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Mechanical vibration theory and calculations
Typology: Lecture notes
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Definition of terminology Modelling of vibrating systems SINGLE DEGREE OF FREEDOM FREE VIBRATING SYSTEMS (WITHOUT DAMPING) Spring-mass systems Pendulum systems (simple and compound) Torsional systems Transverse systems Newton’s method Constant energy method
Spring-mass systems Torsional systems SINGLE DEGREE OF FREEDOM FORCED VIBRATING SYSTEMS WITHOUT DAMPING Spring-mass systems SINGLE DEGREE OF FREEDOM FORCED VIBRATING SYSTEMS WITH DAMPING Spring-mass systems PRESENTATION- VIBRATIONS MEASUREMENT
Mechanical vibration is used to describe the motion of a particle or rigid body which oscillates about a mean position known as the position of equilibrium. The design and operations of high speed engines make the study of mechanical vibrations a must. Most vibrations in machines and structures are undesirable because of the increased stresses and energy losses that accompany them. The ill effects of vibration include: Fatigue failure of structures like aircraft fuselages, buildings etc. Failure of machine components like crankshaft. INTRODUCTION TO MECHANICAL VIBRATIONS
Severe damages due to resonance as in collapsing of bridges, damages in transmission lines, damages to offshore structures. Malfunctioning of sensitive Instruments/ systems such as payloads from vibration of launching rockets. Loss of accuracy of work-pieces due to vibration of machine tools in high precision machining, micro-machining & micro-assembly Discomfort to passengers of land vehicles such as automobiles trains etc. due to uneven road surfaces and rails, unbalanced rotation of rotating masses of components. INTRODUCTION TO MECHANICAL VIBRATIONS
Because of these ill effects of vibration and their accompanying hazards, they cannot be ignored by engineers. Appropriate actions must be taken to either eliminate them where possible or reduce them as much as possible by appropriate designs. This subject of study has become increasingly important in recent years as a result of the current trend towards higher speed, high precision machines and lighter structures capable of withstanding shocks from earthquakes and explosions. INTRODUCTION TO MECHANICAL VIBRATIONS
This trend is bound to continue and therefore the need for an even thorough vibrations analysis tool in the future. Mechanical vibrations generally results when a system/body is displaced from a position of stable equilibrium. The system/body tends to return to the position under the influence of restoring forces which can be; Either elastic /restoring forces as in the case of mass spring systems and other elastic structures Or gravitational forces in the case of pendulum systems. INTRODUCTION TO MECHANICAL VIBRATIONS
All mechanical systems vibrate or exhibit some form of vibrational motion. However, in order to give proper perspective to their behavior, it is first necessary to establish some nomenclature. Several terms are used to describe vibrations, among which we have; oscillatory motion, periodic motion, harmonic motion, period, frequency, amplitude, free vibration, forced vibration, damped vibration, natural and damped frequency, resonance, degrees-of- freedom, and frequency span Oscillatory motion; is any pattern of motion where the system under observation moves back and forth across some mean position, but does not necessarily have any repeating pattern. MECHANICAL VIBRATIONS TERMINOLOGY
Periodic motion is a specific form of oscillatory motion where the motion pattern repeats itself with uniform time interval. Harmonic motion is a specific type of periodic motion where the motion pattern can be modeled by either a sine or cosine curve. This motion is sometimes referred to as Simple Harmonic Motion (SHM). Because the sine and cosine functions technically use angle in radians, the frequency term is expressed for SHMs in units of Radians per second (rad/s). This is sometimes referred to as circular/angular frequency and denoted by the Greek letter Omega ( ) Period is the uniform time interval required for an oscillatory system to complete a full cycle of motion or repeat itself. Period is measured in seconds per cycle and usually denoted by T. MECHANICAL VIBRATIONS TERMINOLOGY
Forced vibration is a type of vibration in which motion is maintained by an externally applied periodic force. Undamped vibration is a type of vibration in which the effect of resistance to motion in the form of friction (coulomb or viscous) is neglected. Damped vibration is a type of vibration in which the effect of resistance to motion in the form of friction (coulomb or viscous) that eventually brings the motion to a halt is considered Degrees-of-freedom is the number of independent coordinates necessary to describe the configuration (state) of a system. MECHANICAL VIBRATIONS TERMINOLOGY
Natural frequency is the frequency at which an undamped system will tend to oscillate due to initial conditions in the absence of any external excitation. In the absence of damping, such a system will oscillate indefinitely Damped natural frequency is a frequency that a damped system will tend to oscillate due to initial conditions in the absence of any external excitation. As a result of the damping, the system the system response will eventually decay to rest. Resonance is the condition of having an external excitation at the natural frequency of the system. In general, this is undesirable since it potentially produce extremely large system response. MECHANICAL VIBRATIONS TERMINOLOGY
Vibrating systems are dynamic in nature. The variables involved such as the excitations (inputs ) and responses (outputs ) are time dependent. The response of a vibrating system generally depends on the initial conditions as well as the external excitations. VIBRATIONS SYSTEM MODELLING
Most practical vibrating systems are very complex, and it is impossible to consider all the details for a mathematical analysis. Only the most important features are considered in the analysis to predict the behaviour of the system under specified input conditions. Often the overall behaviour of the system can be determined by considering even a simple model of the complex physical system. VIBRATIONS SYSTEM MODELLING
Modeling in vibration can be classified into two, which are; Physical models ; This deals with the physical representation of the physics of the system, by making simple approximations based on engineering judgment Mathematical models. This involves the derivation of the governing equations, solution of the equations, and interpretation of the results. The purpose of mathematical modelling is to represent all the important features of the system for the purpose of deriving the mathematical (or analytical/functional) equations governing the system’s behavior. VIBRATIONS SYSTEM MODELLING FRAMEWORK
The mathematical model may be linear or nonlinear, depending on the behaviour of the system components. Linear models permit quick solutions and are simple to handle Nonlinear models on the other hand sometimes reveal certain characteristics of the system that cannot be predicted using linear models A great deal of engineering judgment is needed to come up with a suitable mathematical model of a vibrating system. Sometimes the mathematical model is gradually improved/refined to obtain more accurate results. In this approach, first a very crude or elementary model is used to get a quick insight into the overall behaviour of the system. VIBRATIONS SYSTEM MODELLING FRAMEWORK
