Aeroelasticity (미완성)

 

Chapter 4

 

시험 오답노트

Homogeneous terms are those, which do not vanish when all the unknowns are set equal to zero in a system of equations. ▶ FALSE. ▶ If all unknown become zero, that equation will be yielded nontrivial solutions.

※ inhomogeneous terms : 만약 θ에 대해서 묶어서 문제를 푸는거라면, θ와 관계 없는 terms는 모두 inhomogeneous terms가 된다. 얘네를 보통 우변으로 옮기고 θ로 묶은 애들은 좌변으로 옮기지.

 

Of all the terms in a set of governing differential equations, the homogeneous terms are necessary but not sufficient for a stability analysis ▶ FALSE. ▶ To solve trivial solutions, need the homogeneous terms.

 

Of all the terms in a set of governing differential equation, the inhomogeneous terms are necessary for a response analysis. ▶ TRUE. ▶ inhomogeneous terms indicate 'not harmonic, not pure equation of motion'? (IDK)

 

The presence of nonlinear terms generally makes the answers obtained from an analysis less accurate ▶ FALSE. ▶ ???

 

We have considered only spanwise uniform beams because equations for spanwise nonuniform beams are nonlinear and thus hard to solve. ▶ FALSE. ▶ ??

 

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4.1 Wind-Tunnel Models

 

4.1.1 Wall-Mounted Model (spanwise clamped, like 2D, No flip)

 

4.1.2 Sting-Mounted Model (trailing edge clamped, No flip)

 

4.1.3 Strut-Mounted Model (leading & trailing edge clamped, No flip)

 

4.1.4 Wall-Mounted Model for Application to Aileron Reversal (spanwise clamped, flip exists)

 

4.2 Uniform Lifting Surface

 

4.2.1 Steady-Flow Strip Theory

 

4.2.2 Equilibrium Equation

 

4.2.3 Torsional Divergence

 

4.2.4 Airload Distribution

 

(4.2.5 Aileron Reversal)

 

4.2.6 Sweep Effects

 

4.2.7 Composite Wings And Aeroelastic Trailoring

 

 

 

Chapter 5

 

5.1 Stability Characteristics from Eigenvalue Analysis

Above flutter speed rather than damping out the motions due to small perturbations in the configuration, the air can be said to provide negative damping.

All aerodynamic effects vanish in a vacuum where density vanishes (b is a reference semi-chord)

(real part of the root from the generalized equation, Γk)

Γk < 0 : convergent oscillations = dynamically stable

Γk > 0 : divergent oscillations = dynamically unstable = flutter

Γk = 0 : stability boundary

※ root locus에서 real part가 양수면 불안정, 음수면 안정 ▶ system dynamics

In many published works on flutter analysis, the method outlined in this section based on determination of stability from complex eigenvalues is known as the "p method"; p is frequently termed a "reduced eigenvalue." To provide an accurate prediction of flutter characteristics, the p method must use an aerodynamic theory that accurately represents the loads induced by transient(순간적인) motion of the lifting surface.

 

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5.2 Aeroelastic Analysis of a Typical Section

In this section, steady flow is only considered which means this condition only cares about instantaneous pitch angle, θ.

((( Hㅏ름다운 airfoil의 자태 )))

 

P, reference point : plunge displacement, h, is measured

C, center of mass

Q, aerodynamic center : presumed quarter-chord in subsonic thin-airfoil theory

T, 3/4 chord : important chord-wise location in thin-airfoil theory

static-unbalance parameter :

→ e랑 a랑 단위없음, -1부터 1까지의 범위이고, 음수값일땐 the center of mass is toward the leading edge from the reference point.

 

 

Equation of motion : 

, 

 : 얘네 둘을 위의 식에 대입해서 아래 식으로 만든다.

 

이걸로 묶어줌 :

, 그럼 아래식처럼 됨.

그리고 

이거를 적용., 여기서 p is the unknown dimensionless, complex eigenvalue → "p method"

 

앞 행렬을 determinant 적용시켜서 p에 관해 풀면 아래처럼 두개 나옴 : (ad-bc=0)

 

 → 

r, dimensionless radius of gyration of the section about the reference point P; 

σ, ratio of uncoupled plunge and pitch frequencies (ω)

μ, mass-ratio parameter reflecting the relative importance of the model mass to the mass of the air affected by the model

V, dimensionless freestream speed of the air, "reduced velocity"

 

When flutter occurs, the real part, Γ, of one of the roots is positive and the other is negative.

 

Damping of all modes below the flutter speed is predicted to be zero, which is known to be incorrect.

Steady-flow theory exhibits a coalescence(합체,유착) characterized by the two roots being exactly equal to one another at the point of flutter.

Steady-flow aerodynamic theory, a most significant deficiency(결핍,부족) concerning flutter analysis is that it neglects unsteady effects. To obtain an accurate, to include unsteadiness in the aerodynamic theory.

 

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5.3 Classical Flutter Analysis

Classical flutter analysis can predict the flutter speed and flutter frequency, but it cannot predic values of modal damping and modal frequency away from the flutter condition.

 

5.3.1 One-Degree-of-Freedom Flutter

The plunge degree of freedom is equal to zero.

System equations of motion reduce to one equation :

 

Final equation of flutter boundary : M으로 된 식 두개를

 

잘 나열하고 잘 정리하면

 이렇게 나옴.

 

mθ는 두 파트로 나눠져있는데 M∞를 0으로 임시설정 하고

 

※ reduced frequency, k가 0부터 시작해 점점 증가하면 Im[ mθ (k,0) ] 값이 아래 그래프처럼 됨.

딱 Im[ mθ (k,0) ]=0인 부분이 있는데 저때의 k값이 바로 kF, flutter reduce frequency.

 

5.3.2 Two-Degree-of-Freedom Flutter

Equation of Motion :

 

5.3.1처럼 잘 재정렬하면 

이렇게 나옴.

 

h/b랑 θ에 관해 묶으면

 이렇게 나옴.

 

이 관계식 위에 대입해주고 다시 정리하면

이렇게 나옴. 그리고 determinant 설정해줄때

Re[det(Matrix)]= 0 && Im[det(Matrix)] = 0 이렇게 동시에 만족해야 함.

determinant(Matrix)=0을 만족하는 (real, not imaginary k and omega) k값과 (ωθ/ω)^2값은 flutter condition의 boundary 값. → kF, ωF 구해짐. → Uf 구함.

 

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(((5.4 Engineering Solutions for Flutter

5.4.1 The k Method

5.4.2 The p-k Method)))

 

5.5 Unsteady Aerodynamics

Differently from 5.2, this section deals with angle of attack,

, which includes pitch angle as well as plunge velocity @ reference point P.

Vortices are an integral part of the process of generation of circulatory lift. Basically, there is a difference in the velocities on the upper and lower surfaces of an airfoil.

However, the circulatory forces of steady-flow theories do not include the effects of the vortices shed into the wake.

Helmholts theorem: The total vorticity will always vanish within any closed curve surrounding a particular set of fluid particles.

Three separate physical phenomena

1. direction of the wind changes → the effective a.o.a. changes → lift changes

2. airfoil motion disturbs flow → changes the vortex to be shed at T.E. → unsteady downwash changes → effective a.o.a. changes → lift changes

3. inertial forces are generated due to the air particles near the airfoil surface → apparent-inertia → not a.o.a. is changed but lift and pitching moment are changed

 

5.5.1 Theodorsen's Unsteady Thin-Airfoil Theory (simple harmonic motion)

◈ k method (Z(1+ig)) and Theodorsen's method and classical method (k, σ)

※ unsteady aerodynamics for thin airfoil, small and simple harmonic oscillations, incompressible flow

Lift includes both circulatory(순환,

) and non-circulatory(비순환,

) terms.

Moment includes only non-circulatory

 term.

Theodorsen's Function : 

, k increases → |F(k)| decreases & |G(k)| increases

 

A few things to note :

lift-curve slope is equal to 2π.

non-circulatory terms depend on the acceleration(h'') and angular acceleration(θ'') of the airfoil .

non-circulatory terms are mostly apparent-mass/inertia terms.

 

For steady flow, circulatory lift is linear. (not handle this in current section)

For unsteady flow, no single a.o.a. because the flow direction varies along the chordline as the result of the induced flow varying along the chord.

effective a.o.a. (not really time domain) :

a.o.a. in time domain : α=

 

5.5.2 Finite-State Unsteady Thin-Airfoil Theory of Peters et al. (non simple harmonic motion)

 

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(((5.6 Flutter Prediction via Assumed Modes)))

 

5.7 Flutter Boundary Characteristics

Determination of Flutter boundary : altitude (air density), speed, Mach number.

 

Flutter speed increases as mass ratio (μ) increases. μ = m / (π ρ∞ b²)

1. μ ∝ 1/ρ∞ → mass ratio increases as air density decreases. → flutter speed increases as altitude increases.

any flight vehicle is more susceptible(민감한) to aeroelastic flutter speed @ low altitudes rather than high. → low altitude ∝ low flutter speed, easy to reach this speed in low altitude.

2. μ ∝ m, mass per unit span of the lifting surface // μ ∝ 1/b², half of chord

 

Big dip is observed in the plot of flutter speed versus frequency ratio for the wings of most high-performance aircraft (large mass ratio & positive static unbalances)

center of mass is moved forward of the reference point, flutter speed is high

xθ=0, 즉, CP=0, c.g.가 피벗포인트에 있음.

μ가 낮을 때, 즉 ρ∞가 높을 때 (낮은고도일때), UF가 낮아서 쉽게 flutter speed에 도달하지만

μ가 증가함에 따라 UF도 같이 증가, flutter speed에 도달하기 점점 어려워짐 (??)