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Force-directed graph drawing algorithms assign forces among the set of edges and the set of nodes of a graph drawing.Typically, spring-like attractive forces based on Hooke's law are used to attract pairs of endpoints of the graph's edges towards each other, while simultaneously repulsive forces like those of electrically charged particles based on Coulomb's law are used to separate all pairs ...
In physics, Hooke's law is an empirical law which states that the force (F) needed to extend or compress a spring by some distance (x) scales linearly with respect to that distance—that is, F s = kx, where k is a constant factor characteristic of the spring (i.e., its stiffness), and x is small compared to the total possible deformation of the spring.
When putting two springs in their equilibrium positions in series attached at the end to a block and then displacing it from that equilibrium, each of the springs will experience corresponding displacements x 1 and x 2 for a total displacement of x 1 + x 2. We will be looking for an equation for the force on the block that looks like:
Since the velocity of the object is the derivative of the position graph, the area under the line in the velocity vs. time graph is the displacement of the object. (Velocity is on the y-axis and time on the x-axis. Multiplying the velocity by the time, the time cancels out, and only displacement remains.)
where (,,) are the coordinates of a point in the beam, ,, are the components of the displacement vector in the three coordinate directions, is the angle of rotation of the normal to the mid-surface of the beam, and is the displacement of the mid-surface in the -direction.
Once the global stiffness matrix, displacement vector, and force vector have been constructed, the system can be expressed as a single matrix equation. For each degree of freedom in the structure, either the displacement or the force is known.
Hooke's law gives the relationship of the force exerted by the spring when the spring is compressed or stretched a certain length: = (), where F is the force, k is the spring constant, and x is the displacement of the mass with respect to the equilibrium position. The minus sign in the equation indicates that the force exerted by the spring ...
A displacement field is a vector field of all displacement vectors for all particles in the body, which relates the deformed configuration with the undeformed configuration. The distance between any two particles changes if and only if deformation has occurred. If displacement occurs without deformation, then it is a rigid-body displacement.