Hi,
Just a quick question, what are the differences between modal analysis and dynamic steady-state analysis? The model is the same, the system is undamped and the boundary conditions are the same so the resonant frequencies are very similiar to the modal frequencies as expected. Is one more accurate than the other? Is one cheaper computationally?
Also, is it true to say that a cantilever plate is stiffer than a free-free plate and thus, the natural frequencies would be higher in the nth mode (all properties are constant, only boundary conditions have changed)?
Cheers,
Callum
Hi Callum
To shortly answer your question on the difference between modes and resonance - imagine a system with multiple modes at the same frequency, e.g. a symmetric system. The mode shapes will then mix and complicate analysis of the pattern.
Further, modes are orthogonal, i.e. they do not share energy. Taking advantage of this fact opens up some avenues of thought/analysis that otherwise are closed.
The numerical advantages in the use of modes in FE is that a mode simply is one degree of freedom. Hence, N modes implies N dofs. Computing the response for N modes is way faster than directly solving a large system matrix as long as the number of modes is much lower than the system matrix size.
You find excellent self study material on Experimental Modal Analysis here.
http://qringtech.com/learnmore/recommended-reading/
To answer your question on a free-free versus a beam clamped at one end.
Removing dofs always stiffen a construction.
That said, in evaluating this rule one should account for the fact that rigid body modes are zero stress modes (or they would violate Newton's 1st law). Forgetting this fact may lead us to trade apples for pears.
This observation implies that, indeed, the first bending mode is shifted to a higher frequency than is the case for the clamped system. In a theoretical way, this is not stiffening, but in practice, it does imply a sturdier design - if your system connections can handle the larger amplitude motion of the rigid body mode.
The simple way to prove this is to use a mass-spring-mass system with identical masses, M, connected by a spring K.
- The free-free system rigid body mode would be at 0 Hz, the first flexible mode at sqrt(2K/M)/(2*pi) Hz.
- Clamp one end of this system and the first mode lands at sqrt(K/M)/(2*pi), i.e. ~40% lower in frequency.
Clamping, indeed, raises the first mode from 0 Hz to sqrt(K/M)/(2*pi) Hz. In doing so, the first flexible mode is reduced in frequency.
As a practical example of the above, it is often believed that a sling test of a free-free rotor is conservative on estimating its lowest natural. However, the rotor bending natural frequency drops when placed in its bearings.
I often use the above insight for the design of large machinery skids where effective vibration isolation creates payoff in multiple ways.
Sincerely
Claes
Hi Callum
To shortly answer your question on the difference between modes and resonance - imagine a system with multiple modes at the same frequency, e.g. a symmetric system. The mode shapes will then mix and complicate analysis of the pattern.
Further, modes are orthogonal, i.e. they do not share energy. Taking advantage of this fact opens up some avenues of thought/analysis that otherwise are closed.
The numerical advantages in the use of modes in FE is that a mode simply is one degree of freedom. Hence, N modes implies N dofs. Computing the response for N modes is way faster than directly solving a large system matrix as long as the number of modes is much lower than the system matrix size.
You find excellent self study material on Experimental Modal Analysis here.
http://qringtech.com/learnmore/recommended-reading/
To answer your question on a free-free versus a beam clamped at one end.
Removing dofs always stiffen a construction.
That said, in evaluating this rule one should account for the fact that rigid body modes are zero stress modes (or they would violate Newton's 1st law). Forgetting this fact may lead us to trade apples for pears.
This observation implies that, indeed, the first bending mode is shifted to a higher frequency than is the case for the clamped system. In a theoretical way, this is not stiffening, but in practice, it does imply a sturdier design - if your system connections can handle the larger amplitude motion of the rigid body mode.
The simple way to prove this is to use a mass-spring-mass system with identical masses, M, connected by a spring K.
- The free-free system rigid body mode would be at 0 Hz, the first flexible mode at sqrt(2K/M)/(2*pi) Hz.
- Clamp one end of this system and the first mode lands at sqrt(K/M)/(2*pi), i.e. ~40% lower in frequency.
Clamping, indeed, raises the first mode from 0 Hz to sqrt(K/M)/(2*pi) Hz. In doing so, the first flexible mode is reduced in frequency.
As a practical example of the above, it is often believed that a sling test of a free-free rotor is conservative on estimating its lowest natural. However, the rotor bending natural frequency drops when placed in its bearings.
I often use the above insight for the design of large machinery skids where effective vibration isolation creates payoff in multiple ways.
Sincerely
Claes
In harmonic power systems analysis the imaginary parts of the poles are very close to the resonances frequencies. The modal analysis has some analytical advantages like providing pole and zero sensitivities to a system parameter change. In this area, the modal sensitivity analysis can be used as a cost-effective alternative to solve harmonic distortion problems (for instance, see “Electrical Network Dynamic Models with Application to Modal Analysis of Harmonics”, Electric Power Systems Research, v. 154, p. 433-443, 2018).
Sincerely
Varricchio
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