9780203014554.ch4 - 4 Single-degree-of-freedom systems 4.1...

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4 Single-degree-of-freedom systems 4.1 Introduction This chapter deals with the simplest system capable of vibratory motion: the single-degree-of-freedom (SDOF) system which, in its discrete-parameters form, is often called harmonic oscillator . Despite its apparent simplicity, this system contains and exhibits most of the essential features of vibrating systems and its analysis is a necessary prerequisite to any further investigation in vibration theory and practice. In addition, many complex systems behave and can be considered, under certain circumstances, as SDOF systems, thus considerably simplifying the procedures of measurement and analysis. It is often a matter of the modelling scheme that we choose to adopt for the system under examination and of the degree of approximation that we are willing to accept. The second-order ordinary differential equation which these SDOF obey is commonly found in many branches of physics and engineering—acoustics, mechanical, structural engineering and electronics, to name a few—and all its physical and mathematical aspects are worth considering for their own sake since they are the basis of useful analogies between these different fields. This basic equation, in the general form that is of interest to us, is the equation of motion of the linear harmonic oscillator shown in Fig. 4.1 under an external applied force f(t) . It can be written as (Chapter 3) (4.1) where m, c and k are the mass, the damping and elastic constant of our SDOF system (see also Section 1.4). Unless otherwise specified, we will assume that these quantities are time-independent. When the motion is pure rotational, there is a whole formal analogy with the translational case for which eq (4.1) applies. Obviously, the quantities to consider now are angular displacements, angular velocities and angular accelerations. The analogies are listed in Table 4.1 and, as a consequence of this analogy, only translational systems will be discussed in the following Copyright © 2003 Taylor & Francis Group LLC
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sections. The treatment of rotational systems is obtained by substitution of the appropriate quantities with the consistent units. As a word of caution, it must be said that rotational quantities cannot be measured as easily as translational quantities and therefore the experimental part may turn out to be more critical. 4.2 The harmonic oscillator I: free vibrations When friction forces are absent (or negligible), any elastic system that, in some way, is slightly displaced from its equilibrium position and is subsequently let free by removing the cause of the initial disturbance, executes an oscillatory motion and continues to vibrate forever unless we decide to interfere with it again. This particular condition is called undamped free vibrations . The frequency characteristics of the oscillatory motion depend on the parameters of the system itself, that is, on its mass and elasticity; the amplitude characteristics, on the contrary, depend on the initial conditions and the vibration does not die out because no energy is lost during the motion.
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This note was uploaded on 05/18/2011 for the course MAE 269A taught by Professor Ju during the Spring '11 term at UCLA.

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9780203014554.ch4 - 4 Single-degree-of-freedom systems 4.1...

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