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. Optionally a preload N0 can be specified.
You can also create a new spring by clicking and dragging the mouse.
Springs with spring stiffness greater than 105 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 fi arise in Hz.
To the results
Free oscillations are only possible with certain frequencies, so-called natural frequencies (or eigenfrequencies).
Which mass oscillates how strongly in the x and y directions can be seen in the associated eigenvector.
ui and vi are the displacements at the i-th point in the x and y directions, respectively.
The displayed eigenfrequency ist the natural angular frequency ωi 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 ℝ2.
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 corresponding 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.
Preload Influence
Normally, a spring connected by hinges at both ends cannot transmit transverse forces.
Therefore, it also has no transverse stiffness.
This changes when the influence of preloads is taken into account.
According to second-order theory (equilibrium in a deformed system), there is then a transverse stiffness,
which is proportional to the preload and inversely proportional to the spring length.
This transverse stiffness is independent of the longitudinal stiffness and thus also exists for a rigid rod.
It is therefore also referred to as geometric stiffness.
Only when this effect is taken into account is it possible to model pendulum oscillations and string vibrations.
The preload effect has no influence on the longitudinal stiffness.
System of Differential Equations
The multi-mass oscillators modeled here consist of point masses connected to each other by springs.
For each freely moving node to which a mass is attached, the sum of all forces in two spatial directions can be formulated.
Including the inertia term, equations of the following form result:
mixi•• + ∑ Fxj = 0
miyi•• + ∑ Fyj = 0
The forces Fxj and Fyj can be calculated using the stiffness matrices of the articulated springs and the
displacements of node i and its neighboring nodes.
This then results in a system of differential equations of the form
M x•• + K x = 0.
Here, M is the diagonally populated mass matrix.
K is the stiffness matrix and
x is the displacement vector of all x and y displacements.
Using the approach
x(t) = u eiωt
results a linear, homogeneous system of equations
(- ω2 M + K) u = 0.
The natural frequencies of the system are then determined from the requirement
det (- ω2 M + K) = 0.
The mode shapes of the system are determined by substituting the natural frequencies into the linear, homogeneous system of equations and then solving it.
If a node is supported, its force balance can be eliminated from the system of differential equations.
If no mass is assigned to a node, it must be removed from the system of differential equations by static condensation.
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