We can assume that each node in mesh is a mass point and that this mass has connections with springs to other nodes. If string length differs from original, node will be forced into movement. Angles between connection can be similarly constrained.
Figure 2: Constraints in mesh: node spacing and angles
Figure 2 shows constraints between nodes.
When the length of connection between two nodes is not the desired
one, force is applied at each node. Similarly we have torsion strings
between connections to force the desired angle. Moment M produced by
torsion string is applied at each node as force F= M/(2l), where l
is length between two nodes.
For each
node in mesh governing equation of motion can be written as:
Linear spring constant denotes linear force for shift from
stable position . Since stable position is not know
at beginning this kind of force must be added to external force
. Solution of the homogeneous equation
shows that damper constant should be chosen carefully to achieve
fast convergence to stable position. For aperiodic movement the
following condition must be satisfied:
Note that spring constant has been moved out of equation
2 and only used for dumper constant estimation.
Equivalent needed for equation 4 is obtained from
maximum estimated displacement of node and sum of all
external forces which are connection constraints:
Equation of motion for each node can be written as:
Let be a distance between two nodes. Virtual movement of both
nodes is shown in change of connection length and can be approximated
as
Change of distance is directly connected with dumper speed it must be
applied to both nodes connected.
For 3D space the solution of equation 7 is:
Equation 6 for motion of single node can be rewritten
for all nodes in space state to form system of linear differential
equations
that can be solved with Euler integration method.