Dynamic Relaxation - Main Equations Use

Main Equations Use

Considering Newton's second law of motion (force is mass multiplied by acceleration) in the direction at the th node at time :

Where:

is the residual force

is the nodal mass

is the nodal acceleration

Note that fictitious nodal masses may be chosen to speed up the process of form-finding.

The relationship between the speed, the geometry and the residuals can be obtained by performing a double numerical integration of the acceleration (here in central finite difference form), :

Where:

is the time interval between two updates.

By the principle of equilibrium of forces, the relationship between the residuals and the geometry can be obtained:

where:

is the applied load component

is the tension in link between nodes and

is the length of the link.

The sum must cover the forces in all the connections between the node and other nodes. By repeating the use of the relationship between the residuals and the geometry, and the relationship between the geometry and the residual, the pseudo-dynamic process is simulated.