Nonlinear Translational Spring

Translational spring based on polynomial or table lookup parameterizations

Library

Simscape / Driveline / Couplings & Drives / Springs & Dampers

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Description

The block represents a translational spring with nonlinear force-displacement curve. The spring force 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 force-displacement relationship. The spring force can be symmetric or asymmetric with respect to zero deformation.

The symmetric polynomial parameterization defines spring force according to the expression

$$F=k_{1}x+sign(x)\cdot k_2{x}^2+k_3{x}^3+sign(x)\cdot k_4{x}^4+k_5{x}^5,$$

where:

• F — Spring force

• k₁, k₂, ..., k₅ — Spring coefficients

• x — Relative displacement between ports R and C, $$x=x_{init}+x_R-x_C$$

• x_init — Initial relative displacement

• x_R — Absolute displacement associated with port R

• x_C — Absolute displacement associated with port C

At simulation start (t=0), x_R and x_C are zero, making x equal to x_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 asymmetric polynomial parameterization defines spring force according to the expression:

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

where:

• k₁, k₂, ..., k₅ — Spring tension coefficients

• k₁, k₂, ..., k₅ — 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 damping 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.

Unlike block parameters, variables do not have conditional visibility. The Variables tab lists all the existing block variables. If a variable is not used in the set of equations corresponding to the selected block configuration, the values specified for this variable are ignored.

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 coefficients of polynomial spring function.

Symmetry

Choose between symmetric and two-sided polynomial parameterizations.

• Symmetric — Specify one set of polynomial coefficients governing spring force 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 [100 0 10 0 1]. The default unit is N/m.

• Two-sided — Specify two sets of polynomial coefficients governing spring force, one for positive relative displacements, the other for negative relative displacements.

  Vector of spring tension coefficients

  Enter five-element vector with polynomial spring tension coefficients. Physical units are for the first coefficient.

  The default vector is [100 0 10 0 1]. The default unit is N/m.

  Vector of spring compression coefficients

  Enter five-element vector with polynomial spring compression coefficients. Physical units are for the first coefficient.

  The default vector is [1000 –10 100 0 0]. The default unit is N/m.

 By table lookup

Deformation vector

Enter vector with relative displacement values. the vector requires a minimum number of elements, based on the selected interpolation method — two for Linear, and three for Smooth. The number of elements must match the force vector. If not included in the vector, the block automatically adds a point at the origin, corresponding to zero displacement and zero force. If the vector includes a zero value, then the force vector must contain a corresponding zero value.

Force vector

Enter a vector containing the force 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–force coordinate.

Interpolation Method

Select method used to find intermediate velocity–force 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 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™.