Identification of Amplitude-Dependent Damping Ratio and Flutter Derivatives of a Streamlined Box Girder under Various Wind Angles of Attack

Acknowledgments

Notation

$\mathcal{A}(t_j)$

back-calculated instantaneous amplitude envelope;

A_div,α, A_div,h

largest torsional and vertical amplitudes recorded by the sampling system when the section model is performing a divergent hard-type flutter;

A_exc,α1, A_exc,α2

artificially excited initial torsional amplitudes with different levels;

A_exc,h1, A_exc,h1

artificially excited initial vertical amplitudes with different levels;

A_final,α, A_final,h

final torsional and vertical amplitudes of the section model at each wind speed;

${\mathcal{A}}_l$ $(l=1\sim M)$

segment of the amplitude envelope;

${\mathcal{A}}_s(s=\alpha ,h)$

vibration amplitude in the vertical–torsional direction;

${\mathcal{A}}_t$

tested instantaneous amplitude of a vibration time history with noise included;

A_uα,cal, A_uh,cal

calculated unstable LCOs corresponding to torsional and vertical motions;

A_α,cal, A_h,cal

calculated stable LCOs corresponding to torsional and vertical motions;

${\mathcal{A}}_y$

instantaneous amplitude of motion (i.e., amplitude envelope);

${\mathcal{A}}_{y0}$

steady-state amplitude of a stable LCO;

a₁, a₂, t₀

coefficients of the prescribed function related to the GTR response;

b

half-width of a bridge deck;

E[·]

mathematical expectation;

e

fitting error of the function F;

F

prescribed function used to fit the amplitude envelope of motion;

f_s

sampling frequency;

$H_i^{*}$, $A_i^{*}$ (i

1–4) = amplitude-dependent flutter derivatives;

h, α

vertical and torsional displacements;

I

mass moment of inertia per unit length of the coupled oscillating system;

k

reduced frequency, =ωb/U;

L_se(t), M_se(t)

self-excited forces (lift force and moment, respectively);

M

total number of segments;

m

mass of oscillating system per unit length;

N

sample number of the fitted instantaneous amplitude at time instant t_j;

$N_{{\mathcal{A}}_l}$

data point number of a segment;

N_T

total number of data points;

N_w

number of segments included in a window;

n

logarithm of the instantaneous amplitude of noise;

t

time;

U

mean wind speed;

y

free vibration displacement time history of a motion;

Δt

time increment of each segment;

μ

nondimensional mass of an oscillating system, =ρb²/m;

υ

nondimensional effective mass moment of inertia, =ρb⁴/I;

φ

instantaneous phase of motion;

ρ

air density;

ω

instantaneous circular frequency of motion, = 2πf_y;

ω_h

natural circular frequency of the vertical oscillating system, =2πf_h;

ω₂

modal circular frequency of the coupled oscillating system in the torsional modal branch;

${\tilde{\omega }}_h$

modal frequency of the SDOF vertical oscillating system;

${\omega }_{\alpha }$

natural circular frequency in the torsional direction of the oscillating system, $=2\pi f_{\alpha }$;

Ψ

amplitude ratio between vertical and torsional motions, $={\mathcal{A}}_h/(b{\mathcal{A}}_{\alpha })$;

ψ

phase difference between vertical and torsional motions;

ξ_y

time-varying (or amplitude-dependent) damping ratio;

ξ_y,l(t_j)

damping ratio corresponding to the t_j time instant at the lth window;

ξ_y,mean

mean value of the damping ratio at each time instant;

${\xi }_{\alpha }$, ξ_h

nonlinear torsional and vertical structural damping ratios;

${\tilde{\xi }}_h$

modal damping of the SDOF vertical oscillating system;

ξ₂

modal damping ratio of the coupled oscillating system in the torsional modal branch; and

$\vartheta$, R_d2, ${\overline{\omega }}_1$, ${\overline{\xi }}_1$

parameters related to the coupled flutter closed-form solution.

References