Introduction – Equations of motion G. Dimitriadis 02

Aeroelasticity Lecture 2: Dynamic Aeroelasticity G Dimitriadis Introduction to Aeroelasticity Dynamic Aeroelasticity •! The full equations of motion can be readily solved analytically •! Define x=[h !]T •! Then assemble the equations of motion in the form •! Where M=A+!"b2B Introduction to Aeroelasticity Solution of Equations •! Now the equations of motion are of first order, in the form •! Such equations can be solved by trying a solution of the form •! Where l are the eigenvalues of the system and can be obtained from the characteristic polynomial Introduction to Aeroelasticity Frequency and Damping •! The absolute values of the eigenvalues are the natural frequencies, #n=|$| •! The damping ratios are defined as: %&=-Re($)/#n •! The damping ratios are measures of the amount of damping present in each mode of vibration •! It must be kept in mind that both natural frequencies and damping ratios are functions of airspeed and air density Introduction to Aeroelasticity Variation with airspeed As the airspeed increases, the two natural frequencies approach each other One of the damping ratios increases while the other first increases and then decreases The critical damping ratio becomes zero and then negative Instability ensues This phenomenon is called flutter and the zero damping speed is the flutter speed Introduction to Aeroelasticity Subcritical System response Solve the equations of motion for the time responses of the system from initial conditions ('(0)=5o) Time responses for U=15m/s Both pitch and plunge decay with time Introduction to Aeroelasticity Critical System Response Solve the equations of motion for the time responses of the system from initial conditions ('(0)=5o) Time responses for U=18m/s Both pitch and plunge oscillation amplitudes remain constant Introduction to Aeroelasticity Supercritical Responses Solve the equations of motion for the time responses of the system from initial conditions ('(0)=5o) Time responses for U=20m/s Both pitch and plunge oscillation amplitudes increase with time Introduction to Aeroelasticity Stability Analysis •! The static divergence and flutter speeds can also be obtained directly from the characteristic polynomial •! This can be achieved using the RouthHurwitz stability criterion •! The criterion applies to a polynomial of the form Introduction to Aeroelasticity Routh-Hurwitz (1) •! The system is unstable if –! any of the coefficients is zero or negative while at least one is positive –! There is at least one sign change in the first column of the matrix H •! The matrix H is given by Introduction to Aeroelasticity Routh-Hurwitz (2) •! The condition a0[...]... aeroelastic equations of motion and Matlab: –! Create a computer model of a 2D wing with pitch and plunge degrees of freedom –! Choose values for all the relevant model parameters –! Determine the flutter airspeed and flutter frequency at sea level –! What is the effect of altitude on flutter airspeed and frequency? –! What is the effect of pitch and plunge stiffness on flutter airspeed and frequency? Introduction. .. aeroelastic equations of motion for a pitch-plunge 2D wing are: where Introduction to Aeroelasticity Hint •! For determining the flutter airspeed: –! You can used Routh-Hurwitz; this will only work for a 2 degree -of- freedom system –! Or, preferably, you can use an eigensolution coupled with an indirect search; this is the more general case since it works for any size of aeroelastic system with any number of. .. centre of gravity is at the half-chord Introduction to Aeroelasticity Flexural axis (2) For this NACA 0012 symmetric airfoil, the centre of gravity is ahead of the half-chord The divergence speed becomes lower than the flutter speed at around xf/c=0.5 The NACA 0012 has the same dimensions, structural stiffness and mass as the flat plate of the previous example Introduction to Aeroelasticity Exercise... Watch out for the following issues: –! The pitch-plunge aeroelastic system has two airspeeds at which "=0: U=Uflut and U=0 The Newton-Raphson solution can converge towards either of them –! Newton-Raphson schemes generally converge to the solution closest to the initial guess –! If the initial guess is very far from a solution, the Newton-Raphson scheme will not converge Introduction to Aeroelasticity ... works for any size of aeroelastic system with any number of degrees of freedom –! You can also used a directed search but this is more advanced Introduction to Aeroelasticity Indirect search •! A trial and error search algorithm for pinpointing a bifurcation condition, e.g the flutter condition "=0 The search is performed in terms of a bifurcation parameter, e.g airspeed U •! You start at a low airspeed... •! Two of the solutions are U=+0 and U=-0 •! The other two solutions are U=+Uflut and U=U=-Uflut Introduction to Aeroelasticity Flexural axis (1) The position of the flexural axis has a significant effect on both flutter and static divergence For this flat plate envelope the flutter speed is always lower than the static divergence speed, unless xf/c>0.75 This is due to the fact that the centre of gravity... accuracy, e.g "=-10-6 Introduction to Aeroelasticity Directed search •! The objective is to reach the flutter condition "=0 This is equivalent to: F (U ) = max(Re(! (U ))) = 0 •! This is a nonlinear algebraic equation that can be solved using the Newton-Raphson method: !F !U U 0 "U = #F (U 0 ) •! Where U0 is an initial guess for the flutter speed and !U is a correction to this guess Introduction to Aeroelasticity