NON-LINEAR OSCILLATIONS OF A FLUTTERING PLATE Submitted by SUREKHA REDDY KOLLI-27750803 UVNINDERJEET SINGH MALHANS-40013356 KIRAN NIVRUTTI CHAVANKE-40001151 A Final Report Submitted for MECH 6481 PRINCIPLE OF AEROELASTICITY Concordia University

ABSTRACT The flutter phenomenon is considered to be the most studied and of principle concern in the field of Aeroelasticity. In this project report we address about the Aeroelasticity response and stability for a fluttering plate, which is nonlinear. Initially two dimensional semi-infinite plate undergoing cylindrical bending is modeled. The plate has been considered to have no span wise bending. Von Karman's large deflection plate theory and quasi-steady supersonic piston theory have been employed. The effects of (constant) in plane load and static pressure differential have been included. Galerkin's method was used to reduce the mathematical problem to system of nonlinear ordinary differential equations in time which are solved by numerical integration. A computer code in Maple Software is used to solve the computer oriented numerical method. The results from the code agree reasonably with those from the reference project paper.

INTRODUCTION LITERATURE REVIEW One of the most interesting problems in Aeroelasticity is the Flutter phenomenon. A semi-infinite plate is bounded in one direction, and unbounded in another [35].Consider a Semi Infinite Plate in x-y plane the ‘y’ dimension of plate as doubly infinite that is y = +∞ and y = -∞ then, considering the response of two dimensional plate in a high supersonic flow to a disturbance. In this chapter, for mathematical formulation the Aeroelastic response of two dimensional plates in high supersonic flow is considered. With the concept of linear plate theory, there is a value of air velocity (or dynamic pressure) above which the plate motion is unstable, and the response grows exponentially with time [13]. The nonlinear membrane forces induced by the plate limits the amplitude. In this present paper these nonlinear forces will be included in the analysis via Von Karman's large deflection equations or their two-dimensional equivalent. Quasi-steady (linear) aerodynamic theory is employed. LITERATURE REVIEW The earliest study of flutter seems to have been made by Lanchester[1], Bairstow and Fage [2] in 1916. In 1918, Blasius [3] made some calculations after the failure of the lower wing of Albatross D3 biplane. But the real development of the flutter analysis had to wait for the development of Non-stationary airfoil theory by Kutta and Joukowsky. Glauret[4,5] published data on the force and moment acting on a cylindrical body due to an arbitrary motion. In 1934, Theodorsenís[6] exact solution of a harmonically oscillating wing with a flap was published. The torsion flutter was first found by Glauret in 1929. It is discussed in detail by Smilg[7] . Mathematical formulations of flexure-torsion problem have been broadly described by Mirovitch [15], Thomson [28], Hurty and Rubinstein [29] and Y.C.Fung [12]. Talukedar, Kamle and Yadav [30] have discussed an analytical method for flexure-torsion coupled vibration of vehicles leading to aircraft application.

PROBLEM FORMULATION Assumptions A semi-infinite plate is been considered Two dimensional plate undergoing cylindrical bending is assumed The aerodynamic pressure loading is assumed to be quasi-steady supersonic piston theory Galerkin’s method has been employed to transform the partial differential equation to ordinary differential equation. Assumptions for the boundary conditions for number of modes for Galerkin’s method, Eigen Value Function Modulus Of Elasticity Poisson’s Ratio Plate Thickness Plate Stiffness Air Density Air Velocity Plate Length Dynamic Pressure As, derived in 2.2, the equation of motion for a two-dimensional plate undergoing cylindrical bending (no span wise bending) is Where;

Substitute the above two equations in the equation of motion we get; By using the Galerkin’s method we obtain By multiplying equation with sinsπξ and integrate over the panel length we get;

RESULTS Convergence Study When we are calculating values for 2 number modes, it gives inaccurate results. To find accurate solution we are considering number of 6 modes, for accurate results. For 6 modes we obtain convergence result. As static pressure loading is goes on increasing we should consider more modes for appropriate result. Above mentioned graphs obtained for the Non-Dimensional Limit Cycle Amplitude Vs Non-Dimensional Dynamic Pressure λ for number of modes equal to 4 and 6 respectively.

Non-Dimensional Vs Time Previous graphs are the Non-Dimensional Limit Cycle Amplitude Vs Non- Dimensional Dynamic Pressure (λ=350,375,400,425,450,475,500) for values for number of modes equal to 4 and 6 respectively. Non-Dimensional Vs Time Non-dimensional Limit Cycle Amplitude Vs Non-dimensional Dynamic Pressure Dynamic pressure λ (w/h) 350 0.211 375 0.443 400 0.558 425 0.685 450 0.826 475 0.931 500 1

FLUTTER ANALYSIS Flutter is a dynamic instability of a flight vehicle associated with the interaction of aerodynamic, elastic, and inertial forces. This Graph explains the dynamic pressure λ for air stream velocity (V=1000, 1250, 1500, 1750) Vs non dimensional deflection. This graph is not sufficient to find out the air stream velocity at which flutter occurs. So this graph is introduced as air stream velocity Vs deflection. In graph 8 the nature of non-dimensional deflection is approximately constant up to certain velocity but at the velocity 1250 m/s is resulted in sudden flutter occurrence as 0.029 deflection at x/l=0.75.

CONCLUSION From all the study of flutter to non-linear oscillations of a fluttering plate, it could be concluded that, When considering modes to be two it is observed to have inaccurate results. When there is an increment in the consideration of number of modes from two or four to six, accurate results were obtained. This states that six modes selection would be implied for accurate results. As static pressure loading is goes on increasing we should consider more modes for appropriate result. Especially, in our case for the semi-infinite plate which is infinitely long in y direction which is undergoing cylindrical bending have higher possible occurrence of flutter, where it has been observed to experience flutter at an airstream velocity at 1250m/s and the deflection tends to reach at the point 0.029.   THANK YOU