Engineering | Finite Element Analysis | Stress
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There are a large number of ways to determine eigenvalues & eigenvectors, the best choice depends on the form of the equations being solved. The main methods are the power, subspace, LR, QR, Givens, Householder & Lanczos methods. Each method is usually well documented in advanced Engineering Math texts, except for Lanczos, which is relatively new.
There is a very useful check to evaluate if the eigenvalues have been extracted successfully. This is known as the Sturm sequence check. This is a method where the number of eigenvalues below a certain value can be evaluated, and is useful for finding if there are a large number of low frequency secondary components modes that would result in a long analysis time. It is also useful for indicating blunders with units (mass, length, etc), and can be used with subspace iteration & Lanczos methods as a cut off point (i.e. only extract the first five natural frequencies).
When Modal Analysis is Appropriate
A structure deforms as load is applied. If the load is cyclic, but with a much longer period than that of the fundamental frequency, then it is unlikely that the input will excite a resonance condition. A good rule of thumb is to implement a modal analysis if the forcing frequency is more than one-third the structure's fundamental frequency. However, if the excitation is random or applied suddenly, the problem becomes non-linear & eigenvalue based analyses will not provide correct results, hence a full dynamic analysis is required.
Natural Frequency & Modal-Dynamic Differences
Frequency based analyses perform eigenvalue extraction to calculate the natural frequencies and corresponding mode shapes of a 'free system' (i.e. with no time dependant loads applied). Modal-dynamic analyses are transient in nature. They give the response for the model as a function of time where a cyclic (sinusoidal) load is applied to the structure. Modal-dynamic analyses is also referred to as forced harmonic response analysis. Complex displacements and phase angles are evaluated and deflections & stresses may be calculated at specific times. This analysis type is formulated on the principle of modal superposition, and so a natural frequency analysis must be carried out first. The modal amplitudes are integrated through time & the response is subsequently evaluated. This analysis solution must be linear in nature (in time domain), as superposition & eigenvalue extraction techniques cannot be applied to non-linear time domain applications.
Vibration - modal analysis
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