Eigenfrequency

A natural frequency of a vibrating system is a frequency at which the system after a single excitation can swing as a proper shape.

If such a system from the outside vibrations are imposed, whose frequency coincides with the natural frequency, the system responds with weak damping with especially large amplitudes, which is called resonance.

Degrees of freedom

One writes a system to as many degrees of freedom as each other has independent movement possibilities.

"You will then see that such systems always have a number of natural frequencies, corresponding to the number of degrees of freedom "

A degree of freedom in spring pendulum

As the natural frequency of a system is determined by only one degree of freedom, can be explained using the example of an undamped spring pendulum. A ball of mass m suspended from a coil spring with the spring constant c. This is defined as force per deflection reacts with the spring. According to Newton's second axiom is the acceleration of the sum of all forces acting on the ball, proportionally. Weight and spring force are balanced out in the rest position, so they can be ignored. What remains is a deviation from the static spring force as the only force that must be considered. This force pulls the ball up when this is located below the rest position and pushes the ball down when this is above the rest. So is mass * acceleration equal and opposite to the c times the deflection z (t ), which varies with time t:

This linear homogeneous differential equation can be solved with the following approach:

If one uses the approach in the differential equation, it follows

Which only then for all times t is valid if the coefficient of the sine function by itself is zero.

Is also called the characteristic angular frequency. It is sometimes as large as the undamped natural frequency f0.

When forced to move the spring, at its upper end to the path corresponding to the spring force is no longer of the entire displacement of the ball, but only the difference between the deflection at the opposite end of the spring. The very first equation is therefore transferred in

The homogeneous solution corresponds to the problem described above and provides a free oscillation at the natural frequency is, the amplitude and phase depends on the initial conditions. Your superimposed as particular solution the forced oscillation

Without damping, the amplitude at resonance is infinite. With attenuation that is always present in the real world, reaches the vibration amplitude at the resonance frequency a maximum. With low damping, the natural frequency is only slightly lower than the undamped natural frequency ( frequency characteristics ).

A degree of freedom of a vibrating column of air or an electric wave

The "fixed" end of a shaft corresponding to the open end of an air column in a pipe, because there is the air pressure constant. Conversely, to the "free " end of a shaft the flameproof end of a vibrating column of air.

Waveforms at two fixed ends

Waveforms at two free ends

Multiple degrees of freedom

Systems with several degrees of freedom are described in analogy with a matrix equation:

It is [ M] is the mass matrix, [B ] is the damping matrix, [ C] is the stiffness matrix and { F } is the load vector. An investigation of the free oscillations of the undamped system leads to the general eigenvalue problem

This can be converted into a particular eigenvalue problem, as described in " eigenvalue problem ", to calculate the natural frequencies of the system.

Infinite degrees of freedom

Systems with infinitely many degrees of freedom have infinitely many natural frequencies, for example, a double-sided simply supported beam bending with the bending stiffness and mass per unit length, the deflection results, depending on the time and place of the following differential equation:

The articulated mounting on both sides is satisfied by an integer multiple of half-waves, and the corresponding approach

Yields the undamped natural angular frequencies

Examples

• A bell is struck, then oscillates with the natural frequencies. By damping the vibration over time decays. In this case, higher frequencies are attenuated more rapidly than deeper.
• A tuning fork is designed so that in addition to the lowest natural frequency ( pitch a, 440 Hz) are more hardly excited natural oscillations.
• Even in buildings natural frequencies can be excited. If music really is very quiet at the neighbors, it is possible that the bass is the same frequency with a natural frequency of the building, which manifests itself as a loud rumbling, without the music would be audible as such.
• Drums show a variety of possible resonant frequencies.
• Membranes of speakers. The partials lead to an undesirable disturbance to the playback quality.