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. [S0021-8936(00)00404-9]

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.

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This book presents a theoretical treatment of nonlinear behaviour of solids and structures in such a way that it is suitable for numerical computation, typically using the Finite Element Method. Starting out from elementary concepts, the author systematically uses the principle of virtual work, initially illustrated by truss structures, to give a self-contained and rigorous account of the basic methods. The author illustrates the combination of translations and rotations by finite deformation beam theories in absolute and co-rotation format, and describes the deformation of a three-dimensional continuum in material form. A concise introduction to finite elasticity is followed by an extension to elasto-plastic materials via internal variables and the maximum dissipation principle. Finally, the author presents numerical techniques for solution of the nonlinear global equations and summarises recent results on momentum and energy conserving integration of time-dependent problems. Exercises, examples and algorithms are included throughout.

A common format is developed for a mass and an inerter-based resonant vibration absorber device, operating on the absolute motion and the relative motion at the location of the device, respectively. When using a resonant absorber a specific mode is targeted, but in the calibration of the device it may be important to include the effect of other non-resonant modes. The classic concept of a quasi-static correction term is here generalized to a quasi-dynamic correction with a background inertia term as well as a flexibility term. An explicit design procedure is developed, in which the background effects are included via a flexibility and an inertia coefficient, accounting for the effect of the non-resonant modes. The design procedure starts from a selected level of dynamic amplification and then determines the device parameters for an equivalent dynamic system, in which the background flexibility and inertia effects are introduced subsequently. The inclusion of background effect of the non-resonant modes leads to larger mass, stiffness and damping parameter of the device. Examples illustrate the relation between resonant absorbers based on a tuned mass or a tuned inerter element, and demonstrate the ability to attain balanced calibration of resonant absorbers also for higher modes.