Nonlinear Translational Spring

Translational spring based on polynomial or table lookup parameterizations

Libraries:
Simscape / Driveline / Couplings & Drives / Springs & Dampers

Description

The Nonlinear Translational Spring block represents a translational spring with a nonlinear force-displacement curve. The spring force magnitude is a general function of displacement and does not necessarily need to satisfy Hooke’s law. You can parameterize the force-displacement relationship using a polynomial or a lookup table. The spring force can be symmetric or asymmetric with respect to zero deformation.

The symmetric polynomial parameterization defines spring force according to the expression

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

where:

F is the spring force.
k₁, k₂, ..., k₅ are the spring coefficients.
x is the relative displacement between ports R and C.
x_init is the initial relative displacement.

At simulation start x is 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:

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

where:

k₁, k₂, ..., k₅ are the spring tension coefficients.
k₁, k₂, ..., k₅ are the 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.

Setting Parameterization to By table lookup defines the damping torque based on a set of torque and angular velocity vectors. If you do not enter a datapoint for the origin, the block automatically adds one (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.

Examples

Suspension System Comparison

A testbed containing three sets of springs and dampers excited by the same oscillating velocity source. The first uses the Shock Absorber block and includes linear stiffness and damping. Optional friction and hard stops are not used. The second shock absorber uses a Nonlinear Translational Spring and a Nonlinear Translational Damper specified by symmetric polynomials. The third uses a Variable Translational Spring and Variable Translational Damper. The spring constant is varied during the simulation using open loop control and the damping coefficient adjusts to ensure critical damping is achieved. Closed loop control can be added to simulate an adaptive suspension system.

Open Model

Ports

Conserving

expand all

C — Follower port
mechanical rotational

Mechanical rotational conserving port associated with the follower end.

R — Base port
mechanical rotational

Mechanical rotational conserving port associated with the base end.

Parameters

expand all

Parameterization — Method to characterize spring behavior
`By polynomial` (default) | `By table lookup`

Spring behavior selection. You can choose to parameterize by polynomial or lookup table.

Symmetry — Spring behavior versus zero deformation position
`Symmetric` (default) | `Two-sided`

Spring behavior symmetry about the point of zero deformation. When you select Symmetric, the block applies one polynomial symmetrically about the point of zero deformation. When you select Two-sided, the block applies unique polynomials for tension and compression.

Dependencies

To enable this parameter, set Parameterization to By polynomial.

Vector of spring coefficients — Symmetric deformation polynomial coefficients
`[100, 0, 10, 0, 1]` `N/m` (default) | vector

Polynomial spring rate coefficients. The highest non zero-order element must be positive. Physical units represent the first coefficient, and the block selects the appropriate higher-order units for you. This vector must have a length of 5.

Dependencies

To enable this parameter, set Parameterization to By polynomial and Symmetry to Symmetric.

Vector of spring tension coefficients — Positive two-sided deformation polynomial coefficients
`[100, 0, 10, 0, 1]` `N/m` (default)

Polynomial coefficients for the spring tension. Physical units represent the first coefficient, and the block selects the appropriate higher-order units for you. This vector must have a length of 5.

Dependencies

To enable this parameter, set Parameterization to By polynomial and Symmetry to Two-sided.

Vector of spring compression coefficients — Negative two-sided deformation polynomial coefficients
`[1000, -10, 100, 0, 0]` `N/m` (default)

Polynomial coefficients for the spring compression. Physical units represent the first coefficient, and the block selects the appropriate higher-order units for you. This vector must have a length of 5.

Deformation vector — Deformation due to force table lookup entry
`[-1, -.5, -.3, -.1, .1, .3, .5, 1]` (default)

Relative displacement values with respect to the zero-deformation position for the corresponding elements in the Force vector parameter. This parameter must be the same length as the Force vector parameter. The minimum vector length depends on the interpolation method: two for Linear and three for Smooth. If not included in the vector, the block automatically adds a point at the origin deformation–force coordinate.

Force vector — Applied force table lookup entry
`[-1000, -400, -200, -50, 50, 200, 400, 1000]` (default)

Force values for the corresponding element in the Deformation vector parameter. This parameter must be the same length as the Deformation vector parameter. The minimum vector length is: two for Linear and three for Smooth. The block automatically adds a point at the origin deformation-force coordinate if you do not specify one.

Interpolation method — Method of interpolation between breakpoint values
`Smooth` (default) | `Linear`

Lookup table breakpoint interpolation. The block uses the tablelookup function to model nonlinearity by using vector data to map input values to output values:

Linear — Select this option for the lowest computational cost.
Smooth — Select this option to produce a continuous curve with continuous first-order derivatives.

For more information, see tablelookup.

Extrapolation method — Method of handling input values that fall outside range of breakpoint data set
`Linear` (default) | `Nearest` | `Error`

Lookup table breakpoint extrapolation. This method determines the output value when the input value is outside the range specified in the argument list. The block uses the tablelookup function to model nonlinearity by using vector data to map input values to output values:

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.
Error — Select this option to avoid extrapolating when you want your data to be within the table range. If the input signal is outside the range of the table, the simulation stops and generates an error.

For more information, see tablelookup.

Extended Capabilities

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

Version History

Introduced in R2013a

Nonlinear Translational Spring

Description

Variables

Examples

Suspension System Comparison

Ports

Conserving

C — Follower port mechanical rotational

R — Base port mechanical rotational

Parameters

Parameterization — Method to characterize spring behavior By polynomial (default) | By table lookup

Symmetry — Spring behavior versus zero deformation position Symmetric (default) | Two-sided

Dependencies

Vector of spring coefficients — Symmetric deformation polynomial coefficients [100, 0, 10, 0, 1] N/m (default) | vector

Dependencies

Vector of spring tension coefficients — Positive two-sided deformation polynomial coefficients [100, 0, 10, 0, 1] N/m (default)

Dependencies

Vector of spring compression coefficients — Negative two-sided deformation polynomial coefficients [1000, -10, 100, 0, 0] N/m (default)

Deformation vector — Deformation due to force table lookup entry [-1, -.5, -.3, -.1, .1, .3, .5, 1] (default)

Force vector — Applied force table lookup entry [-1000, -400, -200, -50, 50, 200, 400, 1000] (default)

Interpolation method — Method of interpolation between breakpoint values Smooth (default) | Linear

Extrapolation method — Method of handling input values that fall outside range of breakpoint data set Linear (default) | Nearest | Error

Extended Capabilities

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

Version History

See Also

C — Follower port
mechanical rotational

R — Base port
mechanical rotational

Parameterization — Method to characterize spring behavior
`By polynomial` (default) | `By table lookup`

Symmetry — Spring behavior versus zero deformation position
`Symmetric` (default) | `Two-sided`

Vector of spring coefficients — Symmetric deformation polynomial coefficients
`[100, 0, 10, 0, 1]` `N/m` (default) | vector

Vector of spring tension coefficients — Positive two-sided deformation polynomial coefficients
`[100, 0, 10, 0, 1]` `N/m` (default)

Vector of spring compression coefficients — Negative two-sided deformation polynomial coefficients
`[1000, -10, 100, 0, 0]` `N/m` (default)

Deformation vector — Deformation due to force table lookup entry
`[-1, -.5, -.3, -.1, .1, .3, .5, 1]` (default)

Force vector — Applied force table lookup entry
`[-1000, -400, -200, -50, 50, 200, 400, 1000]` (default)

Interpolation method — Method of interpolation between breakpoint values
`Smooth` (default) | `Linear`

Extrapolation method — Method of handling input values that fall outside range of breakpoint data set
`Linear` (default) | `Nearest` | `Error`

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