A PPLIED M ECHANICS Lecture 03 Slovak University of Technology Faculty of Material Science and Technology in Trnava

FUNDAMENTALS OF VIBRATIONS The first remarkable work from filed of vibration was work of Lord Rayleigh - Theory of Sound (published in 1887) He introduced concept of oscillations of a linear system and showed the existence of natural modes and natural frequencies for discrete as well as continuous systems. This work remains valuable in many ways even though he was concerned with acoustics rather than with structural mechanics.

FUNDAMENTALS OF VIBRATIONS Vibration is in general a periodic motion in time and is used to describe oscillation in mechanical systems. In most cases, the general purpose is to prevent or attenuate the vibrations, because of their detrimental effects, such as fatigue failure of components and generation of noise. However, there are some applications where vibrations are desirable and are usefully employed, as in vibration conveyers, vibrating sieves, etc.

FUNDAMENTALS OF VIBRATIONS Vibration - term describing oscillation in a mechanical system - defined by the frequency (or frequencies) and amplitude. Either the motion of a physical object or structure or, alternatively, an oscillating force applied to a mechanical system is vibration in a generic sense. The time-history of vibration may be considered to be sinusoidal or simple harmonic in form. The frequency is defined in terms of cycles per unit time, and the magnitude in terms of amplitude (the maximum value of a sinusoidal quantity).

FUNDAMENTALS OF VIBRATIONS Vibration may be described as:  deterministic vibration - it follows an established pattern so that the value of the vibration at any designated future time is completely predictable from the past history.  random vibration - its future value is unpredictable except on the basis of probability - defined in statistical terms wherein the probability of occurrence of designated magnitudes and frequencies can be indicated. The analysis involves certain physical concepts that are different from those applied to the analysis of deterministic vibration.

FUNDAMENTALS OF VIBRATIONS The structures have the three fundamental properties which are the inherent characteristics of a structure with which it will resist or oppose vibration. The three fundamental properties are:  m - mass (kg) - represents the inertia of a body to remain in its original state of rest or motion. A force tries to bring about a change in this state of rest or motion, which is resisted by the mass.  k - stiffness (N/m) - there is a certain force required to bend or deflect a structure with a certain distance. This measure of the force required to obtain a certain deflection is called stiffness.  b - damping (Ns/m) - once a force sets a part or structure into motion, the part or structure will have inherent mechanisms to slow down the motion (velocity). This characteristic to reduce the velocity of the motion is called damping.

FUNDAMENTALS OF VIBRATIONS The combined effects to restrain the effect of forces due to mass, stiffness and damping determine how a system will respond to the given external force. If the vibrations due to the external force are much larger than the net sum of the three restraining characteristics, the amount of the resulting vibrations will be higher.

FUNDAMENTALS OF VIBRATIONS Vibrations of mechanical systems may be generally classified into three categories:  Free vibrations - occur only in conservative systems where there is no friction, damping and exciting force. The total mechanical energy, which is due to the initial conditions, is conserved and exchange can take place between the kinetic and potential energies.  Forced vibrations - caused by the external forces, which excite the system. Exciting forces supply energy continuously to compensate for that dissipated by damping.  Self-excited vibrations - periodic oscillations of the limit cycle type and are caused by some nonlinear phenomenon. Energy required to maintain the vibrations is obtained from a non-alternating power source. In this case, the vibrations themselves create the periodic force.

VIBRATION OF SINGLE-DOF SYSTEM Simplest dynamic systems - elastic, dissipating and inertia forces interact. Consists:mass m attached by means of a spring k and a damper b to an immovable support m b k F(t)F(t) x,... x a - translation system, b - torsional system a b

VIBRATION OF SINGLE-DOF SYSTEM The mass is constrained to translational motion in the direction of the x axis - change of position from an initial reference is described fully by the value of a single quantity x - called a single DOF system. Free vibration - mass m is displaced from its equilibrium position and then allowed to vibrate free. Forced vibrations - continuing force F acts upon the mass or the foundation experiences a continuing motion.

VIBRATION OF SINGLE-DOF SYSTEM The equation of motion - model of translation system: The equation of motion - model of torsional system - the mass m is replaced by mass moment of inertia I, the force F(t) by the moment M(t): - torsional damper, - torsional stiffness, - rotation, angular velocity and angular acceleration. btktbtkt

VIBRATION OF SINGLE-DOF SYSTEM The solution of the equation of motion is composed of two parts:  solution of the homogenous equation – so-called free vibration  solution of the non-homogenous equation – with non-zero right side of equation of motion – so-called forced vibration

SINGLE-DOF SYSTEM FREE VIBRATION, WITHOUT DAMPING Mechanical model of a free undamped vibration m k x,... x The equation of motion  0 – natural angular frequency

SINGLE-DOF SYSTEM FREE VIBRATION, WITHOUT DAMPING The characteristic equation Roots of the characteristic equation where A, B - amplitude, - phase angle General solution - constants,

SINGLE-DOF SYSTEM FREE VIBRATION, WITHOUT DAMPING Using the initial condition the constants are and the derivative of solution x with respect to time,

SINGLE-DOF SYSTEM FREE VIBRATION, WITHOUT DAMPING The motion of the system resp. in amplitude form The motion in this case is a harmonic vibration. Period of vibration Linear natural frequency

SINGLE-DOF SYSTEM FREE VIBRATION, WITHOUT DAMPING The displacement x, velocity and acceleration for parameters:

SINGLE-DOF SYSTEM FREE VIBRATION, WITHOUT DAMPING