Thank you ;)

Loss factor is a property of material but influence the beam response depending on its characteristics. Therefore, same material but different beam geometry can result in different beam responses (different total loss factor).

Just to add a couple of remarks:

There is some discussion if the geometry affects the modal damping factor or it is just a non-linear variation of the structural damping factor with the frequency, amplitude or/and strain rate (parameters that are, themselves, geometry dependent).

Typically, for structures (as opposed to single free-free members) most of the damping observed is due to the joints, including the clamping of a beam; in this cases, this overwhelms the material (internal) damping.

The classical reference about this issue and probably the best is Lazan's PhD thesis.

Modal damping (a mathematical quantity) may be geometry dependent, but the internal or material damping factor (not the damping coefficient) is a material property. Natural frequencies are geometry and material dependent and mode shapes are, for homogeneous isotropic materials, only geometry dependent.

Matlab codes only show what the mathematical model at stake tells them to show. Experimental data must be used to develop and validate any model if we want it to describe real life.

That is also why practical clamping does introduce damping, while theoretical - perfect - clamping does not. Real life clamping is never perfect and some movement, as small as it may be, always happens at the interface; since one of the damping mechanisms is itself dependent on inter and intra crystalline shear, the contact between the clamps and the beam also adds damping to the real structures.

By the way, structural damping is better described by the hysteretic model than the viscous one, at least for most metals; for other solid materials either hysteretic or fractional derivative models are usually more accurate describing their behavior.

my 2p.

Yes, it is as good example of a multi-part, multi-material structure (non-homogeneous and non-isotropic) where both materials damping factors are combined in a given geometry. With that example, the modal damping factors will depend on both the geometry and the material damping factors for its components.

Hi

To directly answer your question if the two beams will vibrate exactly the same, a clear answer is that it is most probable that they will not vibrate the same as different cross sections imply different bending stiffness and hence, different natural frequencies.

In theory, you can create combinations that yeild two beams with identical sets of natural freqiencies and damping, i.e. you can get away with it in a computer or analytical siimulation. In practice, this is nearly impossible.

The main difference would likely lie in natural frequencies differing but damping would also differ.

That said, your question is simple and still very wide.

Damping can be lots of things and is often confused with mechanims that involve some form of amplitude reduction. Some also refer to damping as dampening.

The fundamental distinction, I think, lies in damping being a process in which vibration energy is destroyed into heat which, is a nearly irreversible process.

You find a general discussion on the topic of damping here http://qringtech.com/2014/06/22/designed-damping-types-mechanisms-application-limitation/

Hope this helps

Claes

http://www.ircbt.com/ebooks/mechanic/Internal%20Friction%20in%20Metallic%20Materials%20-%20A%20Handbook%20-%20M.%20Blanter,%20et%20al.,%20(Springer,%202007)%20WW.pdf

Dear Professor Golovin

Thanks for the link. It seems to be an excellent reference. I have on/off been looking for something covering Thermoelastic Relaxation which I find in this book. 

Sincerely

Claes

Dear Claes, it is a pleasure for me if this book is of some use. Igor

loss factor (or damping. or internal friction) is in general amplitude dependent property. If cross section is different at constant deflection, the amplitude is different and damping (loss factor) should be different 

Please, look in the Handbook which I recommended: loss factor is tg delta between stress and stain and can be measured if FORCED vibrations are used. For free decay vibration the logdecrement is used. Both measures can be converted to damping (again there are several definitions - internal friction, specific damping index etc) if damping level is NOT too high.

http://ebookee.org/Internal-Friction-in-Metallic-Materials-A-Handbook_142790.html

Loss factor or specific damping factor is defined as the energy dissipated per  radian to the peak potential energy is the cycle. This is a very useful method to compare the damping capacity of different materials. It has to be pointed out that this method is used for materials with very light damping.  For more information, please have a look at the book "Dynamic of Structure", written by Anil K Chopra. 

The question is ill posed, that is, not precisely defined. Let us consider a certain process responsible for the dissipation of mechanical energy in cantilever beams (material). Let us assume that the process is inherently defined by a small activation volume. This is why, damping (loss factor/internal friction/etc.) is the same for different amplitudes.

In general, however, one may expect different answers for different materials and experimental conditions (mode, frequency, effect of clamps, nonlinearities, etc.)

Numerous definitions of damping (dissipation of mechanical energy) are reported in the literature. Search any data base for: logarithmic decrement, internal friction, phase lag, loss angle, damping, damping ratio, etc. (for resonant and subresonant mode).

The loss factor of a material is sensitive to numerous factors. For well defined conditions it may be considered as a semi-constant but in general it varies as a function of the state of the material (defects, microstructure, phase transition, etc.) temperature, frequency of oscillations, ... and sometimes amplitude of oscillations.

A word of warning, dissipation of mechanical energy is frequently estimated by different methods and algorithms. Substantial differences in the estimation results are frequently observed. Such discrepancies may be caused by different computing methods, additive noise, length of the free decaying signal, side effects, nonexponential decay, etc. 

To conclude, an exhaustive answer cannot be provided since your  question is too general.

Yours,

Leszek B. Magalas

My answer was confined to the specific intrinsic properties of a material, that is, the loss factor of the investigated sample. This approach is pertinent to the original question. The answer provided by Subhajit Mondal is also correct. It refers to different experimental situation and different set-up. To sum up, three components should be considered in the most general case.

Materials scientists are interested in the first component. Materials engineers and mechanical engineers are usually interested in the second and/or third component.

Let me add another dimension to the discussion. Is this material/structure loss factor \eta frequency dependent ? You may refer to page 6 of this paper ( Article Experimental demonstration of a dissipative multi-resonator ...

loss factor is a way to quantify a loss of energy that results from any cyclic energy conversion that occurs inside a structure a mechanism or a material. Basically it relates the potential energy with the kinetic energy and corresponds to the ratio between the energy that is lost in this conversion and the energy that is converted,

if one analyses the energy convention that occurs inside a vibrating structure, lost energy is the portion of energy that is converted from the kinetic energy that the structure possesses when their mass points have the maximum velocity (and minimum displacement in a pure harmonic motion) to the potential deformation energy that the structure mass points have when it reaches the maximum deformation and zero velocity. That relation is structural loss factor.

the same process can be observed when a rubber like material is cyclic deformed, and in this case it is a material loss factor, also known as tan (delta).