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:

$T = k_{1} θ + s i g n (θ) \cdot k_{2} θ^{2} + k_{3} θ^{3} + s i g n (θ) \cdot k_{4} θ^{4} + k_{5} θ^{5},$

where:

T — Spring force
k₁, k₂, ...,k₅ — Spring coefficients
θ — Relative displacement between ports R and C, $θ = θ_{i n i t} + θ_{R} - θ_{C}$
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 (b₂,b₄ = 0).

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

$T = {\begin{matrix} k_{1 t} θ + k_{2 t} θ^{2} + k_{3 t} θ^{3} + k_{4 t} θ^{4} + k_{5 t} θ^{5}, & θ \geq 0 \\ k_{1 c} θ - k_{2 c} θ^{2} + k_{3 c} θ^{3} - k_{4 c} θ^{4} + k_{5 c} θ^{5}, & θ < 0 \end{matrix},$

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

To set the priority and initial target values for the block variables prior to simulation, use the Initial Targets section in the block dialog box or Property Inspector. For more information, see Set Priority and Initial Target for Block Variables.

Nominal values provide a way to specify the expected magnitude of a variable in a model. Using system scaling based on nominal values increases the simulation robustness. Nominal values can come from different sources, one of which is the Nominal Values section in the block dialog box or Property Inspector. For more information, see Modify Nominal Values for a Block Variable.

Ports

C: Rotational conserving port
R: Rotational conserving port

Parameters

Parameterization

Select spring parameterization. Options are By polynomial and By table lookup.

By Polynomial

Specify nonlinear spring function in terms of polynomial coefficients.

Symmetry

Choose between symmetric and two-sided polynomial parameterizations.

Symmetric — Specify one set of polynomial coefficients governing spring torque in both tension and compression.
Vector of spring coefficients
Enter five-element vector with polynomial spring coefficients. The highest non-zero order must be positive. Physical units are for the first coefficient.
The default vector is [1 0 0.1 0 0.01]. The default unit is N*m/rad.
Two-sided — Specify two sets of polynomial coefficients governing spring torque, one for positive deformation, the other for negative deformation.
Vector of spring tension coefficients
Enter a five-element vector containing the coefficients of the polynomial spring tension function. The highest order non-zero coefficient must be positive. The specified physical unit is the unit of the first polynomial coefficient.
Vector of spring compression coefficients
Enter a five-element vector containing the coefficients of the polynomial spring compression function. The highest order non-zero coefficient must be positive. The specified physical unit is the unit of the first polynomial coefficient.

By table lookup

Deformation vector

Enter a vector containing the deformation values used in the table lookup. The vector must contain a minimum number of elements based on the interpolation method: two for Linear, and three for Smooth. The vector must also contain the same number of elements as the torque vector.

Torque vector

Enter a vector containing the torque values that correspond to the deformation vector values. The vector must contain a minimum number of elements based on the interpolation method: two for Linear, and three for Smooth. The vector must also contain the same number of elements as the deformation vector. If not included in the vectors, the block automatically adds a point at the (0,0) deformation–torque coordinate.

Interpolation method

Select method used to calculate deformation-torque values between table lookup data points.

Linear — Select this option to get the best performance.
Smooth — Select this option to produce a continuous curve with continuous first-order derivatives.

For more information on interpolation algorithms, see the PS Lookup Table (1D) block reference page.

Extrapolation method

Select method used to calculate deformation-torque values outside the table lookup data range.

Linear — Select this option to produce a curve with continuous first-order derivatives in the extrapolation region and at the boundary with the interpolation region.
Nearest — Select this option to produce an extrapolation that does not go above the highest point in the data or below the lowest point in the data.

For more information on extrapolation algorithms, see the PS Lookup Table (1D) block reference page.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2013a