The eigenmodes of a clamped-free beam may be easily determined from this equation.

This plot shows variation of two functions: -cos(x) and the inverse of cosh(x).

We can notice that there is infinite of solutions, that k_1 equals 1.9 and k_N is very close to

(2N-1)pi/2 for higher modes. The eigenfrequencies omega_N can be derived from the k_N values.

Let us now consider the eigenmodes: The first bending mode gives a global bending

in the same direction for the whole beam The second bending mode leads to one node

and one antinode. An additional vibration node is located at the left clamped end of

the beam. As you can see, these functions are not sinusoidal at all.

The third mode gives two nodes, plus one at the left end, and two antinodes in bending

You have just considered a first set of boundary conditions for the clamped-free case. What

happens for different boundary conditions? Let us consider a free-free condition for

a straight beam. We will now have zero shear force and bending moment at both ends of the

beam! The equation to solve is a bit different:

cos kL cosh kL - 1 = 0. As shown by this graphical solution , it changes

the whole process since k_1=0, it is a bit strange since it corresponds

to omega_1 equals zero Hertz. Its means an infinitely slow motion. The explanation

for this is that we get a rigid body motion due to the free-free boundary conditions!

For higher modes, the k_N values are very close to (2N-1)pi/2 but the mode shapes are

very different than in the clamped-free case! Let us now consider the free-free eigenmodes.

The first eigenmode gives a global translation of the beam since it is a rigid body mode

related to a zero eigenfrequency. The second free-free bending mode leads

to two nodes and one antinode. We can now imagine the free-free flying spaghetti!

The third free-free mode gives three nodes and two antinodes in bending

and so on, for the fourth mode and the higher modes. From both configurations, clamped-free and

free-free, we evidenced the strong influence of the boundary conditions.

Starting from a single beam, we may consider several beams assembled in a so-called frame

structure! This radar tower located in the Ecole Polytechnique

campus is a very nice example of frame structures. The general shape is cylindrical but it is

an assemblage of many prismatic reinforced concrete beams!

The mass, stiffness and inertias of the beams are gathered in mass and stiffness matrices

to solve a large eigenvalue problem! Here is the fifth bending mode of the tower.

As you can see, the entire tower is bending as a whole! The local bending of each beam

may be small but it depends on the stiffness contrast between the posts and the circular

floors! Here is now the tenth eigenmode corresponding

to a torsional motion and showing several nodes and antinodes at the scale of the entire

tower. Finally the 14th mode corresponds to more

complex vibration kinematics leading to strong changes in the tower cross-section!

To summarize this video on beam modes - we determined the bending modes of a beam,

examined the influence of the boundary conditions on the modes,

and obtained the modes shapes which are much different in bending than in tension!

More complex structures can be easily modeled by the Finite Element Method, but it is very

important to know the beam dynamics behind!