Discussion: “Optimum Design of Vibration Absorbers for Structurally Damped Timoshenko Beams” (Esmailzadeh, E., and Jalili, N., 1998, ASME J. Vib. Acoust., 120, pp. 833–841)

In the paper, the mode summation procedure has been applied, and the following forms for the transverse deflection $yx,t$ and the orientation of the beam cross-section $ψx,t$ have been adopted, It is observed that the same modal amplitudes for deflection and section rotation have been assumed. It will be shown that such an assumption results in a contradiction. For clarity of presentation, let’s consider a simple case, where only one vibration absorber system with a single mass, m, stiffness, k, and damping, c is attached to the beam at some location, $x=l.$ Also the only applied force on the beam is assumed to be $gt,$ which is applied by the absorber system. This force can be generated, for example, as a result of an initial condition imposed on the beam. Considering the free-body-diagram of an element of the Timoshenko beam, the equations of motion would be, Now, one can substitute Eqs. (¹) and (²) in (³) and (⁴) to check if assuming similar modal amplitudes for lateral deflection and section rotation is justifiable. Substitution results in, On the other hand, the free vibration analysis, when the same mode summation procedure is applied, gives Substitution of Eqs. (⁷) and (⁸) in (⁵) and (⁶) results in, Equations (⁹) and (¹⁰) are to be valid for all t and $0<x<L.$ Thus one concludes from (¹⁰) that, Substitution of Eq. (¹¹) in (⁹) results in, Equation (¹²) presents a clear contradiction because it would be valid only if $gt$ equals to zero, that is, no absorber system is used at all.