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# A Method to Determine the Exact Time Period of Oscillations of a Bifilar Pendulum

[+] Author Affiliations

Three S Consulting, Irvine, CA

Richard Rodrigues Jettappa

Gas Turbine Research Establishment, Bangaluru, KA, India

Paper No. GT2010-23531, pp. 1097-1102; 6 pages
doi:10.1115/GT2010-23531
From:
• ASME Turbo Expo 2010: Power for Land, Sea, and Air
• Volume 6: Structures and Dynamics, Parts A and B
• Glasgow, UK, June 14–18, 2010
• Conference Sponsors: International Gas Turbine Institute
• ISBN: 978-0-7918-4401-4 | eISBN: 978-0-7918-3872-3

## abstract

A novel method to determine the exact time period of oscillations of a class of non-linear systems is presented. Taking the bifilar pendulum as an example, and employing the conservation of total energy concept, the free oscillations of the system is studied. The governing equation of motion of a bifilar pendulum is non-linear. The integration of this equation to obtain the time period of oscillation is highly complicated and only numerical solution is available. This is because the integral is singular at the extremities of the motion where the velocity will be zero. But, what cannot be achieved by integral calculus can be obtained easily by employing the definition of velocity taught in the high school curriculum. By employing this simple mathematical trick, this intractable equation is recast in a different but exact form. This leads to the identification of what is called the “Geometric Inertia” in bifilar pendulums. This Geometric Inertia is the additional inertia displayed by the system due to the constraint imposed by the two filaments as a result of the geometry of the pendulum. In the proposed method, the total displacement of the system is considered and divided into small equal segments. At the end points of each such segment, the corresponding velocity is calculated from the energy equation. Noting that the velocities are zero at the extremities of the system, an average velocity to each segment is calculated, and this average velocity is positive in each segment. The “delta” time spent by the system in each segment is now calculated by dividing the segment length by the average velocity of that segment. (From, time = displacement/velocity). The linear sum of such “delta” times gives the time period of oscillation. As the number of segments is increased, thereby reducing the segment length, the estimate becomes increasingly accurate. The proposed approach avoids a direct integration of complex, and often singular expressions that complicate the determination of time periods of oscillations of non-linear systems.

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