Mechanical Resonance. Free and forced SHM of a torsional pendulum

Scientific Essay, 2012

9 Pages


Mechanical Resonance

- Free and forced SHM of a torsional pendulum

Laura Imperatori, Murray Edwards College, lsi22 Experiment performed Friday, 27/01/2012 (Practical partner x, Trinity College)


In order to test SHM, the behaviour changes of a torsion pendulum due to different damping factors as well as its changes due to applying an external exciter were observed and compared with the theoretical expectations. The quality factor of the same damping state (with a brake current of 6A in the eddy brakes) was calculated using two different approaches and the resulting values were found to be within o of each one, as Qt =6.0±1.3 and Q2 =7.5±0.7. The first approach was based on measuring the maximum displacement for each successive oscillation and deducing the slope from the plot of the natural logarithm of the amplitude against the number of oscillations. Differently, the second estimate of Q was obtained under forced oscillation conditions by taking the ratio of the experimentally determined resonance amplitude and the amplitude of natural oscillation.

I. Introduction

The aim of the experiment is to measure free, damped and forced SHM in a torsion pendulum. The investigation of damped oscillations led us to the determination of the quality factor Q, while the examination of forced oscillations resulted in observations of resonance phenomena and thus the determination of the resonance frequency. By combining the damped and forced oscillations, it was possible to investigate how the degree of damping affects the amplitude of the response to a sinusoidally varying spring force.

In principle, the quality factor Q can be deduced from any damped SHM oscillating system by measuring the natural frequency and the decay constant / as Q equals [Abbildung in dieser Leseprobe nicht enthalten] Resonance can

also be observed in many different systems, involving simple pendulums and string pendulums. As these measurements are very inaccurate and imprecise, we used a special torsional pendulum developed by German physicist Robert Wichard Pohl (see VI.2). This pendulum is damped with a manually variable eddy current brake that is widely used for didactic purposes demonstrating mechanical resonance to undergraduate Physics students.

Current research also involves the use of torsional pendulums. For example, the Eot-Wash Group at the Center for Experimental Nuclear Physics and Astrophysics of the University of Washington tests the equivalence principle of Einstein's General Theory of Relativity with torsional balances. (See VI.3)

II. Theoretical Background

Simple harmonic oscillation of a system can be caused by any restoring force (or torque) which is directly proportional to linear (or angular) displacement. In the case of a torsion pendulum like the one investigated in this experiment, the restoring torque is supplied by a spring. Along with Newton's Second Law, this suggests a second order differential which describes the oscillating motion of an object of moment of inertia I suspended on a torsion wire of torsion constant r:

Abbildung in dieser Leseprobe nicht enthalten

where[Abbildung in dieser Leseprobe nicht enthalten] is the angular displacement from equilibrium.

The solution is conventionally written:[Abbildung in dieser Leseprobe nicht enthalten] with the natural frequency of oscillation under the assumption that the system has been released from rest at an angle[Abbildung in dieser Leseprobe nicht enthalten].

If there is an opposing damping force proportional to the angular velocity Q, the SHM equation in (1) has to be modified to be

Abbildung in dieser Leseprobe nicht enthalten

where b is a constant.

This can be generalised to be

Abbildung in dieser Leseprobe nicht enthalten

with the decay constant y[Abbildung in dieser Leseprobe nicht enthalten]as a measure of damping and[Abbildung in dieser Leseprobe nicht enthalten]as natural frequency. The motion of the pendulum is now decaying according to:

Abbildung in dieser Leseprobe nicht enthalten

Depending on the parameters B and p, which vary with the magnitude of the damping coefficient b compared with I and r, there are three different resulting cases called underdamped, overdamped and critically damped.

The plot of amplitude against time of these three different cases is shown in figures F1, F2 and F3:

The dotted line in the overdamped case shows an underdamped system with the same decay constant, while in the critically damped case it shows an overdamped system with the same decay constant.

As visible in the above graphs, the only case that oscillates is the underdamped case. Its free response is of the form[Abbildung in dieser Leseprobe nicht enthalten], where [Abbildung in dieser Leseprobe nicht enthalten] with the decay constant/. The slower the decay of the oscillations in relation to the period, the better is an oscillator. Hence, for a good oscillator the quality factor as measure of this has to be significantly greater than[Abbildung in dieser Leseprobe nicht enthalten]

In the case of additionally applying a sinusoidal driving couple A cos(Mt), the overall equation becomes:

Abbildung in dieser Leseprobe nicht enthalten

The general solution to the above equation (2) is the superposition of a transient (or free) and steady-state (or forced) response: the sum of the solution of (1) and a particular solution to (2).

The particular solution can be found by solving equation (2) of the form


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Mechanical Resonance. Free and forced SHM of a torsional pendulum
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Laura Imperatori (Author), 2012, Mechanical Resonance. Free and forced SHM of a torsional pendulum, Munich, GRIN Verlag,


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