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Internal strain energy and resonant/natural frequency: inversely correlated?

Hello all,

I must apologise in advance if this is a silly question—I'm a biologist, not an engineer. I am using FEA for a zoological biomechanics project to model structural performance of certain skeletal elements, and I have observed that, in one particularly spring-like bone, internal strain energy is tightly inversely correlated with resonant/natural frequency (free-free analysis in Abaqus). Is this relationship obvious to an engineer? In particular, is this relationship predicted by the equations describing springs or such things as tuning forks (formula given on this Wikipedia page: 

Just curious to hear if this is a widely recognised relationship in engineering, as the strength of the inverse correlation certainly surprised me.

Many thanks,


ravi_gutti's picture

Hello Roger,


                     Natural frequency is directly proportional to stiffness of the body. This stiffness inturn is directly proportional to the internal strain energy of the body. 

This means when the internal strain energy increases the natural frequency also increases and vice vers.

Anybody correct me if i am wrong...



For a simple linear spring-mass system:

Internal Energy:  W = 1/2 k d^2   where  d = displacement, k = spring stiffness.

Resonance frequency: omega = sqrt(k/m)  where m = mass.


omega^2 = 2/(md^2) W

So what you're seeing could be due to changing m and d (or some inelastic effect).

-- Biswajit


Hi Ravi,

Thanks for the response. Flexural stiffness of a body is inversely proportional to internal strain energy, though, isn't it? So when internal strain energy increases, natural frequency decreases

What equations describing natural frequency can this relationship be seen in? 


ravi_gutti's picture

Hi Roger,

               Sorry for the delay in reply.

Strain energy = 0.5*stress*strain

But Stress = E*Strain

Therefore Strain energy = 0.5*E*Strain*Strain

Flextural rigidity = E*I    (for any beam or rod element)

So when E increases, Flextural rigidity increases.  

And when E increases, as seen in the previous equation, Strain energy also increses.

Since stiffness is directly proportional to Natural frequency, and Strain energy and Stiffness is directly proportional to E,  We can consider Strain energy and Stiffness is directly proportional to one another.


Which finally states that Natural frequncy is directly proportional to Strain energy







Lixiang Yang's picture

\omega_n = \sqrt{1/fm},  f is flexural stiffness.  For multi-freedom system, flexural matrix is the inverse of stiffness matrix.  For single freedom system. K = 1/f;


potential energy 

E = 0.5 u^2/f         

 u is displacement.


Jayadeep U. B.'s picture

Hello Roger,

I feel that your observation is not anything strange; but need more information to make proper comments:

1. Whther the mode shape and deformation pattern you see in the natural frequency analysis and static analysis are exactly matching (except for the magnitudes, which anyway do not have any meaning in natural frequency analysis)?

2. Whether inversely correlated means f = 1/U (f - natural frequency, U - strain energy)?  Or is it some kind of power-law relationship?

3. From your post, I assume that you have done a parametric study for finding the strain energy.  In that case, have you kept the applied forces constant or the deformation constant?  (Ravi' s comment above assumes the deformation to be constant, which may not be correct in your studies).



Hi Jayadeep,

Sorry for taking so long to reply; I didn't get a notification that you'd posted.

I'm not sure what you mean about a parametric study: I just recorded strain energy via the history output in Abaqus. However, you're right that it's a power-law relationship; plotting $U$ vs $f$ on log-log axes produces a perfectly straight line with a slope of approximately -1.5. This may be because the models are a range of morphologies scaled to equal widths (for biological reasons) rather than equal volume or surface area. That would change the slope from -1, wouldn't it?

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