# Joint Spring & Damper

Damped linear oscillator force or torque acting on a joint

Force Elements

## Description

The Joint Spring & Damper block models a damped linear oscillator force acting along a prismatic primitive or a damped linear oscillator torque acting about a revolute primitive. The joint primitives are connected between two bodies, and the force or torque acts between these bodies. The sign of the force or torque is set by the base (B)-to-follower (F) sequence of the bodies. These models represent damped linear translational and torsional springs in the prismatic and revolute cases, respectively.

You connect this block to a Joint at one of the Joint's sensor/actuator ports. (If the Joint lacks a sensor/actuator port, open its dialog and create one.) The Joint represents any mixture of translational and rotational degrees of freedom (DoFs). With the Joint Spring & Damper block, you can then apply any combination of damped linear oscillator forces on any prismatics and damped linear torsion torques on any revolutes.

 Note:   Each Joint Spring & Damper block connected to a revolute primitive adds a normal Simulink® state to your model.This feature does not change the mechanical states of your model.

### Joint Spring and Damper Theory

Connect two Bodies with a Joint having some combination of prismatic and revolute primitives.

 Caution   The Joint Spring & Damper uses a Joint Sensor to measure the degree of freedom in the Joint. These values are measured relative to the home configuration of the DoF, its state before the application of initial condition actuators and assembly of disassembled joints.

#### Translational Case

If x represents the displacement along a prismatic axis, and v = dx/dt is the prismatic DoF's linear speed, then the damped spring force acting along this prismatic and between the Bodies connected by this Joint is

F = -k(x - x0) - bv

The model parameters are the spring constant k, the natural spring length (offset) x0, and the damping constant b. The natural length is the spring's length with no forces acting on it and should be nonnegative: x0 ≥ 0. A stable spring requires k > 0. A damping representing dissipation and respecting the second law of thermodynamics requires b ≥ 0. You can use a negative b to represent energy pumping.

#### Rotational Case

If θ represents the displacement about a revolute axis, and ω = dθ/dt is the revolute DoF's angular speed, then the damped torsion torque acting about this revolute and between the Bodies connected by this Joint is

τ = -k(θ - θ0) - bω

The model parameters are the torsion constant k, the natural torsion angle (offset) θ0, and the damping constant b. The natural angle is the torsion balance's direction with no torques acting on it and can have any sign. A stable torsion requires k > 0. A damping representing dissipation and respecting the second law of thermodynamics requires b ≥ 0. You can use a negative b to represent energy pumping.

## Actuation

The menu lists all the active primitives in the Joint to which the Joint Spring & Damper block is connected. If you connect the Joint Spring & Damper with its dialog open, the primitive list is automatically updated to reflect the connected Joint's primitives.

Primitive

Lists the active primitives in the Joint to which the block is connected. `P` represents a prismatic primitive, `R` a revolute primitive, `S` a spherical primitive, and `W` a weld primitive.

Enable

To enable force or torque actuation on any particular primitive in the Joint, select the Enable check box next to that primitive's name in the Primitive column. You cannot actuate spherical or weld primitives.

Spring Constant k

Enter the spring or torsion constant k, for a prismatic or revolute primitive, respectively. The default is `0`.

The units for k are derived implicitly from your choice of position and force/torque units.

Damper Constant b

Enter the spring or torsion damping constant b, for a prismatic or revolute primitive, respectively. The default is `0`.

The units for b are derived implicitly from your choice of velocity and force/torque units.

Spring Offset x0

Enter the natural spring length x0 or the natural torsion angle θ0, for a prismatic or revolute primitive, respectively. The default is `0`.

Position Units

In the pull-down menu, select linear or angular units for prismatic or revolute primitives, respectively. The default is `m` (meters) or `deg` (degrees).

Velocity Units

In the pull-down menu, select linear or angular velocity units for prismatic or revolute primitives, respectively. The default is `m/s` (meters/second) or `deg/s` (degrees/second).

Force/Torque Units

In the pull-down menu, select force or torque units for prismatic or revolute primitives, respectively. The default is `N` (newtons) or `N*m` (newton-meters).