Random Response Analysis

Cases likely to involve non-deterministic loads are those linked to conditions such as turbulence on an airplane structure, road surface imperfections on a car structure, noise loads on a given structure, and so forth.

Random Response Analysis requires as input, the complex frequency responses from Frequency Response Analysis and Power Spectral Density Functions of the non-deterministic Excitation Source(s). The Complex Frequency Responses can be generated by Direct or Modal Frequency Response Analysis.

Different Load Cases (a and b)

If $H_{x a} (f)$ $H_{x a} (f)$ and $H_{x b} (f)$ $H_{x b} (f)$ are the complex frequency responses (displacement, velocity or acceleration) of the $x$ $x$ ^th degree of freedom, due to Frequency Response Analysis load cases $a$ $a$ and $b$ $b$ respectively, the power spectral density of the response of the $x$ $x$ ^th degree of freedom, $S_{x o} (f)$ $S_{x o} (f)$ is:

S_{x o} (f) = H_{x a}^{*} (f) H_{x b} (f) S_{a b} (f)

$S_{x o} (f) = H_{x a}^{*} (f) H_{x b} (f) S_{a b} (f)$

Where, $S_{a b} (f)$ $S_{a b} (f)$ is the cross power spectral density of two (different, $a \neq b$ $a \neq b$ ) sources, where the individual source $a$ $a$ is the excited load case and $b$ $b$ is the applied load case. This value can possibly be a complex number. $H_{x a}^{*} (f)$ $H_{x a}^{*} (f)$ is the complex conjugate of $H_{x a} (f)$ $H_{x a} (f)$ .

Same Load Case (a)

If $S_{a} (f)$ $S_{a} (f)$ is the spectral density of the individual source (load case $a$ $a$ ), the power spectral density of the response of $x$ $x$ ^th degree of freedom due to the load case $a$ $a$ will be:

S_{x o} (f) = {| H_{x a} (f) |}^{2} S_{a} (f)

$S_{x o} (f) = {| H_{x a} (f) |}^{2} S_{a} (f)$

Combination of Different (a,b) and Same (a,a) Load Cases in a Single Random Response Analysis

If there is a combination of load cases for Random Response Analysis, the total power spectral density of the response will be the summation of the power spectral density of responses due to all individual (same) load cases as well as all cross (different) load cases.

Auto-correlation Function

Consider a time-varying quantity, $y$ $y$ . The auto-correlation function $A_{y} (τ)$ $A_{y} (τ)$ of a time-dependent function $y (t)$ $y (t)$ can be defined by:

A_{y} (τ) = lim T \to \infty \frac{+ T}{2} \int \frac{- T}{2} y (t) y (t + τ) d t

$A_{y} (τ) = \lim_{T \to \infty} \int_{\frac{- T}{2}}^{\frac{+ T}{2}} y (t) y (t + τ) d t$

Where,

$τ$ $τ$: The time lag for Auto-correlation

The variance $σ^{2} (y)$ $σ^{2} (y)$ of the time-dependent function $y (t)$ $y (t)$ is equal to $A_{y} (0)$ $A_{y} (0)$ . The variance $σ^{2} (y)$ $σ^{2} (y)$ can be expressed as a function of power spectral density $S_{y} (f)$ $S_{y} (f)$ , as:

A_{y} (0) = σ^{2} (y) = \infty \int - \infty S_{y} (f) d f

$A_{y} (0) = σ^{2} (y) = \int_{- \infty}^{\infty} S_{y} (f) d f$

The root mean square value ( $y_{R M S}$ $y_{R M S}$ ) of the time-dependent quantity $y (t)$ $y (t)$ can also be written by:

y_{R M S} = \sqrt{{¯ ¯¯¯¯ ¯ y (t)}^{2} + σ^{2} (y)}

$y_{R M S} = \sqrt{{\bar{y (t)}}^{2} + σ^{2} (y)}$

If the mean ( $¯ ¯¯¯¯ ¯ y (t)$ $\bar{y (t)}$ ) of the function is equal to 0, then the RMS value is the square root of the variance. Since the variance is also equal to $A_{y} (0)$ $A_{y} (0)$ , the RMS value can be written as:

y_{R M S} = \sqrt{\infty \int - \infty S_{y} (f) d f}

$y_{R M S} = \sqrt{\int_{- \infty}^{\infty} S_{y} (f) d f}$

RMS of the Response Power Spectral Densities for degree of freedom "x"

The RMS values at each excitation frequency is defined as the cumulative sum of the area under the Power Spectral Density function up to the specified frequency. Based on the equation for $y_{R M S}$ $y_{R M S}$ obtained in the previous section, the RMS value of a response for a particular degree of freedom $x$ $x$ is calculated in the range of excitation frequencies, [ $f_{1}$ $f_{1}$ , $f_{n}$ $f_{n}$ ] as:

{(S_{x} (f))}_{R M S} = \sqrt{f_{n} \int f_{1} S_{x} (f) d f}

${(S_{x} (f))}_{R M S} = \sqrt{\int_{f_{1}}^{f_{n}} S_{x} (f) d f}$

In HyperView, the RMS values are displayed for a Random Response Analysis in a drop-down menu with excitation frequencies. Each selection within this menu displays the sum of cumulative RMS values for the particular response at all previous excitation frequencies (which is the area under the response curve up to the loading frequency of interest). The RMS over frequencies option can be selected to obtain the RMS value of the response in the entire frequency range.

Auto-correlation Function Output for degree of freedom "x"

The RANDT1 Bulk Data Entry can be used to specify the lag time ( $τ$ $τ$ ) used in the calculation of the Auto-correlation function for each response for a particular degree of freedom, $x$ $x$ .

The auto-correlation function and the power spectral density are Fourier transforms of each other. Therefore, the auto-correlation function of a response $S_{x} (f)$ $S_{x} (f)$ can be described as:

A_{x} (τ) = 2 f_{n} \int 0 S_{x} (f) exp (i 2 π f) d f

$A_{x} (τ) = 2 \int_{0}^{f_{n}} S_{x} (f) \exp (i 2 π f) d f$

The Auto-correlation Function is calculated for each time lag value in the specified RANDT1 set over the entire frequency range [0, $f_{n}$ $f_{n}$ ].

Number of Positive Zero Crossing

Random non-deterministic excitation loading on a structure can lead to fatigue failure. The number of fatigue cycles of random vibration is evaluated by multiplying the vibration duration and another parameter called maximum number of positive zero crossing. The maximum number of positive zero crossing is calculated as:

P_{c} = {⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ \frac{f_{n} \int 0 f^{2} S_{x} (f) d f}{f_{n} \int 0 S_{x} (f) d f} ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠}^{0.5}

$P_{c} = {(\frac{\int_{0}^{f_{n}} f^{2} S_{x} (f) d f}{\int_{0}^{f_{n}} S_{x} (f) d f})}^{0.5}$

If XYPLOT, XYPEAK or XYPUNCH, output requests are used, the root mean square value and the maximum number of positive crossing calculated at each excitation frequency will be exported to the *.peak file.