When a system is vibrating in normal mode, at any particular instant of time, the amplitude of expressed as the ration of the amplitude of one of the masses of the system, is known as mode shape co-efficient.
Calculation of natural free-vibration modes (i.e. Eigen vectors) plays an important role in dynamic analyses of structures. These vectors are orthogonal (i.e., and) and they are used to investigate dynamic characteristics of structures. They are exploited in modal analysis in such a way that dynamic equilibrium equations in original system are transformed to a reduced sub-system in which the equilibrium equations are expressed in terms of modal coordinates. During this projection, orthogonal Eigen vectors are used to uncouple equilibrium equations and hence, modal superposition analysis is successfully carried such as in Response Spectra Analysis (RSA) or Modal Time History Analysis (MTHA). Even though selection of free-vibration modes is desirable for modal analysis, it is also noticed in some cases that they are not proper choices where Eigen vectors are orthogonal to dynamic loads. In these cases, these Eigen vectors are not excited under dynamic actions and hence, they do not provide any significant contribution in dynamic responses.
All the international codes has provided the guideline of modes to be used in the analysis. As per IS:1893 Part I, the number of modes to be used in the analysis should be such that the sum of total modal masses of all the modes considered at least 90% of the total seismic mass.