Lab · Live model · 04

Energy between the modes of a beam.

The modes of a linear beam keep their energy. In a geometrically exact beam they exchange it, and the paper shows that the axial modes make that exchange possible. Its smallest model couples the first bending mode with the first axial mode, and gives a bound on the energy that the bending mode keeps on average.

Amplitude, 5 % to 40 % of the length. In Auto the model sets the parameter and runs the cases of the paper; in Hold the slider sets it. The line below the control says what the model does at this moment. With reduced motion the model shows a fixed frame, and the controls still work.

What the drawing shows

The beam bends in its first mode and keeps its length, so its tip curls in. The ticks along it move with the axial mode, drawn six times larger than it is. The grey shapes are the beam at recent instants.

The bars are the share of the energy in each mode at this moment: the accent bar is the bending, the ink bar is the axial mode. The short marks are the running averages since the start of the run. The dashed line is the bound on the average of the bending: the running average stays under it.

The model

This is the simplest quadratic coupling between a bending mode and an axial mode, which is the class of nonlinearity the intrinsic beam equations of the paper have:

\ddot{x_{1}} + ω_{1}^{2} x_{1} + ε x_{1} x_{2} = 0

the bending; the axial force changes its stiffness

\ddot{x_{2}} + ω_{2}^{2} x_{2} + \frac{ε}{2} x_{1}^{2} = 0

the axial mode; the square of the bending stretches it

H = \frac{1}{2} ({\dot{x_{1}}}^{2} + {\dot{x_{2}}}^{2}) + \frac{1}{2} (ω_{1}^{2} x_{1}^{2} + ω_{2}^{2} x_{2}^{2}) + \frac{ε}{2} x_{1}^{2} x_{2}

the energy, which stays constant

E_{1} = \frac{1}{2} ({\dot{x_{1}}}^{2} + ω_{1}^{2} x_{1}^{2}), E_{2} = \frac{1}{2} ({\dot{x_{2}}}^{2} + ω_{2}^{2} x_{2}^{2})

the energy of each mode

\bar{E_{1}} = \underset{T \to \infty}{lim sup} \frac{1}{T} \int_{0}^{T} E_{1} d t \leq C

the bound on the time average of the bending

The sum of the energies is constant, so what the axial mode gains the bending loses. The dashed line is not the bound of the paper: the scene cannot solve a semidefinite program. It runs sixteen starts with the same energy and different shares and phases, takes the largest running average of the bending, and adds the 5 per cent the paper finds between its envelope and its bound. In the paper the axial mode is at 58 or 14 times the frequency of the bending; on the screen it is at 6 times, so that the eye can follow both.

The control

The slider sets the amplitude of the motion, and so the size of the nonlinear terms. At 5 per cent the beam is almost linear and the bending keeps its energy. At 40 per cent it gives almost a third of it away on average. Auto runs four amplitudes in turn, 14 seconds each, and each one restarts the run and the averages.

Where it comes from

The scene draws the background of the paper Capturing & Bounding Nonlinear Modal Energy Transfer for Geometrically Exact Beams using Semidefinite Programming.