Dynamics: Flutter Flashcards
Discuss eigenvalues and stability.
If an eigenvalue has a positive real part, then it is unstable. More specifically:
*When λ is on the positive Re axis (Im=0), the solution is divergent exponential.
*When λ is on the negative Re axis (Im=0), the solution is damped exponential.
*When λ is on the imaginary axis (Re=0), the solution is sinusoid.
*When there is a complex conjugate on the positive real axis, the solution is divergent sinusoid.
*When there is a complex conjugate on the negative real axis, the solution is damped sinusoid.
What are the solutions to the following problem:
Mq_dot_dot + Kq = 0
The eigenvalues are all stable, but not asymptotically stable, because the real part is null. The two frequencies grow separately.
How do we characterise an eigenvalue?
The characteristic polynomial or characteristic equation is: det(Α(λ)) = 0 -> aλ^4 + bλ^2 + c = 0. The solution to this algebraic equation are for complex eigen values, λ = σ + jω, where σ is the real part and ω is the imaginary part. ω = Im(λ) is the damped natural frequency, while ξ = -Re(λ)/|λ| is the damping ratio. If ξ > 0 the system is asymptotically stable, if ξ=0 the system is stable, while for ξ<0, the system is unstable.
How can someone compute the damping ratio?
1) From the logarithmic decrement δ, which take the ln of the value of the solution at t and t+T and calculates the damping ratio ξ using ξ=1/sqrt(1 + (2π/d)^2)
2) Often in aeroelasticity, the damping parameter ξ is used which is equal to: g=Re(λ)/Im(λ). For real eigenvalues g->inf this is not a good approximation. If ξ«1, then g ~ ξ.
What are the solutions if steady aerodynamics are included to the problem Mq_dot_dot + Kq = 0?
The problem is transformed to Mq_dot_dot + (Ks-qKa)q = F0. Taking the determinant of the homogeneous problem, the solution is aλ^4 + bλ^2 + c. The eigenvalues depend on the value of b (if it’s positive or negative). In general, b>0 is valid for q=0 and the first range of values of q, while b < 0 for very large q values.
*If b^2 - 4ac > 0 and b>0, the 4 eigenvalues are complex numbers with the real part equal to zero and the system is stable.
*If b^2-4ac <0, then at least one of the eigenvalues λ will have the real part positive and the system is unstable.
What is the limit of stability?
The limit of stability represents the condition where the system is transformed from being stable into an unstable one. This condition is also identified as flutter. To identify it, it is necessary to find the dynamic pressure at which the discriminant of the characteristic equation changes sign (from positive to negative).
What happens at flutter?
Initially, the eigenvalues are located only on the imaginary axis. As the dynamic pressure increases, the two eigenvalues coalesce, so that both the plunge and pitch oscilate at the same frequency. Also, the damping ratio crosses the x axis and becomes negative, which marks when the instability starts.
If the dynamic pressure is further increased, the two eigenvalues are null, which is the divergence condition.
What is the base of the flutter phenomenon?
The base of the flutter phenomenon is the rise of a phase shift between plunge and pitch. If there is a phase shift between plunge and AoA (or pitch), then there is a phase shift between lift and plunge. The maximum plunge does not correspong to the maximim of pitch, so the maximum of lift. The system generates an hysteretic cycle, so in this case the airflow is pumping energy in the system. This energy increases the amplitude of oscillation, and so the flutter phenomena.
This phase shift generates or dissipates energy which in turn grows the amplitude. The stability or instability is generated by the phase shift.
What is the quasi-steady model?
In the previous model, the lift generated by the plunge movement was not included. The plunge movement generates lift, since it is plunging by a certain speed. Combined with the airspeed, there is a different direction of the final speed vector, meaning a different AoA.
Discuss the eigenvalues of the quasi-steady model.
The eigenvalues are initially purely imaginary. Increasing the dynamic pressure, the two eigenvalues become complex, one lightly damped, while the other increases damping until it reaches a dynamic pressure above which it crosses the right-hand plane (flutter condition).
By increasting the dynamic pressure further, one will go more into the unstable right plane, while the other will tend to fall on the real axis. Then, two branches of real eigenvalues are generated, one in the direction towards zero and one in the negative direction. When the eigenvalue crosses the point of 0, the system will diverge.
There is no need to get coincident frequencies to see a flutter. They tend to get closer but don’t meet.
What are unsteady aerodynamics?
So far, we have used an appromixation of aerodynamics by assuming that the aerodynamics adapt instantaneously to the change of boundary condition, so to the change of pitch or plunge movement. Working under the hypothesis of thin airfoil, whenever there is a change in the angle of attack, the lift will change and therefore there will be a vortex that will be released i n the wake and the intensity of this vortex is proportional to the change of lift.
The varation of lift on the airfoil causes a variation of vorticity released in the wake. The wake will be composed of vortices of variable intensity that are convected in the flow field. Those vortex induce effects on the airfoil that cause a change of lift distribution on it. There is an internal feedback loop in the aerodynamics as well.
What is reduced frequency?
If T = 2π/ω the interval of time necessary to complete a cycle, while τ=c/2U the time that a fluid particle approximately interacts with the airfoil, the a reduced frequency parameter can be introduced: k = T/τ * 2π = ωc/2V. If τ«Τ, then k«_space;and the interaction can be considered static, otherwise dynamic effects could be expected. The wave length also depends on k (η=πc/k). When k«, the wave length is large.
