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1.1. Introduction to Mechanical Vibrations
Vibration is the continuing periodic motion of a particle or a body or system of connected bodies displaced from a specific central reference or equilibrium. Most vibrations are unenviable in machines and structures since they develop increased stresses, energy losses, generate added wear, increase bearing loads, produce fatigue, create passenger uneasiness in vehicles, and absorb energy from the system. Rotating machine parts need careful balancing in order to prevent damage from vibrations.
Vibration occurs when a system is displaced from a position of stable equilibrium. The system tends to return to this equilibrium position under the action of restoring forces (such as the elastic forces, as for a mass attached to a spring, or gravitational forces, as for a simple pendulum). The system keeps moving back and forth across its position of equilibrium. A system is a combination of elements meant to act together to accomplish an objective. For example, an automobile is a system whose elements are the wheels, suspension, car body, and so forth. A static element is one whose output at any given time depends only on the input at that time while a dynamic element is one whose present output depends on past inputs. In the same way we also speak of static and dynamic systems. A static system contains all elements while a dynamic system contains at least one dynamic element.
A physical system undergoing a time-varying interchange or dissipation of energy among or within its elementary storage or dissipative devices is said to be in a dynamic state. All of the elements in general are called passive, i.e., they are incapable of generating net energy. A dynamic system composed of a finite number of storage elements is said to be lumped or discrete, while a system containing elements, which are dense in physical space, is called continuous. The analytical description of the dynamics of the discrete case is a set of ordinary differential equations, while for the continuous case it is a set of partial differential equations. The analytical formation of a dynamic system depends upon the kinematic or geometric constraints and the physical laws governing the behavior of the system.
1.2 . Classification of Vibrations
Vibrations can be classified into three categories: free, forced, and self-excited. Free vibration of a system is vibration that occurs in the absence of external force. An external force that acts on the system causes forced vibrations. In this case, the exciting force continuously supplies energy to the system. Forced vibrations may be either deterministic or random (see Fig. 1.1 and 1.2). Self- excited vibrations are periodic and deterministic oscillations. Under certain conditions, the equilibrium state in such a vibration system becomes unstable, and any disturbance causes the perturbations to grow until some effect limits any further growth. In contrast to forced vibrations, the exciting force is independent of the vibrations and can still persist even when the system is prevented from vibrating.
Fig.1.1 A deterministic (periodic) excitation.
Fig.1.2 Random excitation.
1.3. Elementary Parts of Vibrating Systems
In general, a vibrating system consists of a spring (a means for storing potential energy), a mass or inertia (a means for storing kinetic energy), and a damper (a means by which energy is gradually lost) as shown in Fig. 1.2. An undamped vibrating system involves the transfer of its potential energy to kinetic energy and kinetic energy to potential energy, alternatively. In a damped vibrating system, some energy is dissipated in each cycle of vibration and should be replaced by an external source if a steady state of vibration is to be maintained.
Fig.1.3 Elementary parts of vibrating systems.
1.4. Periodic Motion
When the motion is repeated in equal intervals of time, it is known as periodic motion. Simple harmonic motion is the simplest form of periodic motion. If x(t) represents the displacement of a mass in a vibratory system, the motion can be expressed by the equation
where A is the amplitude of oscillation measured from the equilibrium position of the mass. The repetition time Ï„ is called the period of the oscillation, and its reciprocal, , is called the frequency. Any periodic motion satisfies the relationship
That is Period s/cycle
Frequency cycles/s, or Hz
is called the circular frequency measured in rad/sec.
The velocity and acceleration of a harmonic displacement are also harmonic of the same frequency, but lead the displacement by and radians, respectively. When the acceleration of a particle with rectilinear motion is always proportional to its displacement from a fixed point on the path and is directed towards the fixed point, the particle is said to have simple harmonic motion.
The motion of many vibrating systems in general is not harmonic. In many cases the vibrations are periodic as in the impact force generated by a forging hammer. If is a periodic function with period, its Fourier series representation is given by
Where is the fundamental frequency and are constant coefficients, which are given by:
The exponential form of is given by:
The Fourier coefficients can be determined, using
The harmonic functions or are known as the harmonics of order of the periodic function. The harmonic of order has a period. These harmonics can be plotted as vertical lines in a diagram of amplitude ( and) versus frequency () and is called frequency spectrum.
1.5. Discrete and Continuous Systems
Most of the mechanical and structural systems can be described using a finite number of degrees of freedom. However, there are some systems, especially those include continuous elastic members; have an infinite number of degree of freedom. Most mechanical and structural systems have elastic (deformable) elements or components as members and hence have an infinite number of degrees of freedom. Systems which have a finite number of degrees of freedom are known as discrete or lumped parameter systems, and those systems with an infinite number of degrees of freedom are called continuous or distributed systems.
2.1. Vibration Analysis in General
The outputs of a vibrating system, in general, depend upon the initial conditions, and external excitations. The vibration analysis of a physical system may be summarized by the four steps:
1) Mathematical Modelling of a Physical System
2) Formulation of Governing Equations
3) Mathematical Solution of the Governing Equations
4) Physical interpretation of the result
2.1.1. Mathematical Modelling of a Physical System
The purpose of the mathematical modelling is to determine the existence and nature of the system, its features and aspects, and the physical elements or components involved in the physical system. Necessary assumptions are made to simplify the modelling. Implicit assumptions are used that include:
a) A physical system can be treated as a continuous piece of matter
b) Newton's laws of motion can be applied by assuming that the earth is an internal frame
c) Ignore or neglect the relativistic effects
All components or elements of the physical system are linear. The resulting mathematical model may be linear or non-linear, depending on the given physical system. Generally speaking, all physical systems exhibit non-linear behaviour. Accurate mathematical modelling of any physical system will lead to non-linear differential equations governing the behaviour of the system. Often, these non-linear differential equations have either no solution or difficult to find a solution. Assumptions are made to linearise a system, which permits quick solutions for practical purposes. The advantages of linear models are the following:
i. Their response is proportional to input
ii. Superposition is applicable
iii. They closely approximate the behaviour of many dynamic systems
iv. Their response characteristics can be obtained from the form of system equations without a detailed solution
v. A closed-form solution is often possible
vi. Numerical analysis techniques are well developed, and
vii. They serve as a basis for understanding more complex non-linear system behaviours.
It should, however, be noted that in most non-linear problems it is not possible to obtain closed-form analytic solutions for the equations of motion. Therefore, a computer simulation is often used for the response analysis.
When analyzing the results obtained from the mathematical model, one should realise that the mathematical model is only an approximation to the true or real physical system and therefore the actual behaviour of the system may be different.
2.1.2. Formulation of governing equations
Once the mathematical model is developed, we can apply the basic laws of nature and the principles of dynamics and obtain the differential equations that govern the behaviour of the system. A basic law of nature is a physical law that is applicable to all physical systems irrespective of the material from which the system is constructed. Different materials behave differently under different operating conditions. Constitutive equations provide information about the materials of which a system is made. Application of geometric constraints such as the kinematic relationship between displacement, velocity, and acceleration is often necessary to complete the mathematical modelling of the physical system. The application of geometric constraints is necessary in order to formulate the required boundary and/or initial conditions.
The resulting mathematical model may be linear or non-linear, depending upon the behaviour of the elements or components of the dynamic system.
2.1.3. Mathematical solution of the governing equations
The mathematical modelling of a physical vibrating system results in the formulation of the governing equations of motion. Mathematical modelling of typical systems leads to a system of differential equations of motion. The governing equations of motion of a system are solved to find the response of the system. There are many techniques available for finding the solution, namely, the standard methods for the solution of ordinary differential equations, Laplace transformation methods, matrix methods, and numerical methods. In general, exact analytical solutions are available for many linear dynamic systems, but for only a few non linear systems. Of course, exact analytical solutions are always preferable to numerical or approximate solutions.
2.1.4. Physical interpretation of the results
The solution of the governing equations of motion for the physical system generally gives the performance. To verify the validity of the model, the predicted performance is compared with the experimental results. The model may have to be refined or a new model is developed and a new prediction compared with the experimental results. Physical interpretation of the results is an important and final step in the analysis procedure. In some situations, this may involve (a) drawing general inferences from the mathematical solution, (b) development of design curves, (c) arrive at a simple arithmetic to arrive at a conclusion (for a typical or specific problem), and (d) recommendations regarding the significance of the results and any changes (if any) required or desirable in the system involved.
2.2. Components of Vibrating Systems
Sometimes it requires finding out the equivalent spring stiffness values when a continuous system is attached to a discrete system or when there are a number of spring elements in the system. Stiffness of continuous elastic elements such as rods, beams, and shafts, which produce restoring elastic forces, is obtained from deflection considerations.
The stiffness coefficient of the rod (Fig. 2.1) is given by
The cantilever beam (Fig.2.2) stiffness is
The torsional stiffness of the shaft (Fig.2.3) is
Fig.2.1 Longitudinal vibration of rods.
Fig.2.2 Transverse vibration of cantilever beams.
Fig.2.3 Torsional system.
When there are several springs arranged in parallel as shown in Fig. 2.4, the equivalent spring constant is given by algebraic sum of the stiffness of individual springs. Mathematically,
Fig.2.4 Springs in parallel.
When the springs are arranged in series as shown in Fig. 2.5, the same force is developed in each spring and is equal to the force acting on the mass.
Fig.2.5. Springs in series.
The equivalent stiffness is given by:
Hence, when elastic elements are in series, the reciprocal of the equivalent elastic constant is equal to the reciprocals of the elastic constants of the elements in the original system.
2.2.2 Mass or inertia elements
The mass or inertia element is assumed to be a rigid body. Once the mathematical model of the physical vibrating system is developed, the mass or inertia elements of the system can be easily identified.
2.2.3 Damping elements
In real mechanical systems, there is always energy dissipation in one form or another. The process of energy dissipation is referred to in the study of vibration as damping. A damper is considered to have neither mass nor elasticity. The three main forms of damping are viscous damping, Coulomb or dry-friction damping, and hysteresis damping. The most common type of energy-dissipating element used in vibrations study is the viscous damper, which is also referred to as a dashpot. In viscous damping, the damping force is proportional to the velocity of the body. Coulomb or dry-friction damping occurs when sliding contact that exists between surfaces in contact is dry or have insufficient lubrication. In this case, the damping force is constant in magnitude but opposite in direction to that of the motion. In dry-friction damping energy is dissipated as heat. Solid materials are not perfectly elastic and when they are deformed, energy is absorbed and dissipated by the material. The effect is due to the internal friction due to the relative motion between the internal planes of the material during the deformation process. Such materials are known as viscoelastic solids and the type of damping which they exhibit is called as structural or hysteretic damping, or material or solid damping. In many practical applications, several dashpots are used in combination. It is quite possible to replace these combinations of dashpots by a single dashpot of an equivalent damping coefficient so that the behaviour of the system with the equivalent dashpot is considered identical to the behaviour of the actual system.
2.3. Laplace Transformation Method
The Laplace transformation method can be used for calculating the response of a system to a variety of force excitations, including periodic and non periodic. The Laplace transformation method can treat discontinuous functions with no difficulty and it automatically takes into account the initial conditions. The usefulness of the method lies in the availability of tabulated Laplace transform pairs. From the equations of motion of a single degree of freedom system subjected to a general forcing function, the Laplace transform of the solution is given by
The method of determining given can be considered as an inverse transformation which can be expressed as
FREE VIBRATION OF SINGLE DEGREE OF FREEDOM SYSTEMS
The most basic mechanical system is the single-degree-of-freedom system, which is characterized by the fact that its motion is described by a single variable or coordinates. Such a model is often used as an approximation for a generally more complex system. Excitations can be broadly divided into two types, initial excitations and externally applied forces. The behaviour of a system characterized by the motion caused by these excitations is called as the system response. The motion is generally described by displacements.
3.2 Free Vibration of an Undamped Translation System
The simplest model of a vibrating mechanical system consists of a single mass element which is connected to a rigid support through a linearly elastic mass less spring as shown in Fig. 1.8. The mass is constrained to move only in the vertical direction. The motion of the system is described by a single coordinate x (t) and hence it has one degree of freedom (DOF).
Fig.3.1 Spring Mass System.
The equation of motion for the free vibration of an undamped single degree of freedom system can be rewritten as
3.3 Free Vibration of an Undamped Torsional System
A mass attached to the end of the shaft is a simple torsional system (Fig. 1.9). The mass of the shaft is considered to be small in comparison to the mass of the disk and is therefore neglected.
Fig.3.2 Torsional System.
The torque that produces the twist is given by
Where The polar mass moment of inertia of the shaft ( for a circular shaft of diameter.
shear modulus of the material of the shaft.
Length of the shaft.
The torsional spring constant is defined as
The equation of motion of the system can be written as:
The natural circular frequency of such a torsional system is
The general solution of equation of motion is given by
3.4 Energy Method
Free vibration of systems involves the cyclic interchange of kinetic and potential energy. In undamped free vibrating systems, no energy is dissipated or removed from the system. The kinetic energy T is stored in the mass by virtue of its velocity and the potential energy U is stored in the form of strain energy in elastic deformation. Since the total energy in the system is constant, the principle of conservation of mechanical energy applies. Since the mechanical energy is conserved, the sum of the kinetic energy and potential energy is constant and its rate of change is zero. This principle can be expressed as
Where T and U denote the kinetic and potential energy, respectively. The principle of conservation of energy can be restated by
Where the subscripts 1 and 2 denote two different instances of time when the mass is passing through its static equilibrium position and select as reference for the potential energy. Subscript 2 indicates the time corresponding to the maximum displacement of the mass at this position, we have then
If the system is undergoing harmonic motion, then and denote the maximum values of and, respectively and therefore last equation becomes
It is quite useful in calculating the natural frequency directly.
3.5. Stability of Undamped Linear Systems
The mass/inertia and stiffness parameters have an effect on the stability of an undamped single degree of freedom vibratory system. The mass and stiffness coefficients enter into the characteristic equation which defines the response of the system. Hence, any changes in these coefficients will lead to changes in the system behaviour or response. In this section, the effects of the system inertia and stiffness parameters on the stability of the motion of an undamped single degree of freedom system are examined. It can be shown that by a proper selection of the inertia and stiffness coefficients, the instability of the motion of the system can be avoided. A stable system is one which executes bounded oscillations about the equilibrium position.
3.6. Torsional System with Viscous Damping
The equation of motion for such a system can be written as
Where I is the mass moment of inertia of the disc, is the torsional spring constant (restoring torque for unit angular displacement), and Î¸ is the angular displacement of the disc.
FORCED VIBRATION OF SINGLE DEGREE OF FREEDOM SYSTEMS
A mechanical or structural system is often subjected to external forces or external excitations. The external forces may be harmonic, non-harmonic but periodic, non-periodic but having a defined form or random. The response of the system to such excitations or forces is called forced response. The response of a system to a harmonic excitation is called harmonic response. The non-periodic excitations may have a long or short duration. The response of a system to suddenly applied non-periodic excitations is called transient response. The sources of harmonic excitations are unbalance in rotating machines, forces generated by reciprocating machines, and the motion of the machine itself in certain cases.
4.2 Forced Vibrations of Damped System
Consider a viscously damped single degree of freedom spring mass system shown in Fig. 4.1, subjected to a harmonic function
Fig.4.1 Forced Vibration of single degree of freedom system.
Here is the force amplitude and is the circular frequency of the forcing function. The equations of motion of the system is
The solution of the equation contains two components, complimentary function and particular solution. That is
The particular solution represents the response of the system to the forcing function. The complementary function is called the transient response since in the presence of damping, the solution dies out. The particular integral is known as the steady state solution. The steady state vibration exists long after the transient vibration disappears.
The case that is, when the circular frequency of the forcing function is equal to the circular frequency of the spring-mass system is referred to as resonance. In this case, the displacement goes to infinity for any value of time t. The amplitude of the forced response grows with time as in Fig. 4.2 and will eventually become infinite at which point the spring in the mass-spring system fails in an undesirable manner.
Fig.4.2 Resonance Response
The phenomenon of beating occurs for an undamped forced single degree of freedom spring-mass system when the forcing frequency is close, but not equal, to the system circular frequency. In this case, the amplitude builds up and then diminishes in a regular pattern. The phenomenon of beating can be noticed in cases of audio or sound vibration and in electric power generation when a generator is started.
The forces associated with the vibrations of a machine or a structure will be transmitted to its support structure. These transmitted forces in most instances produce undesirable effects such as noise. Machines and structures are generally mounted on designed flexible supports known as vibration isolators or isolators. In general, the amplitude of vibration reduces with the increasing values of the spring stiffness k and the damping coefficient c. In order to reduce the force transmitted to the support structure, a proper selection of the stiffness and damping coefficients must be made. The transmissibility is defined as the ratio of the maximum transmitted force to the amplitude of the applied force.
4.6. Quality Factor and Bandwidth
The value of the amplitude ratio at resonance is also known as the Q factor or Quality factor of the system in analogy with the term used in electrical engineering applications. That is,
The points and, whereby the amplification factor falls to, are known as half power points, since the power absorbed by the damper responding harmonically at a given forcing frequency is given by
The bandwidth of the system is defined as the difference between the frequencies associated with the half power points and as depicted in Fig. 4.3.
Fig.4.3 Harmonic Response curve showing half power points and bandwidth.
It can be shown that Q-factor can be written as:
The quality factor Q can be used for estimating the equivalent viscous damping in a vibrating system.
4.7. Base Excitation
In many mechanical systems such as vehicles mounted on a moving support or base, the forced vibration of the system is due to the moving support or base. The motion of the support or base causes the forces being transmitted to the mounted equipment. Fig. 1.22 shows a damped single degree of freedom mass-spring system with a moving support or base.
Fig.4.4 Harmonically Excited Base
4.8 Response under Hysteresis Damping
The steady-state motion of a single degree of freedom forced harmonically with hysteresis damping is also harmonic. The steady-state amplitude can then be determined by defining an equivalent viscous damping constant based on equating the energies. The amplitude is given in terms of hysteresis damping coefficient Î² as follows
4.9 General Forcing Conditions and Response
A general forcing function may be periodic or nonperiodic. The ground vibrations of a building structure during an earthquake, the vehicle motion when it hits a pothole, are some examples of general forcing functions. Nonperiodic excitations are referred to as transient. The term transient is used in the sense that nonperiodic excitations are not steady state.
5.1. Unbalance in Rotating Machines
Unbalance in many rotating mechanical systems is a common source of vibration excitation which may often lead to unbalance forces. A common source of such a sinusoidal force is unbalance in a rotating machine or rotor. You may have experienced the effect if you have ever driven a car where the wheels are not balanced; you will have noticed that at a particular speed, the car will shake, sometimes quite violently. At this speed, the rotational speed of the wheels is such as to be close to the natural frequency of the car on its suspension, so that the amplitude becomes a maximum. In this section we will look at how we can describe such a phenomenon more precisely, in a mathematical way. Rotating machines include turbines, electric motors and electric generators, as well as fans, or rotating shafts. Rotating shafts experience a special kind of response, known as "whirl", but we will not be able to consider them in this course.
5.2. Problem Definition
5.2.1 Problem 1
We consider a rotating machine of mass, which is modeled as being mounted on a spring of stiffness, to a fixed support, and that there is a viscous damping in the system, with a damping coefficient, . We will also suppose that the machine is constrained to move vertically, so that this is a single degree-of-freedom system. The rotor is unbalanced if its centre of mass does not coincide with the centre of rotation. Suppose that the unbalance can be represented by a mass at a distance from the centre of rotation ( is also called the eccentricity.) Let the angular speed of rotation of the rotor be. The system is illustrated in Figure 5.1. After a time, the out-of-balance mass will have moved through an angular displacement. If is the displacement of the mass from the equilibrium position after a time t, the displacement of the out-of-balance mass, , will be
The forces acting on the combined mass are and (i.e. they act in the opposite direction to the positive x-direction).
Fig.5.1 Harmonic disturbing force resulting from rotating unbalance.
The equation of motion is then
This can be rearranged to
Applying Laplace transformation, taking initial value at
Performing partial decomposition technique to
Where A, B, D and E are constants
Equating the co-efficient,
Solving the above equations we get,
Where and are
Taking Inverse Laplace of the function
Substituting all the constants
The above equation gives the transient state solution
Here we are considering the same spring-mass system as discussed in the previous problem. In this problem we are performing the analysis relating to mechanical vibration due to the system offering rotating unbalance by applying an external pulsating force to it. Here we will investigate the behaviour of the system by plotting amplitude spectrum and phase spectrum graphs by varying the running frequency,.
When we apply the pulsating force to system described in the fig 5.1, then the equation of motion for the system would be
Where = Amplitude of the pulsating Force
= Frequency of the external force
Applying Laplace transformation, taking initial value at
Taking (sinusoidal steady state)
Changing the above equation into the form
Substituting all the constants
RESULTS AND DISCUSSIONS
For the calculations and for plotting the frequency spectrum graphs the values for constants are taken as follows:
Mass of the rotating machine, M = 20 kg
Unbalanced mass, m = 2 kg
Eccentricity of the unbalanced mass, e=0.05 m
Angular velocity of the unbalanced mass, w1 = 73.33 rad/sec
Frequency of the external force, wo = 146.66 rad/sec
Amplitude of the pulsating force, Fo = 500 N
Spring constant, k = 2000 N/m
Damper constant c = 2000 N-sec/m
Substituting the above constants in the equation
Now we can plot the amplitude spectrum and phase spectrum graphs by varying.
6.2 Results and Discussions:
Transient state solution for the rotating unbalance equation of motion was obtained by applying Laplace transformation.
This shows that at low speed, the amplitude of the motion of the mass (M-m) is nearly 0, while at very high speed, the amplitude becomes constant. This explains why, when your car with unbalanced wheels begins to shake, the shaking reduces and stops when the speed of the car is increased.
Fig.6.1 Bode Plot of the system depicting resonance
It is evident from the bode amplitude plot that resonance occurs at frequency 73.3 rad/sec and 147 rad/sec. System also has negative poles at -1.010 rad/sec and -98.98 rad/sec. But the concept of negative frequency in a real physical system is absurd.
From bode phase plot a phase reversal of 180 degrees is observed. This further confirms that resonance occurs at these particular frequencies.
CHANGE IN PHASE
A jump of around 100db is observed around both the resonant frequencies.
Fig.6.2 Root locus Plot of the system under consideration
Fig.6.3 Nyquist Plot of the system under consideration