# Nonlinear Rotational Spring

Torsional spring based on polynomial or table lookup parameterizations

## Library

Simscape / Driveline / Couplings & Drives / Springs & Dampers

## Description

The block represents a torsional spring with nonlinear torque-displacement curve. The spring torque magnitude is a general function of displacement. It need not satisfy Hooke’s law. Polynomial and table lookup parameterizations provide two ways to specify the torque-displacement relationship. The spring torque can be symmetric or asymmetric with respect to zero deformation.

The symmetric polynomial parameterization defines spring torque according to the expression:

$$T={k}_{1}\theta +sign(\theta )\cdot {k}_{2}{\theta}^{2}+{k}_{3}{\theta}^{3}+sign(\theta )\cdot {k}_{4}{\theta}^{4}+{k}_{5}{\theta}^{5},$$

where:

*T*— Spring force*k*_{1},*k*_{2}, ...,*k*_{5}— Spring coefficients*θ*— Relative displacement between ports R and C, $$\theta ={\theta}_{init}+{\theta}_{R}-{\theta}_{C}$$*θ*_{init}— Initial spring deformation*θ*_{R}— Absolute angular position of port R*θ*_{C}— Absolute angular position of port C

At simulation start (t=0), *θ*_{R} and
*θ*_{C} are zero, making *θ*
equal to *θ*_{init}.

To avoid zero-crossings that slow simulation, eliminate the sign function from the
polynomial expression by specifying an odd polynomial (*b*_{2},*b*_{4}
= 0).

The two-sided polynomial parameterization defines spring torque according to the expression:

$$T=\{\begin{array}{cc}{k}_{1t}\theta +{k}_{2t}{\theta}^{2}+{k}_{3t}{\theta}^{3}+{k}_{4t}{\theta}^{4}+{k}_{5t}{\theta}^{5},& \theta \ge 0\\ {k}_{1c}\theta -{k}_{2c}{\theta}^{2}+{k}_{3c}{\theta}^{3}-{k}_{4c}{\theta}^{4}+{k}_{5c}{\theta}^{5},& \theta <0\end{array},$$

where:

*k*_{1t},*k*_{2t}, ...,*k*_{5t}— Spring tension coefficients*k*_{1c},*k*_{2c}, ...,*k*_{5c}— Spring compression coefficients

Both polynomial parameterizations use a fifth-order polynomial expression. To use a lower-order polynomial, set the unneeded higher-order coefficients to zero. To use a higher-order polynomial, fit to a lower-order polynomial or use the table lookup parameterization.

The table lookup parameterization defines spring torque based on a set of torque and angular velocity vectors. If not specified, the block automatically adds a data point at the origin (zero angular velocity and zero torque).

### Variables

Use the **Variables** tab to set the priority
and initial target values for the block variables before simulating.
For more information, see Set Priority and Initial Target for Block Variables.

## Ports

`C`

Rotational conserving port

`R`

Rotational conserving port

## Parameters

**Parameterization**Select spring parameterization. Options are

`By polynomial`

and`By table lookup`

.

## Extended Capabilities

**Introduced in R2013a**