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We had discussed earlier in Chapter 1 as how to idealize a given real structure into a SDOF syste... more We had discussed earlier in Chapter 1 as how to idealize a given real structure into a SDOF system consisting of a mass connected to a spring and a damper for the purposes of carrying out dynamic analysis. Based on this SDOF system idealization we had also formulated a governing differential equation to describe its motion. The equation of motion given in Chapter 2 for various cases is a linear differential equation of the second order with constant coefficient. The solution of this governing differential equation yields the response of the system which is nothing but the displacement of the mass of the system at any instant of time. The system may or may not be acted upon by an external dynamic force. The form of solution of this governing equation of motion depends upon the mathematical representation for the exciting force. Although the representation of a realistic structure into an idealized SDOF system is an oversimplification, useful results can, however, be obtained from the analysis of the simplified system. We discuss in this chapter systems that are subject to either free vibration or to forced vibration with some kind of excitation. In the case of forced vibration we consider here the time variation of the forcing function. In fact, we have several types of forcing functions such as harmonic, periodic, impulsive, and of general nature. We consider here it is enough to evaluate the response of a SDOF system acted by these exciting functions. 3.2 Undamped Free Vibration The equation of motion for an undamped free vibration of a SDOF is given in Eq. (2.3). It is a second order linear differential equation. The solution of this governing differential equation describes the motion of the structure at any instant of time. The motion of the structure in terms of displacement or velocity or acceleration is called the response of the structure. When a structure is distributed from its equilibrium position, free vibration results in. For this, the mass is given some displacement x(0) and velocity at time t = 0. At the instant of time when the motion is initiated, let x = x(0) and =

The linear elastic single degree of freedom (SDOF) system without damping shown in Fig. 1.4 is us... more The linear elastic single degree of freedom (SDOF) system without damping shown in Fig. 1.4 is used here to illustrate the essential features of the dynamic problem. The properties of this system are assumed to be defined and concentrated at discrete points. This type of idealization is called a lumped mass system. The mass of the system is signified by the symbol m, and the elasticity of the spring stiffness k. Since the system has a single degree of freedom the displacement of the mass and spring components will be the same. Fig. 1.4 will be used to explain and demonstrate the application of the four alternative methods of formulating the equation of motion (EOM) for a dynamic system. 2.2 Free Vibration without Damping We now consider the SDOF shown in Fig. 1.4. Here there is no external force and no loss of energy takes place. During vibration, the spring deforms and accelerates the mass. The resulting displacement of the mass varies with time and is denoted as x(t) at any instant of time. The displacement x(t) is unknown and it is to be evaluated from the properties of the mass and spring. This quantity is also called, in general, a response. It is the response of the system/structure to the disturbance or x(t) excitation. This can be achieved by setting up an equation based on the motion of the mass. For this purpose we make use of Newton's Second Law of Motion. 2.2.1 Newton's Second Law of Motion As per this law, Force = mass × acceleration (2.1) Here acceleration is at any instant of time. Double dots mean the second derivative of x w.r.t. time. RHS of Eq. (2.1) is m or simply m. In order to identify forces induced during the motion, we draw a free body diagram (FBD) of the mass (Fig. 2.1). In this configuration of SDOF, the only force acting on the mass is the spring force or resisting force kx. For the purposes of derivation of the equation, we assume that forces, displacements, velocities, and accelerations acting upward as positive. Therefore substituting various quantities in Eq. (2.1) we get,-kx = m (2.2) Rearranging Eq. (2.2), we get m+ kx = 0 (2.3) Equation (2.3) is the free vibration equation of motion of a SDOF system. The equation of motion is a linear differential equation of second order with constant coefficient. The equation of motion is for the free vibration without damping, since no damper is attached to the system. It is called free vibration because no external load is applied to the system.

In this modern world structures are acted upon frequently by dynamic loads such as earthquake, cy... more In this modern world structures are acted upon frequently by dynamic loads such as earthquake, cyclones, floods, land slides, blasting, etc., in contrast to those constructed a few centuries ago wherein loading on the structures were mainly static in nature. The speciality of static loadings are their variation in magnitudes with time do not change, i.e., their magnitudes remain constant for ever. At that time designers were happy if they designed the structures for static loading only and also found satisfied when the design fulfilled the conditions prescribed by codes of practices prevalent at that time. Subsequently extreme loadings struck the buildings and they were found wanting. When the extreme loading particularly earthquales were widespread throughout the world and their occurrences were repeated resulting in colossal loss of lives and properties, engineers and scientists became wiser and commenced organised research to find a solution to ensure safety to human beings. The outcome of this research is the formulation of texts on dynamics of structures. The present treatise is therefore entitled as Basics of Dynamics and Earthquake Resistant Structures. It covers the fundamental principles of dynamics as applicable to structures and deigns these structures to resist the forces induced by motion of the ground. As various guidelines have been framed by codal authorities of the country to design structures constructed with masonry, reinforced concrete and steel. In India the codal authority is the Bureau of Indian Standard. It has released mainly three codes related to earthquake design, viz., IS: 1890, IS: 13920 and IS: 4320 for designers to follow in their design. In essence the book contains 32 chapters dealing with various aspects of theory of vibration, basics of seismicity, lesson learnt from past earthquakes, related soil properties and its measurement, analysis techniques such as response spectrum approach and time history technique, various aspects of seismic resistance provisions of buildings, codal provisions relevant to buildings constructed in India, details about isolation and vibration control methods, mitigation of earthquake effects and finally quality aspect of materials and construction technique. The book has been prepared based on the syllabus prescribed by Anna University and other universities situated in different parts of the country. This book has been tailored to suit to design engineers and other practising engineers in their profession. The hallmark of this book is that it presents at the end of each chapter Points to Remember section which will be quite beneficial for them to score high marks in 2 marks part of the university question paper.