Stability of Axially Moving Materials by Nikolay Banichuk & Alexander Barsuk & Juha Jeronen & Tero Tuovinen & Pekka Neittaanmäki

Author:Nikolay Banichuk & Alexander Barsuk & Juha Jeronen & Tero Tuovinen & Pekka Neittaanmäki
Language: eng
Format: epub
ISBN: 9783030238032
Publisher: Springer International Publishing

(5.8.114)

which is a standard linear eigenvalue problem for the eigenvalue–eigenvector pair :

(5.8.115)

This may or may not be advantageous, depending on practical details.

The final remaining practical issues are the sorting of the numerically obtained eigenvalues to obtain the lowest modes, and at adjacent values of the problem parameter c, connecting the correct solution points to form continuous curves for visualization, and to aid automatic detection of points of interest.

The sorting is important because the discretization cannot represent all of the countably infinite spectrum of eigenvalues. Typically only the first few numerical modes will be close to the true solutions of the continuum problem. How many modes are accurately obtained, depends on the ability of the discrete basis of W(x) to represent the corresponding eigenfunctions. Practical experience suggests that, in a quadratic eigenvalue problem, even if the basis is chosen optimally, at best only one half of the returned numerical solutions are solutions of the continuum problem. (This can be seen in some simplified cases by using analytically obtained exact eigenfunctions as the basis, leading to a spectral discretization, and then comparing the numerically obtained eigenvalues to the analytical solution for s.)

The ordering issue, on the other hand, arises due to algorithmic reasons. Eigenvalue solvers usually return the solutions in a random order, which may be (and usually is) different at each value of the problem parameter c. To remedy this, see Jeronen [54] for a discussion and algorithms for postprocessing the data.

In the following numerical results, we display the three lowest pairs of nondimensional eigenfrequencies against the nondimensional axial drive velocity c, for fixed values of , and . Considering the focus of this chapter, the values of the problem parameters have been chosen as typical to some paper materials; see Table 5.2 for their values. For the in-plane Poisson ratios, for simplicity, we have taken . In addition, we take , which allows us to introduce the retardation time .

We have chosen the range of the nondimensional axial drive velocity c to investigate the behavior near the first few bifurcation points. The lower end of the range has been set to 1, the critical point of the axially traveling ideal string. It is known that the introduction of bending rigidity stabilizes the system, increasing the critical value of c; hence at the considered systems will be in the initial stable region that begins at . The upper end we have cut slightly before any interactions with the fourth pair of eigenfrequencies occur in the elastic case.Table 5.2Problem parameters used in the numerical examples

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