Damping in low-frequency mechanical pendulums

Measurement Science and Technology, 1993

This paper reports the performance and mechanical properties of an inverted pendulum as a vibration isolator. The advantages of such a system for vibration isolation in an interferometric gravitational wave detector is demonstrated through modelling of the isolator's response. An experimental investigation using a prototype inverted pendulum was also performed where the behaviour of the stability, resonant frequency and 0 factor with varying load mass was examined. The pendulum was also used as a seismometer to examine low-frequency Seismic activity.

2006

Experiments on the oscillatory motion of a suspended bar magnet throws light on the damping effects acting on the pendulum. The viscous drag offered by air was found the be the main contributor for slowing the pendulum down. The nature and magnitude of the damping effects were shown to be strongly dependent on the amplitude.

Journal of Physics A: Mathematical and General, 2002

A simple qualitative physical explanation is suggested for the phenomenon of subharmonic resonances of a rigid planar pendulum whose axis is forced to oscillate with a high frequency in the vertical direction. An approximate quantitative theory based on the suggested approach is developed. The spectral composition of the subharmonic resonances is investigated quantitatively, and the boundaries of these modes in the parameter space are determined. New related modes of regular behaviour are described and explained. The conditions of the inverted pendulum stability are determined with a greater precision than they have been known earlier. A computer program simulating the physical system supports the analytical investigation.

European Journal of Physics

The resonance characteristics of a driven damped harmonic oscillator are well known. Unlike harmonic oscillators which are guided by parabolic potentials, a simple pendulum oscillates under sinusoidal potentials. The problem of an undamped pendulum has been investigated to a great extent. However, the resonance characteristics of a driven damped pendulum have not been reported so far due to the difficulty in solving the problem analytically. In the present work we report the resonance characteristics of a driven damped pendulum calculated numerically. The results are compared with the resonance characteristics of a damped driven harmonic oscillator. The work can be of pedagogic interest too as it reveals the richness of driven damped motion of a simple pendulum in comparison to and how strikingly it differs from the motion of a driven damped harmonic oscillator. We confine our work only to the nonchaotic regime of pendulum motion.

International Journal of Non-Linear Mechanics

In the classical papers (see, e.g. P.L. Kapitsa, Pendulum with vibrating axis of suspension. Usp. Fiz. Nauk 44 1 (1954) 7-20 (in Russian)) motion of pendulum with vibrating suspension axis was considered in the case when frequency of external loading is much higher than the natural frequency of the pendulum in the absence of this loading. The present paper is concerned with the analysis of inverted pendulum's motion at unconventional values of parameters. Case when frequency of external loading and the natural frequency of the pendulum in the absence of this loading are of the same order is studied. Vibration intensity is assumed to be relatively low. A new modification of the method of direct separation of motions (MDSM) is proposed to study the corresponding equation which in the considered case does not contain a small parameter explicitly. The aim is to obtain solutions of this equation in the stability domain. It is revealed that in the considered range of parameters not only the effective stiffness of the system changes due to the external loading, but also its effective mass. Applicability of the proposed approach for solving non-linear equations without small parameter is demonstrated; as an illustration, a damped Duffing equation is considered.

Meccanica

We propose a general model for pendular systems with an arbitrary number of links arranged sequentially. The form of this model is easily adaptable to different settings and operating conditions. The main subject of analysis is a system obtained as a specific case taken from the general analysis, a three-links pendulum with damping subject to periodic perturbation. We performed a theoretical analysis of the frequency response and compared it with results from temporal integration. Moreover, a law was obtained explaining the behavior of the shift of the resonant frequencies due to a change in a parameter.

In our discussion of the damped pendulum case we have dealt with two different types of damping,

Nonlinear Dynamics, 2022

Gaining insight into possible vibratory responses of dynamical systems around their stable equilibria is an essential step, which must be taken before their design and application. The results of such a study can signi cantly help prevent instability in closed-loop stabilized systems through avoiding the excitation of the system in the neighborhood of its resonance. This paper investigates nonlinear oscillations of a Rotary Inverted Pendulum (RIP) with a full-state feedback controller. Lagrange's equations are employed to derive an accurate 2-DoF mathematical model, whose parameter values are extracted by both the measurement and 3D modeling of the real system components. Although the governing equations of a 2-DoF nonlinear system are di cult to solve, performing an analytical solution is of great importance, mostly because, compared to the numerical solution, the analytical solution can function as an accurate pattern. Additionally, the analytical solution is generally more appealing to engineers because their computational costs are less than those of the numerical solution. In this study, the perturbative method of multiple scales is used to obtain an analytical solution to the coupled nonlinear motion equations of the closed-loop system. Moreover, the parameters of the controller are determined, using the results of this solution. The ndings reveal the existence of hardening-and softening-type resonances at the rst and second vibrational modes, respectively. This led to a wide frequency range with moderately large-amplitude vibrations, which must be avoided when adjusting a time-varying set-point for the system. The analytical results of the nonlinear vibration of the RIP are veri ed by experimental measurements, and a very good agreement is observed between the results of both approaches.

European Journal of Physics, 2005

To familiarize undergraduate students with the dynamics of a damped driven harmonic oscillator, a simple pendulum was set up and driven at its suspension point under different damping conditions. From the time domain analysis, the decay constant was estimated and used to predict the frequency response. The simple pendulum was then driven at a series of frequencies near the resonance. By measuring the maximum amplitude at each driving frequency, the frequency response was determined. With one free parameter, which was determined under the first damping condition, the predicted frequency responses showed good agreement with the measured frequency responses under all damping conditions.

European Journal of Physics, 1999

Damped oscillatory motion is one of the most widely studied movements in physics courses. Despite this fact, dry damped oscillatory motion is not commonly discussed in physics textbooks. In this work, we discuss the dry and viscous damped pendulum, in a teaching experiment that can easily be performed by physics or engineering students.