Back in early February, a new internet sensation swept the world via YouTube called the “Harlem Shake”. For those who aren’t familiar with it, you can do some cultural catch-up here.

When I saw these videos, the inner nerd in me started thinking about natural frequencies and resonance which are both based on the standard single degree of freedom (DOF) mass-spring system:

To calculate the natural frequency of this system, the equation below is used where *k* is the spring stiffness in Newtons/meter, m is the mass in kilograms, and ω is the natural frequency in radians/second.

How does this translate to SolidWorks Simulation?

The first thing we would need is SolidWorks Simulation Professional, which includes the frequency analysis study type.

Second we need a model to represent this single DOF mass-spring problem. For the mass we can model a 1m x 1m x 1m cube with a mass density of 1000 kg/m^{3}. For the spring, we can use the “elastic support”, with a stiffness of 100 N/m. And to restrict the motion to the vertical (Y) direction only (the equation above assumes one DOF, rigid body translation in the vertical direction), we will need to add “advanced fixtures” to prevent translation in the X and Z directions.

Finally we need to mesh the problem before running it. Because the geometry is simple and we’re looking for a first-order response, a high mesh density is not critical so we can go with a coarse mesh.

After running the simulation, the first natural frequency is shown to be 0.31622 rad/s. How does this compare with our hand calculations? Well, taking our previous equation we find that:

Our answer is spot on with SolidWorks Simulation! The formula we used also gives us some insight about changing a structure’s first natural frequency. If we increase the stiffness of the structure or decrease the mass, we can raise the natural frequency. If we decrease the stiffness or increase the mass, we can lower the natural frequency.

But what about real world assemblies and components that we want to analyze? With frequency analysis in SolidWorks Simulation, our modeled structures and assemblies behave like a complex combination of this spring-mass system. We assign materials to parts, and these materials specify a density which controls the mass; we specify an elastic modulus which controls the stiffness, and a finite-element version of the resonance equation above is performed across the entire system. For example, if we pattern a few of these mass-spring systems and assign different masses and stiffnesses, we could get something like this….**The Harlem Shake-FEA Edition**:

The next time you have a problem where you need to account for structural resonance, think about stiffness, mass, and SolidWorks Simulation Professional!

Eloise says

Good to see the logic set out so well.