Small oscillations of truss bridge

The bridge consists of massive elastic rods with the same elasticity and mass, which are fixed in massless joints (points) that offer no resistance to rod rotation (i.e., physically they are rotationally mounted in the joints instead of being firmly welded). The entire bridge is fixed at the lower left and lower right points (these are therefore immovable).

The points are allowed to move in (one) plane, meaning the total number of degrees of freedom (and thus also the number of modes in the small oscillation problem) is twice the number of internal points (joints) of the bridge.

The equilibrium position is chosen so that the bridge is exactly "regular" in it (i.e., the deck is horizontal, all triangles in the bridge are equilateral, etc.). This can be achieved by turning off gravity and choosing the rods unloaded in the "regular" position, or with gravity pre-tensioning the rods so that they exactly compensate for the gravitational force in the "regular" position. Due to the linearity of the elastic force, these two situations are equivalent.

Modes are found by the classical small oscillation method in the Lagrangian formalism. The only non-trivial required component is the kinetic energy of an elastic massive rod, which one does not encounter in TEF or VOAF.

Source code: .nb file – everything is in one large code block that just needs to be run. The primarily important variable in it is pocetN, which indicates the number of joints/points on the deck, or the number of deck sections minus one.

Pictures with red-blue arrows graphically represent amplitude ratio vectors – they show in which directions individual bridge points oscillate in the given mode.

And now follow a few selected examples.

Oscillations of bridge with two-section deck

Oscillations of bridge with three-section deck

Oscillations of bridge with eleven-section deck