running...
The science
Torsional flutter. The deck is a section of polar moment I per unit length rotating by theta about its axis, restored by torsional stiffness k, with structural damping zeta and a wind-induced self-excited term (Scanlan form, first order in theta rate):
I theta'' + (2 zeta I w0 - (1/2) rho B^3 a2 U) theta' + k theta = 0, w0 = sqrt(k / I)
The net damping vanishes at the critical (flutter) wind speed
U_cr = 4 zeta I w0 / (rho B^3 a2)
Above it the amplitude grows as exp(sigma t) with sigma = (rho B^3 a2 U / 2 - 2 zeta I w0) / (2 I). A cubic aerodynamic stall term limits the motion to a finite limit cycle, as in the van der Pol oscillator. Stiffer decks raise w0 and U_cr; wider decks raise B and lower U_cr for a fixed a2 in this simple model (real decks are redesigned with trusses, slots and fairings that change a2).
Cable statics. A uniformly loaded cable of span L and sag f is a parabola with horizontal tension H and end tension T_max:
H = w L^2 / (8 f), T_max = sqrt(H^2 + (w L / 2)^2)
Vortex shedding. A bluff deck of depth D also sheds vortices at f_s = St U / D (Strouhal number St about 0.2). Lock-in occurs when f_s matches the structural frequency, giving a smaller, wind-speed-limited vertical or torsional vibration, a different mechanism from flutter.
Honest simplifications. The deck is a single torsional degree of freedom with constant parameters; real flutter couples bending and torsion, uses wind-tunnel flutter derivatives and depends on frequency. The coefficient a2 here is calibrated so that the default deck has U_cr near 19 m/s, similar in size to the Tacoma Narrows event; it is not a measurement of that bridge. Turbulence is a small random torque. The 3D page uses a Verlet deck with a different structure; this 2D model gives the physics a closed form to test.