Resonance curve - the case of pedestrian laoding on structures

Resonance is one of fundamental concepts in structural dynamics. In simple terms, it occurs if a periodic force is applied to the (structural) system at or near one of its natural frequencies. This leads to the amplification of the response relative to the case when no resonance occurs. The consequences of this phenomenon can be severe, from accelerated degradation of the structure, lack of fitness for its purpose, to the catastrophic failure. Therefore, understanding this phenomenon is of critical importance to any engineer.

There are many aids available online helping to explain, by means of visualisations, the occurrance of resonance. However, I am not aware of any which put this phenomenon in the context of human-structure interaction. Because through my personal experience I found students, and civil engineers in general, perceptive to this idea, I decided to share the material I gathered in this respect. Please feel free to use it for non-commercial purposes, remembering to acknowledge the source! :) Here it comes...

A dual outrigger pendulously supported platform was designed and built as part of my BEng final year project [1] conducted at Heriot-Watt University in Edinburgh, Scotland. The platform was inspired by the earlier work by Allan McRobie, Guido Morgenthal, Joan Lasenby and Maurice Ringer [2], who built a large rig accommodating treamills for research on laterally-unstable bridges. Due to its smaller scale, my platform only enabled a single person to step on the spot, i.e. alternately raise their legs but without forward progression. Two features were implemented enabling the damping and mass of the platform to be augmented. This was achieved by incorporating an adjustable friction bearing and a sleeve placed under the deck accommodating heavy plates, respectively.

The platofrm effectively behaved as a single degree of freedom system. This is to say that its arrangement could be deduced by measuring its motion at a single point. This presents an excellent opportunity to consider, at its simplest form, the phenomenon of resonance.

Let us consider the pedestrian stepping on the spot on the platform as a source of external force only. This is to say that the pedestrian-generated force is independent of structural behaviour, which is of course not the case in real life. Nevertheless, this assumption will conceptually simplify the problem and enable the expression defining the response of the platform - here modelled as a single degree of freedom system (SDOF), to be written in simple terms. We will later see, based on a series of videos, that although this simplification is quite crude, the response of the platform to pedestrian loading is captured by the resonance curve rather well.

The resonance curve represents the amplification of the response of SDOF subjected to periodic loading relative to the response to static force with the same amplitude, for various ratios of the forcing frequency to the natural frequency of the SDOF. The response will generally be dependent on the level of damping in the system, and so will the critical forcing frequency. However, to further simplify the problem, we will assume that the damping is constant and small - such as in the case of most civil engineering structures, hence we will consider one resonance curve only, as depicted below.

Resonance curve

The dynamic response of the platform to the pedestrian loading at five different frequency ratios is shown on the videos below. Observe how the maximum vibration amplitude varies with the frequency of the loading, which was enforced with a metronome. Now try to relate these results to the resonance curve previously presented. Any thoughts?

1. Stepping on the spot at 88 beats per minute (or 1.47 Hz):

2. Stepping on the spot at 112 beats per minute (or 1.87 Hz):

3. Stepping on the spot at 144 beats per minute (or 2.4 Hz):

4. Stepping on the spot at 168 beats per minute (or 2.8 Hz):

5. Stepping on the spot at 192 beats per minute (or 3.2 Hz):

[1] Mateusz Bocian, Human-structure interaction on a light footbridge, BEng dissertation, School of the Bulit Environment, Heriot-Watt University, Edinburgh, UK, 2009.
[2] Allan McRobie, Guido Morgenthal, Joan Lasenby and Maurice Ringer, Section model tests on human – structure lock-in, Proceedings of the Institution of Civil Engineers - Bridge Engineering, 156/2, 2003, p. 71-79.