A New Method Development in Time Domain Aeroelastic Vibration

7/25/2019 A New Method Development in Time Domain Aeroelastic Vibration

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A New Method Development in Time DomainSimulation of Nonlinear Aeroelastic Vibrations

Paper Reference Number: 2001-12

Cosmas Pandit Pagwiwoko#and Md Azlin Md Said*

School of Aerospace EngineeringUniversity of Science Malaysia

Engineering Campus

14300 Nibong Tebal, Pulau PinangMalaysia

Tel: +604 5941026Fax: +604 5941027

#

) e-mail: [email protected]

Abstract

Nonlinear aeroelastic behavior is examined. This research extends the effort of severaldynamic modeling that includes the existence of structural non-linear factors such as:

free-play and softening/hardening spring effects. The authors consider the flutterphenomena as a dynamic system constructed in the form of power bond graph. Theexternal loads of the system are modeled as an unsteady aerodynamic filter based on Pad

approximation rational function. By converting the bond graph model into equivalentblock diagram, the analysis of structural response can be carried out directly in time

domain that may enable to model the non-linearity accurately.

Key words:non-linear flutter, system dynamics, bond graph, modeling and simulation.

#) Dr. Cosmas Pandit Pagwiwoko, senior lecturer on Aircraft Structure.*) Assoc.Prof. Md Azlin Md Said, Dean of Faculty.


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I. Introduction

Aeroelasticity is the phenomena resulting from the interaction of structural,inertial and aerodynamic forces. The aerodynamic loads on aircraft wing vary with the

speed of the flow and depend on the structural response. Usually, the unsteady

aerodynamic loads are divided into two parties i.e. the component that supports thestructural movement and the other that resists the movement. The structural damping of

wing structure can be normally neglected, therefore when the aerodynamic and structuralloads are in balance, it will produce a harmonic oscillation. This kind of vibration

happens at certain speed of the flow called the flutter boundary. Above this critical flowspeed, there will be an unbalance of energy that flows into the structure compared withthe dissipated energy, consequently the vibration grows divergently until disintegration of

the structure. In the case of classical flutter, a coupled bending and torsion modes ofvibration are excited at the critical speed. In this critical condition both modes agitate

each other as two frequencies approach one to another in term of flow speed, hence thereis transfer of energy between the two modes so called the internal resonance. Sometimesthe aeroelastic system contains non-linearity factors. These non-linearities result from

unsteady aerodynamic sources, large strain-displacement conditions, or the partial lost ofstructural and control integrity. The such system can render non-linear dynamic response

characteristics like limit cycle oscillation, internal resonance and chaotic motion.Flutter analysis is generally executed by mathematical modeling based on

Lagrange equation and completed by solving the complex determinant of the equation in

frequency domain. However in this paper, the analysis of the problem is carried out in thephysical model rather than the mathematical equation derivation. The aeroelastic problem

is modeled and analyzed by using power bond graph method. In this method, thephenomenon is considered as a dynamic system that consists of interacting sub-systemsand/or components. All the elements within the system are connected each others by

energy bonding where the power flows through. The types of the basic elements thatconstruct the system are: the storing/dissipating energy components, energy transmitters,

and junctions, beside the sources of energy as the external excitations. For the case ofaeroelastic problems where the external excitation is a function of structural response, theenergy source component has the feed back input signals from the model, therefore it can

be considered as a filter or transfer function block. The aerodynamic loads excite thestructure and the same time is induced by structural response. To model this so-called

aero-filter, an approximation rational function in Laplace variable is needed. Thecoefficients of the function can be determined using least-square curve fitting techniqueapplying to unsteady aerodynamic forces that calculated in frequency domain. The

unsteady aerodynamic forces are resolved generally in certain reduced frequencies andmode shapes using singularity method. For the case of 3 dimensional flow, MSC.Nastran

using acceleration double lattice method is applicable.By using the common rules and procedures [2] the power bond diagram can beconstructed for a certain dynamic system and the interaction mechanism (known as

causal stroke) amongst the components can be determined. Once the graph is completed,one can convert it into its equivalent block diagram, hence the system can be simulated

using dynamic simulation software like Simulink-MATLAB. Since the analysis is carried


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out directly in time domain, the structural non-linear factors as function of dynamicresponse can be modeled accurately.

NOMENCLATURE

A aerodynamic coefficient matrixa location of elastic axis relative to mid-chordb half chordc chordC component of capacitorC(k) Theodorsen functionc coefficient of dampinge effort variable

F~

generalized aerodynamic forcesf(t) external force

f flow variableI component of inertia

I mass moment of inertia about elastic axis

K spring constant for pitch degree of freedomKy spring constant for plunge degree of freedomk reduced frequencyk spring stiffness

L liftM momentm lumped massmT total mass of test model due to plungemW mass due to pitch

q dynamic pressureR component of resistorr distance between elastic axis and c.g.

s span of test modelTR component of transformerSE source of effortV~ free stream velocity

y translation degree of freedom{w} degree of freedom vector

rotational degree of freedom generalized damping matrix

generalized stiffness matrix lagging term coefficient{} generalized coordinate mode shape generalized mass matrix

modal matrix

air density


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II. Problem Definition

The idea of this research is to make a software based on power bond graphtechnique to solve the multi-degree of freedom aeroelastic problem with the existence of

non-linear structural factors. To validate this method, the dynamic simulation will be

compared with an experimental wind tunnel results of two degree of freedom airfoil withquadratic/cubic stiffness. In the dynamic modeling of the above physical model, all

variables are taken in modal coordinate to cover the generality of multi-degree offreedom system. Considering that the flow is in 2D, the Theodorsen function [3] will be

used to generate the unsteady aerodynamic loads instead of doublet distribution.

2.1. Review of Bond Graph Technique

For the sake of clarity, the concept of this method will be briefly discussed. In this

method, the various elements are connected by means of power bonds representing dualvariables called effort (e) and flow (f). A bond is represented by a half arrow which at thesame time means the direction of power flow. Within a bond the direction of effort

variable is always in opposite to the flow variable due to the nature, and the product ofboth variables signifies the power magnitude. The arrangement of this causal stroke is

put in order using a common regulation and depends on the configuration of the system.The basic elements used in modeling and simulation of the aeroelastic problems are:

1-port element: Resistor (dissipating energy component), Capacitor and Inertia

(storing energy components).

2-port element: Transformer and Gyrator (components that transmit the energy).

Multi-port element: serial (1) and parallel (0) junctions.

Figure 1. 1-port elements with equivalent block diagrams for Resistor, Capacitor andInertia

R

R

R

R

e

f

e

f

e

f

e

f

1/C

C

C

C

e

f

e

f

e

f

e

f

intg

d/dt

1/I

I

I

I

e

f

e

f

e

f

e

f

intg

d/dt


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Figure 2. 2-port elements with equivalent block diagrams.

Figure 3. Multi-port elements with equivalent block diagrams.

2.2. Structural Dynamics in Bond Graph Modeling

Consider a structural dynamic problem in the form of multi degree of freedom

system as explained in figure 4. The dynamic response of the structure due to the externalloads can be written in the following expression where some limited modes areconsidered and temporal variables are separated from spatial domain:

)().,,();,,( tzytzyw = (1)

whereis natural mode shapes, and is generalized coordinate.

w(x,y,z;t)

Figure 4. Wing structure in the form of multi degree of freedom system.

0

parallel porte

f1

f

e

f x-

-

e

f

e f

e

f

1

1

1

2

e2 2

2

3

3

3

3

1

serial port

e

f1

f

e

f

x -

-

e

f

e f

e

f

1

1

1

2

e2 2

2

3

3

3

3

r

1/r

GY

GY

e

f

e

f

e

f

e

f

r

e e

f f

1/r

e e

f f

m

1/m

TR

TR

e

f

e

f

e

f

e

f

m

e e

f f

1/m

e e

f f


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By respecting the orthogonal condition, the equation of motion can be expressed in modalcoordinate system as a set of linear independent equations:

F~

... =++ &&& (2)where:

fF t.~ = (3)

is the vector of modal force and is the mode shape matrix that can be obtained from

finite element calculation using MSC Nastran for Eigen Value Solution. Parameters , , are the diagonal matrix of generalized mass, damping and stiffness respectively. The setof single degree of freedom systems can be schematized as free body diagrams shown byFigure 5 (below):

1

1

1

)(1 t

n

n

n

)(tn

Figure 5. A set of single degree of freedom systems in modal coordinate.

The above free body diagram can be modeled in power bond graph in the ensemble of thebasic components as shown in Figure 6. The direction of power flow and the arrangement

of causal stroke amongst the components are determined by a certain rule and procedure[2] concerning the nature of the problem.

Figure 6. Power bond graph for single degree of freedom system.

I = C = 1/1

R =

SE

e =f(t) f

1

1

1


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Hence for structural dynamic problem represented in generalized coordinate, can bemodeled in power bond graph as explained in Figure 7, where the natural coordinate

system can be included into the graph by using transformer TR components [6]. These 2-port components signifying the natural mode shapes, transform the external forces into

the modal forces.

11=TR 21 31 1i

1 n

n1

1/1 n/11 1

0 0 00

1f 2f

Figure 7. Structural dynamic problem in power bond graph modeling.

2.3. Aeroelasticity in Bond Graph Modeling

The equation of motion of an aeroelastic system in modal coordinate is expressed

by the equation 2, where the external loads are the function of structural response. Byassuming that the flow is two dimensional, the aerodynamic lift and moment of the wing

can be modeled by the unsteady aerodynamic theory of Theodorsen [3] :

})2

1({2)(. ~~~2 abVyCbVbaVybL ++++= &&&&& (4)

(5)

where C is Theodorsens function that depends on the reduced frequency, k = b/V~ ,

which explains the delay factor of the lifting part of the loads. The functions of (t)andy(t) show the dynamic response of the structure. It is noted that both equations 4 and 5,

represent incompressible, small disturbance unsteady flow. The unsteady aerodynamicforces in the form of generalized coordinate are approximated in Laplace variable using

Pad approximation model [4] as followed:

)(].....[)(

~

1

21

2

0 ss

sBAsAsAqsF

l

m m

m

= +

+++ (6)

})2

1(){2

1(2

})8

1()2

1({.

~2

~

2~

3

&&

&&&&&

abVyaCbV

baVayabM

++++

++=


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where q is the aerodynamic pressure, m are the lagging terms. The matrix coefficientsA0, A1, A2 and Bm are calculated by using least-square technique of curve fitting [5] toapproach the aerodynamic forces that calculated separately in frequency domain. Therational equation above, in time domain analysis, can be considered as a filter that relates

the structural response and the generated loads as shown in Figure 8.

Aero-Filter)(

)(

tM

tL

&&&

&&&

,,

,, yyy

Figure 8. The unsteady aerodynamic filter.

In constructing the aeroelastic model in the form of bond graph, this aero-filter is used as

the exciting component into the structural dynamic model, rather than the energy sourceelements. Thus we have a closed loop system representing the aero-structural interaction

as explained in Figure 9.

Transformer : [Mode Shapes]

Aero-Filter

1 1

0 0

1 n1

1 n

1

1ngeneralized

coordinate

natural coordinate

Figure 9. Aeroelastic problem in power bond graph modeling.

2.4. Case Study

To validate the aeroelastic modeling in bond graph, we apply this method to awind tunnel test set up which has the available experimental results [1]. The flow isconsidered 2 dimensional and incompressible.


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Figure 10. Aeroelastic physical model in the form of two degree of freedom system with

two dimensional incompressible flow.

The physical model of the system is sketched in figure 10. The suspension mechanismallows the model to move only in the in-plane degree of freedom i.e.: translation and

rotation. All of data and physical properties used in the simulation are taken from

reference [1] as follows:

a = -0.460mT = 10.3 kg

mw = 1.662 kgb = 0.1064 m

s = 0.6 mairfoil profile = NACA 0012

K= 2.57 N-m/rad

Ky = 2860 N/mr = 0.0287 m

Ie= 0.0174 kg-m2

Where for linear case, both stiffness parameters are defined as follows:

K= 2.57 ( 1 + 1.33 y2) N-m/rad (7)

Ky = 2860 ( 1 + 0.09 y2) N/m (8)

a.c e.a

c.g

b b

ab

non- l inerar term

r

L

MV~

y

K

Ky


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III. Analysis

The equivalent block diagram for the aeroelastic model can be constructed as the

directions of power flow and causal strokes are known. The corresponding block diagram

converted from Figure 7 for two degree of freedom movement i.e. heaving and pitching isexplained in the following figure:

Figure 11. Equivalent block diagram of structural dynamic problem, converting from

power bond graph.

The dynamic simulation can be executed directly in time domain using Simulink-

MATLAB. In the case of structural non-linearities, based on the equation (7) and (8) theblock diagram for the stiffness is modeled as follows:

-K-ph22b

-K- ph22

-K- ph21b

-K-ph21

-K- ph12b

-K-ph12

-K-ph11b-K- ph11

-K-

gamma1

-K-

gamaa2

0 beta20 beta1

Sum3Sum2

Sum1Sum

Step

Sine Wave

Scope2Scope1

s

1

Integrator5

s

1

Integrator4

s

1

Integrator3

s

1

Integrator2

s

1

Integrator1

s

1

Integrator

-K-

1/mu2

-K-

1/mu1


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Figure 12. Block diagram for non-linear stiffness.

To simulate the flow induced vibration problem, the structural model on Figure 11 needs

to be coupled with the unsteady aerodynamic loads explained by the aero-filter on figure-8. The coefficients of aero-filter are determined by calculating the unsteady aerodynamicloads (equations 4 and 5) for the circulatory components only. The aerodynamic

modeling in the form of Pade approximation function shows a good accuracy aspresented in the Figures 13. The number of lagging terms used in the rational function in

s variable is important to approximate the aerodynamic forces in frequency domain. Inthis case 4 lagging terms were used, i.e. 0.10, 0.15, 0.2, 0.25 .

Figure 13a. Unsteady aerodynamic loads F(1,1) and F(1,2) approximated by Paderational function.

1

Out1Product

u2

Math

Function

-K-

Gain1

-K-

Gain

1Constant

2

In2

1

In1

-0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

-0.75 -0.7 -0.65 -0.6 -0.55 -0.5 -0.45 -0.4

0.08

0.09

0.1

0.11

0.12

0.13

0.14

0.15

0.16


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Figure 13b. Unsteady aerodynamic loads F(2,1) and F(2,2) approximated by Paderational function.

The simulations were conducted at the velocity slightly above the flutter speed i.e. 15.0

m/sec. In this simulation, The aeroelastic vibration was triggered by a unit rotationalimpulse 1 Nm. The structural responses in translation and rotational movement of theairfoil are presented in the Figure 14 for the linear case.

Figure 14. Aeroelastic vibrations of the airfoil in heaving and pitching movements at theair speed of 15.0 m/sec. for the case of linear stiffness.

0 0.5 1 1.5 2 2.5 3 3.5

x 10-3

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

1.8 2 2.2 2.4 2.6 2.8 3 3.2

x 10-3

-7

-6.5

-6

-5.5

-5

-4.5

-4

-3.5x 10

-4


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For the case with the existence of nonlinear stiffness parameters in translation androtation modes, the results of simulation are given in Figure 15.

Figure 15. Aeroelastic vibrations of the airfoil in heaving and pitching modes for thecase of non-linear stiffness parameters, at the flow speed of 15.0 m/second.

IV. Discussion

As explained in reference [1], the test apparatus of the aeroelastic physical model

permits nonlinear pitch and plunge response of a rigid wing section, and the design of thetest apparatus permits independent motion in both degree of freedom. From figure 3 ofreference [1], nonlinear structural response in the experiments is governed by a pair of

cams, which are designed to permit tailored linear or non-linear response. The shape eachcam that adjusts the spring deflection, dictates the nature of the non-linearity. Figure 4 of

reference [1] shows us the free vibration response at the zero speed of flow. From thisfigure the natural frequencies for heaving and pitching modes can be determined. Byusing logarithmic decrement technique, from the time response of free vibrations, the

structural damping factors of the mechanism were predicted and considered as viscousdamping. These parameters were taken into account in the bond graph simulation.

Comparing with Figure 5 of reference [1], from the results of simulation on Simulinkshown in Figure 14, one may conclude that the flutter response in linear case gives agood correlation with experimental results.

In the case of nonlinear stiffness, it is shown in Figure 15 that the aeroelastic vibrationsappear in the form of limit cycle oscillations. By confronting these simulation results with

the experiments in figure 7 of reference [1], one may conclude that the form of theresponse and the magnitude give a sense of similarity.


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V. Conclusion

The power bond graph modeling for aeroelastic system with the existence ofstructural non-linearities has been discussed. Simulation in case of two degree of freedom

system wing section in heaving and pitching modes with 2 dimensional incompressible

flow, shows a good relationship with experimental results. In dynamic modeling, the2DoF system was presented in modal coordinate after being transformed from natural

coordinate, to confirm the generality in multi-degree of freedom structural dynamicproblems. In general, for real continuous structure, the finite element modeling can be

done by using MSC.Nastran to determine the natural frequencies and mode shapes.Since the dynamic modeling is conducted directly in physical problem rather than themathematical equation, this method is practical to use for multi-degree of freedom real

wing structure. And due to the fact that the simulation is executed directly in timedomain, consequently the non-linear factor of structure will be convenient to handle,

beside the high accuracy of the non-linearity model. The inaccuracies of the simulationfor the most part, should be traced from structural dynamic finite element modeling, andunsteady aerodynamic computation using singularities including rational approximation

function.

REFERENCE

Books(1) Dean C. Karnopp et al., System Dynamics: A Unified Approach, John Wiley &

Sons, Inc., 1990.(2) S.T. Nannenberg, Bondgraaf Techniek, Delta Press BV, The Netherlands, 1993.

Journals(3) Todd ONeil & Thomas W. Strganac, Aeroelastic Response of a Rigid Wing

Supported by Nonlinear Springs, Journal of Aircraft, Vol. 35, No. 4, July-August, pp. 616-622, 1998.

(4) Theodorsen, T., General Theory of Aerodynamic Instability and the Mechanism

of Flutter, NACA Rept. 496, 1935.(5) V. Mukhopadhyay , J.R. Newsom, I. Abel, A Methode for Obtaining Reduced-

Order Control Laws for High-Order System Using Optimization Techniques,NASA Technical Paper 1876, 1981.

PhD. Thesis(6) C. PanditPagwiwoko, Vibrations Aeroelastiques des Surfaces Portantes, Ph-D

thesis of Ecole Centrale de Lyon, France, 1991.

Conference Paper

(7) C. PanditPagwiwoko & R. Razali, Dynamic Modeling of Aeroelastic VibrationUsing Power Bond Graph Method, 4th . Pacific International Conference on

Aerospace Science and Technology, Kaohsiung-Taiwan, 21-23 May 2001.

Documents

A New Method Development in Time Domain Aeroelastic Vibration