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MODULE 5 STRUCTURAL DYNAMICS INTRODUCTION Structural analysis is mainly concerned with finding out the behaviour of a structure when subjected to some action. This action can be in the form of load due to the weight of things such as people, furniture, wind, snow, etc. or some other kind of excitation such as an earthquake, shaking of the ground due to a blast nearby, etc. In essence all these loads are dynamic, including the self-weight of the structure because at some point in time these loads were not there. The distinction is made between the dynamic and the static analysis on the basis of whether the applied action has enough acceleration in comparison to the structure's natural frequency. If a load is applied sufficiently slowly, the inertia forces (Newton's second law of motion) can be ignored and the analysis can be simplified as static analysis. Structural dynamics, therefore, is a type of structural analysis which covers the behaviour of structures subjected to dynamic (actions having high acceleration) loading. Dynamic loads include people, wind, waves, traffic, earthquakes, and blasts. Any structure can be subject to dynamic loading. Dynamic analysis can be used to find dynamic displacements, time history, and modal analysis. A dynamic analysis is also related to the inertia forces developed by a structure when it is excited by means of dynamic loads applied suddenly (e.g., wind blasts, explosion, earthquake). A static load is one which varies very slowly. A dynamic load is one which changes with time fairly quickly in comparison to the structure's natural frequency. If it changes slowly, the structure's response may be determined with static analysis, but if it varies quickly (relative to the structure's ability to respond), the response must be determined with a dynamic analysis. Dynamic analysis for simple structures can be carried out manually, but for complex structures finite element analysis can be used to calculate the mode shapes and frequencies. An opensource, lightweight, free software DYSSOLVE can be used to solve basic structural dynamics problems. Importance of vibration in engineering WHAT IS VIBRATION? Vibration can be defined as simply the cyclic or oscillating motion of a machine or machine component from its position of rest. Vibration is a repetitive, periodic, or oscillatory response of a mechanical system. The rate of the vibration cycles is termed “frequency.” Repetitive motions that are somewhat clean and regular, and that occur at relatively low frequencies, are commonly called oscillations, while any repetitive motion, even at high frequencies, with low amplitudes, and having irregular and random behavior falls into the general class of vibration. system and may be representative of its free and natural dynamic behavior. Also, vibrations may be forced onto a system through some form of excitation. The excitation forces may be either generated internally within the dynamic system, or transmitted to the system through an external source. When the frequency of the forcing excitation Forces Involved In any vibrating system, there are a total of four types of forces that need to be taken into account. These are listed here. Inertial Force Spring Force Damping Force Total External Force Inertial Force Inertia Force occurs because acceleration is present. The Spring force arises due to the elasticity of the material. It follows the Hooke's Law and is proportional to the displacement. While these forces are enough to set up harmonic vibrations in a system, most systems also have intrinsic damping even if an external damper is not used. The damping force tends to oppose motion and acts against the velocity in most cases (the exception is negative damping). It generally varies with some power of the velocity though the most useful is viscous damping which varies linearly with velocity. The total External force is the remaining force that acts on the system. It is the force that causes the excitation in the first place and may or may not be present while the system vibrates. coincides with that of the natural motion, the system will respond more vigorously with increased amplitude. This condition is known as resonance, and the associated frequency is called the resonant frequency. There are “good vibrations,” which serve a useful purpose. Also, there are “bad vibrations,” which can be unpleasant or harmful. For many engineering systems, operation at resonance would be undesirable and could be destructive. Suppression or elimination of bad vibrations and generation of desired forms and levels of good vibration are general goals of vibration engineering. Applications of vibration are found in many branches of engineering such as aeronautical and aerospace, civil, manufacturing, mechanical, and even electrical. Usually, an analytical or computer model is needed to analyze the vibration in an engineering system. Models are also useful in the process of design and development of an engineering system for good performance with respect to vibrations. Vibration monitoring, testing, and experimentation are important as well in the design, implementation, maintenance, and repair of engineering systems. Types of vibration There are two general classes of vibrations - free and forced. Free vibration takes place when a system oscillates under the action of forces inherent in the system itself, and when external impressed forces are absent. The system under free vibration will vibrate at one or more of its natural frequencies, which are properties of the dynamic system established by its mass and stiffness distribution. Vibration that takes place under the excitation of external forces is called forced vibration. When the excitation is oscillatory, the system is forced to vibrate at the excitation frequency. If the frequency of excitation coincides with one of the natural frequencies of the system, a condition of resonance is encountered, and dangerously large oscillations may result. The failure of major structures such as bridges, buildings, or airplane wings is an awesome possibility under resonance. Thus, the calculation of the natural frequencies of major importance in the study of vibrations. Vibrating systems are all subject to damping to some degree because energy is dissipated by friction and other resistances. If the damping is small, it has very little influence on the natural frequencies of the system, and hence the calculation for the natural frequencies are generally made on the basis of no damping. On the other hand, damping is of great importance in limiting the amplitude of oscillation at resonance. The number of independent coordinates required to describe the motion of a system is called degrees of freedom of the system. Thus, a free particle undergoing general motion in space will have three degrees of freedom, and a rigid body will have six degrees of freedom, i.e., three components of position and three angles defining its orientation. Furthermore, a continuous elastic body will require an infinite number of coordinates (three for each point on the body) to describe its motion; hence, its degrees of freedom must be infinite. However, in many cases, parts of such bodies may be assumed to be rigid, and the system may be considered to be dynamically equivalent to one having finite degrees of freedom. In fact, a surprisingly large number of vibration problems can be treated with sufficient accuracy by reducing the system to one having a few degrees of freedom. Free vibration occurs when a mechanical system is set off with an initial input and then allowed to vibrate freely. Examples of this type of vibration are pulling a child back on a swing and then letting go or hitting a tuning fork and letting it ring. The mechanical system will then vibrate at one or more of its "natural frequency" and damp down to zero. Forced vibration is when an alternating force or motion is applied to a mechanical system. Examples of this type of vibration include a shaking washing machine due to an imbalance, transportation vibration (caused by truck engine, springs, road, etc.), or the vibration of a building during an earthquake. In forced vibration the frequency of the vibration is the frequency of the force or motion applied, with order of magnitude being dependent on the actual mechanical system. Vibration analysis The fundamentals of vibration analysis can be understood by studying the simple mass–spring– damper model. Indeed, even a complex structure such as an automobile body can be modeled as a "summation" of simple mass–spring–damper models. The mass–spring–damper model is an example of a simple harmonic oscillator. The mathematics used to describe its behavior is identical to other simple harmonic oscillators such as the RLC circuit. Note: In this article the step by step mathematical derivations will not be included, but will focus on the major equations and concepts in vibration analysis. Please refer to the references at the end of the article for detailed derivations. Free vibration without damping To start the investigation of the mass–spring–damper we will assume the damping is negligible and that there is no external force applied to the mass (i.e. free vibration). The force applied to the mass by the spring is proportional to the amount the spring is stretched "x" (we will assume the spring is already compressed due to the weight of the mass). The proportionality constant, k, is the stiffness of the spring and has units of force/distance (e.g. lbf/in or N/m). The negative sign indicates that the force is always opposing the motion of the mass attached to it: The force generated by the mass is proportional to the acceleration of the mass as given by Newton’s second law of motion : The sum of the forces on the mass then generates this ordinary differential equation: Simple harmonic motion of the mass–spring system If we assume that we start the system to vibrate by stretching the spring by the distance of A and letting go, the solution to the above equation that describes the motion of mass is: This solution says that it will oscillate with simple harmonic motion that has an amplitude of A and a frequency of fn. The number fn is one of the most important quantities in vibration analysis and is called the undamped natural frequency. For the simple mass–spring system, fn is defined as: Note: Angular frequency ω (ω=2 π f) with the units of radians per second is often used in equations because it simplifies the equations, but is normally converted to “standard” frequency (units of Hz or equivalently cycles per second) when stating the frequency of a system. If you know the mass and stiffness of the system you can determine the frequency at which the system will vibrate once it is set in motion by an initial disturbance using the above stated formula. Every vibrating system has one or more natural frequencies that it will vibrate at once it is disturbed. This simple relation can be used to understand in general what will happen to a more complex system once we add mass or stiffness. For example, the above formula explains why when a car or truck is fully loaded the suspension will feel ″softer″ than unloaded because the mass has increased and therefore reduced the natural frequency of the system. What causes the system to vibrate: from conservation of energy point of view Vibrational motion could be understood in terms of conservation of energy. In the above example we have extended the spring by a value of x and therefore have stored some potential energy (\tfrac {1}{2} k x^2) in the spring. Once we let go of the spring, the spring tries to return to its un-stretched state (which is the minimum potential energy state) and in the process accelerates the mass. At the point where the spring has reached its un-stretched state all the potential energy that we supplied by stretching it has been transformed into kinetic. The mass then begins to decelerate because it is now compressing the spring and in the process transferring the kinetic energy back to its potential. Thus oscillation of the spring amounts to the transferring back and forth of the kinetic energy into potential energy. In our simple model the mass will continue to oscillate forever at the same magnitude, but in a real system there is always something called damping that dissipates the energy, eventually bringing it to rest. Free vibration with damping Mass Spring Damper Model We now add a "viscous" damper to the model that outputs a force that is proportional to the velocity of the mass. The damping is called viscous because it models the effects of an object within a fluid. The proportionality constant c is called the damping coefficient and has units of Force over velocity (lbf s/ in or N s/m). By summing the forces on the mass we get the following ordinary differential equation: The solution to this equation depends on the amount of damping. If the damping is small enough the system will still vibrate, but eventually, over time, will stop vibrating. This case is called underdamping – this case is of most interest in vibration analysis. If we increase the damping just to the point where the system no longer oscillates we reach the point of critical damping (if the damping is increased past critical damping the system is called overdamped). The value that the damping coefficient needs to reach for critical damping in the mass spring damper model is: To characterize the amount of damping in a system a ratio called the damping ratio (also known as damping factor and % critical damping) is used. This damping ratio is just a ratio of the actual damping over the amount of damping required to reach critical damping. The formula for the damping ratio ( ) of the mass spring damper model is: Damped and undamped natural frequencies The major points to note from the solution are the exponential term and the cosine function. The exponential term defines how quickly the system “damps” down – the larger the damping ratio, the quicker it damps to zero. The cosine function is the oscillating portion of the solution, but the frequency of the oscillations is different from the undamped case. The frequency in this case is called the "damped natural frequency", and is related to the undamped natural frequency by the following formula: The damped natural frequency is less than the undamped natural frequency, but for many practical cases the damping ratio is relatively small and hence the difference is negligible. Therefore the damped and undamped description are often dropped when stating the natural frequency (e.g. with 0.1 damping ratio, the damped natural frequency is only 1% less than the undamped). The plots to the side present how 0.1 and 0.3 damping ratios effect how the system will “ring” down over time. What is often done in practice is to experimentally measure the free vibration after an impact (for example by a hammer) and then determine the natural frequency of the system by measuring the rate of oscillation as well as the damping ratio by measuring the rate of decay. The natural frequency and damping ratio are not only important in free vibration, but also characterize how a system will behave under forced vibration. Harmonic Motion Oscillatory motion may repeat itself regularly, as in the balance wheel of a watch, or display considerable irregularity, as in earthquakes. When the motion is repeated in equal intervals of time T, it is called period motion. The repetition time t is called the period of the oscillation, and its reciprocal, ,is called the frequency. If the motion is designated by the time function x(t), then any periodic motion must satisfy the relationship . Harmonic motion is often represented as the projection on a straight line of a point that is moving on a circle at constant speed, as shown in Fig. 1. With the angular speed of the line o-p designated by w , the displacement x can be written as (1) Harmonic Motion as a Projection of a Point Moving on a Circle The quantity w is generally measured in radians per second, and is referred to as the angular frequency. Because the motion repeats itself in 2p radians, we have the relationship (2) where t and f are the period and frequency of the harmonic motion, usually measured in seconds and cycles per second, respectively. The velocity and acceleration of harmonic motion can be simply determined by differentiation of Eq. 1. Using the dot notation for the derivative, we obtain (3) (4) Simple Harmonic Motion Generally free natural vibrations occur in elastic system when a body moves away from its rest position. The internal forces tend to move the body back to its rest position. The restoring forces are in proportion to the displacement. The acceleration of the body which is directly related to the force on the body is therefore always towards the rest position and is proportional to the displacement of the body from its rest position. The body moves with simple harmonic motion... Simple harmonic motion is most conveniently shown as the projection on the vertical (x) axis of a point rotating in a circular motion (radius a) at a constant angular velocity ω. The tangential velocity of the point = ω a. The acceleration of the rotating point toward the centre of the circle..= ω 2. a .. Free Natural Vibrations D'Alembert's principle D'Alembert's principle, also known as the Lagrange–d'Alembert principle, is a statement of the fundamental classical laws of motion. It is named after its discoverer, the French physicist and mathematician Jean le Rond d'Alembert. The principle states that the sum of the differences between the forces acting on a system of mass particles and the time derivatives of the momenta of the system itself along any virtual displacement consistent with the constraints of the system, is zero. Thus, in symbols d'Alembert's principle is written as following, where is an integer used to indicate (via subscript) a variable corresponding to a particular particle in the system, is the total applied force (excluding constraint forces) on the -th particle, is the mass of the -th particle, is the acceleration of the -th particle, together as product represents the time derivative of the momentum of the -th particle, and is the virtual displacement of the -th particle, consistent with the constraints. It is the dynamic analogue to the principle of virtual work for applied forces in a static system and in fact is more general than Hamilton's principle, avoiding restriction to holonomic systems. A holonomic constraint depends only on the coordinates and time. It does not depend on the velocities. If the negative terms in accelerations are recognized as inertial forces, the statement of d'Alembert's principle becomes The total virtual work of the impressed forces plus the inertial forces vanishes for reversible displacements.[2] This above equation is often called d'Alembert's principle, but it was first written in this variational form by Joseph Louis Lagrange.D'Alembert's contribution was to demonstrate that in the totality of a dynamic system the forces of constraint vanish. That is to say that the generalized forces need not include constraint forces. Single Degree of Freedom System The simplest vibratory system can be described by a single mass connected to a spring (and possibly a dashpot). The mass is allowed to travel only along the spring elongation direction. Such systems are called Single Degree-of-Freedom (SDOF) systems and are shown in the following figure, Equation of Motion for SDOF Systems SDOF vibration can be analyzed by Newton's second law of motion, F = m*a. The analysis can be easily visualized with the aid of a free body diagram, The resulting equation of motion is a second order, non-homogeneous, ordinary differential equation: with the initial conditions, Equation is a linear, time invariant, second order differential equation. Series Combination A typical spring mass system having springs in series combination is shown above. The two springs can be replaced by a equivalent spring having equivalent stiffness equal to k as shown in the figure Springs in series When springs are in series, they experience the same force but under go different deflections. For the two systems to be equivalent, the total static deflection of the original and the equivalent system must be the same. Therefore if the springs are in series combination, the equivalent stiffness is equal to the reciprocal of sum of the reciprocal stiffnesses of individual springs.As an application example, consider the vertical bounce (up-down) motion of a passenger car on a road. Considering one wheel assembly we can develop what is known as a quarter car model as shown in fig. For typical passenger cars, the tyre stiffness is of the order of 200,000N/m while the suspension stiffness is of the order of 20,000N/m. Also, the vehicle mass per wheel (sprung mass) can be taken to be of the order of 250kg while the un-sprung mass (i.e., mass of wheel, axle etc not supported by suspension springs) is less than 50kg. Reciprocal of tyre stiffness is negligible compared to reciprocal of suspension stiffness or in other words, the tyre is very rigid compared to the soft suspension. Hence an equivalent one d.o.f. model can be taken to be as shown in Fig. Fig Quater Car model Fig Spring mass system