Energy Loss in Systems of Stacked Rocking Bodies
Publication: Journal of Engineering Mechanics
Volume 144, Issue 7
Abstract
Free-standing bodies, which rock in response to dynamic base excitations, can be found in a wide range of applications. There are many instances, for example a museum artifact seated on a support pedestal, where both the object and its support are free-standing, resulting in a dynamic system where both bodies can rock with respect to their contact interfaces. This paper analyzes the rich dynamics of systems of two stacked rocking bodies. It focuses on the location of the resultant impulses between bodies during impact, considering their effect on the overall stability of the system. The widely-used assumption that the impulses occur at the future rocking corners is shown to be the least conservative scenario and often leads to underestimating the likelihood of failure. The optimum configuration of two stacked blocks is found to be highly sensitive to the characteristics of the input motion and to the location of the impulses.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
The first two authors would like to acknowledge the financial support of the EC, FP7-PEOPLE-2013, Marie Curie, Career Integration Grant, RERCSGM, project number 618359.
References
Allen, R. H., Oppenheim, I. J., Parker, A. R., and Bielak, J. (1986). “On the dynamic response of rigid body assemblies.” Earthquake Eng. Struct. Dyn., 14(6), 861–876.
Chatzis, M. N., Garcia-Espinosa, M., and Smyth, A. W. (2016). “Examining the energy loss in the inverted pendulum model for rocking bodies.” J. Eng. Mech., 04017013.
Chatzis, M. N., and Smyth, A. W. (2012a). “Modeling of the 3D rocking problem.” Int. J. Non-Linear Mech., 47(4), 85–98.
Chatzis, M. N., and Smyth, A. W. (2012b). “Robust modeling of the rocking problem.” J. Eng. Mech., 247–262.
Chatzis, M. N., and Smyth, A. W. (2012c). “Three-dimensional dynamics of a rigid body with wheels on a moving base.” J. Eng. Mech., 496–511.
Chatzis, M. N., and Smyth, A. W. (2013). “Preliminary investigation of the randomness in the outcome of a die throw.” Proc., 11th Int. Conf. on Structural Safety and Reliability, ICOSSAR 2013, New York.
Clough, R. W., and Penzien, J. (1975). Dynamics of structures, McGraw-Hill, New York.
Dimitrakopoulos, E. G., and Giouvanidis, A. I. (2015). “Seismic response analysis of the planar rocking frame.” J. Eng. Mech., 04015003.
Dormand, J., and Prince, P. (1980). “A family of embedded Runge-Kutta formulae.” J. Comput. Appl. Math., 6(1), 19–26.
ElGawady, M. A., Ma, Q., Butterworth, J. W., and Ingham, J. (2011). “Effects of interface material on the performance of free rocking blocks.” Earthquake Eng. Struct. Dyn., 40(4), 375–392.
FEMA. (2006). “Document control procedures manual.” Washington, DC.
Greenwood, D. T. (2003). Advanced dynamics, Cambridge University Press, New York.
Harvey, P., Jr. (2017). “Behavior of a rocking block resting on a rolling isolation system.” J. Eng. Mech., 04017045.
Housner, G. W. (1963). “The behavior of inverted pendulum structures during earthquakes.” Bull. Seismol. Soc. Am., 53(2), 403–417.
Jennings, P. C., Housner, G. W., and Tsai, N. C. (1968). “Simulated earthquake motions.” California Institute of Technology, Pasadena, CA.
Kounadis, A. N., Papadopoulos, G. J., and Cotsovos, D. M. (2012). “Overturning instability of a two-rigid block system under ground excitation.” ZAMM-J. Appl. Math. Mech./Zeitschrift für Angewandte Mathematik und Mechanik, 92(7), 536–557.
Lipscombe, P. R., and Pellegrino, S. (1993). “Free rocking of prismatic blocks.” J. Eng. Mech., 1387–1410.
Makris, N., and Vassiliou, M. F. (2014). “Are some top-heavy structures more stable?” J. Struct. Eng., 06014001.
MathWorks. (2016). “MATLAB’s ode event location.” ⟨http://uk .mathworks.com/help/matlab/math/ode-event-location.html⟩ (Aug. 28, 2016).
Nikfar, F., and Konstantinidis, D. (2017). “Shake table investigation on the seismic performance of hospital equipment supported on wheels/casters.” Earthquake Eng. Struct. Dyn., 46(2), 243–266.
Palmeri, A., and Makris, N. (2008). “Response analysis of rigid structures rocking on viscoelastic foundation.” Earthquake Eng. Struct. Dyn., 37(7), 1039–1063.
Plaut, R., Fielder, W., and Virgin, L. (1996). “Fractal behavior of an asymmetric rigid block overturning due to harmonic motion of a tilted foundation.” Chaos, Solitons Fractals, 7(2), 177–196.
Psycharis, I. N. (1990). “Dynamic behaviour of rocking two-block assemblies.” Earthquake Eng. Struct. Dyn., 19(4), 555–575.
Psycharis, I. N., and Jennings, P. C. (1983). “Rocking of slender rigid bodies allowed to uplift.” Earthquake Eng. Struct. Dyn., 11(1), 57–76.
Shampine, L. F., and Gordon, M. K. (1975). Computer solution of ordinary differential equations: The initial value problem, WH Freeman, San Francisco.
Shenton, H. W., and Jones, N. P. (1991). “Base excitation of rigid bodies. 1: Formulation.” J. Eng. Mech., 2286–2306.
Shinozuka, M., and Deodatis, G. (1991). “Simulation of stochastic processes by spectral representation.” Am. Soc. Mech. Eng., 44(4), 191–204.
Smyth, A. W., Brewick, P., Greenbaum, R., Chatzis, M., Serotta, A., and Stünkel, I. (2016). “Vibration mitigation and monitoring: A case study of construction in a museum.” J. Am. Inst. Conserv., 55(1), 32–55.
Spanos, P. D., Roussis, P. C., and Politis, N. P. (2001). “Dynamic analysis of stacked rigid blocks.” Soil Dyn. Earthquake Eng., 21(7), 559–578.
Vassiliou, M. F., and Makris, N. (2012). “Analysis of the rocking response of rigid blocks standing free on a seismically isolated base.” Earthquake Eng. Struct. Dyn., 41(2), 177–196.
Voyagaki, E., Psycharis, I., and Mylonakis, G. (2014). “Complex response of a rocking block to a full-cycle pulse.” J. Eng. Mech., 04014024.
Wittich, C. E., and Hutchinson, T. C. (2016). “Experimental modal analysis and seismic mitigation of statue-pedestal systems.” J. Cult. Heritage, 20, 641–648.
Yim, C.-S., Chopra, A. K., and Penzien, J. (1980). “Rocking response of rigid blocks to earthquakes.” Earthquake Eng. Struct. Dyn., 8(6), 565–587.
Zhang, H., Brogliato, B., and Liu, C. (2012). “Study of the planar rocking-block dynamics with Coulomb friction: Critical kinetic angles.” J. Comput. Nonlinear Dyn., 8(2), 021002.
Zhang, J., and Makris, N. (2001). “Rocking response of free-standing blocks under cycloidal pulses.” J. Eng. Mech., 473–483.
Zulli, D., Contento, A., and Di Egidio, A. (2012). “3D model of rigid block with a rectangular base subject to pulse-type excitation.” Int. J. Non-Linear Mech., 47(6), 679–687.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 144 • Issue 7 • July 2018
Copyright
©2018 American Society of Civil Engineers.
History
Received: Jul 28, 2017
Accepted: Oct 28, 2017
Published online: Apr 18, 2018
Published in print: Jul 1, 2018
Discussion open until: Sep 18, 2018
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.