In an ideal system having no damping, the system would vibrate forever (Fig. 2.12). In a real system, where there is always some damping, the amplitude of motion will gradually decrease for each cycle until the structure comes to a complete stop (Fig. 2.13). The system responds in a similar manner if, instead of displacing the mass at the top, a sudden impulse is applied to the base.

Buildings may be analyzed as multidegree-of-freedom (MDOF) systems by lumping story-masses at intervals along the length of a vertically cantilevered pole. During vibration, each mass will deflect in one direction or another. For higher modes of vibration, some masses may move in opposite directions. Or all masses may simultaneously deflect in the same direction as in the fundamental mode. An idealized MDOF system has a number of modes equal to the number of masses. Each mode has its own natural period of vibration with a unique mode shaped by a line connecting the deflected masses. When ground motion is applied to the base of the multimass system, the deflected shape of the system is a combination of all mode shaped, but modes having periods near predominant periods of the base motion will be excited more than the other modes. Each mode of a multimass system can be represented by an equivalent single-mass system having generalized values M and K for mass and stiffness. The generalized values represent the equivalent combined effects of story masses m1, m2, . . . and stiffness k1, k2, . . . . This concept, shown in Fig. 2.14, provides a computational basis for using response spectra based on single-mass systems for analyzing multistoried buildings. Given the period, mode shape, and mass distribution of a multistoried building, we can use the response spectra of a single-degree-of-freedom (SDOF) system for computing the deflected shape, story

Figure 2.12. Undamped free vibrations of a single-degree-of-freedom system.
Renewable Energy 101

Renewable Energy 101

Renewable energy is energy that is generated from sunlight, rain, tides, geothermal heat and wind. These sources are naturally and constantly replenished, which is why they are deemed as renewable. The usage of renewable energy sources is very important when considering the sustainability of the existing energy usage of the world. While there is currently an abundance of non-renewable energy sources, such as nuclear fuels, these energy sources are depleting. In addition to being a non-renewable supply, the non-renewable energy sources release emissions into the air, which has an adverse effect on the environment.

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