An alternative method for plotting dispersion curves

Analysis of non-axisymmetric wave propagation in a homogeneous piezoelectric solid circular cylinder of transversely isotropic material

“…This method produces sharp negative spikes on the surface plot and automatically provides a picture of configuration of the dispersion curves. Similar approach of simultaneous qualitative displaying of the dispersion equation solutions was used by Honarvar et al (2008) that produced a 3-D cross-section of the real part of left hand side of the equation. The main advantage of our approach is that the local minima of the logarithm of modulus of the left hand side of the dispersion equation's matrix give proper approximations of the roots which can be further used as guess values of solutions of the dispersion equation.…”

Controlling the dynamic behavior of piezoceramic cylinders by cross-section geometry

“…In order to obtain the real and imaginary branches of dispersion curves, the fast and "smart" bisection method was used, which decreases the interval and increases the precision when the roots are very close to each other, and hence, it does not miss them. To verify the algorithm, some plots calculated by it for circular cylinders (fourth-order determinants) were compared with the graphs obtained by the method described by Honarvar et al [55], where the dispersion curves are obtained by a two-dimensional cut of a three-dimensional plot in the velocity-frequency plane. Both techniques have produced identical curves.…”

“…These plots are also presented below for some cases, and to evaluate the group velocity, the differentials F k ( , k) and F ( , k) were calculated numerically. The distributions of dispersion curves, phase, and group velocities are presented in the commonly used [18][19][20]23,26,[30][31][32][33]48,54,55] form of dimensionless frequency versus dimensionless wavenumber and dimensionless velocity versus dimensionless frequency, respectively. The angular frequency is normalized in the following way:…”

Analysis of Axisymmetric and Non-Axisymmetric Wave Propagation in a Homogeneous Piezoelectric Solid Circular Cylinder of Transversely Isotropic Material

“…The relevant material parameters for PZT-4 and PZT-7A are given in the Table 1. In order to obtain the dispersion curves we made use of a method similar to the novel method (Honarvar et al, 2008) where the dispersion curves are not produced as a result of solving of the dispersion equation by a traditional iterative find-root algorithm but are obtained by a zero-level cut in the velocity-frequency plane. In this chapter we modify this approach calculating the logarithm of modulus of determinant (37) are also displayed on the surface plots as obvious artefacts.…”