_{b2}=\frac{1}{2}S=5.2887\×{10}^{6}N/m---\left(20\right)$

5th step: verify whether the quiet amount of deformation of rotor during the support stiffness initial value determined exceeds standard.

Adopt finite element method or the quiet distortion according to formula (6) calculating rotor-support-foundation system, make the quiet amount of deformation of rotor be not more than the value of this engine design requirement.

$\mathrm{\δ}=\frac{M\·g}{S_{b1}+S_{b2}}+\frac{m\·g}{s}---\left(21\right)$

In formula, δ is quiet amount of deformation; M is whole rotor quality; G is gravity accleration; M is the quality of dish; S is rotor centroid position lateral stiffness, namely produces the power needed for unit lateral deformation.

6th step: check the mode of rotor during the support stiffness initial value for determining whether to conform to the hot-die state preset.

For selected support stiffness parameter, adopt the mode of transfer matrix method or Finite element arithmetic rotor-support-foundation system.Whether the mode of inspection rotor-support-foundation system conforms to the hot-die state preset in second step.If the mode of rotor-support-foundation system is not inconsistent with the hot-die state preset, then adjust the parameter S of support stiffness _b1and S _b2, the mode of rotor-support-foundation system is conformed to the hot-die state preset in second step.

After the mode of rotor-support-foundation system conforms to the hot-die state preset in second step, check described hot-die state and rotor-support-foundation system just to prop up whether to leave between mode the nargin of 10%.If nargin is inadequate, the adjustment parameter Sb1 of support stiffness and quality, the dimensional parameters of Sb2 or rotor, make hot-die state and rotor-support-foundation system just prop up to leave between mode the nargin of 10%.Need the quiet deformation requirements of satisfied 5th step simultaneously;

7th step: damper is set in supporting place, determine unbalance response and requirement for dynamic balance:

Introduce squeeze film damper in supporting place, calculate the unbalance response of rotor.The damping function that can play according to damper and counterbalance effect, obtain rotor residual unbalance, limitation standard.So far, the Structural Dynamic Design of aeroengine rotor is completed.

The present invention proposes the concept of aeroengine rotor hot-die state, to adapt to the design of aeroengine variable working condition lower rotor part structural dynamics.Described hot-die state refers to engine operation, and the running speed of this motor frequently can pass through the critical speed of rotation of certain or certain several rotor, the mode corresponding to critical speed of rotation of this certain or certain several rotor.Based on this, the present invention establishes method and the step of aeroengine rotor dynamics Design under hot-die state.The core of the present invention to hot-die state lower rotor part dynamics Design optimizes the parameter of rotor and supporting, the mode of rotor when making the hot-die state of rotor-support-foundation system avoid supporting absolute rigidity, play the effect of damper simultaneously, make rotor-support-foundation system under hot-die state, meet the requirement of vibration standard.

The present invention establishes yielding support rotor structure dynamic model, by theory analysis, obtains the vibratory response of rotor-support-foundation system at critical speed of rotation place and the relation of each design parameter.The relation of the vibratory response of described rotor-support-foundation system at critical speed of rotation place and each design parameter comprises the relation of rotor-support-foundation system threshold response and Structural Parameters of its Rotor and the relation of rotor-support-foundation system threshold response and rotor modal parameter.The relation of rotor mode when the flexibly mounted rotor-support-foundation system threshold response of this specification of a model simultaneously and supporting absolute rigidity.The method designed for rotor dynamics and step provide foundation and criterion.

, plural hot-die state in operating range, may be there is in aeroengine rotor complex structure.The present invention propose, utilize flexibly mounted rotor mode and supporting absolute rigidity rotor mode between nargin as Optimal Parameters, physical significance is definitely.

The present invention pushes away the feature than performance aeroengine operation status from height, the concept of hot-die state is proposed, on the basis of rotor dynamics modeling, by the vibratory response characteristic of theory analysis rotor-support-foundation system at critical speed of rotation place and the relation of each design parameter, propose the nargin between flexibly mounted rotor mode and the rotor mode of supporting absolute rigidity as Optimal Parameters, play the effectiveness in vibration suppression of damper, to reach the demanding kinetics of rotor-support-foundation system simultaneously.The rotor dynamics design method that the present invention proposes and step are a set of flow processs through whole rotor design, change traditional first to design the design philosophy checked again.Active designs is carried out, cyclic design to the dynamics of rotor, improves working efficiency, there is important engineering practical value.

Accompanying drawing explanation

Fig. 1 is rotor modeling;

Fig. 2 is the vibration shape of rotor;

Fig. 3 is the vibration shape of rotor;

Fig. 4 is the vibration shape of rotor;

Fig. 5 is the amplitude-versus-frequency curve of rotor;

Fig. 6 is the amplitude-versus-frequency curve of rotor;

Fig. 7 is flow chart of the present invention.In figure: 1. a first order mode; 2. second_mode; 3. three first order modes

Embodiment

In order to adapt to highly to push away designing requirement more dynamic (dynamical) than performance aeroengine rotor, the present embodiment proposes a set of rotor dynamics design method designed through whole engine rotor.

Push away than performance aeroengine for height, the present embodiment proposes a kind of definition of rotor mode, instant heating mode.Described hot-die state refers to engine operation, and the running speed of this motor frequently can pass through the critical speed of rotation of certain or certain several rotor, the mode corresponding to critical speed of rotation of this certain or certain several rotor.

Why be defined as hot-die state, one is that rotor frequently crosses critical speed of rotation, and critical speed of rotation even may become working speed because motor run duration; Two is because under working state, and material behavior, cooperation rigidity and coupling stiffness generation significant change, cause critical speed of rotation to change in a big way, is difficult to rotating speed nargin when ensureing design; Three is because under hot-die state, the criterion of rotor structure dynamics Design is, response limiting when rotor frequently be passed through rotor critical speed under the limits value allowed, and is no longer the rotating speed nargin deliberately ensureing to expect.

The present embodiment is for the dynamic (dynamical) method of designed engines rotor structure.With rotator model as shown in Figure 1 for design object carries out rotor dynamics design.Concrete steps comprise:

The first step: set up rotor dynamics model, obtains the vibratory response of rotor-support-foundation system at critical speed of rotation place and the relation of each design parameter.The relation of the vibratory response of described rotor-support-foundation system at critical speed of rotation place and each design parameter comprises the relation of rotor-support-foundation system threshold response and Structural Parameters of its Rotor and the relation of rotor-support-foundation system threshold response and rotor modal parameter.

The kinetic equations expression matrix form of rotor-support-foundation system is as formula (1):

$M\stackrel{\·\·}{Q}+D\stackrel{\·}{Q}+\mathrm{SQ}=u---\left(1\right)$

In formula,

$M=\left[\begin{array}{cccc}m& 0& 0& 0\\ 0& I& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right],$ M is mass matrix, and m is the quality of dish, and I is the diameter rotary inertia of dish;

$D=\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& -j{I}_p\mathrm{\Ω}& 0& 0\\ 0& 0& -d_{b1}& -d_{b2}\\ 0& 0& {\mathrm{ajd}}_{b1}& -(L-a){\mathrm{jd}}_{b2}\end{array}\right],$ D is damping matrix, and j is imaginary unit, I _pfor the pole axis rotary inertia of dish, Ω is rotary speed of rotator, d _b1, d _b2be respectively the damping constant supported before and after rotor-support-foundation system, L is the span of rotor, and a is the distance of rotor front fulcrum to rotor c.g.;

$S=\left[\begin{array}{cccc}{s}_{11}& -{\mathrm{js}}_{11}& -(1-\frac{a}{L})s_{11}-\frac{s_{12}}{L}& -\frac{a}{L}{s}_{11}+\frac{s_{12}}{L}\\ {\mathrm{js}}_{21}& s_{22}& -j(1-\frac{a}{L})s_{21}-j\frac{s_{22}}{L}& -j\frac{a}{L}{s}_{21}+j\frac{s_{22}}{L}\\ s_{11}& -j{s}_{12}& -(1-\frac{a}{L})s_{11}-\frac{s_{12}}{L}-s_{b1}& -\frac{a}{L}{s}_{11}+\frac{s_{12}}{L}-s_{b2}\\ {\mathrm{js}}_{21}& s_{22}& -j(1-\frac{a}{L})s_{21}-j\frac{s_{22}}{L}+{\mathrm{jas}}_{b1}& -j\frac{a}{L}{s}_{21}+j\frac{s_{22}}{L}-(L-a){\mathrm{js}}_{b2}\end{array}\right],$ S is stiffness matrix, s ₁₁, s ₁₂, s ₁₃and s ₁₄be respectively the stiffness coefficient of rotor, s _b1, s _b2be respectively the stiffness coefficient supported before and after rotor-support-foundation system;

$u=\left\{\begin{array}{c}\begin{array}{cc}\mathrm{m\ϵ}{\mathrm{\Ω}}^2& e^{j(\mathrm{\Ωt}+\mathrm{\β})}\end{array}\\ 0\\ 0\\ 0\end{array}\right\},$ U is the inertial centrifugal force that rotor-support-foundation system bears due to imbalance, and ε is the uneven throw of eccentric of rotor; β is phase angle of unbalance;

q is the response of rotor-support-foundation system, and r is the amount of deflection at rotor c.g. place, for the deflection angle at rotor c.g. place, r _b1, r _b2be respectively the amount of deflection of rotor two supportings place.

Order q ₁it is the response at rotor c.g. place; q ₂the response at rotor bearing place, by matrix expression (1) afterwards two row can obtain:

$Q_1=S_S^{-1}{S}_{\mathrm{bb}}{Q}_2+S_S^{-1}{S}_{\mathrm{bd}}{\stackrel{\·}{Q}}_2---\left(2\right)$

In formula,

$S_s=\left[\begin{array}{cc}{s}_{11}& -j{s}_{12}\\ {\mathrm{js}}_{21}& s_{22}\end{array}\right]$

$S_{\mathrm{bb}}=\left[\begin{array}{cc}(1-\frac{a}{L})s_{11}+\frac{s_{12}}{L}+s_{b1}& \frac{a}{L}{s}_{11}-\frac{s_{12}}{L}+s_{b2}\\ j(1-\frac{a}{L})s_{21}+j\frac{s_{22}}{L}-{\mathrm{jas}}_{b1}& j\frac{a}{L}{s}_{21}-j\frac{s_{22}}{L}+(L-a){\mathrm{js}}_{b2}\end{array}\right]$

$S_S\left\{\begin{array}{ccc}\left[\begin{array}{cc}(1-\frac{a}{L})& \frac{a}{L}\\ \frac{j}{L}& \frac{-j}{L}\end{array}\right]+S_S^{-1}& \left[\begin{array}{cc}1& 1\\ -\mathrm{ja}& j(L-a)\end{array}\right]& \left[\begin{array}{cc}{s}_{b1}& 0\\ 0& s_{b2}\end{array}\right]\end{array}\right\}$

$S_{\mathrm{bd}}=\left[\begin{array}{cc}{d}_{b1}& d_{b2}\\ -{\mathrm{ajd}}_{b1}& (L-a){\mathrm{jd}}_{b2}\end{array}\right]=\left[\begin{array}{cc}1& 1\\ -\mathrm{ja}& j(L-a)\end{array}\right]\left[\begin{array}{cc}{d}_{b1}& 0\\ 0& d_{b2}\end{array}\right]$

Formula (2) is taken back front two row of matrix expression (1), obtains:

$\begin{array}{c}\left[\begin{array}{cc}m& 0\\ 0& I\end{array}\right]S_S^{-1}{S}_{\mathrm{bd}}{\stackrel{\·\·\·}{Q}}_2+\left(\begin{array}{c}\left[\begin{array}{cc}m& 0\\ 0& I\end{array}\right]S_S^{-1}{S}_{\mathrm{bb}}+\left[\begin{array}{cc}0& 0\\ 0& j{I}_p\mathrm{\Ω}\end{array}\right]S_s^{-1}{S}_{\mathrm{bd}}\end{array}\right){\stackrel{\·\·}{Q}}_2\\ +\left(\begin{array}{c}\left[\begin{array}{cc}0& 0\\ 0& -j{I}_p\mathrm{\Ω}\end{array}\right]S_s^{-1}{S}_{\mathrm{bb}}+S_{\mathrm{bd}}\end{array}\right){\stackrel{\·}{Q}}_2+\left[\begin{array}{cc}{S}_{b1}& S_{b2}\\ -{\mathrm{jas}}_{b1}& j(L-a)S_{b2}\end{array}\right]Q_2=u_1\end{array}---\left(3\right)$

In formula,

$S_{\mathrm{bS}}=\left[\begin{array}{cc}-(1-\frac{a}{L})s_{11}-\frac{S_{12}}{L}& -\frac{a}{L}{S}_{11}+\frac{S_{12}}{L}\\ -j(1-\frac{a}{L})S_{21}-j\frac{S_{22}}{L}& -j\frac{a}{L}{S}_{21}+j\frac{s_{22}}{L}\end{array}\right]$

$u_1=\left\{\begin{array}{c}\mathrm{m\ϵ}\\ 0\end{array}\right\}{\mathrm{\Ω}}^2{e}^{j(\mathrm{\Ωt}+\mathrm{\β})}=U_1{\mathrm{\Ω}}^2{e}^{j(\mathrm{\Ωt}+\mathrm{\β})}$

If rotor at the threshold response of supporting place is substitute into equation (3) to obtain:

$\begin{array}{c}\left(\begin{array}{c}-j\left[\begin{array}{cc}m& 0\\ 0& I\end{array}\right]S_s^{-1}{S}_{\mathrm{bd}}{\mathrm{\Ω}}_{\mathrm{cri}}^3-\left[\begin{array}{cc}0& 0\\ 0& -J{I}_p{\mathrm{\Ω}}_{\mathrm{cri}}\end{array}\right]S_s^{-1}{S}_{\mathrm{bd}}{\mathrm{\Ω}}_{\mathrm{cri}}^2+{\mathrm{jS}}_{\mathrm{bd}}{\mathrm{\Ω}}_{\mathrm{cri}}\end{array}\right)Q_{2\mathrm{cr}}+\\ \left(\begin{array}{c}-\left[\begin{array}{cc}m& 0\\ 0& I\end{array}\right]\end{array}{S}_s^{-1}{S}_{\mathrm{bb}}{\mathrm{\Ω}}_{\mathrm{cri}}^2+\left[\begin{array}{cc}0& 0\\ 0& I_p{\mathrm{\Ω}}_{\mathrm{cri}}\end{array}\right]S_s^{-1}{S}_{\mathrm{bb}}{\mathrm{\Ω}}_{\mathrm{cri}}+\left[\begin{array}{cc}{s}_{b1}& s_{b2}\\ -{\mathrm{jas}}_{b1}& j(L-a)s_{b2}\end{array}\right]\right)Q_{2\mathrm{cr}}=U_1{\mathrm{\Ω}}_{\mathrm{cri}}^2\end{array}---\left(4\right)$

In formula, Ω _crifor the critical speed of rotation of rotor, Q _2crfor rotor is in the threshold response amplitude of supporting place, β _2crfor rotor is at the phase angle of supporting place threshold response.

Rotor speed is critical speed of rotation this moment, and the Section 2 on equation (4) left side is 0, then have

$j\left(\begin{array}{c}-\left[\begin{array}{cc}m& 0\\ 0& I-I_p\end{array}\right]\left(S_S^{-1}{S}_{\mathrm{bd}}{\mathrm{\Ω}}_{\mathrm{cri}}^2+S_{\mathrm{bd}}\right)\end{array}\right)Q_{2\mathrm{cr}}=U_1{\mathrm{\Ω}}_{\mathrm{cri}}---\left(5\right)$

Introduce as down conversion

$q_{2\mathrm{cr}}=S_S^{-1}{S}_{\mathrm{bd}}{Q}_{2\mathrm{cr}};Q_{2\mathrm{cr}}=S_{\mathrm{bd}}^{-1}{s}_s{q}_{2\mathrm{cr}}---\left(6\right)$

After substituting into equation (5), obtain

$\left(\begin{array}{c}-\left[\begin{array}{cc}m& 0\\ 0& I-I_p\end{array}\right]{\mathrm{\Ω}}_{\mathrm{cri}}^2+S_s\end{array}\right){\mathrm{jq}}_{2\mathrm{cr}}=U_1{\mathrm{\Ω}}_{\mathrm{cri}}---\left(7\right)$

Can be solved by equation (7)

$q_{2\mathrm{cr}}=-j\frac{1}{\mathrm{\Λ}}\left[\begin{array}{cc}{s}_{22}-(I-I_p){\mathrm{\Ω}}_{\mathrm{cri}}^2& {\mathrm{js}}_{12}\\ -{\mathrm{js}}_{21}& s_{11}-m{\mathrm{\Ω}}_{\mathrm{cri}}^2\end{array}\right]U_1{\mathrm{\Ω}}_{\mathrm{cri}}---\left(8\right)$

In formula, $\mathrm{\Λ}=|S_s-\left[\begin{array}{cc}m& 0\\ 0& I-I_p\end{array}\right]{\mathrm{\Ω}}_{\mathrm{cri}}^2|.$

When supporting absolute rigidity, if the modal matrix of rotor-support-foundation system is

$\widetilde{\left[\mathrm{\Φ}\right]}=\left[{\widetilde{\mathrm{\Φ}}}_1{\widetilde{\mathrm{\Φ}}}_2\right]=\left[\begin{array}{cc}{\widetilde{\mathrm{\ψ}}}_{11}& {\widetilde{\mathrm{\ψ}}}_{12}\\ {\widetilde{\mathrm{\ψ}}}_{21}& {\widetilde{\mathrm{\ψ}}}_{22}\end{array}\right]---\left(9\right)$

In formula for the first first order mode of rotor during rigid support; for the second-order vibration shape of rotor during rigid support.Meanwhile, modal stiffness and modal mass is introduced:

${\widetilde{\left[\mathrm{\Φ}\right]}}^T{S}_s\widetilde{\left[\mathrm{\Φ}\right]}=\left[\begin{array}{cc}{\tilde{S}}_{\mathrm{crl}}& 0\\ 0& {\tilde{S}}_{\mathrm{cr}2}\end{array}\right]---\left(10\right)$

In formula, when being respectively rigid support, the first step mode rigidity of rotor and second-order modal stiffness.

${\widetilde{\left[\mathrm{\Φ}\right]}}^T\left[\begin{array}{cc}m& 0\\ 0& I-I_p\end{array}\right]\widetilde{\left[\mathrm{\Φ}\right]}=\left[\begin{array}{cc}{\tilde{m}}_{\mathrm{crl}}& 0\\ 0& {\tilde{I}}_{\mathrm{cr}2}\end{array}\right]---\left(11\right)$

In formula, when being respectively rigid support, the first step mode quality of rotor and second-order modal mass, wherein subscript cr1 and cr2 represents the first rank and second-order respectively.

In formula (8), the molecule of equal sign right-hand member and denominator are with taking advantage of namely

$q_{2\mathrm{cr}}=-j\frac{\left|{\widetilde{\left[\mathrm{\Φ}\right]}}^T\right|\·\left|\widetilde{\left[\mathrm{\Φ}\right]}\right|}{\left|{\widetilde{\left[\mathrm{\Φ}\right]}}^T\right|\·\mathrm{\Λ}\·\left|\widetilde{\left[\mathrm{\Φ}\right]}\right|}\left[\begin{array}{cc}{s}_{22}-(I-I_p){\mathrm{\Ω}}_{\mathrm{cri}}^2& {\mathrm{js}}_{12}\\ -j{s}_{21}& s_{11}-m{\mathrm{\Ω}}_{\mathrm{cri}}^2\end{array}\right]U_1{\mathrm{\Ω}}_{\mathrm{cri}}---\left(12\right)$

According to the character of determinant, and substitute into formula (10) and (11)

$\begin{array}{c}\left|{\left[\widetilde{\mathrm{\Φ}}\right]}^T\right|\·\mathrm{\Λ}\·\left|\right[\widetilde{\mathrm{\Φ}}\left]\right|=\left|{\left[\widetilde{\mathrm{\Φ}}\right]}^T{S}_s\right[\widetilde{\mathrm{\Φ}}]-{\left[\widetilde{\mathrm{\Φ}}\right]}^T\left[\begin{array}{cc}m& 0\\ 0& I-I_p\end{array}\right][\widetilde{\mathrm{\Φ}}\left]{\mathrm{\Ω}}_{\mathrm{cri}}^2\right|\\ =|\left[\begin{array}{cc}{\tilde{S}}_{\mathrm{cr}1}& 0\\ 0& {\stackrel{0}{S}}_{\mathrm{cr}2}\end{array}\right]-\left[\begin{array}{cc}{\tilde{m}}_{\mathrm{cr}1}& 0\\ 0& {\tilde{I}}_{\mathrm{cr}2}\end{array}\right]{\mathrm{\Ω}}_{\mathrm{cri}}^2|\end{array}---\left(13\right)$

In formula (12), equal sign right-hand member molecule and denominator are with taking advantage of and substitute into formula (13), and abbreviation,

$\begin{array}{c}{q}_{2\mathrm{cr}}=-j\frac{\left|{\left[\widetilde{\mathrm{\Φ}}\right]}^T\right|\·\left|{\left[\begin{array}{cc}{\tilde{m}}_{\mathrm{cr}1}& 0\\ 0& {\tilde{I}}_{\mathrm{cr}2}\end{array}\right]}^{-1}\right|\·\left|\right[\widetilde{\mathrm{\Φ}}\left]\right|}{({\widetilde{\mathrm{\Ω}}}_{\mathrm{cr}1}^2-{\mathrm{\Ω}}_{\mathrm{cri}}^2)({\widetilde{\mathrm{\Ω}}}_{\mathrm{cr}2}^2-{\mathrm{\Ω}}_{\mathrm{cri}}^2)}\left[\begin{array}{cc}{s}_{22}-(I-I_p){\mathrm{\Ω}}_{\mathrm{cri}}^2& -s_{12}\\ -s_{21}& s_{11}-m{\mathrm{\Ω}}_{\mathrm{cri}}^2\end{array}\right]U_1{\mathrm{\Ω}}_{\mathrm{cri}}\\ =-j\frac{\left|\right[\widetilde{\mathrm{\Φ}}]\begin{array}{c}{\left(\begin{array}{c}{[\widetilde{\mathrm{\Φ}]}}^T\left[\begin{array}{cc}m& 0\\ 0& I-I_p\end{array}\right]\left[\widetilde{\mathrm{\Φ\Φ}}\right]\end{array}\right)}^{-1}{\left[\widetilde{\mathrm{\Φ}}\right]}^T|\end{array}}{({\widetilde{\mathrm{\Ω}}}_{\mathrm{cr}1}^2-{\mathrm{\Ω}}_{\mathrm{cri}}^2)({\widetilde{\mathrm{\Ω}}}_{\mathrm{cr}2}^2-{\mathrm{\Ω}}_{\mathrm{cri}}^2)}\left[\begin{array}{cc}{s}_{22}-(I-I_p){\mathrm{\Ω}}_{\mathrm{cri}}^2& -s_{12}\\ -s_{21}& s_{11}-m{\mathrm{\Ω}}_{\mathrm{cri}}^2\end{array}\right]U_1{\mathrm{\Ω}}_{\mathrm{cri}}\\ =-j\frac{1}{m(I-I_p)({\widetilde{\mathrm{\Ω}}}_{\mathrm{cr}1}^2-{\mathrm{\Ω}}_{\mathrm{cri}}^2)({\widetilde{\mathrm{\Ω}}}_{\mathrm{cr}2}^2-{\mathrm{\Ω}}_{\mathrm{cri}}^2)}\left[\begin{array}{cc}{s}_{22}-(I-I_p){\mathrm{\Ω}}_{\mathrm{cri}}^2& -s_{12}\\ -s_{21}& s_{11}-m{\mathrm{\Ω}}_{\mathrm{cri}}^2\end{array}\right]U_1{\mathrm{\Ω}}_{\mathrm{cri}}\end{array}---\left(14\right)$

In formula, when being respectively supporting rigidity, the critical speed of rotation of rotor-support-foundation system.

Formula (14) is substituted into formula (6), namely obtains the vibration of rotor in supporting place:

$Q_{2\mathrm{cr}}=S_{\mathrm{bd}}^{-1}{S}_s{q}_{2\mathrm{cr}}=-j\frac{S_{\mathrm{bd}}^{-1}{S}_s}{m(I-I_p)({\widetilde{\mathrm{\Ω}}}_{\mathrm{crl}}^2-{\mathrm{\Ω}}_{\mathrm{cri}}^2)({\widetilde{\mathrm{\Ω}}}_{\mathrm{cr}2}^2-{\mathrm{\Ω}}_{\mathrm{cri}}^2)}\left[\begin{array}{cc}{s}_{22}-(I-I_p){\mathrm{\Ω}}_{\mathrm{cri}}^2& -s_{12}\\ -s_{21}& s_{11}-m{\mathrm{\Ω}}_{\mathrm{cri}}^2\end{array}\right]U_1{\mathrm{\Ω}}_{\mathrm{cri}}---\left(15\right)$

Introduce modal damping matrix

$S_{\mathrm{bd}}={\left({\left[\mathrm{\Φ}\right]}^T\right)}^{-1}\left[\begin{array}{cc}{2m}_{\mathrm{crl}}{\mathrm{\Ω}}_{\mathrm{crl}}{D}_{\mathrm{crl}}& 0\\ 0& {2I}_{\mathrm{cr}2}{\mathrm{\Ω}}_{\mathrm{cr}2}{D}_{\mathrm{cr}2}\end{array}\right]{\left[\mathrm{\Φ}\right]}^{-1}---\left(16\right)$

In formula, when [Φ] is for bearing elastic, the modal matrix of rotor-support-foundation system, dcr1, dcr2 are respectively the first rank and the second-order modal damping coefficient of rotor-support-foundation system, $D_{\mathrm{crl}}=\frac{d_{\mathrm{crl}}}{{2m}_{\mathrm{crl}}{\mathrm{\Ω}}_{\mathrm{crl}}},D_{\mathrm{cr}2}=\frac{d_{\mathrm{cr}2}}{{2I}_{\mathrm{cr}2}{\mathrm{\Ω}}_{\mathrm{cr}2}}$ Be respectively the first rank and second-order damping ratios, m _cr1, I _cr2be respectively the first rank and second-order modal mass, Ω _cr1, Ω _cr2when being respectively support elasticity, the first rank of rotor and second-order critical speed of rotation.

Wushu (16) substitutes into formula (15), finally obtains the threshold response of rotor-support-foundation system in supporting place

$\begin{array}{c}{Q}_{2\mathrm{cr}}=-j\frac{1}{m(I-I_p)({\widetilde{\mathrm{\Ω}}}_{\mathrm{crl}}-{\mathrm{\Ω}}_{\mathrm{cri}}^2)({\widetilde{\mathrm{\Ω}}}_{\mathrm{cr}2}^2-{\mathrm{\Ω}}_{\mathrm{cri}}^2)}\left[\mathrm{\Φ}\right]\left[\begin{array}{cc}\frac{{\mathrm{\Ω}}_{\mathrm{cri}}}{2m_{\mathrm{crl}}{D}_{\mathrm{crl}}{\mathrm{\Ω}}_{\mathrm{crl}}}& 0\\ 0& \frac{{\mathrm{\Ω}}_{\mathrm{cri}}}{{2I}_{\mathrm{cr}2}{D}_{\mathrm{cr}2}{\mathrm{\Ω}}_{\mathrm{cr}2}}\end{array}\right]{\left[\mathrm{\Φ}\right]}^T.\\ S_s\left[\begin{array}{cc}{s}_{22}-(I-I_p){\mathrm{\Ω}}_{\mathrm{cri}}^2& -s_{12}\\ -s_{21}& s_{11}-m{\mathrm{\Ω}}_{\mathrm{cri}}^2\end{array}\right]U_1\end{array}---\left(17\right)$

In generation, returns the threshold response that formula (2) obtains rotor c.g. place

$Q_{\mathrm{lcr}}=S_s^{-1}{S}_{\mathrm{bb}}{Q}_{2\mathrm{cr}}+S_s^{-1}{S}_{\mathrm{bd}}{\stackrel{\·}{Q}}_{2\mathrm{cr}}---\left(18\right)$

Formula (17), formula (18) describe the vibratory response Q of flexibly mounted rotor-support-foundation system at critical speed of rotation place _1cr, Q _2cr, include the modal parameter of rotor-support-foundation system structural dynamic parameter and rotor-support-foundation system.Formula (17), formula (18) describe flexibly mounted rotor-support-foundation system threshold response and rotor critical speed during supporting absolute rigidity relation.For the design of follow-up rotor dynamics is laid a good foundation.

Second step: the hot-die state during work of setting rotor-support-foundation system.

According to operating rotational speed range and the performance requirement of motor, the hot-die state of setting rotor-support-foundation system.When setting described hot-die state, must meet the following conditions successively: first step mode critical speed of rotation is positioned at below slow train rotating speed, and allow in slow train position; Second-order mode critical speed of rotation and the 3rd rank mode critical speed of rotation are all positioned at below cruising speed, nargin equal 10%; Critical speed of rotation is not established between cruising speed and maximum (top) speed; When the design parameter of rotor requires to establish critical speed of rotation between the cruising speed and maximum (top) speed of rotor-support-foundation system, then quadravalence mode critical speed of rotation is arranged on maximum (top) speed, nargin 8-10%.

In the present embodiment, if the slow train rotating speed of rotor is 6000 revs/min, cruising speed is 13000 revs/min, maximum (top) speed 15000 revs/min.First three rank critical speed of rotation of the rotor that consideration need design, so the first step mode critical speed of rotation of rotor need be arranged on less than 6000 revs/min, second-order mode critical speed of rotation need be arranged on less than 13000 revs/min, nargin 10%, in 13000 revs/min to 15000 revs/min speed range, critical speed of rotation is not set, 3rd rank mode critical speed of rotation is arranged on more than 14000 revs/min, nargin 8-10%.

3rd step: the mode of rotor-support-foundation system when determining to support absolute rigidity.

If each support stiffness of rotor-support-foundation system is infinitely great, transfer matrix method or finite element method is adopted to determine the mode of rotor-support-foundation system with this understanding.Transfer matrix method is adopted to determine the mode of rotor-support-foundation system with this understanding in the present embodiment.

For the rotor in the present embodiment, design parameter is as follows:

Rotor span L=0.81m, shaft diameter 0.055m, coil from front fulcrum distance a=0.54m, dish quality 20kg, diameter rotary inertia I _d=0.032kgm, polar moment of inertia I _p=0.064kgm, the Young's modulus of axle is 2.06 × 10 ¹¹n/m ²., moment of area of inertia I=4.4918 × 10 of shaft section ^-7m ⁴, rotor material density 7850kg/m ³.

Rotor is divided into 18 sections, the two rank critical speed of rotation adopting transfer matrix method to calculate rotor are respectively: 5776 revs/min, 32461 revs/min, its corresponding Mode Shape as shown in Figure 2.

4th step: the initial value choosing support stiffness.The initial value of support stiffness is selected according to the rigidity of supporting absolute rigidity rotor-support-foundation system; Each support stiffness of rotor-support-foundation system should be the 20-50% of rigid support rotor rigidity, adopts finite element method or determines the lateral stiffness of rotor at barycenter place by formula (19).

For the rotor in the present embodiment, rotor centroid place produces the power required for unit deformation, and namely rotor is at the lateral stiffness at barycenter place, is calculated as follows:

$S=\frac{243\mathrm{EI}}{4L^3}=1.0577\×{10}^{7}N---\left(19\right)$

E is the Young's modulus of material, and I is the cross section moments of inertia of rotating shaft.

Get 50% of s as the initial value of rotor-support-foundation system elastic supporting rigidity:

$S_{b1}=S_{b2}=\frac{1}{2}S=5.2887\×{10}^{6}N/m---\left(20\right)$

5th step: verify whether the quiet amount of deformation of rotor during the support stiffness initial value determined exceeds standard.

Adopt finite element method or the quiet distortion according to formula (21) calculating rotor-support-foundation system, make the quiet amount of deformation of rotor be not more than the value of this engine design requirement.Require in the present embodiment that quiet amount of deformation is not more than 0.1mm.

$\mathrm{\δ}=\frac{M\·g}{S_{b1}+S_{b2}}+\frac{m\·g}{s}---\left(21\right)$

In formula, δ is quiet amount of deformation; M is whole rotor quality; G is gravity accleration; M is the quality of dish; S is rotor centroid position lateral stiffness, namely produces the power needed for unit lateral deformation.

For the present embodiment, quiet amount of deformation $\mathrm{\δ}=\frac{M\·g}{s_{b1}+s_{b2}}+\frac{m\·g}{s}=0.0835\mathrm{mm},$ Meet the demands.

6th step: check the support stiffness initial value for determining, whether the mode of rotor conforms to the hot-die state preset.

For selected support stiffness parameter, adopt the mode of transfer matrix method or Finite element arithmetic rotor-support-foundation system; Whether the mode of inspection rotor-support-foundation system conforms to the hot-die state preset in second step; If the mode of rotor-support-foundation system is not inconsistent with the hot-die state preset, then adjust the parameter s of support stiffness _b1and s _b2, the mode of rotor-support-foundation system is conformed to the hot-die state preset in second step;

After the mode of rotor-support-foundation system conforms to the hot-die state preset in second step, check described hot-die state and rotor-support-foundation system just to prop up whether to leave between mode the nargin of 10%; If nargin is inadequate, the parameter S of adjustment support stiffness _b1and S _b2or the quality of rotor, dimensional parameters, hot-die state and rotor-support-foundation system are just propped up to leave between mode the nargin of 10%; Need the quiet deformation requirements of satisfied 5th step simultaneously;

For the present embodiment, when support stiffness is provided by formula (20), equally rotor-support-foundation system is divided into 18 sections, the three rank critical speed of rotation adopting transfer matrix method to calculate are respectively 3880 revs/min, 12133 revs/min, 29862 revs/min, and the Mode Shape of its correspondence as shown in Figure 3.

Through inspection, the second-order mode of the present embodiment rotor is not inconsistent with the hot-die state preset, and namely second-order critical speed of rotation 12133 revs/min is too high, and the first rank and Section of three mode all well meet the requirement of hot-die state.So retainer spring stiffness to be reduced to 40% of s, now the quiet amount of deformation of rotor is meet the demands.The three rank critical speed of rotation calculating now rotor-support-foundation system are respectively 3634 revs/min, 10955 revs/min, 28610 revs/min, and through inspection, rotor-support-foundation system mode now conforms to the hot-die state preset.The Mode Shape of its correspondence as shown in Figure 4.

After reaching the requirement of hot-die state, whether inspection hot-die state has just propped up mode with rotor-support-foundation system is left enough nargin, and it is 10% that the present embodiment requires.Be respectively through the hot-die state of test design rotor and a just mode nargin of rotor-support-foundation system: 37%, 66%, meet the demands.

7th step: arrange damper in supporting place, determines unbalance response and requirement for dynamic balance.

Within the engine, in order to meet demanding kinetics, needing to apply damping at supporting place design squeeze film damper to rotor, especially needing the height repeatedly through critical speed of rotation region to push away and compare performance engine.After introducing squeeze film damper, calculate the unbalance response of rotor.The damping function that can play according to damper and counterbalance effect, finally can provide rotor residual unbalance, limitation standard.

For the present embodiment, rotor is that displacement amplitude is not more than 50 μm at the vibration amplitude limitation standard of supporting place, and apply damper at rear support place, damping constant is 500Ns/m, and rotor amount of unbalance before dynamic balancing is 1 × 10 ^-4kgm, is calculated by transfer matrix method equally, obtain rotor-support-foundation system two bearing places response be respectively: 1 rank: 73.0 μm, 112.2 μm; 2 rank: 86.4 μm, 47.3 μm; 3 rank: 13.63 μm, 23.2 μm, amplitude-versus-frequency curve as shown in Figure 5.Do not meet amplitude standards, need dynamic balancing be done.After dynamic balancing, the amount of unbalance of rotor is reduced to 0.3 × 10 ^-4kgm, now rotor-support-foundation system two bearing places response be respectively: 1 rank: 21.9 μm, 33.7 μm; 2 rank: 25.9 μm, 14.3 μm; 3 rank: 4.1 μm, 7.0 μm, all meet amplitude standards.Amplitude-versus-frequency curve as shown in Figure 6.

By calculating above, for the present embodiment, the 1 rank amplitude at rotor rear support place is relatively large, when the amount of unbalance of rotor reaches 0.44 × 10 ^-4during kgm, the 1 rank amplitude at rotor rear support place has reached standard limit 50 μm, therefore, provides the residual unbalance, standard limit of this rotor for being not more than 0.44 × 10 ^-4kgm.

So far, whole rotor dynamics design process terminates, and by the present embodiment, makes the structural dynamic characteristics of rotor obtain optimization, meets proposed all demanding kinetics.

Claims (1)

1. a design method for aeroengine rotor structural dynamics, is characterized in that, concrete steps are:

The first step: set up rotor dynamics model: obtain the vibratory response of rotor-support-foundation system at critical speed of rotation place and the relation of each design parameter, the relation of the vibratory response of described rotor-support-foundation system at critical speed of rotation place and each design parameter comprises the relation of rotor-support-foundation system threshold response and Structural Parameters of its Rotor and the relation of rotor-support-foundation system threshold response and rotor modal parameter;

The equation matrix expression (1) of rotor dynamics model is:

$M\stackrel{\·\·}{Q}+D\stackrel{\·}{Q}+\mathrm{SQ}=u---\left(1\right)$

In formula (1):

$M=\left[\begin{array}{cccc}m& 0& 0& 0\\ 0& I& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right],$ M is mass matrix, and m is the quality of dish, and I is the diameter rotary inertia of dish;

$D=\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& -j{I}_p\mathrm{\Ω}& 0& 0\\ 0& 0& -d_{b1}& -d_{b2}\\ 0& 0& \mathrm{aj}{d}_{b1}& -(L-a)j{d}_{b2}\end{array}\right],$ D is damping matrix, and j is imaginary unit, I _pfor the pole axis rotary inertia of dish, Ω is rotary speed of rotator, d _b1for the damping constant of rotor-support-foundation system front support, d _b2for the damping constant of rotor-support-foundation system rear support, L is the span of rotor, and a is the distance of rotor front fulcrum to rotor c.g.;

$S=\left[\begin{array}{cccc}{s}_{11}& -j{s}_{12}& -(1-\frac{a}{L})s_{11}-\frac{s_{12}}{L}& -\frac{a}{L}{s}_{11}+\frac{s_{12}}{L}\\ j{s}_{21}& s_{22}& -j(1-\frac{a}{L})s_{21}-j\frac{s_{22}}{L}& -j\frac{a}{L}{s}_{21}+j\frac{s_{22}}{L}\\ s_{11}& -j{s}_{12}& -(1-\frac{a}{L})s_{11}-\frac{s_{12}}{L}-s_{b1}& -\frac{a}{L}{s}_{11}+\frac{s_{12}}{L}-s_{b2}\\ j{s}_{21}& s_{22}& -j(1-\frac{a}{L})s_{21}-j\frac{s_{22}}{L}+\mathrm{ja}{s}_{b1}& -j\frac{a}{L}{s}_{21}+j\frac{s_{22}}{L}-(L-a)j{s}_{b2}\end{array}\right],$ S is stiffness matrix, s ₁₁, s ₁₂, s ₁₃and s ₁₄be respectively the stiffness coefficient of rotor, s _b1for the stiffness coefficient of rotor-support-foundation system front support, s _b2for the stiffness coefficient of rotor-support-foundation system rear support;

$u=\left\{\begin{array}{c}\mathrm{m\ϵ}{\mathrm{\Ω}}^2{e}^{j(\mathrm{\Ωt}+\mathrm{\β})}\\ 0\\ 0\\ 0\end{array}\right\},$ U is the inertial centrifugal force that rotor-support-foundation system bears due to imbalance, and ε is the uneven throw of eccentric of rotor; β is phase angle of unbalance;

q is the response of rotor-support-foundation system, and r is the amount of deflection at rotor c.g. place, for the deflection angle at rotor c.g. place, r _b1for the stiffness coefficient of rotor-support-foundation system front support, r _b2for the stiffness coefficient of rotor-support-foundation system rear support;

Rotor-support-foundation system is at the threshold response of supporting place

$\begin{array}{c}{Q}_{2\mathrm{cr}}=-j\frac{1}{m(I-I_p)({\widetilde{\mathrm{\Ω}}}_{\mathrm{cr}1}^2-{\mathrm{\Ω}}_{\mathrm{cri}}^2)({\widetilde{\mathrm{\Ω}}}_{\mathrm{cr}2}^2-{\mathrm{\Ω}}_{\mathrm{cri}}^2)}\left[\mathrm{\Φ}\right]\left[\begin{array}{cc}\frac{{\mathrm{\Ω}}_{\mathrm{cri}}}{2m_{\mathrm{cr}1}{D}_{\mathrm{cr}1}{\mathrm{\Ω}}_{\mathrm{cr}1}}& 0\\ 0& \frac{{\mathrm{\Ω}}_{\mathrm{cri}}}{2I_{\mathrm{cr}2}{D}_{\mathrm{cr}2}{\mathrm{\Ω}}_{\mathrm{cr}2}}\end{array}\right]{\left[\mathrm{\Φ}\right]}^T\·\\ S_S\left[\begin{array}{cc}{s}_{22}-(I-I_p){\mathrm{\Ω}}_{\mathrm{cri}}^2& -s_{12}\\ -s_{21}& s_{11}-m{\mathrm{\Ω}}_{\mathrm{cri}}^2\end{array}\right]U_1\end{array}---\left(17\right)$

The threshold response at rotor c.g. place

$Q_{1\mathrm{cr}}=S_S^{-1}{S}_{\mathrm{bb}}{Q}_{2\mathrm{cr}}+S_S^{-1}{S}_{\mathrm{bd}}{\stackrel{\·}{Q}}_{2\mathrm{cr}}---\left(18\right)$

Formula (17), formula (18) describe the vibratory response Q of flexibly mounted rotor-support-foundation system at critical speed of rotation place _1cr, Q _2cr, include the modal parameter of rotor-support-foundation system structural dynamic parameter and rotor-support-foundation system; Formula (17), formula (18) describe flexibly mounted rotor-support-foundation system threshold response and rotor critical speed during supporting absolute rigidity relation;

In formula (17) and formula (18),

Q _1crfor the threshold response amplitude at rotor c.g. place; Q _2crfor the threshold response amplitude at rotor bearing place;

Ω _crifor rotor i-th rank critical speed of rotation;

The modal matrix that [Φ] is rotor-support-foundation system, be respectively the first rank and second-order damping ratios, d _cr1, d _cr2be respectively the first rank and the second-order modal damping coefficient of rotor-support-foundation system, m _cr1, I _cr2be respectively the first rank and second-order modal mass, Ω _cr1, Ω _cr2be respectively the first rank and the second-order critical speed of rotation of rotor;

when being respectively supporting rigidity, the critical speed of rotation of rotor-support-foundation system;

$S_S=\left[\begin{array}{cc}{s}_{11}& -j{s}_{12}\\ j{s}_{21}& s_{22}\end{array}\right]$

$\begin{array}{c}{S}_{\mathrm{bb}}=\left[\begin{array}{cc}(1-\frac{a}{L})s_{11}+\frac{s_{12}}{L}+s_{b1}& \frac{a}{L}{s}_{11}-\frac{s_{12}}{L}+s_{b2}\\ j(1-\frac{a}{L})s_{21}+j\frac{s_{22}}{L}-\mathrm{ja}{s}_{b1}& j\frac{a}{L}{s}_{21}-j\frac{s_{22}}{L}+(L-a)j{s}_{b2}\end{array}\right]\\ =S_S\{\left[\begin{array}{cc}(1-\frac{a}{L})& \frac{a}{L}\\ \frac{j}{L}& \frac{-j}{L}\end{array}\right]+S_S^{-1}\left[\begin{array}{cc}1& 1\\ -\mathrm{ja}& j(L-a)\end{array}\right]\left[\begin{array}{cc}{s}_{b1}& 0\\ 0& s_{b2}\end{array}\right]\}\end{array}$

$S_{\mathrm{bd}}=\left[\begin{array}{cc}{d}_{b1}& d_{b2}\\ -\mathrm{aj}{d}_{b1}& (L-a)j{d}_{b2}\end{array}\right]=\left[\begin{array}{cc}1& 1\\ -\mathrm{ja}& j(L-a)\end{array}\right]\left[\begin{array}{cc}{d}_{b1}& 0\\ 0& d_{b2}\end{array}\right]$

U ₁for the amount of unbalance of rotor-support-foundation system;

Ω _crifor rotor i-th rank critical speed of rotation, Q _2crfor rotor is in the threshold response amplitude of supporting place;

Second step: the hot-die state during work of setting rotor-support-foundation system:

According to operating rotational speed range and the performance requirement of motor, the hot-die state of setting rotor-support-foundation system; When setting described hot-die state, must meet the following conditions successively: first step mode critical speed of rotation is positioned at below slow train rotating speed, and allow in slow train position; Second-order mode critical speed of rotation and the 3rd rank mode critical speed of rotation are all positioned at below cruising speed, nargin equal 10%; Critical speed of rotation is not established between cruising speed and maximum (top) speed; When the design parameter of rotor requires to establish critical speed of rotation between the cruising speed and maximum (top) speed of rotor-support-foundation system, then quadravalence mode critical speed of rotation is arranged on maximum (top) speed, nargin 8-10%;

3rd step: the mode of rotor-support-foundation system when determining to support absolute rigidity:

If each support stiffness of rotor-support-foundation system is infinitely great, transfer matrix method or finite element method is adopted to determine the mode of rotor-support-foundation system with this understanding;

4th step: the initial value choosing support stiffness: the initial value selecting support stiffness according to the rigidity of supporting absolute rigidity rotor-support-foundation system; Each support stiffness of rotor-support-foundation system should be the 20-50% of rigid support rotor rigidity, adopts finite element method or determines the lateral stiffness of rotor at barycenter place by formula (19):

$s=\frac{243\mathrm{EI}}{4L^3}---\left(19\right)$

E is the Young's modulus of material, and I is the cross section moments of inertia of rotating shaft;

Get 50% of s as the initial value of rotor-support-foundation system elastic supporting rigidity:

$s_{b1}=s_{b2}=\frac{1}{2}s---\left(20\right)$

5th step: verify whether the quiet amount of deformation of rotor during the support stiffness initial value determined exceeds standard:

Adopt finite element method or the quiet distortion according to formula (21) calculating rotor-support-foundation system, make the quiet amount of deformation of rotor be not more than the value of this engine design requirement;

$\mathrm{\δ}=\frac{M\·g}{s_{b1}+s_{b2}}+\frac{m\·g}{s}---\left(21\right)$

In formula, δ is quiet amount of deformation; M is whole rotor quality; G is gravity accleration; M is the quality of dish; S is rotor centroid position lateral stiffness, namely produces the power needed for unit lateral deformation;

6th step: check the mode of rotor during the support stiffness initial value for determining whether to conform to the hot-die state preset:

For selected support stiffness parameter, adopt the mode of transfer matrix method or Finite element arithmetic rotor-support-foundation system; Whether the mode of inspection rotor-support-foundation system conforms to the hot-die state preset in second step; If the mode of rotor-support-foundation system is not inconsistent with the hot-die state preset, then adjust the parameter s of support stiffness _b1and s _b2, the mode of rotor-support-foundation system is conformed to the hot-die state preset in second step;

After the mode of rotor-support-foundation system conforms to the hot-die state preset in second step, check described hot-die state and rotor-support-foundation system just to prop up whether to leave between mode the nargin of 10%; If nargin is inadequate, the parameter S of adjustment support stiffness _b1and S _b2or the quality of rotor, dimensional parameters, hot-die state and rotor-support-foundation system are just propped up to leave between mode the nargin of 10%;

Need the quiet deformation requirements of satisfied 5th step simultaneously;

7th step: damper is set in supporting place, determine unbalance response and requirement for dynamic balance:

Introduce squeeze film damper in supporting place, calculate the unbalance response of rotor; The damping function that can play according to damper and counterbalance effect, obtain rotor residual unbalance, limitation standard; So far, the Structural Dynamic Design of aeroengine rotor is completed.