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1. (2D+1) pendulum beams: non-diffracting optical spatial wavepackets that simulate quantum pendulum dynamics

   https://doi.org/10.1117/12.2627732  

   Nguyen, Thao; Rodríguez-Fajardo, Valeria; Galvez, Enrique J. (March 2022, Complex Light and Optical Forces XVI, 1201704)

   Andrews, David L.; Galvez, Enrique J.; Rubinsztein-Dunlop, Halina (Ed.)

   The similarity between the 2D Helmholtz equation in elliptical coordinates and the Schr¨odinger equation for the simple mechanical pendulum inspires us to use light to mimic this quantum system. When optical beams are prepared in Mathieu modes, their intensity in the Fourier plane is proportional to the quantum mechanical probability for the pendulum. Previous works have produced a two-dimensional pendulum beam that oscillates as a function of time through the superpositions of Mathieu modes with phases proportional to pendulum energies. Here we create a three-dimensional pendulum wavepacket made of a superposition of Helical Mathieu-Gaussian modes, prepared in such a way that the components of the wave-vectors along the propagation direction are proportional to the pendulum energies. The resulting pattern oscillates or rotates as it propagates, in 3D, with the propagation coordinate playing the role of time. We obtained several different propagating beam patterns for the unbound-rotor and the bound-swinging pendulum cases. We measured the beam intensity as a function of the propagation distance. The integrated beam intensity along elliptical angles plays the role of quantum pendulum probabilities. Our measurements are in excellent agreement with numerical simulations. 

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2. Strong nonreciprocity in a bistable pendulum with contactless coupling to a monostable pendulum

   https://doi.org/10.1007/s11071-025-10992-w  

   Rouleau, Michael; Booker, Zachary; Shi, Chengzhi; Meaud, Julien (February 2025, Nonlinear Dynamics)

   Abstract This article studies the nonreciprocity of a system that consists of a bistable element coupled to a monostable element through a contactless magnetic interaction. To illustrate the concept, the bistable element is physically realized using a pendulum that interacts with a stationary magnet and the monostable element is a classical pendulum. A numerical model is implemented to simulate the nonlinear dynamics of the system. Both simulations and experiments show that the system exhibits a strong amplitude-dependent nonreciprocity in response to initial excitations. At small input amplitudes, the system has an intrawell response with minimal transmission of energy whether the excitation is exerted on the side of the bistable pendulum or on the other side. However, at high input amplitude, a strong nonreciprocal behavior is observed: excitation of the bistable pendulum causes an interwell response which considerably reduces the distance between the two pendulums and allows energy to be efficiently transmitted through the contactless magnetic interaction; excitation of the monostable pendulum does not cause any interwell response and results in limited energy transmission. The combination of bistability and contactless nonlinear interactions allows the system to exhibit very strong amplitude-dependent nonreciprocity, which may be useful in a wide range of applications. 

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3. The Response of an Inerter-Based Dynamic Vibration Absorber With a Parametrically Excited Centrifugal Pendulum

   https://doi.org/10.1115/1.4053789  

   Gupta, Aakash; Tai, Wei-Che (August 2022, Journal of Vibration and Acoustics)

   Abstract The inerter has been integrated into various vibration mitigation devices, whose mass amplification effect could enhance the suppression capabilities of these devices. In the current study, the inerter is integrated with a pendulum vibration absorber, referred to as inerter pendulum vibration absorber (IPVA). To demonstrate its efficacy, the IPVA is integrated with a linear, harmonically forced oscillator seeking vibration mitigation. A theoretical investigation is conducted to understand the nonlinear response of the IPVA. It is shown that the IPVA operates based on a nonlinear energy transfer phenomenon wherein the energy of the linear oscillator transfers to the pendulum vibration absorber as a result of parametric resonance of the pendulum. The parametric instability is predicted by the harmonic balance method along with the Floquet theory. A perturbation analysis shows that a pitchfork bifurcation and period doubling bifurcation are necessary and sufficient conditions for the parametric resonance to occur. An arc-length continuation scheme is used to predict the boundary of parametric instability in the parameter space and verify the perturbation analysis. The effects of various system parameters on the parametric instability are examined. Finally, the IPVA is compared with a linear benchmark and an autoparametric vibration absorber and shows more efficacious vibration suppression. 

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4. A Swing of Beauty: Pendulums, Fluids, Forces, and Computers

   https://doi.org/10.3390/fluids5020048  

   Mongelli, Michael; Battista, Nicholas A. (June 2020, Fluids)

   While pendulums have been around for millennia and have even managed to swing their way into undergraduate curricula, they still offer a breadth of complex dynamics to which some has still yet to have been untapped. To probe into the dynamics, we developed a computational fluid dynamics (CFD) model of a pendulum using the open-source fluid-structure interaction (FSI) software, IB2d. Beyond analyzing the angular displacements, speeds, and forces attained from the FSI model alone, we compared its dynamics to the canonical damped pendulum ordinary differential equation (ODE) model that is familiar to students. We only observed qualitative agreement after a few oscillation cycles, suggesting that there is enhanced fluid drag during our setup’s initial swing, not captured by the ODE’s linearly-proportional-velocity damping term, which arises from the Stokes Drag Law. Moreover, we were also able to investigate what otherwise could not have been explored using the ODE model, that is, the fluid’s response to a swinging pendulum—the system’s underlying fluid dynamics. 

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5. Physical Pendulum without Small Angle Approximation

   https://doi.org/10.1119/PICUP.Exercise.PWSAA  

   Vemuri, Gautam; Gavrin, Andy (November 2020, American Association of Physics Teachers)

   The simple pendulum usually studied by analytic methods invokes the small angle approximation (SAA) so that one can easily reduce the equation of motion to Hooke’s law and thereby obtain the period of the pendulum and other associated quantities. If the approximation is relaxed, the problem becomes analytically intractable and one must resort to computational methods. In this exercise, the pendulum with and without the SAA are compared to allow students to discover what happens to the temporal behavior for larger angles of displacement. The students will also be able to obtain quantitative estimates of what a “small angle” means, and the limits of validity of the SAA. In the computations, students will learn that the second order differential equation that describes the motion of the pendulum can be reduced to two coupled first order differential equations, which can then be solved by the Euler-Cromer algorithm. In this specialized exercise set, students will also learn to use the ODE45 package in MATLAB to solve differential equations and its advantages over Euler-Cromer method, as well as ‘findpeaks’ command in MATLAB. 

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