SimMechanics

Body Spring & Damper

Models a damped linear oscillator force between two bodies

Library

Force Elements

Description

The Body Spring & Damper block models the force of a damped spring acting between two bodies. By Newton's third law, the spring applies equal and opposite forces to the two bodies. You can use this Force Element block to model any linear (Hooke's law) force, with constant coefficients, that acts between a pair of bodies. One of the Bodies can be a Ground.

Caution    Both the spring and the damper force act only along the axis connecting the two bodies. The Body Spring & Damper block does not include the degree of freedom (DoF) needed to model the relative motion of the connected bodies. You need to add a Joint block between the Bodies to represent one, two, or three prismatic primitives. You can use Prismatic blocks or a Custom Joint block to accomplish this. The Body Spring & Damper block and Joints block must form a closed loop together with the two Bodies.

Body Spring and Damper Theory

You connect this block to each Body, A or B, at a Body coordinate system (CS). If r_A and r_B are the positions of these Body CSs, the relative position vector connecting them is r = r_B - r_A. The distance of separation is |r|. The relative velocity is v = dr/dt. Then the vector force that body A exerts on body B is

F = -k(|r| - r₀)(r/|r|) - b(v*r)(r/|r|²)

The first term represents the spring or linear displacement force. The second represents the damper or velocity dissipation force, which acts only along the direction of r. Thus the damper is equivalent to a dashpot, not a viscous medium.

You specify the spring constant k, the natural spring length (offset) r₀, and the damping constant b. The natural length is the length of the spring with no forces acting on it and physically must be nonnegative: r₀ 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.

The Spring and Damper Force in Singular Cases

In certain cases, the force formula breaks down, and SimMechanics uses special rules to determine the force.

• If r₀ and v 0, and r = 0 at some instant, both terms in the force become singular. The spring force is reprojected along the velocity vector. That is, v/|v| replaces r/|r| in both terms, once in the first term and twice in the second. If the state r = 0 does not persist for more than an instant, this replacement has no effect on the motion.
• If r₀ 0, and both r and v = 0 at some instant, the force direction is undefined. The simulation stops with an error.

To avoid singularities in the initial state of motion, be sure to set the bodies' initial conditions of position and velocity to physically sensible values.

Dialog Box and Parameters

The dialog box has two active areas, Parameters and Units.

Parameters

Spring constant (k)

Enter the linear spring force constant k. The default is 0.

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

Damper constant (b)

Enter the linear damping force constant b. The default is 0.

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

Spring natural length (r0)

Enter the spring's natural length (offset) r₀. The default is 0.

Units

Position

In the pull-down menu, choose units for the relative position vector r. The default is m (meters).

Velocity

In the pull-down menu, choose units for the relative velocity vector v. The default is m/s (meters/second).

Force

In the pull-down menu, choose units for the spring-damper force F acting between the bodies. The default is N (Newtons).

See Also

Body, Body Actuator, Body Sensor, Custom Joint, Ground, Joint Initial Condition Actuator, Joint Spring & Damper, Prismatic

See Modeling Force Elements in the Modeling Mechanical Systems chapter.

Body Sensor

Bushing