Trigonometric

A Trigonometric Time Series defines a time–history where the load factor $\lambda$ varies sinusoidally in time. In Alpaca4d (following OpenSees) it is typically defined as:

$$\lambda(t) = c_\text{factor} \, \sin\!\Big( \tfrac{2\pi}{T} (t - t_\text{shift}) \Big)$$

over a specified active interval $t_\text{start} \le t \le t_\text{end}$, where:

• $c_\text{factor}$ is the amplitude,

• $T$ is the period,

• $t_\text{shift}$ is a phase shift in time.

This is useful for modelling harmonic / cyclic loading such as machine vibrations or idealised base excitations.

🔧 Grasshopper component

The Trigonometric Time Series (Alpaca4d) component constructs a periodic (sine) time series with user‑controlled time window, period and amplitude.

• Inputs

  • TStart: Start time $t_\text{start}$ $[s]$.

    • Type: Number

    • Default: 0.0

    • Effect: For $t < t_\text{start}$ the load factor is zero.

  • TEnd: End time $t_\text{end}$ $[s]$.

    • Type: Number

    • Requirement: $t_\text{end} > t_\text{start}$. For $$$t > t_\text{end} $$ the load factor is usually zero.

  • Period: Period $ T $ of the sinusoidal function $[s]$.

    • Type: Number

    • Effect: Controls the frequency of oscillation.

  • Shift: Time shift $t_\text{shift}$ $[s]$.

    • Type: Number

    • Default: 0.0

    • Effect: Moves the sine wave left/right in time (phase shift).

  • LoadFactor: Amplitude $c_\text{factor}$ of the sine wave.

    • Type: Number

    • Default: 1.0.

• Outputs

  • TimeSeries: Alpaca4d Trigonometric time series object.

  • Graph: A list of values representing the sinusoidal time–history between TStart and TEnd.

📈 When to use a trigonometric time series

• Use it when

  • You need harmonic / cyclic loading with a well‑defined frequency (e.g. machine vibration, idealised sinusoidal base motion).

  • You want to control the number of cycles in a given time window via TStart, TEnd and Period.

  • You are performing parametric studies on frequency–dependent response.

• Do not use it when

  • The excitation is non‑periodic or irregular (e.g. real earthquake records) → use a Path time series.

  • The loading is simply constant or linearly varying → use Constant or Linear time series.

🔗 Relation to OpenSees

Alpaca4d’s trigonometric time series corresponds to the OpenSees Trig timeSeries:

timeSeries Trig $tag $tStart $tEnd $period -shift $shift -factor $cFactor

timeSeries('Trig', tag, tStart, tEnd, period,
           '-shift',  shift,
           '-factor', cFactor)

where:

• TStart, TEnd, Period, Shift, LoadFactor in the Grasshopper component map directly to tStart, tEnd, period, shift, cFactor in the OpenSees commands.