Frequency equation and resonant frequencies of free-free Timoshenko beams with unequal end masses

Transverse vibration of free-free Timoshenko beams carrying concentrated masses at two ends is investigated. An exact frequency equation is derived and resonant frequencies are calculated. A Fredholm integral equation approach is formulated to obtain an explicit expression for the fundamental frequency. The fundamental frequencies obtained using the analytical and approximate methods are compared to demonstrate the accuracy of the approximate method. The influences of attached masses, rotational inertia and shear rigidity on the resonant frequencies are discussed. Results show that the Timoshenko beam theory is necessary at higher order frequencies due to the error caused by other theories like Euler-Bernoulli, shear or Rayleigh beams. These results are useful to design energy harvesters and mass sensors.