## The frequency of free oscillation of a damped simple pendulum with large amplitude depends on its amplitude unlike the amplitude-independent frequency of oscillation of a damped simple harmonic osc.

Free Oscillations of a Damped Simple Pendulum: An Analog Simulation Experiment

## Experiments with a damped simple pendulum were carried out and modelled for large amplitude motion with physics students in a mechanics laboratory course of the first year of the university.

Using a large amplitude pendulum in a learning cycle strategy

## This study aimed to investigate the chaos behavior resulted from damped driven nonlinear simple pendulum motion, which was simulated by using Mathematica.

Investigation of chaos behaviour on damped and driven nonlinear simple pendulum motion simulated by mathematica

## The response of the nonlinear simple pendulum was used for benchmarking the boundary conditions for each of the four response regimes and the test criterion was demonstrated using relevant examples.

Energy-Based Criterion for Testing the Nonlinear Response Strength of Strong Nonlinear Oscillators

## We propose a simple pendulum model that accurately captures the velocity fluctuations of the particles in still fluid and find that differences in the falling style might be explained by a closer alignment between the particle's pitch angle and its velocity vector.

Settling behaviour of thin curved particles in quiescent fluid and turbulence

## In particular, a number of classical non-linear oscillators, such as the simple pendulum model, the van der Pol circuital model and the Duffing oscillator class are recalled from the dedicated literature and are extended to evolve on manifold-type state spaces.

Synthetic nonlinear second-order oscillators on Riemannian manifolds and their numerical simulation

## This study aimed to investigate the chaos behavior resulted from damped driven nonlinear simple pendulum motion, which was simulated by using Mathematica.

Investigation of chaos behaviour on damped and driven nonlinear simple pendulum motion simulated by mathematica

## Three nonlinear oscillators including restoring force, the simple pendulum motion, the cubic Duffing oscillator, the Sine-Gordon equation are offered to clarify the effectiveness and usefulness of the proposed technique.

The Rank Upgrading Technique for a Harmonic Restoring Force of Nonlinear Oscillators

10.1016/J.JOULE.2021.04.016

## Herein, we report an active resonance triboelectric nanogenerator (AR-TENG) system fabricated by using simple pendulum, tumbler, and flexible ring TENG, which can harvest omnidirectional and frequency varying water-wave energy through excellent structural design and resonance effect.

Active resonance triboelectric nanogenerator for harvesting omnidirectional water-wave energy

10.1109/DDCLS52934.2021.9455503

## Numeral and simple pendulum examples examine the results and simulations show the effectiveness.

A 2-Dimensional Linear Non-minimum Phase Estimator Based on Dynamic Exceeded Regressor

## We propose a simple pendulum model that accurately captures the velocity fluctuations of the particles in still fluid and find that differences in the falling style might be explained by a closer alignment between the particle's pitch angle and its velocity vector.

Settling behaviour of thin curved particles in quiescent fluid and turbulence

10.1504/ijhvs.2021.10040934

## The testing rig is used to validate a simple pendulum-based, simplified three degree-of-freedom mathematical model of the response of a partially filled container to lateral accelerations.

Experimental-theoretical modelling of the lateral sloshing in rail tankers

10.1177/16878140211050801

## Nondimensional equations are obtained by defining nondimensional parameters like; natural frequency, blade rotation, slenderness ratio, and simple pendulum frequency.

Parametric study of flapwise and edgewise vibration of horizontal axis wind turbine blades

10.1088/2040-8986/abe393

## When expressed in elliptical coordinates the solution of the angular part is the same as the Schrödinger equation for the simple pendulum.

Pendulum beams: optical modes that simulate the quantum pendulum

10.1109/ICRA48506.2021.9561871

## A case study of applying LQR to a simple pendulum and simulations comparing the basins of attraction of minimal- and maximal-coordinate LQR controllers suggest that maximal-coordinate LQR achieves greater robustness and improved performance compared to minimal-coordinate LQR when applied to nonlinear systems.

Linear-Quadratic Optimal Control in Maximal Coordinates

10.1088/1742-6596/1936/1/012017

## In the first, simple SMS, such as the mass-spring system on a plane or the simple pendulum, are analysed.

Interactive Mechanical Systems Using Mathematica

10.1007/S11071-021-06479-Z

## However, systems examined to date have largely comprised simple pendulums limited to planar motion and to correspondingly limited degrees of excitational freedom.

The dynamics of an omnidirectional pendulum harvester

## In this paper, based on conventional photothermal shrinkable material or photothermal expansive material, a simple pendulum is proposed.

Self-Sustained Oscillation of a Photothermal-Responsive Pendulum under Steady Illumination

10.1109/LCSYS.2020.3046682

## Tuning rules to achieve the desired settling time are explicitly derived and illustrated in an experimental study of a simple pendulum to track a reference trajectory with periodic velocity jumps.

Prescribed-Time Stabilization of Controllable Planar Systems Using Switched State Feedback

10.1088/1742-6596/1811/1/012014

## This study aimed to investigate the chaos behavior resulted from damped driven nonlinear simple pendulum motion, which was simulated by using Mathematica.

Investigation of chaos behaviour on damped and driven nonlinear simple pendulum motion simulated by mathematica

10.23919/ACC50511.2021.9483418

## A noticeable finding from the study of the necessary conditions is that the flight path angle of the optimal trajectory obeys the simple pendulum dynamics.

Nonlinear Optimal Missile Guidance for Stationary Target Interception with Pendulum Motion Perspective

10.1109/AIIoT52608.2021.9454228

## To overcome the limitations of the high-order differential equations and the loop gain, the waveforms of the physical periodic motions are expressed by helix functions at time variation, and the characteristics of complex functions are used to examine the behaviors of the transmission spaces and the transmission networks in the different motion models including the Earth's motion, the simple pendulum systems, and the electronic systems.

Study of Behaviors of Motion Models in High-Order Systems

10.1002/9781119604570.ch6

## Contents Preface to the first edition xiii Preface to the second edition xvii 1 Differential and Difference Equations 1 10 Differential Equation Problems 1 100 Introduction to differential equations 1 101 The Kepler problem 4 102 A problem arising from the method of lines 7 103 The simple pendulum 10 104 A chemical kinetics problem 14 105 The Van der Pol equation and limit cycles 16 106 The Lotka-Volterra problem and periodic orbits 18 107 The Euler equations of rigid body rotation 20 11 Differential Equation Theory 22 110 Existence and uniqueness of solutions 22 111 Linear systems of differential equations 24 112 Stiff differential equations 26 12 Further Evolutionary Problems 28 120 Many-body gravitational problems 28 121 Delay problems and discontinuous solutions 31 122 Problems evolving on a sphere 32 123 Further Hamiltonian problems 34 124 Further differential-algebraic problems 36 13 Difference Equation Problems 38 130 Introduction to difference equations 38 131 A linear problem 38 132 The Fibonacci difference equation 40 133 Three quadratic problems 40 134 Iterative solutions of a polynomial equation 41 135 The arithmetic-geometric mean 43 CONTENTS 14 Difference Equation Theory 44 140 Linear difference equations 44 141 Constant coefficients 45 142 Powers of matrices 46 Numerical Differential Equation Methods 51 20 The Euler Method 51 200 Introduction to the Euler methods 51 201 Some numerical experiments 54 202 Calculations with stepsize control 58 203 Calculations with mildly stiff problems 60 204 Calculations with the implicit Euler method 63 21 Analysis of the Euler Method 65 210 Formulation of the Euler method 65 211 Local truncation error 66 212 Global truncation error 66 213 Convergence of the Euler method 68 214 Order of convergence 69 215 Asymptotic error formula 72 216 Stability characteristics 74 217 Local truncation error estimation 79 218 Rounding error 80 22 Generalizations of the Euler Method 85 220 Introduction 85 221 More computations in a step 86 222 Greater dependence on previous values 87 223 Use of higher derivatives 88 224 Multistep-multistage-multiderivative methods 90 225 Implicit methods 91 226 Local error estimates 91 23 Runge-Kutta Methods 93 230 Historical introduction 93 231 Second order methods 93 232 The coefficient tableau 94 233 Third order methods 95 234 Introduction to order conditions 95 235 Fourth order methods 98 236 Higher orders 99 237 Implicit Runge-Kutta methods 99 238 Stability characteristics 100 239 Numerical examples 103 CONTENTS vii 24 Linear Multistep Methods 105 ….

Numerical Methods for Ordinary Differential Equations

10.21307/ijssis-2021-014

## We aim to develop a computer-based simple pendulum experiment set consisting of a simple pendulum, infrared phototransistor, and Arduino board for calculating the gravitational acceleration (g).

Development of a computer-based simple pendulum experiment set for teaching and learning physics

10.1007/978-981-15-9209-6_1

## The liquid free surface oscillation angle is simulated based upon the principles of the simple pendulum analogy for sloshing.

A Simplified Analytical Approach on the Dynamic Pressures in Cylindrical Vertical Tanks

10.1038/s41598-021-92646-6

## In this paper, a pendulum model is represented by a mechanical system that consists of a simple pendulum suspended on a spring, which is permitted oscillations in a plane.

A new pendulum motion with a suspended point near infinity

## For the period T(α) of a simple pendulum with the length L and the amplitude (the initial elongation) α ∈ (0, π), a strictly increasing sequence Tn(α) is constructed such that the relations T1(α)=2Lgπ−2+1ϵln1+ϵ1−ϵ+π4−23ϵ2,Tn+1(α)=Tn(α)+2Lgπwn+12−22n+3ϵ2n+2,$$\begin{array}{c} \displaystyle T_1(\alpha)=2\sqrt{\frac{L}{g}}\left[\pi-2+\frac{1}{\epsilon} \ln\left(\frac{1+\epsilon}{1-\epsilon}\right)+\left(\frac{\pi}{4}-\frac{2}{3}\right)\epsilon^2\right],\\ \displaystyle T_{n+1}(\alpha)=T_n(\alpha)+2\sqrt{\frac{L}{g}}\left(\pi w_{n+1}^2 - \frac{2}{2n+3}\right)\epsilon^{2n+2}, \end{array}$$ and 0<T(α)−Tn(α)T(α)<2ϵ2n+2π(2n+1),$$\begin{array}{} \displaystyle 0 \lt \frac{T(\alpha)-T_n(\alpha)}{T(\alpha)} \lt \frac{2\epsilon^{2n+2}}{\pi(2n+1)}\,, \end{array}$$ holds true, for α ∈ (0, π), n ∈ ℕ, wn:=∏k=1n2k−12k$\begin{array}{} \displaystyle w_n:=\prod_{k=1}^n\frac{2k-1}{2k} \end{array}$ (the nth Wallis’ ratio) and ϵ = sin(α/2).

How is the period of a simple pendulum growing with increasing amplitude?

10.1088/1757-899X/1188/1/012008

## The combination of the nonlinear rheological models to the simple pendulum in the conditions of the forced oscillations simulates the action of the let-off motion with weights.

Relaxation function experimental study of the warp strain at slow and regular weaving motion

10.22055/JACM.2020.35454.2660

## Three nonlinear oscillators including restoring force, the simple pendulum motion, the cubic Duffing oscillator, the Sine-Gordon equation are offered to clarify the effectiveness and usefulness of the proposed technique.

The Rank Upgrading Technique for a Harmonic Restoring Force of Nonlinear Oscillators

10.1007/S11071-021-06274-W

## The dynamic behaviors of the unperturbed system with irrational nonlinearity bear significant similarities to the coupling of a simple pendulum and the smooth and discontinuous (SD) oscillator with the coexistence of the standard homoclinic orbits of Duffing type and pendulum type and the coexistence of the nonstandard homoclinic orbits of SD type and pendulum type in the smooth and discontinuous case, respectively.

An approximate technique to test chaotic region in a rotating pendulum system with bistable characteristics

10.1088/1742-6596/1982/1/012146

## Compared with the common simple pendulum method and falling body method, the theoretical results show that our measurement method has higher precision and is feasible to accurately measure the acceleration of gravity.

Systematic Research on Measuring Acceleration of Gravity by Laser Interferometry

## In particular, a number of classical non-linear oscillators, such as the simple pendulum model, the van der Pol circuital model and the Duffing oscillator class are recalled from the dedicated literature and are extended to evolve on manifold-type state spaces.

Synthetic nonlinear second-order oscillators on Riemannian manifolds and their numerical simulation

## A nonlinear example of a simple pendulum also serves as a benchmark to illustrate the potential of the proposed approach.

Uncertainty Quantification of Differential Algebraic Equations Using Polynomial Chaos

## The response of the nonlinear simple pendulum was used for benchmarking the boundary conditions for each of the four response regimes and the test criterion was demonstrated using relevant examples.

Energy-Based Criterion for Testing the Nonlinear Response Strength of Strong Nonlinear Oscillators

10.35819/remat2021v7i1id4164

## In Physics, one of the applications is the simple pendulum that has oscillation independent of the mass, when it is constant.

Equações diferenciais aplicadas ao pêndulo com massa dependente do tempo: estudo de massa com variação exponencial e polinomial

10.1088/1742-6596/2049/1/012065

## This study aims to produce a digital simple pendulum prototype.

Development of Digital Simple Pendulum Learning Media

## In this paper, a method to measure the gravitational acceleration by oscillation of a simple pendulum, using Arduino board, is presented.

Teaching Physics by Arduino during COVID-19 Pandemic: Oscillation of a simple pendulum

10.1016/j.ymssp.2020.107011

## The present study demonstrates how to perform a artificial intelligence assisted transport operation under variable conditions with hybrid control methods on a simple pendulum.

An intelligent serial connected hybrid control method for gantry cranes

10.23919/ACC50511.2021.9483079

## Tuning rules to achieve the desired settling time are explicitly derived and illustrated in an experimental study of a simple pendulum to track a reference trajectory with periodic velocity jumps.

Prescribed-Time Stabilization of Controllable Planar Systems Using Switched State Feedback

10.5194/BG-18-4059-2021

## vertically distributed mass), whereas broadleaves were better approximated by the simple pendulum model (i.

The motion of trees in the wind: a data synthesis

10.1007/978-3-030-75653-6_4

## The study of simple pendulum has a long history.

Oscillations of a Pendulum Wrapping on Two Cylinders

10.1109/ICESC51422.2021.9532731

## To design the antenna, a simple pendulum shape structure is considered.

Design of Triple Band Slot Antenna for RF Energy Harvesting

10.1088/1742-6596/1806/1/012133

## The main purpose of this study is to figure out the effectivity of virtual laboratory on improving student’s conceptual understanding related to simple pendulum topic using PhET simulation.

Virtual experiment of simple pendulum to improve student’s conceptual understanding

10.1007/S00707-020-02916-Z

## , a simple pendulum, is analyzed with a multiple scale method thanks to supposed assumptions about the excitation.

Nonlinear passive control of a pendulum submitted to base excitations

10.1007/S00419-019-01548-7

## More specifically, five basic structural models are considered: (a) the simple pendulum (SP), (b) the rigid inverted pendulum (RIP), (c) the flexural inverted pendulum (FIP), (d) the rigid rocking block (RRB), and (e) the flexural rocking block (FRB).

Revisiting the fundamental structural dynamic systems: the effect of low gravity

10.23919/ECC.2019.8795821

## In this paper, we present the dynamics of the simple pendulum by using the fractional-order derivatives.

Stability and Trajectories Analysis of a Fractional Generalization of Simple Pendulum Dynamic Equation

10.31349/REVMEXFISE.65.213

## We have considered two cases, a repulsive and attractive electric interactions as perturbations to the classical simple pendulum model.

Dynamics for an electric pendulum

## Such test was used to validate a simplified theoretical approach consisting of a two degree-of-freedom double pendulum mechanical system, where a simple pendulum, representing the sloshing cargo, is articulated to the spring-supported vehicle chassis, which is modelled as an inverted torsional pendulum.

Experimental Modeling of Railway Car Wheel-Forces During Turning

## The aim of this research is to develop teacher guidance material on 12th grade, in the subject of simple pendulum in physics lesson.

Development of teacher guide materials based on Logger Pro sensors for simple pendulum

10.1109/ISACS48493.2019.9068890

## In the first case, the main objective of the simple pendulum test is to determine the oscillation period (T), then the determination of the acceleration due by gravity (g).

Smartphones-based in educational practice works: a case study of mechanical vibration practical works

10.1016/j.heliyon.2019.e02827

## A laboratory activity based on the calculation of the oscillation period of a simple pendulum was done with 12 prospective teachers at the University of Extremadura, who were specializing in Biology/Geology, Physics/Chemistry, and Mathematics.

Improving the self-regulation in prospective science teachers: the case of the calculus of the period of a simple pendulum

10.1088/1361-6552/AB2011

## The setup is made up of a simple pendulum apparatus, an ultrasonic position sensor, and an Arduino Uno board.

Measurement of the gravitational acceleration using a simple pendulum apparatus, ultrasonic sensor, and Arduino

10.1016/J.CHAOS.2019.01.028

## In this paper, we introduce some practical applications of systems of FDAEs in physics such as a simple pendulum in a Newtonian fluid and electrical circuit containing a new practical element namely fractors.

System of fractional differential algebraic equations with applications

10.1142/s266133951950015x

## The frequency of free oscillation of a damped simple pendulum with large amplitude depends on its amplitude unlike the amplitude-independent frequency of oscillation of a damped simple harmonic osc.

Free Oscillations of a Damped Simple Pendulum: An Analog Simulation Experiment

10.1007/S13349-019-00344-9

## To simulate the structural response to the bell ringing, the tower and the bell are modelled as a single degree of freedom system and an unforced and undamped simple pendulum, respectively.

Dynamic monitoring and evaluation of bell ringing effects for the structural assessment of a masonry bell tower

10.1201/9780203750087-2

## It was found that for a simple pendulum, linearized equations were accurate for small values of θ and for damped oscillators, linearized equations were accurate after long time periods.

The nonlinear pendulum

## For the sake of simplicity, we model the swing by a simple pendulum, but allowing amplitudes beyond 90°.

Playground Physics: What Happens When Trying a 360° Swing?

## The Foucault pendulum is a simple pendulum basically set in a non-inertial reference frame.

Simulación numérica del péndulo de Foucault con Octave

## In this paper, two single-sided PTMDs (spring steel-type PTMD and simple pendulum-type PTMD) were designed and fabricated.

Experimental Study on Vibration Control of Suspended Piping System by Single-Sided Pounding Tuned Mass Damper

10.1080/0020739x.2019.1692934

## In differential equations textbooks, the motion of a simple pendulum for small-amplitude oscillations is analyzed.

Motion of a simple pendulum: a digital technology approach

## A simple pendulum is vertically excited on its support and consequently exhibiting oscillations and rotations.

Energy harvesting through pendulum motion and DC generators

## For instance, the smartphone-based acceleration sensor has been employed in a rather quick measurement of the acceleration due to gravity and in analyzing simple pendulum phenomena.

Measuring a spring constant with a smartphone magnetic field sensor

10.1088/1361-6552/AAEA9D

## In the field of physics teaching, the simple pendulum provides a very simple way to study the harmonic oscillator, the source of some fundamental concepts such as periodicity, relaxation time and anharmonicity.

A simple pendulum studied with a low-cost wireless acquisition board

10.1109/ICRA.2019.8793651

## This paper proposes an analysis of walking balance in terms of movements of stance and swing legs based on an inverted pendulum and a simple pendulum.

Balance Map Analysis as a Measure of Walking Balance Based on Pendulum-Like Leg Movements

## e equilibrium position), similar to a simple pendulum.

Application of a two dipole model to PSR J1640-4631, a pulsar with an anomalous braking index.

10.1088/1742-6596/1286/1/012036

## Experiments with a damped simple pendulum were carried out and modelled for large amplitude motion with physics students in a mechanics laboratory course of the first year of the university.

Using a large amplitude pendulum in a learning cycle strategy

10.1007/s10957-018-1424-0

## It also finds applicability on the dynamics of the simple pendulum in a resistive medium.

Variational and Optimal Control Approaches for the Second-Order Herglotz Problem on Spheres

10.1088/1742-6596/1223/1/012004

## As the students have already known the kinematics, dynamics and energy analysis of a simple pendulum in harmonic case, we tried to find analogies between pendulum and compass.

From harmonic oscillation to chaotic motion of a compass

## As an example of the benefit of computer program in the classes room, the results of the comparison of the value of the gravitational acceleration from the data analysis with Tracker are presented; those results are compared to the value of the gravitational acceleration obtained during the development of traditional teaching experiments such as the simple pendulum.

Uso de Tracker para Análisis y Modelado de Datos Experimentales en Laboratorios Tradicionales de Fı́sica

10.1016/J.PROMFG.2019.06.010

## This paper attempts to model a damped driven simple pendulum as dynamical system and solve the model using Undetermined Coefficients Method.

Analytical modelling of driven pendulum with small angular displacement using Undetermined Coefficients Method

10.1299/TRANSJSME.18-00467

## Swing mode 1 has softening as in the simple pendulum, but Swinging-bar mode 2 makes softening/hardening mainly depending on the configuration.

A nonlinear analysis on the three oscillation modes of a bifilar suspension pendulum (1) (Formulation of nonlinear oscillation problem and primary resonance analysis of one degree of freedom nonlinear inertia type forced damping systems)

10.33308/26674874.2019332108

## In both groups, four ADI activities were conducted for a period such as the Oscillation of Simple Pendulum, Factors Affecting the Surface Tension, Determining the Cubes with Same Appearance and Twirly.

Öğretmen Adaylarının Yazılı Argümanlarının Gelişiminde Akran ve Öğretmen Değerlendirmesinin Etkisinin Karşılaştırılması

## We show that (1) residual variation, the variation not explained by the cantilever beam approximation, in fundamental frequencies of broadleaf trees is driven by their architecture; (2) slender trees behave like a simple pendulum, with a single natural frequency dominating their motion, which makes them vulnerable to wind damage and (3) the presence of leaves decreases both the fundamental frequency and the damping ratio.

An architectural understanding of natural sway frequencies in trees