Figure 2.66. Analytical models for single-degree-of-freedom stem: (a) model in horizontal position; (b) model in vertical position.

The spring constant or spring stiffness k denotes the load required to produce unit extension of the spring. If W is measured in kips and the extension in inches, the spring stiffness will have a dimension of kips per inch. The weight W comes to rest after the spring has extended by the length x. Equation (2.51) expresses the familiar static equilibrium condition between the internal force in the spring and the externally applied force W.

If a vertical force is applied or removed suddenly, vibrations of the system are produced. Such vibrations, maintained by the elastic force in the spring alone, are called free or natural vibrations. The weight moves up and down, and therefore is subjected to an acceleration x given by the second derivative of displacement x, with respect to time t. At any instant t, there are three forces acting on the body: the dynamic force equal to the product of the body mass and its acceleration, the gravity force W acting downward, and the force in the spring equal to W + kx for the position of weight shown in Fig. 2.67. These are in a state of dynamic equilibrium given by the relation

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