Resonance: The Maths 1 INTRODUCTION
Mike Tsoi (Year 13, Peel)
When designing structures, one of the most important issues to consider is resonance. What is resonance? In layman’s terms, resonance describes the gradual increase in the oscillation amplitude of an object over time. When designing a bridge, engineers take resonance into account - if resonance occurs, then the bridge would sway back and forth with increasing amplitude and the oscillation would get more violent. Inevitably, the bridge would snap, as was the case for the Tacoma Narrows Bridge on November 7th 1940. Also, relatively recently in 2016, jumping fans made a football stadium resonate [1]. Fortunately, it didn’t collapse.
2 HOW DOES RESONANCE OCCUR? When the frequency of a periodically applied force (which will now be referred to as the driving frequency) is equal to the natural frequency of the object, the force is most efficient at transferring energy to the object. The energy the object receives is converted to kinetic energy, so the oscillation amplitude increases over time.
3 THE MATHS BEHIND RESONANCE
The oscillation follows a linear second order non-homogeneous differential equation:
(1) where a, b, c, F and ω are constants such that a > 0, b ≥ 0, c > 0, F > 0, ω > 0. ω is the driving frequency, ÿ is the acceleration at any given time, ẏ is the velocity at any given time, y is the displacement at any given time, and t is the time since the oscillation started. We can solve (1) completely, but the solution - and hence the behaviour of the oscillation - depends on b. b is the damping coefficient. Let’s say that the system is undamped, i.e. b = 0. Thus, (1) becomes
(2) For convenience, define
where ω0 is the natural frequency of the oscillating object. To solve (2), we must consider two cases: when ω ≠ ω0 and ω = ω0 3.1 FIRST CASE (ω ≠ ω0) When ω ≠ ω0, the auxiliary equation is
am2 + c = 0
The solutions to the auxiliary equation are imaginary.
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