Systems of Linear Algebraic Equations

The static mechanical spring-mass system illustrated in Figure 1.1 consists of three masses Inl to 1n3&s; having weights WI to W3 , interconnected by five linear springs K I to K s . In the configuration illustrated on the left, the three masses are supported by forces F I to F3 equal to weights WI to W3, respectively, so that the five springs are in a stable static equilibrium configuration. When the supporting forces F j to F3 are removed, the masses move downward and reach a new static equilibrium configuration, denoted by Xl> X2&s; and X3&s; where XI&s; X2&s; and X3 are measured from the original locations ofthe corresponding masses. Free-body diagrams of the three masses are presented at the bottom of Figure 1.1. Perfonning a static force balance on the three masses yields the following system of three linear algebraic equations: When values of K I to K s and WI to W3 are specified, the equilibrium displacements XI to X3 can be detennined by solving Eq. (1.1).