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 lookup-table
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:

where:

T — Spring force

k_{1}, k_{2},
...,k_{5} — Spring coefficients

θ — Relative displacement
between ports R and 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}.

Specifying an odd polynomial (b_{2},b_{4} =
0) eliminates the sign function from the polynomial
expression. This avoids zero-crossings that slow down simulation.

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

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 lookup table parameterization.

The lookup table 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).

Dialog Box and Parameters

Parameterization

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

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.

Enter a vector containing the deformation values used in the
lookup table. The vector must contain a minimum number of elements
based on the interpolation method: two for Linear,
and three for Cubic or Spline.
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 Cubic or Spline.
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
lookup-table data points.

Linear — Interpolate
using first-order polynomial. Two points or more required.

Cubic — Interpolate
using third-order polynomial. Three points or more are required.

Spline — Interpolate
using piecewise polynomial. Three points or more are required.

Extrapolation method

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

From last two points —
Extrapolate by extending the straight line between the last two data
points.

From last point —
Extrapolate by extending the horizontal straight light passing through
the last data point.