Byrtus, Miroslav

Smolík, Luboš

Steinbach, Václav

DSpace at University of West Bohemia

StudentSkávědecká konference 2022Non-linear dynamics of kinematically excited bistable pendulumVáclav Steinbach1, Miroslav Byrtus2, Luboš Smolı́k3IntroductionIn many devices, such as switches, clasps and closures, mechanisms having stable equi-libria in two distinct positions are desired. As an example, to achieve a bistable configurationof a mechanical system, buckling of a beam has been mentioned by Kovacic (2011). Here,we deal with dynamical properties of a bistable pendulum that is kinematically excited. Dy-namic properties are investigated both computationally and experimentally using high-speedcamera Smolik (2020) to capture the large scale motion of the pendulum in different opera-tional regimes.System description and dynamical analysisPresented system consists of physical pendulum and two identical permanent magnets.The first one is placed at the free end of the pendulum and the second one is fixed with themoving frame to which is the pendulum connected by a revolute joint, see Fig. 1 right. Theorientation of the poles of the magnets causes the magnets repel each other. Therefore, twostable and one unstable equilibria exist. The dynamical behaviour is studied both experimentallyFigure 1: Setup of the experiment (left) and trajectory of the end point at ω = 21.6 rad/s (right)and computationally. The equation of motion can be witten in following normalized form, aswas derived in Steinbach (2021)φ̈+ 2DΩ + Ω2 sinφ = fml cosψ sinφ+ u(t)Ω2gω2 sinφ, (1)1 Václav Steinbach; student of study program Applied Mechanics - Dynamics of contructions and mechatronics,e-mail: vstein@students.zcu.cz2 Miroslav Byrtus; Associate Professor at the Department of mechanics, University of West Bohemia, e-mail:mbyrtus@kme.zcu.cz3 Luboš Smolı́k; Research and Testing Institute Plzen; Tylova 1581/46 301 00 Plzeň, Czech Republic; smo-lik@vzuplzen.cz21where the kinematic excitation is assumed in a form u(t) = U cosωt, l represents the lengthof the pendulum and the mentioned angular quantities are defined in Fig. 1 right. Normalizedmagnetic force fm, based on Gilbert model presented in Vokoun (2009), utilizes followingquatities: distance between poles of magnets p, area of magnet cross-section S, length of magnetd, permeability of vacuum µ0, magnetic induction B0, which can be expressed as B0 =µ02M ,where M is magnetization of magnets.Dynamic properties of this system were investigated using numerical integration in timedomain for different excitation frequencies. Based on that bifurcation diagram was created,see Fig. 2 left. It helps to visualize qualitative changes in the system response. Some chaoticresponses were found and for illustration, chaotic attractor is visualized using Poincaré’s sectionin Fig. 2 right.0 5 10 15 20 25 30-80-60-40-20020406080-60 -50 -40 -30 -20 -10 0 10 20 30-4-20246810Figure 2: Bifurcation diagram of pendulum angular displacement (left), chaotic attractor rep-resented by Poincaré’s section corresponding to excitation frequency ω = 13.7 (right)The demonstrator design is intertwined with modelling. Therefore, many parameters ofthe model were chosen with respect to physical limitations of the demonstrator. Kinematic ex-citation is realized by means of a DC motor. Since the motor is capable of rotational motiononly, a Scotch Yoke mechanism was employed to convert rotational motion to a linear reversibleone, see Fig. 1 left. The motion of the demonstrator was captured using an IRIS M high-speedcamera and Genaray AK-230 monolight LED reflector. The recordings were captured with aframerate of 256 fps in HD resolution (1280× 720 px) and the trajectories of chosen referencepoints were tracked and compared with those gained using computational model based on equa-tion (1).This publication was supported by the project SGS-2022-008.ReferencesKovacic I., Brennan M.J. (2011) The Duffing Equation. Nonlinear Oscillators and their Be-haviour, Wiley 2011.Vokoun, D., Beleggia, M., Heller, L. L., Sittner, P. (2009). Magnetostatic interactions andforces between cylindrical permanent magnets. Journal of Magnetism and Magnetic Ma-terials, 321(22), 3758-3763.Smolik, L.: Vyuzitı́ vysokorychlostni kamery v technicke diagnostice, Jaderna energie, SpecialIssue October 2020, 145-147.Steinbach, V.: Vibration of bistable mechanical systems, bachleor thesis, University of WestBohemia.22

Non-linear dynamics of kinematically excited bistable pendulum

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Non-linear dynamics of kinematically excited bistable pendulum

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