A Spring represents a single component force acting between two bodies over a distance and in a particular direction.

Figure 4.1: Kinematics of Spring Force

The direction of the force is along the line segment connecting the action and base markers and the magnitude of the force is linearly dependent on the relative displacement and velocity of the markers. The relative displacement and velocity between the base and action markers are calculated using the following equations:

(4–1)

(4–2)

where the vectors , , , and are the absolute displacements and velocity of the action and base markers, respectively.

A Spring can represent spring, damping and constant actuating forces. The spring force is calculated using the stiffness coefficient and distance and the damping force is calculated using the damping coefficient and time derivative of the distance. The constant actuating force can be defined by a preload. The linear force acting on the action marker can be written as follows:

(4–3)

where the distance and its time derivative and can be calculated from the relative displacement and its time derivative as follows.

(4–4)

(4–5)

If the force is positive, the markers experience a repelling force along this line, and if the force is negative, the markers experience an attracting force.

Figure 4.2: Spring and Damping Force Curves

The non-linear force acting on the action marker can be written as follows:

(4–6)

where and are spring force curve and damping force curve as shown in Figure 4.2: Spring and Damping Force Curves, respectively. These can be measured by experimental method or by finite element analysis.

The reacting force on the base marker can be calculated as follows.

(4–7)

The definitions of the parameters in Equation 4–3 and Equation 4–6 are shown in the table below.

Figure 4.3: Spring parameters

Symbol	Description	Dimension
	Spring coefficient. This is an axial stiffness and can be measured by experimental method or flexible body simulation.	Force/Length
	Damping coefficient. This affects dynamic stiffness and can be measured by experimental method.	Force*Time/Length
	Free length. If the distance between two markers approaches this value, the spring force approaches zero.	Length
	Preload. When this value is positive, the spring is under compression. When it is negative, the spring is under tension.	Force
	Spring force curve. The x-axis of the curve must be the deformation in length units and the y-axis of the curve must be force.	Force
	Damping force curve. The x-axis of the curve must be the time derivative of the deformation in length per time units and the y-axis of the curve must be force.	Force