The stages associated with problem definition and the interpretation of the results in design are not discussed in this book; early experience is usually gained in project work. It is on the techniques for use in stages two and three, modeling and the analysis of the model, that emphasis is laid. The models examined are mathematical models involving linear differential equations with constant coefficients. In devising the models and discovering their behaviour, familiarity with the statics of elastic systems and the dynamics of particles, together with the theory of linear differential equations, vector algebra and matrix algebra has been assumed. It has also been assumed that systems of units present no difficulty to readers at this stage, for although SI units have been used in the numerical examples exercises, it should be obvious that any consistent set of units may be ascribed to the quantities involved.
This work consists of three chapters. Each of them is dedicated to vibrations of different concrete systems, such as one degree of freedom systems, systems with more than one degree of freedom and systems having distributed mass and elasticity. Equation of vibrations of each of these systems and their solutions are known to have specific forms. That is why these systems are described separately in this work.
The first chapter is dedicated to vibration of linear mechanical systems having one degree of freedom as has been mentioned above.
Vibratory phenomena associated with complicated systems may be understood by essential studying the behavior of simple systems. The simplest type of a system has one degree of freedom, that is the motion of the system may be described by one independent coordinate.
The investigation of vibrations of these systems has been started in the work with analyzing equations of motion for any simple system.
Newton’s laws are formulated there and each simple system under consideration is illustrated with figures.
The vibratory phenomena consisting of a free vibration and forced vibration have been mentioned in this chapter. And corresponding main equation of motion are given there.
Problems of energy dissipation are paid attention to. The energy associated with the oscillation, potential and kinetic, is gradually converted to other forms of energy, such as heat or sound. It leads to a gradual decrease of the amplitude of a freely vibrating system.
The independence of the small vibrations about equilibrium position from gravity force has been discussed in the article.
The solution of the received equation of motion is described in the second part. The cases of free and of forced vibrations are considered separately.
Three different solutions of these equations are described. The motion is periodic for one of them non-oscillatory or aperiodic for two others. Responses of a simple system illustrating the above mentioned damping coefficient influence on the vibratory motion are given in this part.
All above mentioned for solution of the equation of motion is referred to the free vibration case. More complicated case of the solution of equations such as forced vibration one is considered further.
The complete solution of this differential equation consists of a transient part corresponding to free vibrations together with constant amplitude or steady state part. The responses of a simple system for this case are illustrated too.
The special attention is paid to the Lissajous’ figures, which are displayed on the screen of an oscilloscope for the different cases of the forced vibrations.
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