a mechanical engineering toolbox
source code - https://github.com/nagordon/mechpy
documentation - https://nagordon.github.io/mechpy/web/
Neal Gordon
2017-02-20
Jul 1, 2015
modal analysis is similar to frequency analysis. In frequency analysis a complex signal is resolved into a set of simple sine waves with individual frequency and amplitude and phase parameters. In modal analysis, a complex deflection pattern of a vibrating structure is resolved into a set of simple mode shapes with the same individual parameters.
Most systems are actually multiple degrees of freedom (MDOF) and have some non-linearity, but can be simplified with a superposition of SDOF linear systems
Newtons law states that acceleration is a function of the applied force and the mass of the object, or $$ [inertial forces] + [Dissipative forces] + [Restoring Forces] = [External Forces] \\ m\ddot{x} + c\dot{x} + kx = f(t) \\ \zeta<1 is\ underdamped \\ $$
some other dynamic characteristics are $$ \omega = frequency \\ \zeta = damping \\ \{\phi\} = mode shape \\ \omega^{2}_{n}=\frac{k}{m} = natural frequency \\ \zeta = \frac{c}{\sqrt{2km}} \\ H(\omega)=Frequency\ Response \\ \phi(\omega)=Phase $$
Where there is energy dissipation, there is damping. The system can be broken into the system inputs/excitation, a system G(s), and the output response, in Laplace or space
The transfer function is a math model defining the input/output relationship of a physical system. Another definition is the Laplace transform ( $\mathcal{L}$) of the output divided by the Laplace transform of the input.
The frequency response function (FRF) is defined in a similar manner such that FRF is the fourier transform ($ \mathcal{F} $) of the input divided by the fourier transform of the output
$$ Transfer\ Function=\frac{Output}{Input} \\ G(s) = \frac{Y(s)}{X(s)} $$These relationships can be further explained by the modal test process. The measurements taken during a test are frequency response function measurements. The parameter estimation routines are curve fits in the Laplace domain and result in transfer functions.
Frequency Response Matrix
$$ \begin{bmatrix} H_{11} & H_{12} & \cdots & H_{1n} \\ H_{21} & H_{22} & \cdots & H_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ H_{n1} & H_{n2} & \cdots & H_{nn} \end{bmatrix} $$The signal-analysis approach is done by measuring vibration with accelerometers and determine the frequency spectrum. The other moethod is a system -analysis where a dual-channel FFT anlayzer is used to measure the ratio of the response to the input giving the frequency response function (FRF)
a modal model allows the analysis of structural systems
a mode shape is a deflection-pattern associated with a particular modal frequency or pole location. It is not tangible or easily observed. The actual displacement of the structure will be a sum of all the mode shapes. A harmonic exitation close to the modal frequency, 95% of the displacement may be due to the particular modeshape
Modal Descriptions Assumes Linearity
Superposition of the component waves will result in the final wave. A swept sinosoid will give the same result as a broadband excitation
Homogeneity is when a measured FRF is independent of excitation level
Reciprocity implies that the FRF measured between any two DOFs is independent of which of them for excitation or response
small deflections - cannot predict buckling or catastrophic failure
casual - the structure will not vibrate before it is excited
stable - the vibrations will die out when the excitation is removd
time-invariant - the dynamic characteristics will not change during the measurments
[Physical Coordinates] = [Modal Matrix][Modal Coordinates]
$$ [x] = [\phi][q] $$
In [ ]: