Translational spring based on polynomial or lookup table parameterizations

Couplings & Drives/Springs & Dampers

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 lookup-table 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

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

*F*— Spring force*k*_{1},*k*_{2}, ...,*k*_{5}— 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 (*b*_{2},*b*_{4} =
0).

The asymmetric polynomial parameterization defines spring force according to the expression:

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

*k*_{1},*k*_{2}, ...,*k*_{5}— Spring tension coefficients*k*_{1},*k*_{2}, ...,*k*_{5}— 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 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).

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 (Simscape).

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.

`C`

Rotational conserving port

`R`

Rotational conserving port

**Parameterization**Select spring parameterization. Options are

`By polynomial`

and`By lookup table`

.

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