The mechanical properties of anisotropic nanoparticles like gold nanorods (AuNRs) and silver nanorods (AgNRs) are different from those of isotropic shapes such as nanospheres. We probed the coherent lattice oscillations of nanoparticles by following the modulation of the plasmonic band frequency using ultrafast laser spectroscopy. We found that while the frequency of the longitudinal vibration mode of AgNRs is higher than that of AuNRs of similar dimensions, similarly sized gold and silver nanospheres have similar lattice vibration frequencies. Lattice vibrations calculated by finite element modeling showed good agreement with the experimental results for both AgNRs and AuNRs. The accuracy of the calculations was improved by using actual pentagonal shapes rather than cylinders that did not agree well with the experimental results. As the plasmon energy is transferred into lattice vibrations, the temperature of the nanoparticle necessarily increases as a result of this electron-phonon relaxation process. This results in a decrease in the Young's modulus that was accounted for in the calculations. Calculations showed that the tips of the nanorods are "softer" than the rest of the nanorod. Because the tips comprise a larger portion of the overall rod in the smaller rods, the smaller rods were more affected by the tip effects.