# Vibration Analysis of Multi-Degree-of-Freedom Systems

The program on this page calculates *natural frequencies* and *mode shapes* of Multiple Degree of Freedom Systems (MDOF).

The mode shapes are visualized. Free vibrations in each mode are simulated.

Mass points and mass-free springs are used to build models.

Rigid bodies with moment of inertia can be considered by rigidly connecting 2 mass points.

## To the model data

A model to be examined is defined by:

- All points at which springs end or should be connected to each other.

When entering the point coordinates you can access the coordinates of other points via x1,y1,x2,y2 etc.

The mouse can be used: double click to create new points, click drag to move existing points.
- A list of spring data. For each spring its 2 end points must be specified.

The spring stiffness is also specified.

You can also create a new spring by clicking and dragging with the mouse.

Springs with spring stiffness greater than 10^{5} times the smallest spring stiffness are assumed to be *rigid connections*.
- A list of masses. A mass can (but does not necessarily has to) be located at each spring connection point used.

Here masses are mass points. Mass moments of inertia are made possible by the rigid coupling of several mass points.

A rigid body with mass moment of inertia J and mass m can be modeled by 2 rigidly connected masses m/2 at a distance of 2*(J/m)^{1/2}.
- Information regarding bearings. Here you can tie point-related in both the x and y directions.

It is best to use the following units when entering data: masses in *kg* stiffness in *N/m*.

Then the output natural frequencies f_{i} arise in *Hz*.
## To the results

*Free* oscillations are only possible with certain frequencies, the so-called *natural frequencies* (or eigenfrequencies).

Which mass oscillates how strongly in the x and y directions can be seen in the associated *eigenvector*.

u_{i} and v_{i} are the displacements at the i-th point in the x and y directions, respectively.

The displayed eigenfrequency ist the natural angular frequency ω_{i} devided by 2·π.

The number of possible natural vibration forms and natural frequencies is equal to the total number of degrees of freedom of movement in the system.

Every free or elastically bound mass point initially has 2 degrees of freedom of movement in **ℝ**^{2}.

This results in a maximum of 2·n degrees of freedom of movement for a system with n mass points.

This maximum total number of degrees of freedom of movement decreases due to bearings and the use of rigid connecting elements.

A dual-mass oscillator has 2 natural frequencies if the movements are guided along one axis and both masses are movable.

However, if the two masses are movable in *both* directions, there are 4 natural frequencies.

If, on the other hand, the two masses are movable in *both* directions and *rigidly coupled* to each other, there are 3 natural frequencies.

If a system has options for movement that do not cause any spring force, there are mode shapes with a natural frequency of 0.

The system can then adjust itself without any forces according to this eigenform, but does not oscillate.

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