Engineering | Finite Element Analysis | Stress
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Eignevalues are otherwise known as latent roots and characteristic values, the square root of the eigenvalue is known as a natural frequency or resonant frequency. There is also a number of terms used to describe mode shapes, they are also known as eigenvectors, normal modes, characteristic vectors or latent vectors.
Design Applications of Frequency Analysis
In many practical problems the natural frequencies and mode shapes are all that are
required. Designers use modal analyses to determine if there are any natural frequencies
within the range of operation. Alternatively, measured mode shapes and natural frequencies
of a structure can be compared with those predicted by FEA in a condition monitoring
program to verify structural integrity.
There are also situations where the response
of the structure to a particular forcing excitation is required. This is usually
found using a technique known as modal superposition. The overall response is described
in terms of a sum of modal responses, with the contribution of a particular mode
given by the proximity of the forcing frequency to the natural frequency and the
amount of damping present in the system. The response is dominated by modes close
to the excitation frequency and therefore the modal series is often truncated to
reduce computation. Modal superposition methods can only be applied in applications
with a harmonic excitation, otherwise the response becomes non-linear & cannot be
solved using the eigenvalue extraction approach.
The results from a forced harmonic
analysis can be used to determine whether the displacement of a particular structure
is within acceptable limits. By calculating the stress induced by the vibration it
is also possible to predict the fatigue life of a particular component.
How Eigenvalues are Extracted
Having discretised the component (continuum) into elements and described the variation
of the displacement within each element, the kinetic and potential energy of the
structure are the calculated to find the natural frequencies and mode shapes in terms
of various nodal values. Numerically this equates to solving an eigenvalue problem
expressed in terms of mass and stiffness matrices. You will remember from elementary
engineering mathematics that once the eigenvalues are known, the eigenvectors may
be evaluated.
Vibration - eigenvalues
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