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.
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.
Energy balance equations are established for the Newmark time integration algorithm, and for the derived algorithms with algorithmic damping introduced via averaging, the so-called α-methods. The energy balance equations form a sequence applicable to: Newmark integration of the undamped equations of motion, an extended form including structural damping, and finally the generalized form including structural as well as algorithmic damping. In all three cases the expression for energy, appearing in the balance equation, is the mechanical energy plus some additional terms generated by the discretization by the algorithm. The magnitude and character of these terms as well as the associated damping terms are discussed in relation to energy conservation and stability of the algorithms. It is demonstrated that the additional terms in the energy lead to periodic fluctuations of the mechanical energy and are the cause of the phenomenon of response 'overshoot', previously observed empirically in the application of Newmark based algorithms to high frequency components. It is also demonstrated that the stability limit of the explicit Newmark algorithm is reached, when the stiffness term in the algorithmic energy vanishes, and that energy fluctuations take place for integration intervals close to the stability limit.In this equation M, C and K are the mass, damping and stiffness matrices of the system, and f (t) is the external load vector, conjugate to the displacements u(t). A central problem to which much effort has been devoted over the last fifty years is the numerical integration of this equation of motion. The preferred methods are of the single-step type consisting of updating the displacement, velocity and acceleration vectors u,u,ü at current time t n to the time t n+1 = t n + h, a small time interval h later. It is desirable that the algorithm has at least second order accuracy, and because the spatial discretization used in structural dynamics often leads to inclusion of high-frequency modes in the model, it is also desirable to have unconditional stability.There are two basically different approaches to the development of single-step time integration algorithms: collocation of the equation of motion at selected points in time in connection with a set of difference relations between the displacement, velocity and acceleration vectors, or the the use of an integrated form of the equations of motion whereby the acceleration is eliminated and
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