> For the complete documentation index, see [llms.txt](https://alpaca4d.gitbook.io/docs/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://alpaca4d.gitbook.io/docs/basics/time-history/linear.md).

# Linear

A **Linear Time Series** defines a time–history where the load factor $$\lambda$$ varies **linearly with time**:

$$
\lambda(t) = c\_\text{factor} , t
$$

where $c\_\text{factor}$ is a constant slope. This is useful when you want loads to **ramp up (or down, if negative)** at a constant rate as the analysis time increases.

## 🔧 Grasshopper component

The `Linear Time Series (Alpaca4d)` component creates an Alpaca4d linear time series that can be connected to load pattern or excitation components.

* **Input**
  * **LinearFactor**: Linear scale factor $$c\_\text{factor}$$ that multiplies time $$t$$.
    * Type: Number
    * Default: `1.0`
    * Effect: The load factor increases (or decreases) proportionally to time: doubling the analysis time doubles the load factor.
* **Outputs**
  * **TimeSeries**: Alpaca4d `Linear` time series object, to be plugged into components that require a time series.
  * **Graph**: A list of values representing the time series, typically visualised as a straight line increasing with time.

## 📈 When to use a linear time series

* **Use it when**
  * You want a **ramp load** that grows from zero to a target value over a given duration.
  * You need to **gradually apply loads** to avoid sudden jumps (e.g. quasi‑static ramping in nonlinear analysis).
  * You are modelling a **linearly increasing excitation intensity**.
* **Do not use it when**
  * The load should be **constant in time** → use a **Constant** time series.
  * The load follows a **recorded or arbitrary signal** → use a **Path / Time History** time series.
  * The load is **cyclic or periodic** → use a **Trigonometric** time series.

## 🔗 Relation to OpenSees

Alpaca4d’s linear time series is conceptually equivalent to the OpenSees `Linear` timeSeries:

```tcl
timeSeries Linear $tag -factor $cFactor
```

```python
timeSeries('Linear', tag, '-factor', cFactor)
```

where `cFactor` corresponds to the **LinearFactor** input in the Alpaca4d Grasshopper component.


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