CN110837678A - Binary airfoil frequency domain flutter model modeling method based on multi-body system transfer matrix method - Google Patents
Binary airfoil frequency domain flutter model modeling method based on multi-body system transfer matrix method Download PDFInfo
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Abstract
The invention provides a binary airfoil frequency domain flutter model modeling method based on a multi-body system transfer matrix method, which is characterized in that firstly, a rudder system is reasonably simplified, so that the simplified model is convenient for structural dynamics analysis; secondly, establishing a multi-body dynamic model by using a multi-body system transfer matrix method according to the simplified model, determining transfer matrices of all parts of the simplified model, and assembling an assembly transfer matrix; then, the rigid center in the transfer matrix is coincided with the mass center, and a characteristic equation of the total transfer matrix is solved to obtain the pure bending and pure torsion frequencies of the system; finally, establishing a binary flutter frequency domain model of the rudder system, and performing frequency domain analysis on the binary flutter frequency domain model by using a U-g method to obtain the flutter speed and frequency of the rudder system; the method adopts MSTMM to calculate the key parameters required by the binary flutter model, establishes the binary flutter frequency domain model of the rudder system, facilitates the research of the influence rule of the structural parameters of the rudder system of the underwater vehicle on the water elasticity of the rudder system, and provides reference for the design of vibration and noise reduction of the rudder system in engineering.
Description
Technical Field
The invention relates to a model modeling method, in particular to a binary airfoil frequency domain flutter model modeling method based on a multi-body system transfer matrix method, and belongs to the field of vibration and control.
Background
In the field of ocean engineering, ship components such as hydrofoils of hydrofoil boats, elevators and tail rudders in submarines are used as complex multi-rigid-flexible dynamic systems, and if structural parameters are not properly designed, the flutter phenomenon can be generated under the action of fluid force. The flutter may cause structural damage, or a sustained weak vibration phenomenon occurs to induce water noise, reducing the concealment of the underwater vehicle. In order to explore whether the rudder system structure is safe enough, the influence of the structure parameters on the water elasticity of the rudder system must be studied. A large number of scholars and experts use binary models to study the aeroelasticity and water elasticity problems of wings and hydrofoils, thereby greatly simplifying the analysis process and being easy to realize engineering. For a rudder system of an underwater vehicle, if the structure design is not reasonable, linear classical flutter, hydrostatic elastic dispersion to cause structural damage or water noise can be caused and the concealment of the underwater vehicle is reduced. Therefore, the research on the influence rule of the rudder system structural parameters on the linear flutter has certain significance.
The two-dimensional model is generally used for the principle research of the water elasticity of the marine structure system. The prior researches on the rudder system are mostly based on a binary flutter model, but hardly mention how to establish the rudder system as the binary flutter model and how to obtain important parameters of the binary flutter model. In addition, the influence of structural gap nonlinearity on the rudder system of the underwater vehicle is rarely involved in the previous research. The multi-body system transfer matrix method (MSTMM) was established by Rankine and team thereof for multi-body dynamics analysis. The method can realize the intrinsic vibration characteristic and dynamic response of the linear multi-body system and the dynamic research of the nonlinear, time-varying, large-motion, controlled and general multi-body system. The method does not need a system overall dynamics equation, has high programming degree and low system matrix order, can realize the rapid modeling and rapid calculation of a complex system, and has high calculation efficiency and convenient application to engineering. The modeling idea of the binary flutter model is provided based on the MSTMM, and key dynamic parameters required by the binary flutter model, namely pure bending and pure torsion frequencies of a rudder system, are calculated by adopting the MSTMM. The method has the advantages that the binary linear flutter model of the rudder system is established, the influence rule of the structural parameters of the rudder system on the water elasticity of the rudder system is conveniently researched, and reference is provided for the design of the rudder system in engineering.
Disclosure of Invention
The invention aims to provide a binary airfoil frequency domain flutter model modeling method based on a multi-body system transfer matrix method, which is convenient for researching the influence rule of the structural parameters of a rudder system of an underwater vehicle on the hydroelasticity of the rudder system.
The purpose of the invention is realized as follows: a binary airfoil frequency domain flutter model modeling method based on a multi-body system transfer matrix method comprises the following steps:
the method comprises the following steps: from the angle of a rudder blade of the rudder system, simplifying the rudder system;
step two: deducing a transfer matrix of each bending-torsion coupling beam based on MSTMM, assembling a system total transfer equation, and establishing a multi-body dynamic model with a simplified structure;
step three: obtaining the pure bending and pure torsion frequencies of the system, and using the pure bending and pure torsion frequencies in a next step flutter model;
step four: and establishing a binary linear flutter model of the rudder system, and solving the model based on a U-g method to obtain frequency domain response.
As a further limitation of the invention, the steps are simplified for the rudder system, the hydrodynamic forces acting on the two rudder blades of the rudder system are completely identical; the steering system provides torque to the rudder blade so that the rudder blade can rotate; therefore, the rudder system can be simplified into a torsion spring connected with a rudder blade to capture the dynamic behavior of the whole rudder system; the rudder blade is divided into a plurality of sections of bending coupling beams with different parameters, so that a simplified structure of the rudder system is formed, and further analysis is facilitated.
As a further limitation of the present invention, the second step specifically comprises the following steps:
firstly, establishing a vibration differential equation of the bending torsion coupling beam:
wherein m is the mass per unit length, IαThe unit length moment of inertia, EI bending rigidity, GJ torsion rigidity, x rudder blade axial displacement, y rudder blade bending direction displacement, thetaxIs the torsion angle, t is time, b is half of the chord length of the rudder blade, xαDistance from center of mass to elastic axis, and when center of mass is in positive Z direction, xαIs positive;
the transfer matrix U of the cranked coupling beam can be determined according to the formula (1)i；
Assembling the multi-section rudder blade crankle coupling beam transfer matrix, and establishing a total transfer equation of a rudder system:
Zn,n+1＝UallZ0,1(3)
in the formula of Uall＝Un···U3U2U1，U1,U2,U3···UnRespectively represents the transmission matrix of each unit of the rudder system, and can fully consider the specific structural details, Z, of the rudder system0,1Representing the state vector of the input of the rudder system in modal coordinates, Zn,n+1Representing a state vector of an output end of the rudder system under the modal coordinate;the state vector is written as [ X, Y, Θz,Mz,Qx,Qy,Θx,Mx]TX is a modal coordinate array corresponding to displacement along coordinate axis X, and Y is a modal coordinate array corresponding to displacement along coordinate axis Y, thetazA modal coordinate array corresponding to the angular displacement at the point relative to the equilibrium position in the z-axis, MzIs a modal coordinate array, Q, corresponding to the moment in z along a coordinate axisxIs a modal coordinate array, Q, corresponding to the internal force along the coordinate axis xyIs a modal coordinate array corresponding to the internal force along a coordinate axis y, thetaxA modal coordinate array corresponding to the angular displacement of the point relative to the equilibrium position with respect to the x-axis, MxIs a modal coordinate array corresponding to the moment in the coordinate axis x.
As a further limitation of the invention, the third step is to obtain the pure bending and pure torsion frequencies of the rudder system, and the distances from the mass center to the elastic axis in the transfer equation of the bending-torsion coupling beam are all changed to 0 (x)αApproximately equals to 0), namely, the coupling term in the bending and twisting coupling beam is removed; determining boundary conditions, solving the characteristic equation of the modified total transfer matrix, and calculating the non-coupling frequency of the simplified model of the rudder system.
As a further limitation of the invention, the step four of establishing the two-degree-of-freedom rudder blade hydro-elastic control equation is as follows:
wherein m is the mass of the rudder blade with unit extension and xαIs the distance from the center of mass to the elastic axis; b is half of the chord length of the rudder blade;
is the moment of inertia per unit span length relative to the elastic axis; r isαThe turning radius of the hydrofoil to the rigid center, L is the lift force, and the direction of orientation is the positive direction; hydrodynamic torque T of the elastic shaftαThen head-up is positive, α represents pitch motion, h represents heave motion, k representshRepresenting hydraulic spring rate, kαRepresents the torsional stiffness; dissipative structural damping force
ωh、ωαG is the non-coupling pure bending, pure torsion frequency and artificial damping of the rudder system respectively; the pure bending and pure torsion frequencies obtained in the last step are substituted into an equation to obtain kh、kαAnd (4) solving the flutter model to perform frequency domain analysis to obtain the flutter speed of the binary hydrofoil.
Compared with the prior art, the invention has the following remarkable advantages:
(1) the invention reasonably simplifies the rudder system, improves the calculation efficiency, captures the real dynamic characteristics of the rudder system and is suitable for engineering calculation. The dynamic model of the bending-torsion coupled beam established based on the MSTMM expands the application of a multi-body system transfer matrix library and a multi-body system transfer matrix method in fluid-solid coupling dynamics.
(2) The invention provides a method for simplifying a rudder system into a binary flutter model in detail, and few documents in a large amount of research work of predecessors mention a method for simplifying the rudder system, three-dimensional hydrofoils such as wings and the like or the wing system into the binary flutter model. A method for obtaining pure bending and pure torsion frequencies of the rudder system based on MSTMM calculation is provided, and basic key dynamic parameters are provided for building a binary linear flutter model of the rudder system.
(3) The method has the advantages of quick and simple modeling, no need of establishing a system overall dynamics equation, low matrix order, high calculation speed, convenience in considering the influence of each component of the system on the vibration characteristic of the rudder system and the like, can quickly calculate the result, researches the influence of the structural parameters on the vibration characteristic of the rudder system, and provides reference for quick modeling and simulation of a similar binary airfoil flutter model in engineering.
The invention can be applied to the vibration and noise reduction design of the underwater rudder system considering the water elasticity.
Drawings
FIG. 1 is a flow chart of binary airfoil frequency domain flutter model modeling based on a multi-body system transfer matrix method.
Figure 2 is a simplified model schematic of a rudder system.
FIG. 3 is a schematic diagram of a binary hydrofoil linear flutter model of the rudder system.
Figure 4 is the result of the rudder system binary hydrofoil linear flutter calculation.
Detailed Description
The following detailed description of embodiments of the invention refers to the accompanying drawings.
Example 1
Establishing a binary linear flutter model of the rudder system, and carrying out frequency domain analysis on the model; taking an airfoil at a rudder blade 3/4, and obtaining the calculation parameters through three-dimensional geometric analysis software as follows: distance x from center of mass to elastic axisα0.2889, the radius of gyration r of hydrofoil to rigid centerα 2The chord length of the rudder blade is half b of 0.405, the mass ratio mu is 0.403, and the ratio a of the distance between the center of the rudder blade and the midpoint of the chord line to the half chord length is-0.48; the pure bending and pure torsion frequencies of the rudder system are obtained through the simulation of a multi-body system transfer matrix method as follows: uncoupled pure bending frequency omega of rudder systemk21.094Hz, uncoupled pure torsional frequency omega of the rudder systemα18.931Hz, ratio of pure bending and pure twisting frequency
The specific implementation steps are as follows.
The method comprises the following steps: from the angle of the stress of the rudder blade, the hydrodynamic forces acting on the two rudder blades of the rudder system are completely consistent; the steering system provides torque to the rudder blade, so that the rudder blade can rotate, and the rudder system can be simplified into a torsion spring connected with one rudder blade to capture the dynamic behavior of the whole rudder system; then the rudder blade is divided into 7 sections of the cranked coupling beams with different parameters, a simplified structure from a complete machine system of the three-dimensional rudder system to the rudder system is formed and is shown in figure 2, and further analysis is facilitated.
Step two: establishing a dynamic equation for each divided section of the crankle coupling beam, and deducing a transfer matrix U of each crankle coupling beam unit by using MSTMMiAnd assembling an assembly transmission matrix and establishing a multi-body dynamic model of the system.
And transfer equation Z0＝UiZI，ZiRepresenting the state vector of the input of the cell in modal coordinates, Z0Representing the state vector at the output of the unit in modal coordinates.
Deducing a transmission matrix of the rudder blade bending and twisting coupling beam, and establishing a vibration differential equation of the bending and twisting coupling beam:
wherein m is the mass per unit length, IαThe unit length moment of inertia, EI bending rigidity, GJ torsion rigidity, x rudder blade axial displacement, y rudder blade bending direction displacement, thetaxIs the torsion angle, t is time, b is half of the chord length of the rudder blade, xαDistance from center of mass to elastic axis, and when center of mass is in positive Z direction, xαIs positive.
Order to
y(x,t)＝Y(x)sinωt,θx(x,t)＝Θx(x)sinωt(2)
In the formula, ω is a circle frequency.
Bringing (2) into (1), (1) becomes:
elimination of Y (x) or theta in the formula (3)x(x) To obtain
In the formula,
w ═ Y or Θ (5)
Introducing dimensionless lengths
ξ＝x/L (6)
And L in the formula (6) is the length of the divided unit crankle coupling beam.
Formula (4) can be rewritten into a dimensionless form
(D6+aD4-bD2-abc)W＝0 (7)
Wherein,
the general solution of the sixth order differential equation (7) can be expressed as
W(ξ)＝C1coshαξ+C2sinhαξ+C3cosβξ+C4sinβξ+C5cosγξ+C6sinγξ (9)
In the formula C1-C6Is constant, and
q＝b+a2/3
w (ξ) in the formula (9) represents the bending displacement Y and the torsion angle thetaxSolutions at different constants. Therefore, the temperature of the molten metal is controlled,
Y(ξ)＝A1coshαξ+A2sinhαξ+A3cosβξ+A4sinβξ+A5cosγξ+A6sinγξ (11)
Θx(ξ)＝B1coshαξ+B2sinhαξ+B3cosβξ+B4sinβξ+B5cosγξ+B6sinγξ (12)
in the formula A1-A6And B1-B6Are two different sets of constants.
Substituting equations (11) and (12) for equation (3) can determine the constants as follows:
in the formula,
the bending angle theta can be obtained from the formulas (11) and (12)z(ξ), bending moment Mz(ξ), shear force Qy(ξ) and Torque MxExpression of (ξ):
thus, it is possible to obtain
That is, Z (ξ) ═ V (ξ) · a, a ═ A1,A2,A3,A4,A5,A6]TWhere Z (ξ) is the state vector, i.e., the left term of the equation (19) equation, B (ξ) represents the first matrix to the right of the equation (19). Where ξ is 0, equation (19) can be written as Z according to equation (19) and the multi-system transfer matrix methodIForm V (0) a. letting ξ be 1 gives ZO＝V(1)·a＝V(1)V(0)-1ZI＝UiZI。
So that the transfer matrix of the cranked coupling beam is
Ui＝V(1)·V-1(0) (20)
Wherein,
when the rigidity of the hydraulic spring is Kh＝4×108N/m, the equivalent torsional spring stiffness of the corresponding control system is Kα_actuator＝1.23×107N.m/rad. Based on MSTMM, the state vector is written as [ X, Y, Θz,Mz,Qx,Qy,Θx,Mx]TX is a modal coordinate array corresponding to displacement along coordinate axis X, and Y is a modal coordinate array corresponding to displacement along coordinate axis Y, thetazA modal coordinate array corresponding to the angular displacement at the point relative to the equilibrium position in the z-axis, MzIs a modal coordinate array, Q, corresponding to the moment in z along a coordinate axisxIs a modal coordinate array, Q, corresponding to the internal force along the coordinate axis xyIs a modal coordinate array corresponding to the internal force along a coordinate axis y, thetaxA modal coordinate array corresponding to the angular displacement of the point relative to the equilibrium position with respect to the x-axis, MxIs a modal coordinate array corresponding to the moment in the coordinate axis x.
In MSTMM, the overall transfer equation is:
step three: in order to solve the pure bending and pure torsion frequencies of the rudder system, the distances from the mass center to the elastic axis in the transmission equation of the bending-torsion coupling beam are all changed to be 0 (x)α0), i.e. the coupling terms in the flextensionally coupled beam are removed. Determining transfer matrix boundary conditions as follows: z1,0＝[0,0,0,Mz1,Qx1,Qy1,0,Mx1]T，Z9,0＝[X9,Y9,Θz9,0,0,0,Θx9,0]T. Solving for modified (x)αApproximately equals 0), the pure bending frequency omega of the model after the rudder system is simplified can be calculatedhPure torsional frequency omegaαFor use in the next step of modeling.
Step four: as shown in fig. 3, a binary linear flutter model of the rudder system is established, and first, a section at the extension position of 3/4 rudder blades is taken, and a two-degree-of-freedom rudder blade hydro-elastic control equation is established as follows:
in the formula, m is the mass of the unit extended rudder blade; b is half of the chord length of the rudder blade;
is the moment of inertia per unit span length relative to the elastic axis; r isαIs the turning radius of the hydrofoil to the rigid core; l is a lifting force, and the direction of orientation is a positive direction; hydrodynamic torque T of the elastic shaftαThen head-up is positive, α represents pitch motion, h represents heave motion, k representshRepresenting hydraulic spring rate, kαRepresents the torsional stiffness; according to the U-g method, a dissipative structural damping force is introduced
ωh、ωαAnd g are the non-coupling pure bending, pure torsion frequency and artificial damping of the rudder system, which are calculated in the previous step.
According to Theodersen's unsteady fluid theory in the frequency domain, a binary hydrofoil in an incompressible flow has a lift L (positive upward) and a pitching moment T to the center of gravity of the hydrofoil unit when it makes simple harmonic motion and pitching motion at an angular frequency omegaα(head-up in windward is positive) are respectively:
where U is the incoming flow velocity, ρ is the density of the incompressible fluid, a is the ratio of the distance of the center of the steel to the midpoint of the chord line to the half chord length and is positive after the midpoint, and C (k) is the Theodorsen function whose value is related to the reduced frequency
In this case, U is the incoming flow velocity, ω is the angular frequency, h is the heave displacement, and α is the pitch displacement.
For the convenience of derivation, the above formula is rearranged into a more concise form:
in the formula,
LAh、LAU、LAαthe additional mass of the water flow is caused by a lifting force component which is irrelevant to the ring volume distribution of the rudder blade; l isCh、LCU、LCαIs a lift component related to the distribution of the cyclic quantity and is mainly shown to be related to the slope of a lift line.
According to the U-g method, when the incoming flow velocity is equal to the flutter velocity, the hydrofoil makes simple harmonic motion, so that
First, assuming that the structural damping of the system is zero, artificial structural damping is introduced. Equation (22) can be written by equation derivation as follows:
in the formula,
ρwaterand μ are the density and mass ratio of the fluid, respectively. OmegahAnd ωαRespectively the uncoupled heave and pitch natural frequencies,is the ratio of the pure bending frequency and the pure twisting frequency of the rudder system.
Because A is1、A4All are composed of
The term, equation (27) can be written as solving a generalized eigenvalue problem, the eigenvalue being
Therefore, there are:
the model is solved and the obtained binary hydrofoil linear flutter result is shown in fig. 4. The calculation result shows that the rudder system can reach the critical state of flutter only by adding negative damping under the working condition. It is stated that the damping of the system itself is sufficient to prevent its flutter, which is not linear classical flutter.
Similarly, the method establishes the binary airfoil linear flutter models of different blades and only needs to modify specific parameters in the simplified models and the system.
The present invention is not limited to the above-mentioned embodiments, and based on the technical solutions disclosed in the present invention, those skilled in the art can make some substitutions and modifications to some technical features without creative efforts according to the disclosed technical contents, and these substitutions and modifications are all within the protection scope of the present invention.
Claims (5)
1. A binary airfoil frequency domain flutter model modeling method based on a multi-body system transfer matrix method is characterized by comprising the following steps:
the method comprises the following steps: from the angle of a rudder blade of the rudder system, simplifying the rudder system;
step two: deducing a transfer matrix of each bending-torsion coupling beam based on MSTMM, assembling a system total transfer equation, and establishing a multi-body dynamic model with a simplified structure;
step three: obtaining the pure bending and pure torsion frequencies of the system, and using the pure bending and pure torsion frequencies in a next step flutter model;
step four: and establishing a binary linear flutter model of the rudder system, and solving the model based on a U-g method to obtain frequency domain response.
2. The binary airfoil frequency domain flutter model modeling method based on the multi-body system transfer matrix method according to claim 1, characterized in that the steps of simplifying a rudder system are as follows: the rudder system is simplified into a torsion spring which is connected with a rudder blade to capture the dynamic behavior of the whole rudder system, and the rudder blade is divided into a plurality of sections of bending coupling beams with different parameters to form a simplified structure of the rudder system.
3. The binary airfoil frequency domain flutter model modeling method based on the multi-body system transfer matrix method according to claim 2, wherein the second step specifically comprises the following steps:
firstly, establishing a vibration differential equation of the bending torsion coupling beam:
wherein m is the mass per unit length, IαThe unit length moment of inertia, EI bending rigidity, GJ torsion rigidity, x rudder blade axial displacement, y rudder blade bending direction displacement, thetaxIs the torsion angle, t is time, b is half of the chord length of the rudder blade, xαDistance from center of mass to elastic axis, and when center of mass is in positive Z direction, xαIs positive;
the transfer matrix U of the cranked coupling beam can be determined according to the formula (1)i；
Assembling the multi-section rudder blade crankle coupling beam transfer matrix, and establishing a total transfer equation of a rudder system:
Zn,n+1＝UallZ0,1(3)
in the formula of Uall＝Un…U3U2U1，U1,U2,U3…UnRespectively represents the transmission matrix of each unit of the rudder system, and can fully consider the specific structural details, Z, of the rudder system0,1Representing the state vector of the input of the rudder system in modal coordinates, Zn,n+1Representing a state vector of an output end of the rudder system under the modal coordinate; the state vector is written as [ X, Y, Θz,Mz,Qx,Qy,Θx,Mx]TX is a modal coordinate array corresponding to displacement along coordinate axis X, and Y is a modal coordinate array corresponding to displacement along coordinate axis Y, thetazA modal coordinate array corresponding to the angular displacement at the point relative to the equilibrium position in the z-axis, MzIs a modal coordinate array, Q, corresponding to the moment in z along a coordinate axisxIs a modal coordinate array, Q, corresponding to the internal force along the coordinate axis xyIs a modal coordinate array corresponding to the internal force along a coordinate axis y, thetaxA modal coordinate array corresponding to the angular displacement of the point relative to the equilibrium position with respect to the x-axis, MxIs a modal coordinate array corresponding to the moment in the coordinate axis x.
4. The modeling method of binary airfoil frequency domain flutter model based on multi-body system transfer matrix method according to claim 3, characterized in thatIn the third step, the pure bending and pure torsion frequencies of the rudder system are solved, and the distances from the mass center to the elastic axis in the transmission equation of the bending-torsion coupling beam are all changed to be 0 (x)αApproximately equals to 0), namely, the coupling term in the bending and twisting coupling beam is removed; determining boundary conditions, solving the characteristic equation of the modified total transfer matrix, and calculating the non-coupling frequency of the simplified model of the rudder system.
5. The modeling method of the binary airfoil frequency domain flutter model based on the multi-body system transfer matrix method according to claim 4, wherein the step four of establishing the two-degree-of-freedom rudder blade hydro-elastic control equation is as follows:
wherein m is the mass of the rudder blade with unit extension and xαIs the distance from the center of mass to the elastic axis; b is half of the chord length of the rudder blade;
is the moment of inertia per unit span length relative to the elastic axis; r isαThe turning radius of the hydrofoil to the rigid center, L is the lift force, and the direction of orientation is the positive direction; hydrodynamic torque T of the elastic shaftαThen head-up is positive, α represents pitch motion, h represents heave motion, k representshRepresenting hydraulic spring rate, kαRepresents the torsional stiffness; dissipative structural damping force
ωh、ωαG is the non-coupling pure bending, pure torsion frequency and artificial damping of the rudder system respectively; the pure bending and pure torsion frequencies obtained in the last step are substituted into an equation to obtain kh、kαAnd (4) solving the flutter model to perform frequency domain analysis to obtain the flutter speed of the binary hydrofoil.
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| CN113434961A (en) * | 2021-06-29 | 2021-09-24 | 北京理工大学 | One-dimensional composite material airfoil fluid-solid coupling characteristic prediction method based on beam theory |
| CN114117842A (en) * | 2021-11-04 | 2022-03-01 | 中国运载火箭技术研究院 | Nonlinear thermal modal analysis method for control rudder structure |
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