Dorothea A. Klip

Formulas for Phase Velocity and Damping of Longitudinal Waves in Thick‐Walled Viscoelastic Tubes

The frequency equation for longitudinal waves in a viscoelastic tube of infinite length filled with a viscous fluid is solved for the complex wavenumber, k(=k1+ik2). Thus for two modes, Youngs mode and Lambs mode, an explicit expression for k is obtained which yields the phase velocity c(=ω/k1, ω=radial frequency) and the damping constant k2. The formulas for k, c, and k2 are valid for wall thicknesses up to the size of the inner radius of the tube. The approximation as compared with results obtained numerically with an IBM 7040 digital computer from the original frequency equation involves an error which in almost all cases is considerably less than 1.5%. For Lambs mode, as it was found earlier for the torsional mode, an increase of the fluid viscosity increases the damping at the higher frequencies but decreases the damping at the lower frequencies. The theoretical results are compared with those of experiments in which the longitudinal waves are generated in the fluid and the fluid in its turn moves...

The independence on preload and afterload of the contraction variable for the heart muscle. Thermodynamic and clinical consequences

The time dependence of the contraction variable (Qc) for dog hearts was found to be largely independent of preload and afterload. This appears to make its clinical determination by pressure‐volume measurements not unfeasible and could have important diagnostic consequences. The independence of the function Qc=?c(t) on the deformation variable (Q) led also to the derivation of some thermodynamic formulas for this contractile material.

Phase velocity and damping of torsional waves in viscous‐fluid‐filled stretched viscoelastic tubes

Using the theory of elasticity for large deformations and a strain energy function of the form W=½Φ(I1 −3)+½Ψ(I2 −3), where Φ and Ψ are constants, formulas for the phase velocity and damping of torsional waves of small amplitude in stretched viscoelastic isotropic vessels of arbitrary wall thickness and filled with a viscous fluid under pressure have been derived and experimentally tested (frequency range 0.3–33 Hz; viscosity of the fluids between 0.01 and 13 P; pressure between 0 and 4×105 dyn cm−2; stretches up to twice the original length) in vertically or horizontally suspended latex tubes and in such tubes floating on a pillow of pressurized air. The agreement between theory and experiment is satisfactory. The phase velocity increases with the stretch, but decreases with an increase of the fluid viscosity. The damping decreases with an increase of the stretch. At the higher frequencies the damping increases with an increase of the viscosity of the fluid, but at the lower frequencies the opposite is t...

Static and Dynamic Experiments on the Stretch and Frequency Dependence of Elastic and Viscoelastic Coefficients of Latex Tubes

Assuming for the wall material (supposed incompressible) of a latex tube a strain‐energy function of the form W = ½Φ (I1−3) +½Ψ (I2−3), (Φ and Ψ constants, i.e., independent of the deformation) a stress‐strain relationship for simple stretch, an equation of motion for small torsional vibrations, and a frequency equation for torsional waves of small amplitude are derived. In the equations of motion Φ and Ψ are replaced by Voigt operators Φ+Φvi (∂/∂t) and Ψ+Ψvi (∂/∂t). Experiments with simple large‐stretch free torsional oscillations of small amplitude and torsional waves of small amplitude superposed on a large stretch showed that the postulates predict the behavior adequately. Φ and Ψ were found to be not only independent of the stretch (1≤λ≤2.3) but also independent of the frequency (0≤f≤33 cps). ωΦvi and ωΨvi appeared at the lower frequencies (free vibrations) to be independent of the stretch and of the frequency. At the higher frequencies a frequency and stretch dependence could not with certainty be d...

Completeness of zero curve tracing for analytic functions

Abstract Empirical data for the polynomial confirm the efficiency of exploring function structure as a means of isolating the zeros of a scalar analytic function defined on a disk (Klip, 1985). In exceptional cases the tracing is not complete which means that certain arcs may not have been traced. A mathematical basis for the algorithm which investigates and restores completeness, the so-called completeness algorithm, is presented. The main theorem is that completeness of the tracing, as verified at the isolated zeros, is sufficient for completeness of the tracing. Its proof evolves along various lemmas, one of which provides a new condition for the location of the critical points. The paper concludes with a brief description of the completeness algorithm.

An energy density function for papillary muscle encompassing both elastic and contractile properties

Not only for the the heart, but also for papillary muscle a contraction variable κ can be defined (in analogy with the deformation variable λ) in a phenomenological approach that makes use of the finding that two of the three constants (c2 and c3) in the isochronics of such a muscle can be considered functions of the third one (κ). This fact provides four constants (k1, k2, k3, and k4) of an energy density function. A tentative bridge to the heart (for clinical application) is laid; four analogous constants of the same order of magnitude can be measured there.

Different polynomial representations and their interaction in the portable algebra system PORT-ALG

In recent years a system for the handling of large multivariate polynomials has been developed at the University of Alabama in Birmingham. The coding is generally done in Fortran IV; some special features are coded in PL/1, for the IBM systems 360 or 370. No machine coding is involved.

An analytic approach to the solution of non-linear equations

Abstract Isolating the zeros of a scalar function analytic on a disk by means of exploring function structure was shown to be very efficient (Klip, 1985). This will be exemplified in this paper on the hand of data obtained with certain test polynomials. It was shown by Klip (1987) that the structure of the zero curves for the polynomial leads to a new formulation of the location of the critical points. It will be proven that the number of critical points in a branch region of order m equals m − 1. Completeness of the tracing is an important issue. We give an account of the design of the completeness algorithm and discuss the role of a small listprocessing system in registering temporarily stored data as well as modifications of the pathways.

Isolation of the Zeros of a Complex Polynomial by Exploring Function Structure Uniqueness of the Solution Set Established

The presented algorithm for simultaneous polynomial root isolation is based on the discrete tracing of the zero curves of the real (and imaginary) part of the function. Being guided by the structure of the function as presented by these curves is essential in attaining the desired characteristics of robustness, reliability and global optimal computational complexity. A mathematical condition for uniqueness of the tracing is optionally verified and -if not fulfilled in exceptional cases – uniqueness is established. The method immediately applies to transcendental functions in one complex variable in a region of analyticity.