Simple ‘log formulae’ for pendulum motion valid for any amplitude

The magnitude of the damping force of the mathematical pendulum swinging on a medium is usually proportional to the speed of the pendulum. In this research, the pendulum oscillation parameters oscillating on an air medium under the influence of a magnetic field of 1.8 G will be investigated. In the initial stage, the effect of the magnetic force on the damping coefficient of ferrite pendulum oscillations with an initial deviation of 15 degrees observed. Furthermore, the study continued with varying the angle of deviation from 5 degrees to 25 degrees. The results of the data fitting amplitude (xi) at various swing times (ti) are using to analyze the effect of the angle of deviation on the maximum amplitude. The results showed that for the deviation angle of 15o the coefficient of damping of the medium affected by the magnetic force was 0,0022 greater than the coefficient of air damping 0,00006. It affects the amplitude, which decreases faster than the pendulum amplitude without the influence of magnetic force. Variation in the angle of deviation also affects the amplitude of the pendulum. In the deviation angle below 10, the pendulum motion is more influenced by the magnetic force, whereas in the deviation above 10, the pendulum motion is more dominated by gravity.

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Modeling a Knuckle-Boom Crane Control to Reduce Pendulum Motion Using the Moving Frame Method

Volume 4: Dynamics, Vibration, and Control ◽

10.1115/imece2019-10436 ◽

2019 ◽

Author(s):

Maren Eriksen Eia ◽

Elise Mari Vigre ◽

Thorstein Ravneberg Rykkje

Keyword(s):

Equations Of Motion ◽

Moving Frame ◽

Web Pages ◽

Moving Frames ◽

Pendulum Motion ◽

Moving Frame Method ◽

Frame Method ◽

The Masses ◽

And Control ◽

Boom Crane

Abstract A Knuckle Boom Crane is a pedestal-mounted, slew-bearing crane with a joint in the middle of the distal arm; i.e. boom. This distal boom articulates at the ‘knuckle (i.e.: joint)’ and that allows it to fold back like a finger. This is an ideal configuration for a crane on a ship where storage space is a premium. This project researches the motion and control of a ship mounted knuckle boom crane to minimize the pendulum motion of a hanging load. To do this, the project leverages the Moving Frame Method (MFM). The MFM draws upon Lie group theory — SO(3) and SE(3) — and Cartan’s Moving Frames. This, together with a compact notation from geometrical physics, makes it possible to extract the equations of motion, expeditiously. The work reported here accounts for the masses and geometry of all components, interactive motor couples and prepares for buoyancy forces and added mass on the ship. The equations of motion are solved numerically using a 4th order Runge Kutta (RK4), while solving for the rotation matrix for the ship using the Cayley-Hamilton theorem and Rodriguez’s formula for each timestep. This work displays the motion on 3D web pages, viewable on mobile devices.

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