Frequency Formulas

Closed-form solutions for natural frequency exist for a variety of simple structural systems, including cables, beams, and plates. You can compare these closed-form solutions to experimental results to learn more about physical systems. If you find simple systems in the world, measure their natural frequencies and compare to an appropriate closed-form analytical solution. Then, start questioning the quality of the individual properties and measurements in the formula. If the system is too complex for an analytical solution, prepare a numerical model. 

Simple Spring-Mass Systems

If a physical system has a lumped mass supported by a relatively light spring, then the spring-mass model can be very effective. It is the starting point for a broader study of vibrations and also an excellent approximation for lightly damped physical systems, which is many of those we encounter in our physical world. It also points to the fundamental concept that a system has a higher natural frequency if it is stiff and light and a lower natural frequency if it is flexible and heavy. 

Tip-Loaded Cantilever Beam

A relatively stiff cantilever beam with a tip mass that weighs substantially more than the beam can be easily prepared in the lab and its stiffness is easily derived or looked up in beam deflection tables. 

Simply Supported Beam with Concentrated Mass at Mid-Span

A simply supported beam can be loaded at midspan as a demonstration. If the mass of the beam is small relative to the mass of the load placed at midspan, the stiffness of the system, like the cantilever above, can be found in beam deflection tables. 

Prismatic Cantilever Beam

A cantilever beam loaded with its self weight is a great introduction to continuous systems (that is systems with distributed mass and stiffness) and multiple modes of vibration. If the loading of the prismatic beam is its uniformly distributed self weight, and you measure with relatively light objects (like the accelerometers in mobile phones!), or using non-contact methods (like LiDAR or a mobile phone magnetometer!), you have a system that is very consistent with this analytical model. The first mode of vibration is demonstrated in the steel cantilever beam experiment that is used to demonstrate all the phone-based vibration measurements methods here. 

Prismatic Simply-Supported Beam

A uniform simply-supported beam is another great introduction to continuous systems (that is systems with distributed mass and stiffness) and multiple modes of vibration. If the loading of the prismatic beam is its uniformly distributed self weight, and you measure with relatively light objects (like the accelerometers in mobile phones!), or using non-contact methods (like LiDAR or a mobile phone magnetometer!), you have a system that is very consistent with this analytical model. This model is the basis for the introduction to modal analysis I use in my classes. It's also the basis for the walk the plank demonstration. 

Taught-String Model of a Cable in Tension

The natural frequency of cables carrying substantial tension along their length, for which bending stiffness is negligible, can be expressed with a wonderfully simple formula. The mode shapes are the same as for the simply supported beam, but the stiffness is derived from the tension in the element, rather than the bending stiffness of a beam that does not carry tension. This simple model applies in many situations, including the straight cables of bridges, bicycle spokes, and guitar strings. I demonstrate its use in an experiment on the back stay of a historical bridge in Weimar, Germany. 

Modified Taught-String Model of a Cable in Tension

The natural frequency of cables carrying substantial tension along their length, for which bending stiffness is NOT negligible, can be expressed with a combination of the above two models. The mode shapes are the same, but the stiffness is derived from both the tension in the element and its bending characteristics. This relatively simple model applies in many situations when the length of the cable is relatively short or its proportions are relatively stocky. It also assumes that the end conditions are pinned-pinned, as evidenced by the similarity to the simply-supported beam model. 

Beam-Column in Compression

While tension applied to a beam results in an increased natural frequency, as compression is applied the natural frequency is reduced, becoming zero at the Euler critical buckling load. 

Single-Story Moment Frame Structures

The natural frequency of simple moment frame structures is dependent on the lateral stiffness of the moment frame. Common end conditions for the columns are either fixed-fixed or fixed-pinned. Actual end conditions in real structures may vary considerably from these theoretical values. 

References

Virgin, Lawrence N. (2007) Vibration of Axially Loaded Structures. Cambridge University Press.