An approximate solution of the steady forced vibration of a system of one degree of freedom under the influence of various types of damping1

Abstract

In this paper the author presents a general method of obtaining approximate solutions of the steady forced vibration of a damped system of one degree of freedom for the case of sinusoidally varying disturbing forces. The approximation consists in expressing all the damping terms of the original differential equation by a single equivalent damping term, proportional to the first power of the velocity of motion. In the case of a system influenced by a centrifugal disturbing force and damped by constant friction and by friction proportional to the first power of the velocity, experimental evidence is in good agreement with the approximate solution. While the paper deals solely with a system of one degree of freedom, there is no reason aside from complexity why the approximate solution cannot be applied to systems of several degrees of freedom.

A simplified analysis method for estimating the peak responses of recentering structures with viscous damping systems

“…and then continually improved by other researchers. Some prior studies proposed that using the effective damping ratio, which was initially proposed by Jacobsen, 48 in the Rosenblueth method may lead to underestimation of inelastic displacement 33–35 and will then generate unsafe design results. Also, Gang Xu et al.…”

An equivalent damping ratio that accounts for ground-motion spectral shape for displacement-based design of flag-shaped hysteresis systems

“…By convention, ξ el is normally assumed to be 5% for reinforced concrete structures, and ξ hyst is determined by making the energy dissipated in the nonlinear system (Figure 3a) and viscously damped elastic system (Figure 3b) equal. Jacobsen (1930) used the area-based approach to calculate ξ hyst , as shown in Equation 2 and illustrated in Figure 3a and b.…”

“…Self-centering systems with flag-shaped hysteresis play a critical role in earthquake resilience as they are proportioned to nearly eliminate residual drifts and improve structural repairability after earthquakes (Li and Wang, 2022;Thonstad et al, 2017Thonstad et al, , 2021Xiao et al, 2022aXiao et al, , 2022bYang et al, 2022;Zhong and Christopoulos, 2022). Several area-based equations have been proposed to estimate EDR for self-centering systems (Blandon and Priestley, 2005;Gulkan and Sozen, 1974;Jacobsen, 1930;Marriott, 2009;Priestley et al, 2007), but the assumption of harmonic excitation in the area-based strategy leads to overestimated EDR (Twigden, 2016), and no correlation factor has yet been established specifically for flag-shaped hysteresis systems. Only a few researchers have used the regressionbased strategy for flag-shaped systems, and they were all limited to a flag-shaped hysteresis with fixed hysteresis parameters (Blandon and Priestley, 2005;Dwairi, 2004), which cannot meet the more general needs of DDBD.…”

“…The previously developed methods to estimate EDR mainly fall into three categories: the area-based strategy, the regression-based strategy, and the interpolation strategy. The area-based strategy was initially proposed by Jacobsen (1930). The EDR was selected so that the energy dissipated for the nonlinear system and viscously damped system would be the same for complete loops of the hysteretic models under steady-state harmonic excitations.…”

“…They found that Jacobsen’s approach tends to overestimate EDR, and they proposed modified equations that incorporate period-dependency. To address the deficiencies of the steady-state harmonic excitation approach (Jacobsen, 1930), Priestley et al (2007) and Marriott (2009) proposed factors for correlating elastic viscous damping and hysteretic damping.…”