# Complex Results

Results in some analysis types come as complex numbers.

`z = x + iy`

`x`

is the real part of`z`

.`y`

is the imaginary part of`z`

.

Complex number can also be written in the polar format:

`x`

is the real part of`z`

.`y`

is the imaginary part of`z`

.

Complex number can also be written in the polar format:

where

, is the magnitude of z.

and

, is the phase of z.

Where , is the angular velocity and t is time. In HyperLife, you can select the increment of ωt (called "angle") from the Animation Controls panel.

## Complex Result in HyperLife

In modal animation mode, there is selection to choose a complex filter.

- mag * cos (ωt- phase)
- the response with varying angle or ωt (in degree).
- mag
- magnitude (r) of the complex result
- phase
- phase of the complex number
- real
- real part (x) of the complex number
- imaginary
- imaginary part (y) of the complex number

Other functionalities can be viewed in Contour panel.

## Invariants for a Complex Vector or Tensor

Complex results of invariants of a vector (like magnitude of displacement) or a tensor (like von-Mises value of stress) can only be calculated at a specific angle from the response of each components at that angle. For example, complex responses of x, y and z components of displacement are:

For a specific angle ωt, the magnitude of displacement is calculated as:

Note that the response of an invariant is not a complex number. It can only be calculated at a specific angle. When an invariant is selected for modal animation, HyperLife will calculate the value at each specified angle increment for each response point separately.

## Plotting Complex Curves

You can view the curve plot from the Measure panel.

## View the Contour of a Complex Response at a Certain Angle

- On the Animation toolbar, click .
- From the Animation Controls panel, input the desired angle into the Current angle text field.
- Press the Enter key on the keyboard.

## View the Contour Animation of a Complex Response

- On the Animation toolbar, click .
- From the Animation Controls panel, input the desired angle increment into the Increment by text field.
- Click .