In certain cases, radiation heat transfer is important to include in one’s calculations. Radiation heat transfer is nonlinear because the heat flux is proportional to the temperature to the fourth power. I find two frequent simplifications for radiation problems quite useful. The first approximation is to linearize and create an effective convection coefficient. The second approximation is the effective emissivity for two bodies which are transferring heat between each other via radiation.
One somewhat surprising result in the vibration of beams is that the natural frequencies of a free-free beam are identical to those of a clamped-clamped beam. Our intuition may tell us that the fixed-fixed beam is stiffer and should have higher natural frequencies but while the fixed beam is indeed statically stiffer, the natural frequencies are identical. As we show below, fixed and free boundary conditions result in the equivalent characteristic equations to find the natural frequency because they differ simply by two differentiations. Similarly, pinned and sliding boundary conditions also have the same equivalence. As a result many pairs of boundary conditions can result in identical sets of natural frequencies but with clearly different mode shapes.