CN110008543A - A kind of emulation mode for considering neutral axis of the beam and rotating beam dynamic response being influenced - Google Patents
A kind of emulation mode for considering neutral axis of the beam and rotating beam dynamic response being influenced Download PDFInfo
- Publication number
- CN110008543A CN110008543A CN201910217117.8A CN201910217117A CN110008543A CN 110008543 A CN110008543 A CN 110008543A CN 201910217117 A CN201910217117 A CN 201910217117A CN 110008543 A CN110008543 A CN 110008543A
- Authority
- CN
- China
- Prior art keywords
- flexible beam
- deformation
- flexible
- neutral axis
- equation
- Prior art date
- Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
- Granted
Links
Classifications
-
- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F30/00—Computer-aided design [CAD]
- G06F30/20—Design optimisation, verification or simulation
Landscapes
- Engineering & Computer Science (AREA)
- Physics & Mathematics (AREA)
- Theoretical Computer Science (AREA)
- Computer Hardware Design (AREA)
- Evolutionary Computation (AREA)
- Geometry (AREA)
- General Engineering & Computer Science (AREA)
- General Physics & Mathematics (AREA)
- Testing Of Devices, Machine Parts, Or Other Structures Thereof (AREA)
- Management, Administration, Business Operations System, And Electronic Commerce (AREA)
Abstract
The present invention relates to a kind of emulation modes that consideration neutral axis of the beam influences rotating beam dynamic response, by assuming that special circumstances of flexible beam under the conditions of pure bending are derived the position equation of cross-section center axis.Lagrangian (Lagrange) equation of second class of model use, in conjunction with hypothesis modal method, it is derived the Rigid-flexible Coupling Dynamics equation comprising Feng Keli (von K á rm á n) geometrical non-linearity, has calculated rotation center rigid body (Hub) -- the endpoint of flexible FGM beam responds and bending strain.The present invention can provide certain design basis for delicate fields such as aerospace, automobiles.
Description
Technical field
The invention belongs to dynamics of multibody systems to model field, and it is loud to rotating beam dynamics to be related to a kind of consideration neutral axis of the beam
The emulation mode that should be influenced.
Background technique
MATLAB be MathWorks company, the U.S. produce business mathematics software, for algorithm development, data visualization,
The advanced techniques computational language and interactive environment that data analysis and numerical value calculate mainly include MATLAB and Simulink
Two large divisions.Dynamic Modeling and emulation is carried out to multi-body system based on MATLAB widely to be used by scholars.
With the fast development of science and technology, the attribute of traditional homogenous material is no longer satisfied some special environment, such as
High temperature and pressure etc., the functionally graded material (FGMs) that thus scholars one after another propose Japan material scientist have carried out extensively
Research, due to its material high temperature resistant, lightweight, high intensity the features such as, be widely used in aerospace, micro nano structure,
Automotive field etc..Due to the single uniform continuity material of tradition, position of neutral axis at its geometry mass center, however FGMs due to
Its material is no longer symmetrical, therefore its neutral axis is changed, and the change of neutral axis no doubt can generate one to the property of its structure
Fixed influence, therefore study its mechanical property and also become a kind of necessity that today's society pursues precision.
So the present invention is based on the researchs of forefathers, it is contemplated that the change of neutral axis, based on MATLAB to Hub-- flexibility FGM
Girder system system carried out dynamics calculation and output schematic diagram from the system beam to the bending strain in the direction z so as to engineering science and technology people
Member's research and application.
Summary of the invention
The purpose of the present invention is to provide a kind of emulation mode for being influenced on rotating beam dynamic response of consideration neutral axis of the beam,
Using dynamics of multibody systems and assume modal method as theoretical basis, it is therefore intended that a wide range of rotation Hub-- flexibility FGM beam of analysis
Rigid-flexible Coupling Dynamics response.
The technical solution for realizing the aim of the invention is as follows: a kind of consideration neutral axis of the beam influences rotating beam dynamic response
Emulation mode, comprising the following steps:
Step 1 determines Rigid Base, sets material gradient direction, geometric parameter and the material parameter of flexible beam, establishes tool
The Hub- Rigid chain polymer of functional gradient, is transferred to step 2；
Step 2, the Non-linear coupling amount for considering flexible beam deformation are obtained in conjunction with von K á rm á n strain and Hooke's law
The strain energy and kinetic energy of Hub- Rigid chain polymer with functionally gradient, and determine the position for leading to beam section neutral axis by material
It sets, is transferred to step 3；
Step 3 is established the rigid-soft of the Hub- flexibility FGM girder system system with functionally gradient by Second Kind Lagrange Equation
The coupled dynamical equation is transferred to step 4 with assuming that it is discrete that modal method carries out coupling dynamics kinetics equation；
Step 4, acquire with functionally gradient Hub- flexibility FGM girder system system in each moment space configuration coordinate.
Compared with existing multi-body system solving system, remarkable advantage is the present invention:
(1) variation of flexible beam position of neutral axis caused by functionally graded material is considered, so that Hub- flexibility FGM beam
Model more closer to reality.
(2) three-dimensional modeling has been carried out to Hub-- flexibility FGM beam model, while has considered tangential bending and flapwise bending pair
It is axially stretched to influence.
(3) due to considering Non-linear coupling amount and high order coupling terms, allow the system under high speed rotation operating condition
Correctly solved.
Detailed description of the invention
Fig. 1 is three-dimensional rotation Hub-- flexibility FGM beam model schematic diagram of the invention.
Fig. 2 is the schematic diagram of flexible beam deformation and floating coordinate system relative to inertial coodinate system.
Fig. 3 is flexibility FGM beam cross section and the configuration schematic diagram that cross section neutral axis changes.
Fig. 4 is flexibility FGM beam-ends point deformation course figure.
Fig. 5 is bending strain course figure in upper surface when flexibility FGM neutral axis of the beam is overlapped and is not overlapped with middle plane.
Fig. 6 is the flow chart for the emulation mode that the present invention considers that neutral axis of the beam influences rotating beam dynamic response.
Specific embodiment
In conjunction with Fig. 1, Fig. 2, Fig. 3 and Fig. 6, a kind of emulation side for considering neutral axis of the beam and rotating beam dynamic response being influenced
Method, comprising the following steps:
Step 1 determines Rigid Base, sets material gradient direction, geometric parameter and the material parameter of flexible beam, establishes tool
The Hub- Rigid chain polymer of functional gradient, is transferred to step 2；
Consider Hub-- flexibility FGM girder system system, Rigid Base is cylinder, radius R；One end of flexible beam and Rigid Base
The total length of consolidation, flexible beam is L, width b, is highly h, establishes the floating of flexible beam in flexible beam and Rigid Base junction
Moving coordinate system oxyz establishes inertial coodinate system OXYZ at Rigid Base center, and floating coordinate system turns relative to inertial coodinate system
Angle is θ, and the rotary inertia that Rigid Base does fixed-axis rotation is Joh, and the Hub- flexibility FGM girder system system with functionally gradient is with angle speed
Degree θ rotates around inertial coodinate system OZ axis, and flexible beam carries out gradient distribution along thickness direction, and flexible beam upper surface is by pure
Metal composition, lower surface are made of pure ceramics, and material gradient obeys following distribution
Wherein VcAnd VmIndicate volume fraction, N indicates power exponent, and E (z) represents the elasticity modulus of material, and ρ (z) represents material
The density of material, subscript c represent ceramics, and subscript m represents metal material；As coordinate z=-h/2, flexible beam upper surface is by pure metal
Composition, as z=h/2, flexible beam lower surface is made of pure ceramics；As N=10e10, flexible beam is made of pure metal, works as N
When=0, flexible beam is made of pure ceramics.
Step 2, the Non-linear coupling amount for considering flexible beam deformation are obtained in conjunction with von K á rm á n strain and Hooke's law
The strain energy and kinetic energy of Hub- Rigid chain polymer with functionally gradient, and determine the position for leading to beam section neutral axis by material
It sets, is transferred to step 3；
Non-linear coupling amount is considered, using the geometrical relationship between deformation, obtains following equation
Wherein deformation field ux(x, t) indicates deformation of the flexible beam along x-axis, uy(x, t) indicates flexible beam along the change of y-axis
Shape, uzDeformation of (x, the t) flexible beam along z-axis, usThe deformation of (x, t) expression flexible beam axially, i.e. linear deformation, v (x,
It t) is the tangential deformation for indicating flexible beam, w (x, t) indicates the flapwise deformation of flexible beam, and y and z indicate certain on flexible beam cross section
A little arrive the distance of neutral axis；hvAnd hwCoupling amount is expressed as follows:
Wherein hvIndicate the tangential second nonlinear coupling amount for deforming and causing axial shortening, hwIndicate that flapwise deformation causes axis
To the second nonlinear coupling amount of shortening, t indicates the time；
Based on Feng Keli nonlinear strain displacement theory:
Deformation formula substitution above formula can be obtained into axial strain εxxAre as follows:
It can be obtained by Hooke's law again:
σxx=Pxxεxx (7)
Wherein σxxFor axial stress, PxxFor stiffness coefficient, PxxIts function expression is different in different papers, as follows:
Here, P is takenxx=E (z).
In initial strainUnder possess the FGM beam that length is L and sectional area is A, primary stress is expressed as follows:
The axial force F generated by primary stress0Are as follows:
Herein,Indicate neutral axis to flexible beam lower surface distance,It indicates that flexible beam section takes up an official post to anticipate a little to arrive
The distance of flexible beam lower surface, therefore flexible beam section takes up an official post meaning a little to the distance z of neutral axis are as follows:
Due to being zero in the initial axial force referring to configuration underbeam, in order to determine neutral axis of the beam to beam lower surface distance's
Size can will be expressed as follows by the equilibrium equation of pure bending flexible beam in the x-direction
Thus the position equation of neutral axis of the beam is obtained are as follows:
As long as being aware of elasticity modulusDistribution function, the position of FGM neutral axis of the beam can pass through equation above
It is determined.
Axial strain and axial stress are brought into strain energy formulation again:
Wherein V indicates the volume of beam, i, j, and k indicates the x along floating coordinate system, and y, the direction z can be with above equation is crossed
Obtain strain energy U are as follows:
Wherein G0, G1, G2It is expressed as follows:
In order to further obtain the expression formula of the Hub-- flexibility FGM girder system system kinetic energy with functionally gradient, first flexible beam
On the position vector r at any point be expressed as
R=(R+x+ux)i+(y+uy)j+(z+uz)k (17)
Above formula is obtained into any point to time derivation and obtains velocity vector v
Finally, the kinetic energy expression T of Hub-- flexibility FGMs beam can be expressed as follows
Formula (18), which is brought into (19) available kinetic energy T, is
Wherein I0, I1, I2It is expressed as follows:
In formula (20),WithIndicate second nonlinear coupling amount hvAnd hwTo the derivative of time, the formula of single underscore
Subrepresentation is first approximation coupling terms, what the formula of double underline indicated is high order coupling terms.The Hub- of functionally gradient is flexible
FGM girder system is united under the circumference of the high-speed rotation, and zero degree Rigid-flexible Coupling Model will no longer restrain, and first-order approximation coupling model is reaching
It is also no longer restrained in the case where certain revolving speed, and high order coupling model can restrain, and be consistent with actual conditions.
Step 3 is established the rigid-soft of the Hub- flexibility FGM girder system system with functionally gradient by Second Kind Lagrange Equation
The coupled dynamical equation is transferred to step 4 with assuming that it is discrete that modal method carries out coupling dynamics kinetics equation；
By assuming that modal method, the linear deformation u of arbitrary point on flexible micro- beams(x, t), tangential deformation v (x, t), flapwise
Deformation w (x, t) is expressed as
In formula, φx(x)∈R1×N、φy(x)∈R1×NAnd φz(x)∈R1×NThe respectively extensional vibration of flexible beam, tangential
The row vector of the mode function of vibration and flapwise vibration, A (t) ∈ RN×1、B(t)∈RN×1With C (t) ∈ RN×1Respectively longitudinal vibration
The modal coordinate column vector of dynamic, tangential vibration and flapwise vibration.By second nonlinear coupling amount hvAnd hwIt is discrete to obtain:
Wherein Hv(x)∈RN×NAnd Hw(x)∈RN×NFor coupling shape function
Herein, φ 'y, φ 'zIndicate φy, φzTo the local derviation of x.
Take q=(AT, BT, CT)TAs generalized coordinates, with Second Kind Lagrange Equation
Obtain the coupling dynamics kinetics equation of rotation Hub-- flexibility FGM beam:
Wherein M is mass matrix,Expression generalized acceleration battle array, Q expression generalized force battle array, M,It can be indicated such as with Q
Under:
Wherein M11, M12, M13, M21, M22, M23, M31, M32, M33It is the submatrix in mass matrix,Indicate wide
Adopted acceleration, QA, QB, QCIndicate generalized force, and M11, M12, M13, M21, M22, M23, M31, M32, M33And QA, QB, QCContaining related
In the primary and high order coupling terms of second nonlinear coupling amount.
Step 4, acquire with functionally gradient Hub- flexibility FGM girder system system in each moment space configuration coordinate.
Use the Iteration of Fourth order Runge-Kutta for
Wherein yk, yk+1It indicates and node xk, xk+1Corresponding functional value, f (xk, xk+1) indicate the required differential equation,
K1, K2, K3, K4Indicate that iteration symbol, the H in Iteration are expressed as follows
H=xk+1-xk (31)
Wherein H indicates the step-length between two nodes.
WhereinIndicate corresponding functional value when half step-length,Indicate corresponding functional value, Δ table when step-length is H
Show that the difference of the functional value when functional value and step-length H when half step-length H/2 can carry out as follows to carry out the selection of step-length
Operation
Wherein ε indicates taken accuracy value, is calculated repeatedly above-mentioned formula, until Δ < ε, is taken at this time by half
Step-length in final calculating is required step-length.
Embodiment 1
Case study on implementation of the present invention discloses a kind of emulation mode that consideration neutral axis of the beam influences rotating beam dynamic response,
The specific method is as follows:
Step 1, setting Rigid Base geometric parameter, the geometric parameter and material parameter of beam are established as depicted in figs. 1 and 2
Rotation Hub-- flexibility FGM girder system system, Rigid Base geometric parameter, the geometric parameter of curved beam and material parameter, as shown in table 1.
Assuming that Rigid Base is with certain angular speedIt is rotated around own rotation axis, angular speed expression formula is as follows
Wherein taking emulation cycle is T=4s, simulation time t=6s, angular speed maximum value ω0=4rad/s.
1 Rigid Base of table, the geometric parameter and material parameter of beam
Step 2, the Non-linear coupling amount for considering flexible beam deformation are obtained in conjunction with von K á rm á n strain and Hooke's law
The strain energy and kinetic energy of Hub- Rigid chain polymer with functionally gradient, and determine the position for leading to beam section neutral axis by material
It sets, such as Fig. 3, is transferred to step 3；
Step 3 is established the rigid-soft of the Hub- flexibility FGM girder system system with functionally gradient by Second Kind Lagrange Equation
The coupled dynamical equation is transferred to step 4 with assuming that it is discrete that modal method carries out coupling dynamics kinetics equation；
Step 4, acquire with functionally gradient Hub- flexibility FGM girder system system in each moment space configuration coordinate.
Step 5, MATLAB export flexibility FGM beam-ends point deformation figure, as shown in figure 4, output flexibility FGM neutral axis of the beam and
Bending strain figure in upper surface when middle plane is overlapped and is not overlapped, as shown in Figure 5.
Claims (5)
1. a kind of emulation mode for considering neutral axis of the beam and being influenced on rotating beam dynamic response, which is characterized in that including following step
It is rapid:
Step 1 determines Rigid Base, sets material gradient direction, geometric parameter and the material parameter of flexible beam, and establishing has function
The Hub- Rigid chain polymer of energy gradient, is transferred to step 2；
Step 2, the Non-linear coupling amount for considering flexible beam deformation are had in conjunction with von K á rm á n strain and Hooke's law
The strain energy and kinetic energy of the Hub- Rigid chain polymer of functionally gradient, and determine the position for leading to beam section neutral axis by material,
It is transferred to step 3；
Step 3 is established the coupling dynamics that there is the Hub- flexibility FGM girder system of functionally gradient to unite by Second Kind Lagrange Equation
Kinetics equation is transferred to step 4 with assuming that it is discrete that modal method carries out coupling dynamics kinetics equation；
Step 4, acquire with functionally gradient Hub- flexibility FGM girder system system in each moment space configuration coordinate.
2. the emulation mode according to claim 1 for considering neutral axis of the beam and being influenced on rotating beam dynamic response, feature
Be: in step 1, Rigid Base is cylinder, radius R；One end of flexible beam and Rigid Base consolidate, the total length of flexible beam
It is highly h for L, width b, the floating coordinate system oxyz of flexible beam is established in flexible beam and Rigid Base junction, at center
Inertial coodinate system OXYZ is established at rigid body center, and floating coordinate system is θ relative to the corner of inertial coodinate system, and Rigid Base does dead axle
The rotary inertia of rotation is Joh, and the Hub- flexibility FGM girder system with functionally gradient is united with angular speedAround inertial coodinate system OZ
Axis rotates, and flexible beam carries out gradient distribution along thickness direction, and flexible beam upper surface is made of pure metal, and lower surface is by pure pottery
Porcelain composition, material gradient obey following distribution
P (z)=VmPm+VcPc (2)
Wherein VcAnd VmIndicating volume fraction, N indicates power exponent, and P (z) represents the density p (z) of material, elastic modulus E (z), under
Mark c represents ceramics, and subscript m represents metal material；As coordinate z=-h/2, flexible beam upper surface is made of pure metal, works as z=
When h/2, flexible beam lower surface is made of pure ceramics；As N=10e10, flexible beam is made of metal, as N=0, flexible beam
It is made of ceramics.
3. the emulation mode according to claim 1 for considering neutral axis of the beam and being influenced on rotating beam dynamic response, feature
It is: in step 2, it is contemplated that Non-linear coupling amount obtains following equation using the geometrical relationship between deformation
Wherein deformation field ux(x, t) indicates deformation of the flexible beam along x-axis, uy(x, t) indicates deformation of the flexible beam along y-axis, uz
Deformation of (x, the t) flexible beam along z-axis, us(x, t) indicates the deformation of flexible beam axially, i.e. linear deformation, and v (x, t) is
Indicate the tangential deformation of flexible beam, w (x, t) indicates the flapwise deformation of flexible beam, and y and z indicate certain point on flexible beam cross section
To the distance of neutral axis；hvAnd hwCoupling amount is expressed as follows:
Wherein hvIndicate the tangential second nonlinear coupling amount for deforming and causing axial shortening, hwIndicate that flapwise deformation causes axial contracting
Short second nonlinear coupling amount, t indicate the time；
Based on Feng Keli nonlinear strain displacement theory:
Deformation formula substitution above formula can be obtained into axial strain εxxAre as follows:
It can be obtained by Hooke's law again:
σxx=Pxxεxx (7)
Wherein PxxFor stiffness coefficient, σxxFor axial stress；
In initial strainUnder possess the FGM beam that length is L and sectional area is A, primary stress is expressed as follows:
The axial force F generated by primary stress0Are as follows:
Herein,Indicate neutral axis to flexible beam lower surface distance,Indicate that flexible beam section takes up an official post meaning a little to flexible beam
The distance of lower surface, therefore flexible beam section takes up an official post meaning a little to the distance z of neutral axis are as follows:
By the equilibrium equation by pure bending flexible beam in the x-direction, the position equation of neutral axis of the beam is obtained are as follows:
Axial strain and axial stress are brought into strain energy formulation:
Wherein V indicates the volume of flexible beam, i, j, and k indicates the x along floating coordinate system, and y, the direction z is obtained by above equation
Strain energy U are as follows:
Wherein G0,G1,G2It is expressed as follows:
In order to further obtain the expression formula of the Hub-- flexibility FGM girder system system kinetic energy with functionally gradient, first on flexible beam
The position vector r at any point is expressed as
R=(R+x+ux)i+(y+uy)j+(z+uz)k (15)
Above formula is obtained into any point to time derivation and obtains velocity vector v
Finally, the kinetic energy expression T of Hub-- flexibility FGMs beam is expressed as follows
4. the emulation mode according to claim 1 for considering neutral axis of the beam and being influenced on rotating beam dynamic response, feature
It is: in step 3, by assuming that modal method, the linear deformation u of arbitrary point on flexible beams(x, t), tangential deformation v (x, t), the wing
It is expressed as to deformation w (x, t)
In formula, φx(x)∈R1×N、φy(x)∈R1×NAnd φz(x)∈R1×NThe respectively extensional vibration of flexible beam, tangential vibration
With the row vector of the mode function of flapwise vibration, A (t) ∈ RN×1、B(t)∈RN×1With C (t) ∈ RN×1Respectively extensional vibration, string
The modal coordinate column vector vibrated to vibration and flapwise；
Take q=(AT,BT,CT)TAs generalized coordinates, with Second Kind Lagrange Equation
Obtain the coupling dynamics kinetics equation of rotation Hub-- flexibility FGM beam:
Wherein M is mass matrix,Indicate generalized acceleration battle array, Q indicates generalized force battle array.
5. the emulation mode according to claim 1 for considering neutral axis of the beam and being influenced on rotating beam dynamic response, feature
Be: in step 4, use the Iteration of Fourth order Runge-Kutta for
Wherein yk,yk+1It indicates and node xk,xk+1Corresponding functional value, f (xk,xk+1) indicate the required differential equation, K1,K2,
K3,K4Indicate that iteration symbol, the H in Iteration are expressed as follows
H=xk+1-xk (22)
Wherein H indicates the step-length between two nodes.
Priority Applications (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN201910217117.8A CN110008543B (en) | 2019-03-21 | 2019-03-21 | Simulation method considering influence of beam neutral axis on dynamic response of rotating beam |
Applications Claiming Priority (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN201910217117.8A CN110008543B (en) | 2019-03-21 | 2019-03-21 | Simulation method considering influence of beam neutral axis on dynamic response of rotating beam |
Publications (2)
| Publication Number | Publication Date |
|---|---|
| CN110008543A true CN110008543A (en) | 2019-07-12 |
| CN110008543B CN110008543B (en) | 2022-09-13 |
Family
ID=67167628
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CN201910217117.8A Active CN110008543B (en) | 2019-03-21 | 2019-03-21 | Simulation method considering influence of beam neutral axis on dynamic response of rotating beam |
Country Status (1)
| Country | Link |
|---|---|
| CN (1) | CN110008543B (en) |
Cited By (1)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN113536595A (en) * | 2021-08-10 | 2021-10-22 | 西北工业大学 | High-precision dynamics modeling method for spatial large-scale rigid-flexible coupling system |
Citations (2)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN108388699A (en) * | 2018-01-29 | 2018-08-10 | 南京理工大学 | Rigid Base-FGM tapered beam system end dynamic response computational methods |
| CN108593961A (en) * | 2018-04-08 | 2018-09-28 | 西安交通大学 | A kind of low g value raster pattern mems accelerometer |
-
2019
- 2019-03-21 CN CN201910217117.8A patent/CN110008543B/en active Active
Patent Citations (2)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN108388699A (en) * | 2018-01-29 | 2018-08-10 | 南京理工大学 | Rigid Base-FGM tapered beam system end dynamic response computational methods |
| CN108593961A (en) * | 2018-04-08 | 2018-09-28 | 西安交通大学 | A kind of low g value raster pattern mems accelerometer |
Cited By (2)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN113536595A (en) * | 2021-08-10 | 2021-10-22 | 西北工业大学 | High-precision dynamics modeling method for spatial large-scale rigid-flexible coupling system |
| CN113536595B (en) * | 2021-08-10 | 2022-09-13 | 西北工业大学 | High-precision dynamics modeling method for spatial large-scale rigid-flexible coupling system |
Also Published As
| Publication number | Publication date |
|---|---|
| CN110008543B (en) | 2022-09-13 |
Similar Documents
| Publication | Publication Date | Title |
|---|---|---|
| Yu et al. | Generalized Timoshenko theory of the variational asymptotic beam sectional analysis | |
| Campanelli et al. | Performance of the incremental and non-incremental finite element formulations in flexible multibody problems | |
| CN107895069A (en) | A kind of fluid structurecoupling Numerical Predicting Method based on composite structure | |
| Bayoumy et al. | A continuum based three-dimensional modeling of wind turbine blades | |
| CN104281730B (en) | A kind of finite element method of the plate and shell structure dynamic response of large rotational deformation | |
| Xie et al. | Static aeroelastic analysis including geometric nonlinearities based on reduced order model | |
| CN102866637A (en) | Quadratic order-reduction based method for simulating unsteady aerodynamic force of aerofoil with operation surface | |
| Attar et al. | Modeling delta wing limit-cycle oscillations using a high-fidelity structural model | |
| Ritter et al. | Static and dynamic simulations of the Pazy wing aeroelastic benchmark by nonlinear potential aerodynamics and detailed FE model | |
| CN104268329A (en) | Joint simulation method for electro-hydraulic and piezoelectric driving macro and micro parallel mechanism | |
| Miyashita et al. | Folding angle regulation by curved crease design for self-assembling origami propellers | |
| Hilger et al. | Nonlinear aeroelastic simulations and stability analysis of the pazy wing aeroelastic benchmark | |
| Wang et al. | Structural dynamics model of multisegmented folding wings: theory and experiment | |
| Ritter et al. | Static and dynamic aeroelastic validation of a flexible forward swept composite wing | |
| Ritter et al. | Comparison of nonlinear aeroelastic methods for maneuver simulation of very flexible aircraft | |
| CN108182330A (en) | A kind of method that the response of flexible rectangular thin plate Rigid-flexible Coupling Dynamics is calculated based on B-spline | |
| CN106096293B (en) | A kind of big rotation composite material extending arm heat-driven oscillation prediction technique | |
| Li et al. | Geometrically nonlinear model of piezoelectric wind energy harvesters based on vortex-induced vibration and galloping | |
| CN110008543A (en) | A kind of emulation mode for considering neutral axis of the beam and rotating beam dynamic response being influenced | |
| CN107766686B (en) | Simulation method for calculating FGM thin plate rigid-flexible coupling dynamic response based on MATLAB | |
| CN105160092B (en) | A kind of thermal environment interpolation method suitable for thermal protection system transient temperature field computation | |
| CN109902418A (en) | A kind of calculating EACLD Rigid Base-Flexural cantilever model dynamic response emulation mode | |
| Koohi et al. | Nonlinear aeroelastic analysis of a composite wing by finite element method | |
| Li et al. | Dynamic modeling of a fish tail actuated by IPMC actuator based on the absolute nodal coordinate formulation | |
| Wu et al. | A recursive formulation based on corotational frame for flexible planar beams with large displacement |
Legal Events
| Date | Code | Title | Description |
|---|---|---|---|
| PB01 | Publication | ||
| PB01 | Publication | ||
| SE01 | Entry into force of request for substantive examination | ||
| SE01 | Entry into force of request for substantive examination | ||
| GR01 | Patent grant | ||
| GR01 | Patent grant |