# [Miles' Equation]{.smallcaps}

## BASICS OF MILES' EQUATION.

-   **[The following equation is attributed to John W.
    Miles.]{.smallcaps}**

$$G_{\text{RMS}}\  = \ \sqrt{\frac{\pi}{2}f_{n}Q\lbrack\text{ASD}_{\text{input}}\rbrack}$$

> Where: $f_{n}\  = \ \text{natural}\ \text{frequency}$
>
> $Q\  = \ \frac{1}{2\zeta}\  = \ \text{transmissibility}\ \text{at}\ f_{n}$
> *(ζ is the critical damping ratio)*
>
> $\text{ASD}_{\text{input}}\  = \ \text{input}\ \text{Acceleration}\ \text{Spectral}\ \text{Density}\ \text{at}\ f_{n}\ \text{in}\ \frac{g^{2}}{\text{Hz}}$

In 1954, Miles developed his version of this equation for G~RMS~ (root
mean square acceleration) as he was researching fatigue failure of
aircraft structural components caused by jet engine vibration and gust
loading. Miles simplified his research by modeling a system using one
degree of freedom only. He also applied statistical advances that had
been made at the time. While his goal was to analyze the stress of a
component, the equation can be rearranged and used to determine, among
others, displacement, force, and, in our case, acceleration.

-   **[Single Degree Of Freedom System]{.smallcaps}** -- Miles' Equation
    is derived using a single degree of freedom (SDOF) system (lightly
    damped), consisting of a mass, spring and damper, that is excited by
    a constant-level "white noise" random vibration input from 0 Hz to
    infinity. Miles' Equation is thus technically applicable only to a
    SDOF system.\
    \
    This figure shows a typical representation of a SDOF oscillator. The
    mass (m) is attached to the spring (stiffness k) and the damper
    (damping c). The system is forced by the random vibration
    function (F) in the y-direction only.

-   **[SDOF System Response Plot]{.smallcaps}** -- The plot shows the
    input and response of a SDOF system. Miles' Equation calculates the
    square root of the area under the response curve, providing us with
    the well known G~RMS~ value.

-   **[Response Parameters]{.smallcaps}** -- Miles' Equation can also be
    used to predict other response parameters such as stress or
    displacement. For example, the displacement equation for Y~RMS~ is
    given here.

## USES OF MILES' EQUATION

-   **[Design]{.smallcaps}** -- During the design of a part, if enough
    analysis has been performed to determine the part has a predominant
    resonant frequency, then Miles' Equation can be used to estimate the
    loads due to random vibration. Just calculate the G~RMS~ value and
    multiply it by 3. That's the "three sigma" load. Keep in mind, this
    is conservative.

-   **[Testing]{.smallcaps}** -- Accelerations due to random vibration
    at resonant frequencies in a multiple degree of freedom system can
    be approximated using Miles' Equation. This will indicate how much
    of the overall RMS acceleration is occurring at a resonant peak of
    interest compared to the complete frequency spectrum.

## MILES' EQUATION PITFALLS

-   **[Miles' Equation Does NOT Work in Reverse]{.smallcaps}** --
    Accelerations cannot be determined during random vibration testing
    using Miles' Equation. An upper bound on loads can be calculated
    using the 3σ value, but that's about it. In other words, a part
    designed to 3σ equivalent static loads will survive a random
    vibration test. However, Miles' Equation cannot be used to predict
    the failure of a part designed to less than 3σ levels.

-   **[Miles' Equation Does NOT Give an Equivalent Static
    Load]{.smallcaps}** -- Calculating the G~RMS~ value at a resonant
    peak after a random vibration test and multiplying it by the test
    article mass does not mean that the test article was subjected to
    that same, equivalent static load. It simply provides a statistical
    calculation of the peak load for a SDOF system. The actual loading
    on a multiple DOF system due to random input depends on the response
    of multiple modes, the mode shapes and the amount of effective mass
    participating in each mode. Static testing must still be done.

-   **[Miles' Equation may not be Conservative for a Shaped Input
    Spectrum]{.smallcaps}** -- Miles' Equation is based on the response
    of a SDOF system subjected to a *flat* random input. If Miles'
    Equation is used to calculate the G~RMS~ response to a shaped input
    spectrum, the result will not accurately reflect the rigid body
    response below the resonant frequency. If the shaped input spectrum
    has high ASD levels below the resonant frequency, then Mile's
    Equation may significantly underpredict the G~RMS~ response. The
    following plot shows this condition. Note that the Miles' assumption
    has a response of 31.8 G~RMS~ while the actual SDOF response is much
    greater at 39.7 G~RMS~. This demonstrates that Miles' Equation is
    best used when the input ASD is flat or nearly so.

## REFERENCES:

-   John W. Miles, *On Structural Fatigue Under Random Loading*, Journal
    of the Aeronautical Sciences, pg. 753, November, 1954.

-   Dave S. Steinberg, *Vibration Analysis for Electronic Equipment*,
    Wiley-Interscience, New York

-   Thomas P. Sarafin (editor), *Spacecraft Structures and Mechanisms:
    From Concept to Launch*, Microcosm, Inc., Torrance, CA, 1995

## PREPARED BY:

**[Ryan Simmons, Scott Gordon, Bob Coladonato]{.smallcaps}**,

NASA Goddard Space Flight Center, May 2001