zeta : numeric : Non-negative damping ratio, with default 0.05.

beta : numeric : First non-negative constant which determines the discretization scheme for the equation of motion, with default 0.25.

gamma : numeric : Second non-negative constant which determines the discretization scheme for the equation of motion, with default 0.5.

periodfrequency : identical(period, frequency) : One of the names period (default) or frequency.

xscale : identical(log, linear) : One of the names log or linear (default).

yscale : identical(log, linear) : One of the names log or linear (default).

output : The type of output. The supported options are:

absoluteaccelerationdata: Vector, of datatype float[8], containing the absolute acceleration data.

absoluteaccelerationplot: Plot of the absolute acceleration versus period.

accelerationdata: Vector, of datatype float[8], containing the acceleration data.

accelerationplot: Plot of the acceleration versus time.

displacementdata: Vector, of datatype float[8], containing the displacement data.

displacementplot: Plot of the displacement versus time.

perioddata: Vector, of datatype float[8], containing the period data.

pseudoaccelerationdata: Vector, of datatype float[8], containing the pseudospectral acceleration data.

pseudoaccelerationplot: Plot of the pseudospectral acceleration versus period.

pseudovelocitydata: Vector, of datatype float[8], containing the pseudospectral velocity data.

pseudovelocityplot: Plot of the pseudospectral velocity versus period.

relativedisplacementdata: Vector, of datatype float[8], containing the relative displacement response spectrum data.

relativedisplacementplot: Plot of the relative displacement response spectrum versus period.

relativevelocitydata: Vector, of datatype float[8], containing the relative velocity response spectrum data.

relativevelocityplot: Plot of the relative velocity response spectrum versus period.

timedata: float[8] Vector of the time data.

velocitydata: Vector, of datatype float[8], containing the velocity data.

velocityplot: Plot of the velocity versus time.

record: Returns a record with the previous options. This is the default.

list of any of the above options: Returns an expression sequence with the corresponding outputs, in the same order.

A response spectrum is a plot of how a structure or system responds to varying frequencies of ground motion or input excitation. It is commonly used in structural engineering and earthquake engineering to assess the potential response of a structure to seismic events.

A response spectrum is generated by calculating the maximum response of a structure to different frequencies of ground motion. An acceleration time history is used to perturb a single degree of freedom harmonic oscillator. Typically period is plotted on the $x$-axis against acceleration, velocity or position on the $y$-axis.

This procedure computes the response spectrum of an input accelerogram (time-acceleration history). It numerically solves the differential equation for a harmonic oscillator with one degree of freedom.

The parameters $\mathrm{\beta }$, $\mathrm{\gamma }$, and $\mathrm{\zeta }$ are the same as those described in the Newmark-beta Method.

$\mathrm{with}\left(\mathrm{SignalProcessing}\right)\:$

$\mathrm{datafile}≔\mathrm{FileTools}:-\mathrm{JoinPath}\left(\left[\mathrm{kernelopts}\left('\mathrm{datadir}'\right)\,''datasets''\,''el-centro\_NS.txt''\right]\right)\:$

$\mathrm{data}≔\mathrm{ImportMatrix}\left(\mathrm{datafile}\,'\mathrm{delimiter}'=''''\right)\:$

$\mathrm{results}≔\mathrm{ResponseSpectrum}\left(\mathrm{data}\,0.02\,0.01\,5\,'\mathrm{\zeta }'=0.02\,'\mathrm{\beta }'=0.25\,'\mathrm{\gamma }'=0.5\right)\:$

$\mathrm{results}:-\mathrm{absoluteaccelerationplot}$

$\mathrm{results}:-\mathrm{relativevelocityplot}$

$\mathrm{results}:-\mathrm{relativedisplacementplot}$

$\mathrm{results}:-\mathrm{pseudovelocityplot}$

$\mathrm{results}:-\mathrm{pseudoaccelerationplot}$

$\mathrm{results}:-\mathrm{displacementplot}$

$\mathrm{results}:-\mathrm{velocityplot}$

$\mathrm{results}:-\mathrm{accelerationplot}$

$\mathrm{results}:-\mathrm{perioddata}$

$\begin{array}{c}\left[\begin{array}{c}0.0100000000000000\\ 0.0200000000000000\\ 0.0300000000000000\\ 0.0400000000000000\\ 0.0500000000000000\\ 0.0600000000000000\\ 0.0700000000000000\\ 0.0800000000000000\\ 0.0900000000000000\\ 0.100000000000000\\ ⋮\end{array}\right]\\ \hfill \text{500 element Vector[column]}\end{array}$

$\mathrm{results}:-\mathrm{absoluteaccelerationdata}$

$\begin{array}{c}\left[\begin{array}{c}3.14927271191529\\ 3.46090061117149\\ 3.43063275429584\\ 4.11315124613268\\ 4.90707270638583\\ 5.88166270446522\\ 5.80056751481156\\ 6.00724471387179\\ 6.63219941750928\\ 6.67043185549458\\ ⋮\end{array}\right]\\ \hfill \text{500 element Vector[column]}\end{array}$

$\mathrm{results}:-\mathrm{relativevelocitydata}$

$\begin{array}{c}\left[\begin{array}{c}0.000521384452815372\\ 0.00226968541709548\\ 0.00671955649372630\\ 0.0139973386238865\\ 0.0266995629221143\\ 0.0376959034968711\\ 0.0426156496543002\\ 0.0459874400175774\\ 0.0665364650049295\\ 0.0898241053504349\\ ⋮\end{array}\right]\\ \hfill \text{500 element Vector[column]}\end{array}$

$\mathrm{results}:-\mathrm{relativedisplacementdata}$

$\begin{array}{c}\left[\begin{array}{c}7.97748506689015\times {10}^{\mathrm{-6}}\\ 0.0000349332734251345\\ 0.0000714663351525557\\ 0.000139174640856899\\ 0.000272414246922823\\ 0.000413002526181314\\ 0.000479423707224402\\ 0.000760760406816611\\ 0.000890989935065240\\ 0.00125610883541503\\ ⋮\end{array}\right]\\ \hfill \text{500 element Vector[column]}\end{array}$

$\mathrm{results}:-\mathrm{pseudovelocitydata}$

$\begin{array}{c}\left[\begin{array}{c}0.00501240169605288\\ 0.0109746115158246\\ 0.0149678742329503\\ 0.0218615014641015\\ 0.0342325838746374\\ 0.0432495234055081\\ 0.0430329713306562\\ 0.0597499826299262\\ 0.0622028318760756\\ 0.0789236457889815\\ ⋮\end{array}\right]\\ \hfill \text{500 element Vector[column]}\end{array}$

$\mathrm{results}:-\mathrm{pseudoaccelerationdata}$

$\begin{array}{c}\left[\begin{array}{c}3.14938486903215\\ 3.44777589141166\\ 3.13486424867285\\ 3.43399661980319\\ 4.30179336055830\\ 4.52907950006680\\ 3.86263047412942\\ 4.69275266205735\\ 4.34257688120801\\ 4.95891891610375\\ ⋮\end{array}\right]\\ \hfill \text{500 element Vector[column]}\end{array}$

$\mathrm{results}:-\mathrm{velocitydata}$

$\begin{array}{c}\left[\begin{array}{c}0.\\ 0.000975114000000000\\ 0.00142931700000000\\ 0.00194630400000000\\ 0.00310977000000000\\ 0.00491971500000000\\ 0.00665510400000000\\ 0.00759588300000000\\ 0.00774205200000000\\ 0.00797749200000000\\ ⋮\end{array}\right]\\ \hfill \text{1560 element Vector[column]}\end{array}$

$\mathrm{results}:-\mathrm{displacementdata}$

$\begin{array}{c}\left[\begin{array}{c}0.\\ 9.75114000000000\times {10}^{\mathrm{-6}}\\ 0.0000337954500000000\\ 0.0000675516600000000\\ 0.000118112400000000\\ 0.000198407250000000\\ 0.000314155440000000\\ 0.000456665310000000\\ 0.000610044660000000\\ 0.000767240100000000\\ ⋮\end{array}\right]\\ \hfill \text{1560 element Vector[column]}\end{array}$

Aeran, Ashish and Hirpa G. Lemu. "Time Integration Schemes in Dynamic Problems: Effect of Damping on Numerical Stability and Accuracy". International Workshop of Advanced Manufacturing and Automation (IWAMA), pp. 213-220. Atlantis Press, 2016.

"Newmark-beta Method", Wikipedia. https://en.wikipedia.org/wiki/Newmark-beta_method

"Response Spectrum", Wikipedia. https://en.wikipedia.org/wiki/Response_spectrum

The SignalProcessing[ResponseSpectrum] command was introduced in Maple 2024.

For more information on Maple 2024 changes, see Updates in Maple 2024.