# Sensitivity Calculation

The sensitivity of an output *y* (*u*) to an input *u* is defined as
the change in *y* due to a unit change in *u*.

Around an operating point $({y}_{0},{u}_{0})$ one can write the Taylor’s series as:

The quantity
$\left(\frac{\partial y}{\partial u}\right)$
is called the first order sensitivity of the
quantity *y* to the input *u*.

Sensitivity analysis is the study of how the change or uncertainty in the output of a
mathematical model or system (*y*) can be apportioned to different sources of
change or uncertainty in its inputs (*u*).

For Design Sensitivity Analysis, the quantity (*u*) is the design *b*. Here
we are asking the question: How does the response *y* change for a given change
in the design *b*.

In a multibody simulation, the response
$y$
is typically a function of the system states
$x$
and
$\dot{x}$
, and, perhaps, explicitly on the design *b*.
The system states
$x$
consist of (a) Displacements, (b) Velocities, (c)
Lagrange Multipliers (or constraint reaction forces), (d) User defined differential
equations, (e) User defined algebraic equations originating from Variables and
LSE/GSE/TFSISO outputs, and, (f) Internally created intermediate states that
simplify computation.

In mathematical terms:

The equations of motion provide an implicit relationship between (*x* and*
b*) and (
$\dot{x}$
and (*x*,* b*)). The quantities
$\frac{\partial x}{\partial b}\text{,}\frac{\partial \dot{x}}{\partial b}\text{and}\frac{\partial y}{\partial b}$
need to be computed first. Once these are known, the
design sensitivity,
$\frac{\Delta y}{\Delta b}$
, can be computed.

The calculation of $\frac{\Delta y}{\Delta b}$ is called Design Sensitivity Analysis (DSA) in MotionSolve. This is a new analysis method in MotionSolve. It always accompanies a regular analysis, such as static analysis, quasi-static analysis, kinematic analysis or dynamic analysis.

The job of the regular analysis is to compute the states *x, ẋ * and the
outputs *y* for a given design *b*.

Once these are known, the DSA analysis will compute the sensitivity,
$\frac{\Delta y}{\Delta b}$
. When there are *N*_{y} responses and
*N*_{b} design variables,
$\left[\frac{\Delta y}{\Delta b}\right]$
is a matrix of dimension *N*_{y} x
*N*_{b}.

There are three well-known methods for computing design sensitivity: Finite Differencing, Direct Differentiation, and Adjoint Approach.