Abstract
The chapter deals with the analytical investigation of the physical pendulum mounted on the spring-damper suspension. The pendulum exhibits three degrees of freedom in plane motion. Three types of external loading and viscous damping are considered. The method of multiple scales is used to solve the initial value problem defined through Lagrange formalism. The proposed analytical method allows for the prediction of resonance conditions. The amplitude-frequency response curves have been determined for the external resonance, and their stability has been assessed using the Routh–Hurwitz criterion. The modulation equations of the amplitudes and phases are the basis for studying the impact of the chosen parameters on the internal resonance 1:2.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Parks, H.V., Faller, J.E.: A simple pendulum laser interferometer for determining the gravitational constant. Philos Trans A Math Phys Eng Sci. 372(2026), 20140024 (2014)
Lefrancois, S., Gosselin, C.: Point-to-point motion control of a pendulum-like 3-dof underactuated cable-driven robot. In: Proceedings of the 2010 IEEE International Conference on Robotics and Automation, ICRA 2010, 3–7, pp. 5187–5193 (2010)
Loram, I.D., Lakie, M.: Human balancing of an inverted pendulum: position control by small, ballistic-like, throw and catch movements. J. Physiol. 540(Pt 3), 1111–1124 (2002)
Starosvetsky, Y., Gendelman, O.V.: Dynamics of a strongly nonlinear vibration absorber coupled to a harmonically excited two-degree-of-freedom system. J. Sound Vib. 312, 234–256 (2008)
Sado, D.: Regular and Chaotic Vibration of Some Systems with Pendula. WNT, Warsaw (2010). (in Polish)
Starosta, R., Sypniewska-Kamińska, G., Awrejcewicz, J.: Plane motion of a rigid body suspended on nonlinear spring-damper. In: Problems of Nonlinear Mechanics and Physics of Materials, pp. 157–170. Springer, Switzerland (2018)
Awrejcewicz, J., Starosta, R., Sypniewska-Kamińska, G.: Asymptotic analysis of resonances in nonlinear vibrations of the 3-dof pendulum. Differ. Equ. Dyn. Syst. 21(1&2), 123–140 (2013)
Manevitch, L.I., Musienko, A.I.: Limiting phase trajectories and energy exchange between anharmonic oscillator and external force. Nonlinear Dynam. 58, 633–642 (2009)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Awrejcewicz, J., Starosta, R., Sypniewska-Kamińska, G. (2020). External and Internal Resonances in a Mass-Spring-Damper System with 3-dof. In: Lacarbonara, W., Balachandran, B., Ma, J., Tenreiro Machado, J., Stepan, G. (eds) Nonlinear Dynamics of Structures, Systems and Devices. Springer, Cham. https://doi.org/10.1007/978-3-030-34713-0_17
Download citation
DOI: https://doi.org/10.1007/978-3-030-34713-0_17
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-34712-3
Online ISBN: 978-3-030-34713-0
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)