The knowledge of the vibration behavior of a system is very important
for an engineer. Because in this case it is possible to take influence on
the system in order to prevent resonance phenomena or to use vibrations in
an effective way. For this purpose, it is necessary to consider the natural
vibrations of a system and to calculate its natural frequencies. With an
increasing complexity of the systems the exakt way of calculating is no
longer practical so that methods for approximation are used. Therefore,
this thesis covers different methods for calculating the natural
frequencies. Only free, undamped vibrations of Euler-Bernoulli beams in
conservative systems are considered. First, the thesis takes a look at the
theory of beam vibrations and shows a method for exactly calculating the
natural frequencies. After this, different methods of approximation are
portrayed: the Rayleigh-quotient, the method of Rayleigh-Ritz, the finite
element method, the formulas of Southwell and Dunkerley, and the
approximation using a multibody system. Afterwards, the methods are applied
to three introductory examples and examined, where the Rayleigh-quotient
and the method of Rayleigh-Ritz are focussed. As the precision of the
approximations of these methods depends crucially on the choice of the
ansatz functions, this thesis deals with new possibilities in order to
extend the varieties of the ansatz functions for both methods. For this
reason, the hyperbolic functions, the static bending lines and a piecewise
handling of more complex systems are considered. First, these possibilities
for an ansatz function are examined with the Rayleigh-quotient. If the
possibilities work for this method, they are tested with the method of
Rayleigh-Ritz. Finally, two complex examples are considered in order to use
the previous results and to examine their practicability.