Equivalent Modal Damping Ratios Research Articles

Towards a seismic design method for plane steel frames using equivalent modal damping ratios

A method for seismic design of plane steel moment resisting frames based on the use of equivalent modal damping ratios is developed. The method determines the design base shear of the structure through spectrum analysis using rationally obtained equivalent modal damping ratios instead of the crude strength reduction (behavior) factor. An equivalent linear structure, which retains the mass and initial stiffness of the original non-linear structure and takes into account geometrical non-linearity and inelasticity in the form of equivalent, time-invariant, modal damping ratios is established. The equivalent damping ratios for the first few significant modes are numerically computed by first iteratively forming a frequency response transfer function modulus until it satisfies certain smoothness criteria and then by solving a set of non-linear algebraic equations. Thus, design equations providing equivalent damping ratios as functions of period and allowable deformation and damage are constructed using extensive numerical data coming from plane steel moment resisting frames excited by various seismic motions. These equations can be used in conjunction with a design spectrum, appropriately constructed for high damping values, and modal synthesis tools to calculate the seismic design forces of the structure. The proposed method is illustrated by numerical examples. It is concluded that unlike the usual approach of seismic codes employing a single common value of the strength reduction factor value for all modes, the proposed approach working with deformation and damage dependent equivalent modal damping ratios leads to more accurate and deformation and damage controlled results.

Investigation of vibration mitigation of stay cables incorporated with superelastic shapememory alloy dampers

This paper addresses the vibration mitigation of stay cables by using superelastic shapememory alloy (SMA) dampers. A closed form solution of the additional equivalent modaldamping ratio of a combined stay cable/SMA damper system is formulated when thecombined stay cable/SMA damper system vibrates with a single mode. The responses of astay cable model on a cable-stayed bridge model with/without one SMA damper subjectedto sweeping sinusoidal excitations are numerically investigated. An experimentalinvestigation on the stay cable model incorporated with superelastic SMA spring dampersis finally carried out to verify the effectiveness of the proposed approach and thecomputational results.

Vibration mitigation of a stay cable with one shape memory alloy damper

The paper investigates the vibration mitigation of a stay cable provided with one shape memory alloy (SMA) damper, which dissipates energy through its superelastic behavior. The closed-form solution of the additional equivalent modal damping ratio to the stay cable by the SMA damper is theoretically analyzed, which indicates the relationships of the additional equivalent modal damping ratio and both the parameters and the locations of the SMA damper when the stay cable–SMA damper system vibrates with only a single mode. The responses and the additional equivalent modal damping ratios of the stay cable with one SMA damper attached at different locations subjected to harmonic excitation in plane are also calculated with single-mode vibration only and with coupled multi-mode vibration. The analytical results show that the SMA damper can mitigate the vibration of the cable significantly and the control effectiveness is influenced by SMA damper parameters and locations. Copyright © 2004 John Wiley & Sons, Ltd.

Evaluation of equivalent damping ratio of a structure with added dampers

The purpose of this study is to propose a new method for evaluating the equivalent damping ratio of a structure with supplemental damping devices to assess the vibration control effect quantitatively. The modal-energy-formed Lyapunov function is defined first, and the Riccati matrix and damping rate parameter are derived. Then from its analogy with the definition of a viscous modal damping ratio, the equivalent damping ratio is defined. The proposed approach is applied to estimate the equivalent damping ratio of a structure with various added damping devices. Then a closed-form solution is derived to obtain an equivalent modal damping ratio without carrying out numerical analysis. Results from the numerical analysis verify that the proposed method provides an equivalent damping ratio of a structure equipped with nonlinear as well as linear damping devices quite accurately.

Mental results and dynamic parameters for the Penguin Vibration Damper (PVD) for wind and earthquake loading

Penguin Engineering Ltd has developed a compact, efficient, hysteretic damping device, the Penguin Vibration Damper (PVD). Experimental test results show that the PVD can provide a significant amount of damping at displacements as small as 50 micro-metres.
 The hysteresis behaviour of the PVD can be described well either by a model having a linear spring in parallel with a viscous dashpot, or by a bi-linear model, with the parameters of both models being displacement-amplitude dependent. For large displacements, the bi-linear model gives an accurate representation of the PVD's hysteresis loops, and the parameters for the bi-linear model can be taken as constants. Non-linear models, such as the hyperbolic, Ramberg-Osgood and multi-surface plasticity models, can also be used and have an advantage of displacement-amplitude-independent parameters. However, it can be shown that nonlinear models do not correctly predict the amount of damping that a PVD provides at large displacement even though the equivalent spring coefficient can be well approximated. When the PVDs are expected to undergo large displacements, it is possibly best to use a simple bi-linear model in dynamic nonlinear structural analyses, because the bi-linear model with suitably selected parameters can produce the correct amount of damping derived from the experimental data.
 The changes of the PVD's dynamic behaviour are small after a fatigue test of 144 000 cycles with a displacement amplitude of 2 mm.
 An analysis of a 6-storey reinforced concrete moment resisting frame is used to demonstrate the effect of the damper. Equivalent first modal damping ratios are estimated for various levels of earthquake excitations. The example shows that the dampers can provide a large amount of damping to the structure and enhance the structural capacity, for resisting earthquakes, by 50-100%.

MODAL ANALYSIS OF SOFT-SOIL SITES INCLUDING RADIATION DAMPING

For the one-dimensional analysis of soft-soil layers on an elastic half-space, a general form of analytical solution is developed for converting radiation damping due to energy leaking back to the half-space into equivalent modal damping, allowing the modal analysis technique to be extended to a site where radiation damping has to be accounted for. Closed-form solutions for equivalent modal damping ratios and effective modal participation factors are developed for a single layer with a shear wave velocity distribution varying from constant to linearly increasing with depth. Compact and recursive forms of solutions for equivalent modal damping ratios are developed for a system with an arbitrary number of homogeneous layers on an elastic half-space. Comparisons with numerical solutions show that the modal solutions are accurate. The nominal frequency of a site, i.e. the inverse of four times the total shear wave travel time through the layers, is an important parameter for estimating the high mode frequencies. A parameter study shows that for the same impedance ratio of the bottom layer to the elastic half-space, a system of soil layers with an increasing soil rigidity with depth has, in general, larger peak modal amplifications at the ground surface than does a single homogeneous layer on an elastic half-space, while a system with a decreasing soil rigidity with depth has smaller modal peak amplifications. © 1997 by John Wiley & Sons, Ltd.

Analytical determination of equivalent modal damping ratios of a composite tower in wind-induced vibrations

We present two methods to determine the equivalent modal damping ratios of a composite tower under wind-induced vibrations. Different damping characteristics arise from the construction of the tower with two materials; for the lower main part, say of reinforced concrete and an upper part, say of steel. The first method employs a detailed numerical integration procedure in which the static condensed stiffness matrices are established by the transfer matrix method. The damping matrices of the upper and the lower substructures are modelled with the Rayleigh damping formulation. The second method makes use of a simplified approximation of two lumped masses. Computation using the two methods is demonstrated with a numerical example for a composite T.V. tower of a total height 459m. Results show that, while the detailed method gives good accuracy but demands intensive computation, the approximate method involves simple calculations and produces data of enough precision for engineering applications.

Equivalent Modal Damping Ratios of a Composite Tube-Type Tall Building to Dynamic Wind Loading

This article discusses computing methods of the equivalent modal damping ratio for a composite tube-type tall building. The equivalent damping ratio of a composite structure under wind-induced vibrations encompasses the combined effects from the different values of damping, mass, and stiffness of the external and internal tubes. A detailed method incorporating Rayleigh damping and the transfer matrix method is proposed for the determination of the equivalent modal damping ratio. A simplified method for engineering applications is also proposed. Results for a 40-story building are presented exemplifying the computation and relationships between the properties of the external and internal tubes.

Equivalent Modal Damping Ratios of a Composite Tube-Type Tall Building to Dynamic Wind Loading

This article discusses computing methods of the equivalent modal damping ratio for a composite tube-type tall building. The equivalent damping ratio of a composite structure under wind-induced vibrations encompasses the combined effects from the different values of damping, mass, and stiffness of the external and internal tubes. A detailed method incorporating Rayleigh damping and the transfer matrix method is proposed for the determination of the equivalent modal damping ratio. A simplified method for engineering applications is also proposed. Results for a 40-story building are presented exemplifying the computation and relationships between the properties of the external and internal tubes.

Estimation methods of equivalent modal damping ratio of tall buildings with soil — structure interaction (in Chinese) : Shang Shiyang, Earthquake Engineering & Engineering Vibration, 14(1), 1994, pp 83–88

Estimation methods of equivalent modal damping ratio of tall buildings with soil — structure interaction (in Chinese) : Shang Shiyang, Earthquake Engineering & Engineering Vibration, 14(1), 1994, pp 83–88