Designer’s Guide to Advanced Vibration Analysis, Part II

Last month’s column discussed the basics of simulating resonant or natural vibration frequencies as an introduction to performing advanced vibration analysis. Recall that advanced vibration-analysis problems can be solved with what are called the normal modes and the Mile’s Equation approaches.

In detail, the normal-modes method suits large, complex, multidegree of freedom systems (>100 DOFs) that are typically analyzed using FEA techniques. This approach introduces a transformation of coordinate approximations to decouple the differential equations of motion and make the problem easier to solve.

The Mile’s Equation approach, on the other hand, involves a simplified linear-static approximation for FEA models of large complicated systems that contain an excessively large number of DOFs, which makes an explicit computer solution difficult to obtain. The approach is based on statistical analyses of induced acceleration spectra with a three-sigma distribution. The software approximates an equivalent g loading using the power-spectral density (PSD) criteria at the resonant frequency in each orthogonal direction of interest. This equivalent g load is sometimes referred to as the random-vibration load factor (RVLF).

Following is a summary of the use of these approaches to approximate the response caused by a random vibration input on the simple 2-DOF spring-mass system.

In the normal-modes approach, summing the forces in the vertical direction gives these equations of motion, which represent the free vibration, natural frequencies of the undamped system:

 For this sample problem, m₁ = m₂ = 90 lb-in./sec2; k₁ = 125,000 lb/in.; and k₂ = 200,000 lb/in.

Also, viscous damping = β = c/c0 = 0.012; the PSD input excitation f (Ω) = 0.20 g²/Hz.

Next comes solving for the natural frequencies (eigenvalues) and associated mode shapes (eigenvectors).

The column matrix of natural frequencies, {ω_n}, is:

And normalizing the eigenvectors matrix, [Ø_r] brings:

Or, in terms of cycles/sec:

Without going through all the matrix arithmetic:
The modal-participation factors {Γ_r} is:

And the generalized mass matrix, [M_r], is:

Next comes approximating the mean-squared response of the 2-DOF system with this summation:

In contrast, use the Mile’s Equation to compute the g_rms value, sometimes called a random-vibration load factor (RVLF), based on the fundamental resonant frequency of a single-degree-of-freedom spring-mass system. The Mile’s Equation is:

where f_n = fundamental natural frequency = 3.861 Hz; Q = transmissibility (amplification) at resonance = 1/(2×β) = 1/(2×0.012) = 41.67; and Ω = input power spectral density (PSD) = 0.20 g²/Hz.
Substituting gives g_rms = 7.109.

Next, estimate the static deflections at each mass:

 For a sanity check, an FEA approach would entail developing an FE idealization of the 2-DOF spring-mass system to approximate the static deflections and resonant-vibration frequencies, processing the dynamic response due to random vibration, and comparing results.