Structural Steel

Vibration of Single-Degree-of-Freedom Systems

Several dynamic characteristics of a structure can be illustrated by studying single-degreeof-
freedom systems. Such a system may represent the motion of a beam with a weight at center span and subjected to a time-dependent concentrated load P(t) (Fig. 3.100a). It also may approximate the lateral response of a vertically loaded portal frame constructed of flexible columns, fully restrained connections, and a rigid beam that is also subjected to a time-dependent force P(t) (Fig. 3.100b).
In either case, the system may be modeled by a single mass that is connected to a weightless spring and subjected to time-dependent or dynamic force P(t) (Fig. 3.100c). The magnitude of the mass m is equal to the given weight W divided by the acceleration of gravity g = 386.4 in /sec^2. For this model, the weight of structural members is assumed negligible compared with the load W. By definition, the stiffness k of the spring is equal to the force required to produce a unit deflection of the mass. For the beam, a load of 48EI/ L^3 is required at center span to produce a vertical unit deflection; thus k  48EI/L3, where E is the modulus of elasticity, psi; I the moment of inertia, in4; and L the span of the beam, in. For the frame, a load of 2 x 12EI/h^3 produces a horizontal unit deflection; thus k  24EI/h^3, where I is the moment of inertia of each column, in4, and h is the column height,

in. In both cases, the system is presumed to be loaded within the elastic range. Deflections
are assumed to be relatively small.
At any instant of time, the dynamic force P(t) is resisted by both the spring force and the inertia force resisting acceleration of the mass (Fig. 3.100d). Hence, by d’Alembert’s principle (Art. 3.7), dynamic equilibrium of the body requires

For undamped free vibration, the natural frequency, period, and circular frequency depend only on the system stiffness and mass. They are independent of applied loads or other disturbances.