Chapter 6 Introduction to MDOF System (original) (raw)
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Chapter 3 Response of Structures to Dynamic Loading 3.1 Introduction
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 =
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Chapter 2 Formulation of Equation of Motion 2.1 Introduction
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.