Steen Krenk

Vibrations of a taut cable with an external damper

A solution is presented of the problem of vibrations of a taut cable equipped with a concentrated viscous damper. The solution is expressed in terms of damped complex-valued modes, leading to a transcendental equation for the complex eigenfrequencies. A simple iterative solution of the frequency equation for all complex eigenfrequencies is proposed. The damping ratio of the vibration modes, determined from the argument of the complex eigenfrequency, are typically determined to within one percent in two iterations. An accurate asymptotic approximation of the damping ratio of the lower modes is obtained. This formula permits explicit determination of the optimal location of the viscous damper, depending on its damping parameter.

Frequency Analysis of the Tuned Mass Damper

The damping properties of the viscous tuned mass damper are characterized by dynamic amplification analysis as well as identification of the locus of the complex natural frequencies. Optimal damping is identified by a combined analysis of the dynamic amplification of the motion of the structural mass as well as the relative motion of the damper mass. The resulting optimal damper parameter is about 15% higher than the classic value, and results in improved properties for the motion of the damper mass. The free vibration properties are characterized by analyzing the locus of the natural frequencies in the complex plane. It is demonstrated that for optimal frequency tuning the damping ratio of both vibration modes are equal and approximately half the damping ratio of the applied damper, when the damping is below a critical value corresponding to a bifurcation point. This limiting value corresponds to maximum modal damping and serves as an upper limit for damping to be applied in practice.

Elastic Wave Scattering by a Circular Crack

The scattering of waves by a circular crack in an elastic medium is solved by a direct integral equation method. The solution method is based on expansion of stresses and displacements on the crack surface in terms of trigonometric functions and orthogonal polynomials. The expansion coefficients are related through an infinite matrix, and by contour integration the matrix elements are expressed in terms of finite integrals. The scattered far field is expressed explicitly in terms of simple functions and the displacement expansion coefficients. The system of equations is solved numerically, and extensive results are given both in the form of maps of the scattered far field and as scattering cross sections. Neither the method nor the specific results are restricted by any assumptions of symmetry.

Energy release rate of symmetric adhesive joints

Abstract A formula is derived for the mixed mode energy release rate of adhesive joints in terms of local stress concentrations σmax and τmax determined by beam theory. The formula G= 1 2 t E′ a σ 2 max 1 2 t G a τ 2 max is equivalent to a similar expression from two-dimensional elasticity theory in terms of the stress intensity factors K I and K II . The equivalence permits explicit determination of the two stress intensity factors K I and K II in terms of the finite stress concentrations σmax and τmax determined by technical beam theory. The reinterpretation of the stress concentration from beam theory in terms of stress intensity factors enables the use of simple beam-type calculations for fracture predictions without the need for a detailed three-dimensional analysis. While the stress concentrations (σmax and τmax increase with decreasing thickness t of the adhesive layer, the energy release rate ℷ is nearly independent of t . The centrally loaded symmetric lap joint is analysed in detail and explicit results for the mixed mode energy release rate ℷ and the corresponding load phase angle Ψ = tan −1 (K II /K I ) are given. For long joints Ψ = 49° , independent of adhesive properties, increasing for shorter joints.

Dispersion-corrected explicit integration of the wave equation

A simple dispersion correction is introduced into the centered difference explicit time integration scheme for wave propagation problems. The extra term introduces a change of the wave speed, depending on local wave characteristics. A rigorous theoretical analysis is provided for the scalar wave equation on a regular mesh, and a notable improvement in the ability to conserve wave fronts is demonstrated by two numerical examples. The correction is reinterpreted to apply to a typical finite element formulation in terms of mass and stiffness matrices, resulting in a general dispersion correction of the explicit centered difference time integration algorithm for acoustic and elastic wave propagation problems.

On the Elastic Constants of Plane Orthotropic Elasticity

The four independent material parameters of plane orthotropic elasti city are introduced as the effective stiffness, the effective Poisson ratio, the stiffness ratio and the shear parameter. It is proved that stress boundary value problems with zero resulting force on internal contours lead to stress fields that are independent of the effective stiffness and the effective Poisson ratio, and a general transformation is described which is equivalent to a change of the stiffness ratio. These properties suggest the importance of the remaining shear parameter, that has the interesting property of being invariant with respect to a 90 degree rotation of the principal axes of the material.

Complex modes and frequencies in damped structural vibrations

It is demonstrated that a state space formulation of the equation of motion of damped structural elements like cables and beams leads to a symmetric eigenvalue problem if the stiffness and damping operators are self-adjoint, and that this is typically the case in the absence of gyroscopic forces. The corresponding theory of complex modal analysis of continuous systems is developed and illustrated in relation to optimal damping and impulse response of cables and beams with discrete viscous dampers.

Vibrations of a shallow cable with a viscous damper

The optimal tuning and effect in terms of modal damping of a viscous damper mounted near the end of a shallow cable are investigated. The damping properties of free vibrations are extracted from the complex wavenumber. The full solution for the lower modes is evaluated numerically, and an explicit and rather accurate analytical approximation is obtained, generalizing recent results for a taut cable. It is found that the effect of the damper on the nearly antisymmetric modes is independent of the sag and the stiffness parameter. In contrast, the nearly symmetric modes develop regions of reduced motion near the ends, with increasing cable stiffness, and this reduces the effect of the viscous damper. Explicit results are obtained for the modal damping ratio and for optimal tuning of the damper.

The stress distribution in an infinite anisotropic plate with co-linear cracks

Abstract A general solution of the plane problem of a finite number of co-linear cracks in an anisotropic material is presented. The solution is obtained by reducing the problem to four very simple Riemann-Hilbert problems. From the solution it is concluded that if the loads acting on the cracks have the resultant zero for each of the cracks, then the normal and shear stresses created on the line of the cracks are independent of the elastic constants. Expressions for the stress intensity factors are derived, and some examples are presented.

On the elastic strip with an internal crack

Abstract The paper presents a method to deal with an inclined crack in an elastic strip. No assumptions of symmetry are made. The method involves the solutions for a cracked plane and an uncracked strip and results in two coupled singular integral equations with finite interval of integration. A crack in a half-plane arises as a limiting case. For internal cracks the integral equations are of a standard type and do not present any numerical difficulties. Results are presented for loads according to the technical beam theory.