## 1 Introduction

In a recent paper by E. Esmailzadeh and N. Jalili the design of optimal vibration absorbers for Timoshenko beams was discussed. While the paper is interesting, the solution seems to be questionable, because it does not satisfy the governing differential equation of motion. There has been no reference made to the corresponding experimental results, or results obtained through the application of other analytical methods. Therefore, the application of the results presented in the paper seems to be misleading.

## 2 Analysis

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,
$yx,t=∑i=1nYixs˙qbit$
(1)

$ψx,t=∑i=1nΨixs˙qbit$
(2)
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,
$ρA ∂2y∂t2−kAG∂2y∂x2−∂ψ∂x=gts˙δx−l$
(3)

$EI ∂2ψ∂x2+kAG∂y∂x−ψρI ∂2ψ∂t2=0$
(4)
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,
$ρA∑i=1nq¨bitYix−kAG∑i=1nqbitYi″x−Ψi′x=gts˙δx−l$
(5)

$∑i=1nqbits˙[EIΨi″x+kAGYi′x−Ψix]−ρI∑i=1nq¨bis˙Ψix=0$
(6)
On the other hand, the free vibration analysis, when the same mode summation procedure is applied, gives
$−ρAωi2s˙Yix−kAGs˙Yi″x−Ψi′x=0$
(7)

$EIΨi″x+kAGs˙Yi′x−Ψix=−ωi2ρIs˙Ψix$
(8)
Substitution of Eqs. (7) and (8) in (5) and (6) results in,
$ρA∑i=1nYixs˙[q¨bit+ωi2s˙qbit]=gts˙δx−l$
(9)

$ρI∑i=1nΨixs˙[q¨bit+ωi2s˙qbit]=0$
(10)
Equations (9) and (10) are to be valid for all t and $0 Thus one concludes from (10) that,
$q¨bit+ωi2s˙qbit=0$
(11)
Substitution of Eq. (11) in (9) results in,
$0=gts˙δx−l$
(12)
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.

## 3 Conclusion

The method used in the paper for generating a solution for the optimal design of vibration absorbers for Timoshenko beams, is questionable. The assumption of same modal amplitudes for deflection and section rotation results in a severe contradiction. Since the results obtained by the application of this method have not been compared with any other method or experimental data, one cannot rely on the results presented in the paper.

Corresponding author.