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Available online at www.sciencedirect.com Procedia Engineering 00 (2017) 000–000
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Procedia Engineering 199 (2017) 495–500
X International Conference on Structural Dynamics, EURODYN 2017
Structural damping estimation using a simple equation based on the equivalent viscous damping concept Ronwaldo Emmanuel R. Aquinoa*, Yukio Tamurab a RWDI,
RWDI, 600 Southgate Drive, Guelph N1G 4P6, Ontario, Canada b Beijing Jaotong University, Beijing, China
Abstract Earlier research papers by the authors have presented a simple equation, derived using the equivalent viscous damping concept, to characterize the amplitude-dependency of modal damping. While initially intended for damping estimation on account of stick-slip surfaces acting on main structural systems of buildings within the linear range, the current paper discusses how this same equation can be used to describe damping in general given the force-displacement relationship of the element or structure of interest. This is done by modifying and extending the original derived equation to allow the calculation of amplitude-dependent damping given the force-deformation characteristic of the particular element or structure. The paper then presents amplitudedependent damping estimates for a structure with a given modal pushover (i.e. force-displacement) curve generated by static nonlinear analysis as an example application of this simple formula. The effect of the resolution of the force-displacement curve on the damping estimate is discussed by contrasting the initial estimate against simplified force-displacement curves. Limitations and applications of this procedure in wind and seismic design are briefly discussed. © 2017 The Authors. Published by Elsevier Ltd. © 2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the organizing committee of EURODYN 2017. Peer-review under responsibility of the organizing committee of EURODYN 2017. Keywords: equivalent viscous damping concept; amplitude-dependent damping; stick-slip mechanism; force-displacement characteristic;
1. Introduction Earlier research papers by the authors [1,2,3] discuss a simple equation for characterizing the amplitudedependency of modal damping. The equation is derived based on the equivalent viscous damping concept (EVDC). While initially intended for damping estimation on account of stick-slip surfaces acting on primary structural
* Corresponding author. Tel.: +1-519-823-1311 ; fax: +1-519-823-1316. E-mail address:
[email protected] 1877-7058 © 2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the organizing committee of EURODYN 2017.
1877-7058 © 2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the organizing committee of EURODYN 2017. 10.1016/j.proeng.2017.09.135
Ronwaldo Emmanuel R. Aquino et al. / Procedia Engineering 199 (2017) 495–500 Aquino & Tamura / Procedia Engineering 00 (2017) 000–000
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systems of buildings within the linear range, the current paper discusses how this same equation can be used to describe damping in general given the force-displacement (F-D) relationship of the element or structure of interest. This is done by modifying and extending the original derived equation to allow the calculation of amplitudedependent damping (ADD) given the F-D characteristic of the particular element or structure. The paper then presents ADD estimates for a structure with a given modal pushover (i.e. F-D) curve generated by static nonlinear analysis as an example application of this simple formula. The effect of the resolution of the F-D curve on the damping estimate is discussed by contrasting the initial estimate against simplified pushover curves. Limitations and applications of this procedure in wind and seismic design are also discussed. 2. Summary of Previous Research In the earlier work, the effect of one stick-slip surface (SSS) within a single-degree-of-freedom (SDOF) system meant to represent a mode was first studied. This is summarized in Fig. 1. Figs. 1a and b show the “stuck” and “slipping” phase of the motion, defined by where x is relative to dc, and the corresponding value of Q. Fig. 1c shows the mass-spring-dashpot representation of the base structure, as well as the spring representation of the secondary element and the friction surface representation at the point of contact. Fig. 1d shows the F-D characteristic of the friction surface. For simplicity, it is assumed that does not change from the “stuck” phase to the “slipping” phase. Fig. 1e shows a nonlinear spring representation of the contact-surface-secondary-element (referred to as “stick-slip component” or SSC), which has a F-D characteristic shown in Fig. 1f. Fig. 1g shows the whole structure’s F-D characteristic (WS-FD). It is established that FS-WD should be used when applying EVDC. After proper application of EVDC, the following simplified equation was obtained: (1) X b
d c d c 2 1 1 H1 kb X X X 1 k0 dc
Nomenclature cb dc i mb k0 kb kc ktot x x’ F Fb Fci H( ) N’ Q Qc S X
b
base structure damping coefficient slip trigger displacement, beyond which slipping will occur at the contact surface, equal to Qc/k0 index in multi-stick-slip-surface systems, applied to k0 and dc base structure mass secondary element stiffness base structure stiffness effective stick-slip bilinear stiffness whole system effective stiffness base structure displacement (time-variant) relative displacement between friction and stiffness element of stick-slip component (time-variant) force in whole system at amplitude X force in basic system at amplitude X inflection point in F-D curve, corresponding to dci Heaviside step function normal force at contact surface friction force at the contact surface between basic system and secondary element friction “capacity”, beyond which slipping will occur at the contact surface total number of stick-slip surfaces in a multi-stick-slip-surface system base structure displacement amplitude coefficient of friction at the contact surface whole system damping ratio base structure damping ratio, corresponding to cb
Ronwaldo Emmanuel R. Aquino et al. / Procedia Engineering 199 (2017) 495–500 Aquino & Tamura / Procedia Engineering 00 (2017) 000–000
a ½kb
e
½kb ½cb
b ½kb
mb
Q kb
k0
d
mb
kc kb
½cb
½kb ½cb
Q Force-displacement at stick-slip friction surface
cb
xx'
f
mb x + cb x + ( kb + kc ) x = 0 kc = k0 H ( dc − X ) Force-displacement of whole system (linear main structure + nonlinear SSC)
k0
Q=Qc=N’
Qc
cb
g
mb
x ≥dc
k0
½cb
c
Q
