Beyond the Linear Damping Model for Mechanical Harmonic Oscillators

The characteristics of drive-free oscillations of a damped simple pendulum under sinusoidal potential force field differ from those of the damped harmonic oscillations. The frequency of oscillation of a large amplitude simple pendulum decreases with increasing amplitude. Many prototype mechanical simple pendulum have been fabricated with precision and studied earlier in view of introducing them in undergraduate physics laboratories. However, fabrication and maintenance of suc...

The motion of a pendulum is described as Simple Harmonic Motion (SHM) in case the initial displacement given is small. If we relax this condition then we observe the deviation from the SHM. The equation of motion is non-linear and thus difficult to explain to under-graduate students. This manuscript tries to simplify things.

We have designed, built and operated a physical pendulum which allows one to demonstrate experimentally the behaviour of the pendulum under any equation of motion for such a device for any initial conditions. All parameters in the equation of motion can be defined by the user. The potential of the apparatus reaches from demonstrating simple undamped harmonic oscillations to complex chaotic behaviour of the pendulum. The position data of the pendulum as well as derived kinemat...

The nearly-incessant free oscillations of the Earth (not the larger, long-lived normal modes seen following intense quakes) were first observed by accident in the record of a tilt-sensitive instrument designed to study surface physics. Later tiltmeter studies demonstrated the usefulness of autocorrelation for the routine study of these normally short-lived eigenmodes. More recent studies suggest that many tilt-sensitive instruments are capable of observing these oscillations-...

Many damped mechanical systems oscillate with increasing frequency as the amplitude decreases. One popular example is Euler's Disk, where the point of contact rotates with increasing rapidity as the energy is dissipated. We study a simple mechanical pendulum that exhibits this behaviour.

In this paper we present a study of the non-linear effects of anharmonicity of the potential of the simple pendulum. In a theoretical reminder we highlight that anharmonicity of the potential generates additional harmonics and the non-isochronism of oscillations. These phenomena are all the more important as we move away from the oscillations at small angles, which represent the domain of validity of the harmonic approximation. The measurement is apprehended by means of the a...

Theoretically, solutions of the damped harmonic oscillator asymptotically approach equilibrium, i.e., the zero energy state, without ever reaching it exactly, and the critically damped solution approaches equilibrium faster than the underdamped or the overdamped solution. Experimentally, the systems described with this model reach equilibrium when the system's energy has dropped below some threshold corresponding to the energy resolution of the measuring apparatus. We show th...

This paper is concerned with identifying the instantaneous modal parameters of forced oscillatory systems with response-dependent generalized inertia (mass, inductance, or equivalent) based on their measured dynamics. An identification method is proposed, which is a variation of the "FORCEVIB" method. The method utilizes analytic signal representation and the properties of the Hilbert transform to obtain an analytic relationship between a system's natural frequency and dampin...

The small angle approximation often fails to explain experimental data, does not even predict if a plane pendulum's period increases or decreases with increasing amplitude. We make a perturbation ansatz for the Conserved Energy Surfaces of a one-dimensional, parity-symmetric, anharmonic oscillator. A simple, novel algorithm produces the equations of motion and the period of oscillation to arbitrary precision. The Jacobian elliptic functions appear as a special case. Thrift ex...

We present an analytical description of the large-amplitude stationary oscillations of the finite discrete system of harmonically-coupled pendulums without any restrictions to their amplitudes (excluding a vicinity of $\pi$). Although this model has numerous applications in different fields of physics it was studied earlier in the infinite limit only. The developed approach allows to find the dispersion relations for arbitrary amplitudes of the nonlinear normal modes. We unde...