A Yard of iPods

The concept of a mode shape can be demonstrated effectively with a yard stick loaded with seven iPods running the Vibration Analysis app and displaying the frequency spectrum in real time. The fundamental natural frequency (3.99 Hz) is the same for each iPod and the amplitude of acceleration that is measured at each iPod can be plotted to show the mode shape of the beam, a half sine wave. 

Want to compare these results with an analytical model of this beam? Weigh the seven iPods and assume they are uniformly distributed to the yard stick. Measure the cross-section (7/32" x 1 3/8"), calculate the moment of inertia (0.001199 in4) and use a reasonable modulus of elasticity for the yard stick material, assumed to be Pine (600,000 psi). The span of the beam is 34" and the weight of a single iPod is 3.1 oz or 0.194 lb. 

Neglecting the weight of the beam and distributing the weight of 7 iPods over 34" gives us 0.0399 lb/in. Dividing by gravity (386.4 in/sec2) yields a distributed mass of 0.0001034 lb-sec2/in/in. 

We can use the natural frequency formula for a straight prismatic simply-supported beam for the first mode:

Evaluating with the values described above, we find

This compares reasonably well with the 3.99 Hz measurement, given the substantial uncertainty in the parameters of this system. The boundary conditions and loading of this structure are poorly controlled. The material is unknown and therefor the modulus of elasticity is purely a guess. The iPods themselves could stiffen the beam locally, which is challenging to model. Thus, a rigorous model validation is not useful here, but the yard of iPods functions well as a simple demonstration of mode shapes and frequencies.