Bending waves cause structural-borne noise.

Remember when we were kids how we’d use a ruler to flick a piece of paper? That’s a way to harness the power of a bending moment! Bending moments form the basis of a bending wave, and we encounter these waves all around us every day without ever realising.

Bending waves exist in solid objects. They can only exist with things that have a width, length and height. Another example is when someone jumps off a diving board into the pool. The diving board usually vibrates momentarily after the person makes the jump.

Commonly, we encounter bending waves when we find a cantilever structure such as below, e.g. a plank with one edge fixed, and another edge free.

Theory

Concept 1: compressed and expanded structural elements.

Take for example a simple rectangular beam. The most important aspect of a bending wave (or a moment) is the compression of the elements in the inner portion of the beam, and expansion on the outer portion of the beam, as illustrated below.

Concept 2: wave speed dependent on cross-sectional area as well as frequency.

Bending wave speed equation for an infinitesimal element can be represented as:

The velocity of a bending wave (phase velocity) is dependent on: 1) frequency 2) bending stiffness and 3) density. The big thing here is that bending stiffness is dependent on the cross-sectional area of the beam in addition to the elasticity of its material.

There is an additional factor here which is that depending which way you bend the object, the axis makes a difference. The resultant stiffness of the object is different depending which way you bend it. We can revisit our ruler example, you can bend it easily one way, e.g. it is soft therefore lower bending stiffness around that axis, but you can’t bend it easily the other way, i.e. it is harder therefore higher bending stiffness. Higher bending stiffness results in a higher wave speed, but the frequency has a higher effect on wave speed (due to the dominating square root in equation above). If you want to have your own deep dive into this equation, please consult Cremer — Structural Vibration, one of my favourite books about the subject, and provides simplified equation for the case of a plate.

Example: natural frequency of a cantilever.

Back to the diving board example. The person jumping off the board has a mass m, which causes the diving board to deflect by amount delta, as illustrated. One neat trick we can do is we can calculate the natural frequency of the board by only knowing delta, as follows:

This is really cool because it is similar to calculating the static deflection on a spring. There are some simple assumptions made here, but ultimately, a cantilever does behave like a spring in this case. (Credits to Prof. J. Kim Vandiver who came up with this example, link).

Applications: Bending waves can be effective noise radiators and are used in products.

Under the right conditions, bending waves travelling in solid objects can generate noise e.g. convert to air pressure waves. A common real world use case for this is flat panel loudspeakers (also known as distributed-mode or slim profile loudspeakers). Have you ever heard the sound coming from a flat panel loudspeaker?

Especially thin panels could act as efficient noise radiators with of course many limitations such as frequency. Sometimes this could be problematic even in buildings, for example non-structural partition walls. High energy vibration can travel from a source such as an underground train or a heavy vehicle passing by, through the building and gets converted to noise inside the apartment or house.

Thank you for reading our article, for any questions please email info@vibration-consultants.