When solving a PDE using the method of separation of variables, some higher modes are eventually truncated and not considered.
How significant is the effect of higher modes that are not considered?
Some believe adding a constant feed-through term will solve the problem, is there any other more accurate method to compensate this negative effect of truncation?
*. Considering the fact that this is an engineering problem not a pure theoretical math problem.
If the solution to the PDE can indeed be expressed as the multiplication of two terms F(x)*G(y), then the method of separation of variables is analytically correct.
If, to solve the PDE, you use a series to express the solution in a finite orthonormal basis, then, you will have a truncation error.
The truncation error depends on the number of terms you are considering and how well the "higher modes" are represented by the terms.
If the solution to the PDE can indeed be expressed as the multiplication of two terms F(x)*G(y), then the method of separation of variables is analytically correct.
If, to solve the PDE, you use a series to express the solution in a finite orthonormal basis, then, you will have a truncation error.
The truncation error depends on the number of terms you are considering and how well the "higher modes" are represented by the terms.
If separation of variables T(x,y)=F(x)\phi(y) leads to a Sturm- Liouville eigenvalue problem, which implies that the eigenvalues are real and can be ordered, i.e., lambda1
Hi
To answer your question directly, at least for the case of structural vibration, getting accurate phase can be quite tricky.
As simple rules of thumb
- 5 degree or less phase errors require inclusion of modes up to 10x the highest analysis frequency.
- 1 dB amplitude error requires inclusion of modes up to 3x the highest analysis frequency.
This is why, in substructuring, things like static modes and residues are used.
It may perhaps be useful to borrow some practical ideas on how to reduce the effect from higher order modes, the way it is done in substructuring analysis?
A related discussion on this topic is found here: https://www.researchgate.net/post/How_can_I_make_mode_shape_expansion_for_plate_structures-using_the_best_way_for_this_problem
Sincerely
Claes
Let me address this controversial question within the context of vibration analysis of continuous systems (hyperbolic PDE). When we represent the system dynamics in the modal space, we use a finite (truncated) set of modes (commonly called significant modes), which include the low-frequency subsystem. The accuracy of the solution is mainly dependent on the percentage of the system's kinetic energy contained in this selected set of modes. The difficult question has always been the following: What is the appropriate number of significant modes? In fact, there is no a priori means to judge the dimension of the significant frequency subsystem. Yet, there are some indirect measures to give some insight into this problem. (1) if the excitation is smooth, then very likely most of the system kinetic energy will be picked up by the first few modes. the term "few" may widely vary for different systems. A couple of simulation runs would lead to the appropriate number of modes as judged by convergence. (2) the tricky case is the related to the impulsive excitation; i.e. due to impact for instance. In this case, higher modes are normally excited to pick up proportionally and appreciable percentage of the system's kinetic energy. In this case, more higher modes need to be included in the model. Since there is no readily available information as to which higher modes will be excited, one must perform few preliminary runs to check convergence as the size of the frequency spectrum is increased. It is also possible to find out which higher frequencies may be excited by the impulsive force if the frequency spectrum of the excitation is available.
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