Introduction – Equations of motion G. Dimitriadis 04

Aeroelasticity Lecture 4: Theodorsen for non-sinusoidal motion G Dimitriadis Introduction to Aeroelasticity Time domain responses •! Theodorsen analysis requires that the equations of motion are only valid at zero airspeed or at the flutter condition •! They are also valid in the case of forced sinusoidal excitation •! We can calculate the response of an 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 #"1& $ 'F0 %x f ( •! This equation is of the form H(!)q0=F, where H-1(!) is the Frequency Response Function Introduction to Aeroelasticity FRF for pitch-plunge system !'!!( ! Introduction to Aeroelasticity ! " # $ )*+,-+./012345 % 632!5!&617.1" & ! !& &! ! " # $ )*+,-+./012345 % &! ! " # $ )*+,-+./012345 % &! !'( !'!9 !'!" !'!& ! " !" !'!# FRF of " The first mode is present as an antiresonance :8;132!5!&17.18 The two modes are clearly present !'!& :8;132!5!&17.1" FRF of h 632!5!&617.18 !'!&( ! !!'( !& !&'( !" !"'( !9 ! " # $ )*+,-+./012345 % &! Working with the FRF •! If the force is non-sinusoidal, F0=F0(!) •! The system’s response to such a force is obtained as q0(!)=H(!)-1F(!) •! If F(!)=1 then the inverse Fourier Transform of q0(!) is the system’s impulse response •! The impulse response can also be used to perform stability analysis Introduction to Aeroelasticity Impulse response of pitch-plunge airfoil !) !# 9:&),;/[...]... domain is using Roger’s Approximation •! The frequency-dependent part of equations (2), Q(jk), is approximated as: 2 nl Q( jk ) = A 0 + A1 jk + A 2 ( jk ) + # A 2+n n =1 jk jk + " n •! Where nl is the number of aerodynamic lags and "n are aerodynamic lag coefficients Introduction to Aeroelasticity Roger’s EOMs •! The equations of motion of the complete aeroelastic system then become: $ "M "1C "M "1K "M... 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 •! Bisplinghoff, Ashley and Halfman wing •! FEM with 12 nodes and 72 dof Introduction to Aeroelasticity First 5 modes of BAH wing Introduction to Aeroelasticity GTA Example •! Here is a very simple aeroelastic model... rectangle for the wall Introduction to Aeroelasticity Flutter plots for SST First 9 flexible modes Clear flutter mechanism between first and third mode Introduction to Aeroelasticity Practical Session •! You are required to design a pitch-plunge flat plate with the following characteristics –! –! –! –! Chord length: 0.4m Material: aluminium Maximum flight altitude: 2000m (air density of 1kg/m3) Maximum...Basics (3) •! Therefore, the equations contain terms that depend on frequency •! The basis of the p-k method is to define •! Then, the equations of motion become 1 $ 2 & 2 p M s + K s " #U Q( p) q = 0 % ' 2 •! Where q=[h "]T Introduction to Aeroelasticity Using p! •! Using the p notation, the Q(p) matrix becomes: 2 ( p 2$ p& "2#cC... real parts of the p values will have the correct value Introduction to Aeroelasticity 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 1 $ 2 & p M s + K s " #U 2Q( jk ) q = 0 % ' 2 (2) •! Since it is known that the aerodynamic matrix is only a function of frequency (not of damping) •! Again, k=!b/U Introduction. .. not on flight condition Introduction to Aeroelasticity The p-k solution •! The solution of these equations is iterative •! We guess a value for the frequency ! (and hence k) and then we calculate p from the 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... degrees of freedom Introduction to Aeroelasticity 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 V