Introduction – Equations of motion G. Dimitriadis 04

Time domain responses equations of motion are only valid at zero airspeed or at the flutter condition.. Introduction to AeroelasticityFrequency Response Function •!. Introduction to Aer

Time domain responses

equations of motion are only valid at zero airspeed or at the flutter condition

sinusoidal excitation

aeroelastic system with Theodorsen

aerodynamics to any excitation force

Introduction to Aeroelasticity

Frequency Response Function

• ! Imagine that we excite the pitch-plunge airfoil

at the leading edge with a force F0expj!t

• ! The equations of motion become

•! This equation is of the form H(!)q0=F, where

H-1(!) is the Frequency Response Function

FRF for pitch-plunge system

The two modes

are clearly present

FRF of "

The first mode is

present as an

anti-resonance

Introduction to Aeroelasticity

Working with the FRF

response

perform stability analysis

Impulse response of pitch-plunge airfoil

! " # $ % &! &" &# &$ &%

Introduction to Aeroelasticity

Damped sinusoidal motion

• ! The previous discussion shows that:

– ! Theodorsen aerodynamics are only valid for

sinusoidal motion – ! Yet Theodorsen aerodynamics can be used to

calculate damped impulse responses

• ! Stability analysis is slow and and can be less accurate when performed on impulse

responses

• ! We need a method for calculating the

damping at all airspeeds directly from the

equations of motion

The p-k Method

technique for obtaining aeroelastic

solutions

has become the industrial standard

been designed using the p-k method

Introduction to Aeroelasticity

Basics

equations of motion in the standard

form

forces of the form

With k=!b/U

Basics (2)

response is sinusoidal, since the

Theodorsen lift is equal to

Introduction to Aeroelasticity

Basics (3)

that depend on frequency

Introduction to Aeroelasticity

The p-method

• ! The p-method consists of solving this

eigenvalue problem for p

• ! It’s a nonlinear eigenvalue problem but

polynomial so it can be solved

•! The p values will generally be complex

• ! There is no guarantee that the real parts of

The p-k method

• ! The p-k method is more sophisticated than the p-method in that it performs frequency matching

• ! The equations solved are

• ! Since it is known that the aerodynamic

matrix is only a function of frequency (not

Introduction to Aeroelasticity

Application to 2-dof model

are:

The p-k solution

• ! The solution of these equations is iterative

• ! We guess a value for the frequency ! (and

resulting eigenvalue problem

•! The norm of p should be equal to !

• ! If it is not, we change the value of ! until the

scheme converges

• ! This is called frequency matching

Introduction to Aeroelasticity

Frequency matching

k-method results

Introduction to Aeroelasticity

Results

•! Where n l is the number of aerodynamic lags

Practical Aeroelasticity

•! For an aircraft, the matrix Q(jk) is obtained using a

panel method-based aerodynamic model

• ! The modelling is usually performed by means of commercial packages, such as MSC.Nastran or Z- Aero

•! For a chosen set of k values, e.g k1, k2, !, k m, the corresponding Q matrices are returned

•! The Q matrices are then used in conjunction with

the p-k method to obtain the flutter solution or

time-domain responses

•! The values of Q at intermediate k values are

obtained by interpolation

Introduction to Aeroelasticity

BAH Example

First 5 modes of BAH wing

Introduction to Aeroelasticity

GTA Example

a Generic Transport Aircraft

Finite element model: Bar elements

with 678 degrees of freedom Aerodynamic model: 2500 doublet lattice panels

Flutter plots for GTA

First 7 flexible modes

Clear flutter mechanism

between first and third

mode (first wing bending

and aileron deflection)

Introduction to Aeroelasticity

Time domain plots

for the GTA

Introduction to Aeroelasticity

Flutter plots

for SST

First 9 flexible modes

Clear flutter mechanism

between first and third

mode

the above specifications and does not flutter inside the flight

envelope

–! The plunge spring cannot exceed 5000N/m

– ! The pitch spring cannot exceed 4000Nm/rad