08 Flutter Phenomenon PART 2 SECTION 2:

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08 Flutter Phenomenon PART 2 SECTION 2: CONTINUATION WITH THE FORMULATION OF THE UNSTEADY AERODYNAMIC FORCES + GENERAL FORMULATION + 2D TYPICAL SECTION IN INCOMPRESSIBLE FLOW REVIEW OF FLUTTER

1 08 Flutter Phenomenon PART 2 SECTION 2: CONTINUATION WITH THE FORMULATION OF THE UNSTEADY AERODYNAMIC FORCES + GENERAL FORMULATION + 2D TYPICAL SECTION IN INCOMPRESSIBLE FLOW REVIEW OF FLUTTER EQUATION WITH AERO FORCES IN THE FREQUENCY DOMAIN Vibraciones y Aeroelasticidad Dpto. de Vehículos Aeroespaciales P. García-Fogeda Núñez & F. Arévalo Lozano ESCUELA TÉCNICA SUPERIOR DE INGENIERÍA AERONÁUTICA Y DEL ESPACIO UNIVERSIDAD POLITÉCNICA DE MADRID

2 INVISCID + IRROTATIONAL FLOW POTENTIAL FLOW Once viscous effects are neglected Ref.: Chapter 2 of Fundamentals of Unsteady Aerodynamics Ülgen Gülçat

3 COMPRESSIBLE FLOW DIFFERENTIAL EQUATION Mass-conservation equation: Equations of motion w/o viscosity: 1 2 Assumption of Irrotational Flow 1) 2)

4 LINEARIZATION : STEADY + UNSTEADY PROBLEMS COMPRESSIBLE FLOW DIFFERENTIAL EQUATION LINEARIZATION (STEADY+UNSTEADY) Small perturbation method: Thickness & Steady problem ( 0 ) + Unsteady problem ( 1 ) STEADY PROBLEM UNSTEADY PROBLEM

5 COMPRESSIBLE FLOW BOUNDARY CONDITION Equation of the airfoil surface: Condition of flow in permanent contact with the surface: Unsteady-flow condition

6 COMPRESSIBLE FLOW FORMULATION OF THE PRESSURE COEFFICIENT

7 1935: Theodore Theodorsen (t)

8 UNSTEADY INCOMPRESSIBLE FLOW DIFFERENTIAL EQUATION + BOUNDARY CONDITIONS Boundary condition: (*) WING IS A FLAT PLATE Far Field: Kutta condition at trailing edge: (*) ATTACHED FLOW HYPOTHESIS VALID FOR SMALL MOTIONS

9 ELEMENTARY SOLUTIONS OF THE 2D LAPLACE EQUATION 2D VORTICES AT THE AIRFOIL AND THE WAKE A 2D Vortex singularity dx is an elementary solution of 2 1 =0 as the potential d 1 written below that is induced by the 2D vortex satisfies the Laplace equation: Vortices are distributed along the 2D section and the wake:

10 OVERVIEW OF THE THEODORSEN S APPROACH (*) FLAT WAKE IS IMPLICITLY ASSUMED Goldstein

11 C p1 AS FUNCTION OF THE RUNNING CIRCULATION a

12 C p1 AS FUNCTION OF THE AIRFOIL MOTION C k F k i G k for the boundary condition Note: H n2 ( ) k are Hankel functions of the second kind that are expressed in terms of Bessel (2) functions of the first and second type as H k J k i Y k n n n

13 GENERALIZED FORCES AS INTEGRAL OF C p1 Boundary condition (vertical speed): Vertical displacement (to obtain ): Substituting in the equation for of previous slide: The integration of leads to the generalized forces:

14 CALCULATION OF GENERALIZED FORCES TYPICAL SECTION WITH UPDATED NOMENCLATURE

15 3 DOFs TYPICAL SECTION : h(t), (t), and (t) NONCIRCULATORY TERMS = APPARENT-MASS CIRCULATORY TERMS

16 T1 to T14: GEOMETRICAL CHARACTERISTICS OF SECTION

17 QUASI-STEADY FORMULATION Motion at low reduced frequency, i.e., k 1. Fung (1983) approach: Unsteadiness is retained exclusively in the boundary condition over the section. Approach valid for incompressible or subsonic flow. The vortices distribution is written as:

18 QUASI-STEADY FORMULATION Vertical motion (plunging or heaving): Rotation/Torsion Control surface rotation

19 Escuela Técnica Superior de Ingeniería Aeronáutica y del Espacio UNIVERSIDAD POLITÉCNICA DE MADRID

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