CN102722612A - Helicopter rotor airframe coupling system model and application thereof - Google Patents
Helicopter rotor airframe coupling system model and application thereof Download PDFInfo
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Abstract
The invention discloses a helicopter rotor airframe coupling system model and application thereof, which are used for analyzing the dynamic stability judgment of articulated, hingeless and bearingless rotor helicopter systems and belong to the technical field of the design of a helicopter. In the invention, an equivalent paddle rigid model with an equivalent hinge overhanging beam or a 15-degree-of-freedom beam element model is used as a rotor paddle; a rigid model with the pitching and rolling degrees of freedom or a 15-degree-of-freedom beam element model is used as an airframe structure; and the aerodynamic force of the section of the paddle is calculated by utilizing the lifting line theory. A paddle motion equation, an airframe motion equation and an aerodynamic force model equation are subjected to simultaneous processing to form a motion equation set so as to obtain a motion equation of a helicopter rotor airframe coupling system; and the motion equation is solved to obtain the dynamic stability of the helicopter system. The helicopter rotor airframe coupling system model disclosed by the invention has higher reliability; and the dynamic stability of the helicopter system can be accurately analyzed by adopting a time domain analysis method.
Description
Technical field
The invention belongs to the helicopter design field, be specifically related to lifting airscrew body coupling system model, can be used for judging the dynamic stability of radial type, no hinge formula and bearingless rotor lifting airscrew body coupled system.
Background technology
Helicopter System is a complex dynamic system, and the dynamic stability problem is one of basic problem of its dynamics Design, and wherein ground all is the problem that the deviser mainly considers with aerial resonance problem all the time.The main vibration source of helicopter is varied, mainly contains main rotor system, tail-rotor and engine etc.Helicopter System is extremely complicated, mainly comprises rotor and body etc.Not only rotor blade inner exist wave, shimmy and coupling such as reverse, and also have coupling between rotor and the body, be exactly the autovibration that the coupling between the shimmy and body vibration produces owing to helicopter rotor blade like ground resonance.In order to improve the helicopter performance, avoid going straight up to motor-driven wild effect, the air worthiness regulation of helicopter has been carried out specified in more detail to this.
Because how Helicopter System and dynamic stability the complex nature of the problem thereof judge that the helicopter dynamic stability problem of considering rotor and airframe systems coupling becomes urgent problem.Understand helicopter performance main method in depth experimental study and theoretical analysis are arranged; Though experimental technique can accurately be assessed the dynamic stability of Helicopter System; But the experimental study cost is high and not economical and practical in the design starting stage, and therefore need setting up accurately, structure, aerodynamic force and coupling model thereof come the accurate description helicopter.The helicopter analytical technology has obtained bigger progress, but reliably accurately the vibration prediction problem remain a great challenge.Traditional helicopter vibration analysis mainly is divided into two stages, and the hypothesis propeller hub is fixed in the phase one, through finding the solution the load that complicated structure and Aerodynamic Model obtain rotor blade and propeller hub; Thereby in subordinate phase, adopt the propeller hub load that calculates to be applied to the vibration of prediction helicopter fuselage in the helicopter finite element model.But in the Helicopter System of reality, the vibration of body equally also can influence the motion of rotor blade, but this method is not considered the influence of body to rotor, therefore is difficult to accurately predict the dynamic stability of helicopter.
Summary of the invention
The present invention is directed to the deficiency of existing lifting airscrew body coupled system analytical model; A kind of new model that is used to judge radial type, no hinge formula and bearingless rotor Helicopter System dynamic stability has been proposed; Described model has higher computational accuracy; Rotor blade can adopt blade equivalence rigid model with equivalent hinge overhanging beam or the elastic beam model of element with 15 degree of freedom of the bending of waving, shimmy bending, axis deformation and elastic torsion in this model, and the structural model of body then adopts the perhaps elastic beam model of element of 15 degree of freedom of rigid model with pitching and lift-over degree of freedom.Utilize lift line theory to calculate propeller-blade section aerodynamic force.Dynamic inflow model, expansion Pitt/Peters dynamic inflow model and permanently go into flow model and be used for judging the helicopter dynamic stability problem under the low frequency state, under the disturbance motion situation and under the standing state situation respectively.Set up the body movement equation according to d'Alembert principle, thereby can obtain the dynamic stability model of lifting airscrew and body coupled system.The present invention can adopt Eigenvalue Analysis method and time-domain analysis method to find the solution lifting airscrew and body coupled system equation, obtains modal damping, thereby reaches the purpose of accurate judgement lifting airscrew and body coupled system dynamic stability.
The present invention also provides a kind of method for building up of above-mentioned model; With the equation of motion group of the rotor blade equation of motion, body movement equation and Aerodynamic Model equations simultaneousness composition lifting airscrew and body coupled system, promptly obtain the equation of motion of lifting airscrew and body coupled system.For above-mentioned lifting airscrew and body coupling system model; The present invention also provides a kind of above-mentioned application of model, promptly uses described model, adopts eigenvalue Method or temporal analysis to find the solution; Obtain the dynamic stability of lifting airscrew and body coupled system; Lifting airscrew and body coupling system model and application thereof that the present invention proposes can obtain the dynamic stability of lifting airscrew and body coupled system, and have higher computational accuracy.
Description of drawings
Fig. 1 is lifting airscrew and body coupled system synoptic diagram among the present invention;
The blade equivalence rigid model that Fig. 2 adopts for the present invention;
The rotor blade structural model that Fig. 3 forms for the 15 degree of freedom beam elements that the present invention adopts;
Elasticity body and rotor blade model that Fig. 4 adopts for the present invention;
Fig. 5 is an Eigenvalue Analysis method flow diagram of the present invention;
Fig. 6 is a time-domain analysis method flow diagram of the present invention;
Among the figure:
1, blade; 2, propeller hub; 3, body;
4, spring constraint; 5, rotor center; 6, rigid body blade;
7, beam element; 8, elasticity body model.
Embodiment
Below in conjunction with accompanying drawing and embodiment a kind of lifting airscrew body coupling system model that the present invention proposes is elaborated.
The present invention provides a kind of employing rotor and body coupling system model and method for building up thereof, uses this model and can judge lifting airscrew and body coupled system dynamic stability.
Be illustrated in figure 1 as lifting airscrew and body coupled system synoptic diagram; 1 rotation of engine drives rotor blade, blade 1 produces vibratory response at rotary course, and passes to propeller hub 2; Propeller hub 2 passes to body 3 with oscillating load more then; Body 3 produces vibration deformations, and produces feedback responses can for propeller hub 2, and finally influences waving and motion such as shimmy of blade.The present invention adopts Eigenvalue Analysis method or time-domain analysis method to find the solution this model, thereby can accurately judge the dynamic stability of lifting airscrew and body coupled system through making up rotor and body coupling system model.
Complete lifting airscrew and body coupling system model comprise the part such as structural model and Aerodynamic Model of structural model, the body of rotor, can set up the equation of motion of lifting airscrew and body coupled system through the equation of motion of each model,
Specific as follows:
(1) structural model of rotor.
When lifting airscrew and body coupling system model mainly be used for carrying out lifting airscrew and body parameter (as gyroplane rotate speed with advance than etc.) research and analysis, only considered basic blade flapping and shimmy mode this moment in the structural model of rotor is set up.Blade as shown in Figure 2 equivalence rigid model, blade 1 is connected 5 places, rotor center, with flapping hinge with lead lag hinge be assumed to be spring constraint 4, therefore do not have the rigid body blade 6 that hinge formula or bearingless rotor blade 1 are equivalent to the hinged spring constraint 4 of biasing.This blade equivalence rigid model can be used for simulating radial type, no hinge formula and bearingless rotor.In the structural model of described rotor, considered to wave and the coupling of shimmy structure, adopt the structural model of this rotor can accurately analyze the parameter of rotor and body (as gyroplane rotate speed and advance than etc.) to the influence of rotor and body coupled system.
And when needing to confirm lifting airscrew and the aeroelastic stability of body coupled system when preceding flying and aerial resonance stability, then need adopt the structural model of more accurate rotor.The present invention is thought of as one with rotor blade and has comprised the elastic beam of waving bending, shimmy bending, axis deformation and elastic torsion, and this model can react the motion state of true blade.As shown in Figure 3; Blade 1 on the propeller hub 2 is discrete to be several beam elements 7 with being connected; Displacement of the lines and angular displacement between two adjacent beam elements 7 are continuous, and each beam element 7 has two exterior nodes and three interior nodes, and two exterior nodes have six-freedom degree respectively; Three interior nodes all have only one degree of freedom, so each beam element 7 has 15 degree of freedom.Adopt this rotor model can accurately divide gassing bullet stability and aerial resonance stability.
(2) structural model of body.
For lifting airscrew and body coupled system; Consider in this coupled system is analyzed; The elasticity effect of helicopter structure is smaller, therefore with the helicopter fuselage structure as a rigid model, and considered its pitching and rolling movement; Hovering and before flying under the situation, rotor and body model rotate around center of gravity separately; The center of rotation of body mode instantaneous center of rotation as body then adopted in the ground resonance analysis.
In order to obtain the vibration characteristics of body in lifting airscrew body coupling system model; The present invention also provides a kind of structural model of more accurate body, and is as shown in Figure 4, and body is reduced to elasticity body model 8; Elasticity body model 8 tops link to each other with rotor blade 1; And elasticity body model 8 also adopts beam element 7 structures of 15 degree of freedom above-mentioned, and body 3 quality are evenly distributed on the beam element 7, and body 3 is made up of several beam elements 7; This moment, body 3 was considered translation and the rotational freedom on three directions on three directions, the distortion of analog machine body structure more accurately.
(3) Aerodynamic Model:
The present invention adopts lift line theory to calculate propeller-blade section aerodynamic force.Adopt different Aerodynamic Model to different state, if the aerial resonance problem of helicopter under the assessment low frequency state adopts the dynamic inflow model; And under the disturbance motion situation, with the expansion Pitt/Peters dynamic inflow model that relates to non-permanent aerodynamic effect; Then adopt the permanent flow model of going into for standing state at last.These three kinds of models are prior art, and the more any induced velocity v on the rotor blade once only is described here, and it can represent following form:
v＝v
0+v
s(r+e)sin(ψ
k-ξ
k)+v
c(r+e)cos(ψ
k-ξ
k)（1）
Wherein e is zero dimension equivalent hinge overhanging beam, r for the blade arbitrary section to lead lag hinge distance, v
0Be the average induced velocity on oar dish plane, v
sAnd v
cBe the induced velocity that pneumatic lift-over and pitching by rotor produce, ξ
kBe the shimmy displacement of k sheet blade, ψ
kIt is the position angle of k sheet blade.
(4) rotor blade and body coupling model:
At first set up waving and the shimmy equation of motion that waits of rotor blade, implement many blades coordinate conversion (MCT) then and convert the blade freedom of motion of rotating coordinate system into rotor mass motion degree of freedom, the present invention can adopt implicit expression or explicit MCT method.
Adopt d'Alembert principle to set up the body movement equation then.
With the equation of motion group of the blade equation of motion, body movement equation and Aerodynamic Model equations simultaneousness composition lifting airscrew and body coupled system, promptly obtain the equation of motion of lifting airscrew and body coupled system at last.This moment, rotor and body were under same fixed coordinate system.
Set up the kinetic model of lifting airscrew/body coupled system thus.
Having set up the kinetic model of lifting airscrew/body coupled system, next is exactly the dynamic stability of finding the solution lifting airscrew and body coupled system, and the model that the present invention proposes can adopt the method for Eigenvalue Analysis or time-domain analysis to confirm dynamic stability.
Be illustrated in figure 5 as the Eigenvalue Analysis method flow diagram, need to adopt the microvariations hypothesis of equilibrium position, concrete steps are following:
(1) will obtain at first that the standing state rotor is waved, the equilibrium value of shimmy and body, i.e. the permanent response of lifting airscrew and body coupled system is adopted in concrete solution procedure and is removed the amount relevant with the time earlier, finds the solution balance equation then and gets final product.
(2) obtained after the equilibrium value by a last step; Suppose that lifting airscrew and body coupled system are equipped with microvariations at balance position; Carry out dynamic stability according to the disturbance motion equation then and find the solution, but it is different with the dynamic stability equation solution process that before flies under the situation to hover.
When judging the dynamic stability of floating state, because perturbation equation is the constant coefficient differential equation, can be through finding the solution the mode eigenwert that proper value of matrix obtains system; The imaginary part of eigenwert is a model frequency; Real part then is a modal damping, if real part is negative, the expression modal damping is a negative value; System is stable, otherwise is unsettled.
And lifting airscrew when flying before judging and body coupled system dynamic stability the time; Because perturbation equation is the differential equation that contains periodic coefficient; Can adopt the Floquet theory to find the solution; Transfer matrix method can be adopted in the concrete computation process, the modal damping and the model frequency of system can be obtained equally.Thereby reach the purpose of judging lifting airscrew and body coupled system dynamic stability.
Except the Eigenvalue Analysis method, lifting airscrew and body coupling system model that the present invention proposes can also adopt time-domain analysis method as shown in Figure 6 to find the solution, and key step is following:
The first step according to the differential equation of motion of lifting airscrew of being set up and body coupled system, is carried out integration to this differential equation, thereby can be obtained each degree of freedom response curve in time in time domain;
Second step, the last time-domain response curve that obtains of step is carried out Fourier transform (FFT) obtain frequency response curve, from frequency response curve, can obtain the frequency of each mode easily;
In the 3rd step, adopt methods such as moving rectangular window, envelope logarithmic decrement or Prony curve fitting can obtain the damping of mode at last.Thereby accomplished the process of time-domain analysis, obtained the modal damping of lifting airscrew and body coupled system and the dynamic stability that model frequency is judged system.
Embodiment
In order to verify that the present invention proposes lifting airscrew and body coupling system model and effectiveness of application thereof, adopts this embodiment to verify.The lifting airscrew that adopts and the model parameter of body coupled system are: 3 blades, rotor radius are 0.81m, section chord length 0.0419m, and the section aerofoil profile is NACA 2301, and equivalent hinge overhang is 0.0851m, and blade is 0.0173kg.m to the moment of inertia of hinge
2, blade flapping and shimmy natural frequency are respectively 3.14Hz and 6.70Hz, and shimmy structural damping is than 0.52%, and body pitching moment of inertia, restriction rigidity and damping ratio are respectively 0.607kg.m
2, 11.20kg.m.rad
-1With 3.2%, body rolling moment of inertia, damping ratio and restriction rigidity are respectively 0.177kg.m
2, 6.19kg.m.rad
-1With 0.929%.
The lifting airscrew and the body coupled system kinetic model that adopt the present invention to propose; Rotor adopts blade equivalence rigid model; Body then adopts the rigid model of having considered pitching and lift-over; Aerodynamic Model adopts Pitt/Peters dynamic inflow model, adopts Eigenvalue Analysis method and time-domain analysis method to find the solution respectively.In this example, adopting total elongation is 9 °, and gyroplane rotate speed is 700r/min, and can obtain shimmy back type modal damping value is 0.502s
-1, trial value is 0.512s
-1, error is merely 2%.The real part of the eigenwert that employing Eigenvalue Analysis method is tried to achieve is 0.410s
-1, error is 18.3%.The model and the analytical approach that can draw the present invention's proposition thus are accurately and reliably, and the time-domain analysis method has higher computational accuracy than eigenwert analytical approach.
Claims (5)
1. lifting airscrew body coupling system model; It is characterized in that: described model comprises the structural model of rotor, the structural model and the Aerodynamic Model of body; Set up the equation of motion of lifting airscrew and body coupled system through the equation of motion of each model, find the solution the dynamic stability that the said equation of motion obtains lifting airscrew and body coupled system;
The structural model of described rotor:
When lifting airscrew and body coupling system model are used for carrying out the parameter study and the analysis of lifting airscrew and body, only considered basic blade flapping and shimmy mode this moment in the structural model of rotor is set up, adopt blade equivalence rigid model; And when needing to confirm lifting airscrew and the aeroelastic stability of body coupled system when preceding flying and aerial resonance stability; The structural model of the rotor that then adopts is several beam elements for the blade discrete that will be connected on the propeller hub; Displacement of the lines and angular displacement between two adjacent beam elements are continuous; Each beam element has two exterior nodes and three interior nodes; Two exterior nodes have six-freedom degree respectively, and three interior nodes all have only one degree of freedom, so each beam element has 15 degree of freedom; The structural model of described body:
The helicopter fuselage structure pitching and rolling movement rigid model have been considered as one, have hovered and before flying under the situation that rotor and body model rotate around center of gravity separately; The center of rotation of body mode instantaneous center of rotation as body then adopted in the ground resonance analysis; When needs are confirmed the vibration characteristics of body at lifting airscrew and body coupled system; Then body adopts elasticity body model; Elasticity body model also adopts the beam element structure of 15 degree of freedom; The body quality is evenly distributed on the beam element, and body is made up of several beam elements, and this moment, body was considered translation and the rotational freedom on three directions on three directions; Described Aerodynamic Model:
Utilize lift line theory to calculate propeller-blade section aerodynamic force, adopt different Aerodynamic Model, if the aerial resonance problem of helicopter under the assessment low frequency state adopts the dynamic inflow model to different state; And under the disturbance motion situation, with the expansion Pitt/Peters dynamic inflow model that relates to non-permanent aerodynamic effect; Then adopt the permanent flow model of going into for standing state at last.
2. a kind of lifting airscrew body coupling system model according to claim 1; It is characterized in that: described blade equivalence rigid model is: blade is connected the rotor center; With flapping hinge with lead lag hinge be assumed to be spring constraint, no hinge formula or bearingless rotor blade are equivalent to the rigid body blade of the hinged spring constraint of setovering.
3. the method for building up of the described lifting airscrew body of claim 1 coupling system model is characterized in that:
At first set up waving and the lagging motion equation of rotor blade, implement many blades coordinate conversion then and convert the blade freedom of motion of rotating coordinate system into rotor mass motion degree of freedom;
Adopt d'Alembert principle to set up the body movement equation then;
The equation of motion group of at last the rotor blade equation of motion, body movement equation and Aerodynamic Model equations simultaneousness being formed lifting airscrew and body coupled system; Promptly obtain the equation of motion of lifting airscrew and body coupled system; Rotor and body have been set up the kinetic model of lifting airscrew body coupled system thus under same fixed coordinate system at this moment.
4. the application of the described lifting airscrew body of claim 1 coupling system model; It is characterized in that: adopt the Eigenvalue Analysis method to find the solution the kinetic model of lifting airscrew body coupled system; Obtain the dynamic stability of lifting airscrew and body coupled system, concrete steps are following:
(1) will obtain at first that the standing state rotor is waved, the equilibrium value of shimmy and body, i.e. the permanent response of lifting airscrew and body coupled system is adopted in concrete solution procedure and is removed the amount relevant with the time earlier, finds the solution balance equation then;
(2) adopt the microvariations of equilibrium position to suppose; Suppose that lifting airscrew and body coupled system are equipped with microvariations at balance position; Carries out dynamic stability according to the disturbance motion equation then and find the solution, but it is different with the dynamic stability equation solution process that before flies under the situation to hover:
When judging the dynamic stability of floating state, because perturbation equation is the constant coefficient differential equation, through finding the solution the mode eigenwert that proper value of matrix obtains system; The imaginary part of eigenwert is a model frequency; Real part then is a modal damping, if real part is negative, the expression modal damping is a negative value; System is stable, otherwise is unsettled;
Lifting airscrew when flying before the judgement and body coupled system dynamic stability the time because perturbation equation is the differential equation that contains periodic coefficient, adopt the Floquet theory to find the solution, obtain the modal damping and the model frequency of system.
5. the application of the described lifting airscrew body of claim 1 coupling system model; It is characterized in that: adopt the time-domain analysis method to find the solution the kinetic model of lifting airscrew body coupled system; Obtain the dynamic stability of lifting airscrew and body coupled system, concrete steps are following:
The first step according to the differential equation of motion of lifting airscrew of being set up and body coupled system, is carried out integration to this differential equation in time domain, thereby obtains each degree of freedom response curve in time;
Second step, the last time-domain response curve that obtains of step is carried out Fourier transform obtain frequency response curve, from frequency response curve, obtain the frequency of each mode;
The 3rd step; Adopt at last and move the damping that rectangular window, envelope logarithmic decrement or Prony curve-fitting method obtain mode; Thereby accomplished the process of time-domain analysis, obtained the modal damping of lifting airscrew and body coupled system and the dynamic stability that model frequency is judged system.
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| CN104787357A (en) * | 2015-04-27 | 2015-07-22 | 中国直升机设计研究所 | Design method for preventing ground resonance of bearingless rotary-wing helicopter |
| CN104890867A (en) * | 2015-04-27 | 2015-09-09 | 中国直升机设计研究所 | Design method capable of expanding air resonance security boundary of helicopter |
| CN106156382A (en) * | 2015-04-03 | 2016-11-23 | 哈尔滨飞机工业集团有限责任公司 | A kind of helicopter ship resonance method for analyzing and modeling |
| CN108839817A (en) * | 2018-06-26 | 2018-11-20 | 中国直升机设计研究所 | A kind of bearingless rotor ground resonance test method |
| CN109522637A (en) * | 2018-11-09 | 2019-03-26 | 中国直升机设计研究所 | Helicopter slides or slides the analysis method of ground resonance under race state |
| CN112149232A (en) * | 2020-09-29 | 2020-12-29 | 南京航空航天大学 | Heavy helicopter flight dynamics rigid-elastic coupling modeling method |
| CN112597582A (en) * | 2020-12-11 | 2021-04-02 | 中国直升机设计研究所 | Longitudinal helicopter rotor and fuselage coupling stability modeling method |
| CN112632695A (en) * | 2020-12-11 | 2021-04-09 | 中国直升机设计研究所 | Ground resonance modeling and analyzing method for transverse double-rotor helicopter |
| CN114084375A (en) * | 2021-11-19 | 2022-02-25 | 中国直升机设计研究所 | Method for calculating coupling natural frequency of rotor system installation test bed |
| CN116127613A (en) * | 2023-04-14 | 2023-05-16 | 北京航空航天大学 | Method for analyzing coupling dynamic stability of rotor wing organism with viscoelastic shimmy damper |
| CN116757124A (en) * | 2023-08-17 | 2023-09-15 | 北京航空航天大学 | Method and system for analyzing structural dynamic stability of bearingless helicopter |
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| CN106156382A (en) * | 2015-04-03 | 2016-11-23 | 哈尔滨飞机工业集团有限责任公司 | A kind of helicopter ship resonance method for analyzing and modeling |
| CN104890867A (en) * | 2015-04-27 | 2015-09-09 | 中国直升机设计研究所 | Design method capable of expanding air resonance security boundary of helicopter |
| CN104787357B (en) * | 2015-04-27 | 2018-05-18 | 中国直升机设计研究所 | One kind prevents bearingless rotor helicopter ground resonance design method |
| CN104787357A (en) * | 2015-04-27 | 2015-07-22 | 中国直升机设计研究所 | Design method for preventing ground resonance of bearingless rotary-wing helicopter |
| CN108839817B (en) * | 2018-06-26 | 2021-08-13 | 中国直升机设计研究所 | Bearing-free rotor ground resonance test method |
| CN108839817A (en) * | 2018-06-26 | 2018-11-20 | 中国直升机设计研究所 | A kind of bearingless rotor ground resonance test method |
| CN109522637B (en) * | 2018-11-09 | 2022-12-30 | 中国直升机设计研究所 | Analysis method for ground resonance of helicopter in sliding or running state |
| CN109522637A (en) * | 2018-11-09 | 2019-03-26 | 中国直升机设计研究所 | Helicopter slides or slides the analysis method of ground resonance under race state |
| CN112149232A (en) * | 2020-09-29 | 2020-12-29 | 南京航空航天大学 | Heavy helicopter flight dynamics rigid-elastic coupling modeling method |
| CN112597582A (en) * | 2020-12-11 | 2021-04-02 | 中国直升机设计研究所 | Longitudinal helicopter rotor and fuselage coupling stability modeling method |
| CN112632695A (en) * | 2020-12-11 | 2021-04-09 | 中国直升机设计研究所 | Ground resonance modeling and analyzing method for transverse double-rotor helicopter |
| CN112632695B (en) * | 2020-12-11 | 2022-12-06 | 中国直升机设计研究所 | Ground resonance modeling and analyzing method for transverse double-rotor helicopter |
| CN114084375A (en) * | 2021-11-19 | 2022-02-25 | 中国直升机设计研究所 | Method for calculating coupling natural frequency of rotor system installation test bed |
| CN114084375B (en) * | 2021-11-19 | 2023-04-28 | 中国直升机设计研究所 | Coupling natural frequency calculation method for rotor wing system mounting test bed |
| CN116127613A (en) * | 2023-04-14 | 2023-05-16 | 北京航空航天大学 | Method for analyzing coupling dynamic stability of rotor wing organism with viscoelastic shimmy damper |
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