Steel Cantilever

The various methods outlined on this website are demonstrated with measurements of the fundamental frequency of vibration of a simple steel cantilever beam. The results from each method are within 3% of one another, demonstrating the utility of any one method and the potential of all of them. 

Precision Accelerometer - 2.80 Hz

The precision accelerometer system that serves as the benchmark for comparison of all other methods identified a fundamental frequency of 2.80 Hz. It identified additional modal frequencies when the beam was excited by random tapping along its length and left to vibrate freely. In this case, additional peaks appeared in the FFT at 17.1 Hz, 47.5 Hz, and 93.2 Hz. 

Analytical Results

It should be noted that for this beam, using the prismatic continuous cantilever solution, the first natural frequency is 2.97 Hz, which is higher than the measured values. The fun of measuring the vibrations of physical systems comes when you try to explain how they differ from analytical or numerical results. Are differences related to actual physical differences in material, section geometry, length, or support conditions? Or are they the result of a model with inaccurate assumptions? Answering these questions can give you wonderful insight into the behavior and modeling of structural systems. 

The modal frequencies for this very well described cantilever beam can be calculated using an analytical model:

The modal frequencies are thus calculated to be

• Mode 1: 3.00 Hz (2.71 Hz measured with precision accelerometer)

• Mode 2: 18.77 Hz (17.1 Hz measured with precision accelerometer)

• Mode 3: 52.64 Hz (47.5 Hz measured with precision accelerometer)

• Mode 4: 103.23 Hz (93.2 Hz measured with precision accelerometer)

• Mode 5: Outside the bounds of our measurement system

The lateral mode is a first flexural frequency with bending about the strong axis of the beam and is calculated to be 24.02 Hz. 

Other Modes

Additional modes that ought to be explored for this system include lateral vibration (measured as 21.3 Hz by phone-based accelerometer) and torsion (measured as 17.2 Hz by phone-based accelerometer). These modes should show up in a numerical model, but must be specifically excited to be measured physically. 

Numerical Results

A MASTAN model, developed to analyze this beam, produced the following results

Mode 1: Period = 0.337 s Frequency = 2.97 Hz

Mode 2: Period = 0.0537 s Frequency = 18.6 Hz

Mode 3: Period = 0.042081 s Frequency = 23.763 Hz (lateral)

Mode 4: Period = 0.019192 s Frequency = 52.11 Hz 

Mode 5: Period = 0.0097958 s Frequency = 102.08 Hz

The mode numbers no longer match the analytical ones, because a lateral mode appears as the third mode in this analysis in 3D space employing beam elements. 

Note that this model was not able to identify the torsional mode. While the beam elements used in the model include torsional stiffness, the mass is lumped at the nodes along the length of the beam. The mass is not distributed through the cross section and therefor cannot be considered. A more detailed 3D numerical model, or specific analytical model, is necessary to verify this mode and frequency. 

Comparison of Measured, Analytical, and Numerical

A comparison of the measurements, analytical results, and numerically modeled results shows us that the analytical and numerical results are very similar, while the natural frequencies from the physical system are all lower. 

If we refer to the relationship of frequency, mass, and stiffness, we might wonder if we either have a heavier beam than we anticipated. Or perhaps its more flexible than we assumed? This could be the result of flexibility in the fixed support, an inaccurate measurement of the cross section, an inaccurate assumption of the material's modulus of elasticity, or some combination of all of these. Thinking more about these sources of error should encourage us to measure more and continue questioning the assumptions of our analytical and numerical models!