The majority of the theory presented in this section is based on Chapter 8 in Theory of Bridge Aerodynamics, by E. Strømmen [40].
There are four different types of aerodynamic instability phenomena that can occur for suspension bridges. These are:
• Static Divergence
• Galloping
• Dynamic Stability in Torsion
• Flutter
The lowest wind speed at which one of these instabilities occur is referred to as the crit-ical wind speed, Vcr, or the aerodynamic stability limit. As previously mentioned, these instabilities are directly related to the self-excited forces of the bridge girder, which are described by the aerodynamic derivatives. Equation 2.7 can be rewritten to enlighten the effect of the aerodynamic damping and stiffnessCae,Kae, in the equation of motion:
M˜0η(t) + (¨ C˜0−C˜ae(V, ω)) ˙η(t) + (K˜0−K˜ae(V, ω))η(t) = 0 (2.19) This structural system is velocity- and frequency-dependent. The characteristic eigen-value problem from equation 2.19 have several formulations, and this thesis presents the formulation from by Einar Strømmen. Here, the response of a modal dynamic system is described by the response function whereI is the identity matrix, ζ is a diagonal matrix ofζi containing the damping ratio for the respective mode, ϕi, andωi is the in-wind resonance frequency for the respective mode,ϕi. The members of the matrices κaeand ζae are defined as
The stability limit of such a system is expressed by the critical wind speed,Vcrand a
corre-number of modes of the system,NmodxNmod. The solution of the eigenvalue problem will result inNmodstability limits for the respective mode shapes,ϕi. As Eˆ contains complex quantities, 2.24 implies the simultaneous conditions of 2.25 and 2.26.
Observing the eigenvalues resulting from solving 2.24, they consist of pairs ofωand values of V. For a static limit where ω = 0, it is simply defined by a critical wind speed, Vcr. Considering the dynamic stability, the response is assumed narrow-banded around either the in-wind preference frequency or the resonance frequency for a certain mode or mode combination. Out of all of the eigenvalues resulting from solving 2.24, the lowest value of Vcr with the associatedωr will determine the overall critical wind speed.
The four instability phenomena are described in the following subsections. In common for them all is that it is either the vertical motionrz, or the torsional motionrθ, or both, that are the dominating response quantities inr(x, t) = [ry rz rθ]T. Further, a decent estimate for the stability limit can be provided with only two modes. However, the thesis has included several modes, in order to investigate the stability limit in a greater detail.
It is still useful to express analytical solutions only considering the first two modes ϕ1 and ϕ2 and their associated eigen-frequenciesω1 and ω2. One of these modes containing a dominant component of φz and the other containing a dominant component of φθ. Simplifying the calculations, the modes can be described as
ϕ1(x)≈[0 φz 0]T (2.27)
ϕ2(x)≈[0 0 φθ]T (2.28)
By implementing this in the impedance matrix, it can be reduced to Eˆη(ωr, Vcr) =
whereωrare the resonance frequency. Establishing this reduced impedance function is use-ful when reviewing the four different types of structural behaviour close to the instability limit.
2.5.1 Static Divergence
The mode shape of static divergence in torsion can be simplified as
ϕ2(x)≈[0 0 φθ]T (2.39)
when considering the static instability limit,ωr= 0. Implementing this into 2.30 yields Eˆη(ωr= 0, Vcr) = 1−κaeθθ (2.40) This is not a dynamic problem, so the quasi-static value of A∗3 can be applied. As 2.40 goes to zero when κaeθθ = 1, the critical wind speed for static divergence can be derived from 2.33, as:
The mode shape for galloping has the lowest eigen-frequency, ω1 = ωz, with a main component ofφz, and can be simplified as
ϕ1(x)≈[0 φz 0]T (2.42)
As the resonance frequency related to the galloping mode is ωz(Vcr), this can be inserted into equation 2.30, to reduce the impedance matrix to
Eˆη(ωr, Vcr) = 1−κaezz −(ωr/ωz)2+ 2i(ζz−ζaezz)ωr/ωz (2.43) where κaezz and ζaezz is defined in 2.31 and 2.35, respectively. By setting both the real and imaginary part of 2.43 equal to zero the dynamic stability limit can be identified
ωr=ωz 1 + ρB2
However, in a dynamic instability the respective eigen-frequency is the lowest whereφθ is the dominating component, ω2 =ωθ. Applying this into 2.30, the impedance function is reduced to
Eˆη(ωr, Vcr) = 1−κaeθθ−(ωr/ωθ)2+ 2i(ζθ−ζaeθθ)ωr/ωθ (2.47) whereκaeθθ and ζaeθθ is defined in 2.33 and 2.37, respectively. Again, by setting the real and imaginary part of 2.47 equal to zero, the dynamic stability limit can be identified by
ωr =ωθ 1 +ρB4
From this, one can observe that instability in pure torsion will only occur if A∗2 attains positive values.
2.5.4 Flutter
Flutter is the dynamic instability phenomenon where rz and rθ couples. This coupling occurs via κaeθz and κaezθ in the impedance function 2.30, and is therefore most likely to occur between modes ϕ1 and ϕ2 with main components of φz and φθ, respectively.
Further,rz andrθ couples and obtain the same resonant frequency
ωr=ωz =ωθ (2.50)
Computing the critical wind speed for this case becomes easier if the impedance function is split into four parts
With all quantities given in 2.31-2.37. Again, both the real and imaginary part of 2.51 are set to zero to identify the critical wind speed.
Re As the ADs are dependent onωr and Vcr, solving 2.56 and 2.57 demands iterations until convergence. If the ratio ωr/ωz is higher than 1.5, Selberg’s formula van be useful to provide a first estimate of the flutter stability [40] [32].
Vcr = 0.6Bωθ·
However, for the estimation of critical wind speed in this thesis, a multi-mode approach is performed.
2.5.5 Multi-modal Flutter
While the bi-modal approach does provide reasonable approximations of the critical wind speed, it is widely recognized that a multi-modal approach provides more accurate results.
Also, when analysing the instability behaviour, a higher number of DOFs, in this case modes, will improve the accuracy of the prediction.
The multi-modal approach does not make assumptions regarding the mode shapes, so all three components are included:
ϕi(x) = [φy φz φθ]T (2.59)
As all three components are included, the expressions forκaeij and ζaeij in equations 2.21 and 2.22 are expanded. This is a tedious exercise and is omitted for the theory presented in this thesis.
By assuming a solution on formη= Φeλt one can rewrite equation of motion 2.19 as the following
λ2M˜0+λ(C˜0−C˜ae) + (K˜0−K˜ae)
ϕ=0 (2.60)
The resulting complex eigenvalues of the corresponding eigenvalue problem is on the form λn=−ζnωn+−iωnp
1−ζ2 (2.61)