Technology & Engineering
Forced Vibration
Forced vibration refers to the oscillation of a system under the influence of an external force or excitation. This force can be periodic, such as a harmonic force, or non-periodic, and it causes the system to vibrate at a frequency determined by the force. Forced vibration is a key concept in understanding the behavior of structures and mechanical systems under external influences.
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8 Key excerpts on "Forced Vibration"
eBook - PDFVibration
Fundamentals and Practice, Second Edition
- Clarence W. de Silva(Author)
- 2006(Publication Date)
- CRC Press(Publisher)
1 Vibration Engineering 1.1 Introduction Vibration is the repetitive, periodic, or oscillatory response of a mechanical system. The rate of vibration cycles is termed ‘‘frequency.’’ Repetitive motions that are without aberrations and are regular and occur at relatively low frequencies are commonly called ‘‘oscillations’’ whereas any repetitive motion, even at high frequencies, with low amplitudes and having irregular and random behavior falls into the general class of vibration. Nevertheless, the terms vibration and oscillation are often used interchange-ably, as is done in this book. Vibrations can occur naturally in an engineering systems and will be representative of their free and natural dynamic behavior. Vibrations also may be forced onto a system through some form of excitation. The excitation forces may be either generated internally within the dynamic system or imparted on the system through an external source. When the frequency of the forcing excitation 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.’’ Vibrations can be ‘‘good’’ or ‘‘bad,’’ the former serving a useful purpose and the latter having unpleasant or harmful effects. For many engineering systems, operation at a resonance would be undesirable and could be destructive. It is important to study human’s responses to vibrations. Suppression or elimination of bad vibrations and generation of desired forms and levels of good vibration are the general goals of vibration engineering. This book deals with the analysis , observation , and modification of vibrations in engineer-ing systems. Applications of vibration are found in many branches of engineering such as aeronautics and aerospace, civil, manufacturing, mechanical, mechatronics, and even electrical and electronics.
eBook - PDFDesign Engineer's Reference Guide
Mathematics, Mechanics, and Thermodynamics
- Keith L. Richards(Author)
- 2014(Publication Date)
- CRC Press(Publisher)
89 Mechanical Vibrations 6.1 INTRODUCTION Mechanical vibrations are defined as oscillations in mechanical dynamic systems and are the motions of a particle or a body or system of connected bodies that have been displaced from a posi-tion of equilibrium. The majority of vibrations are undesirable in machines or structures as they can result in increased stresses, causing increased wear such as fretting and increased bearing loads. Mechanical fatigue can also result from vibrations and rotating machine parts including aero engine parts and will need careful balancing in order to prevent any damage resulting from vibrations. Although most vibration problems are undesirable, such as the Tacoma Narrow Bridge failure in the United States in the 1940, and innumerable airframe failures resulting from vibration-induced fatigue, some mechanical systems such as the Beal free-piston Stirling engine rely on the vibration characteristics of the system to function correctly. In the mining and quarrying industries, these rely on sifting different sized particles using vibrating screed beds. In the manufacturing industry, vibra-tion conveyors are used to convey components from one machining process to another. Vibrations can be classified into four basic categories: 1. Free 2. Forced 3. Self-excited 4. Random Free vibration of a system is vibration that occurs in the absence of any external force. External force acting on a system will cause Forced Vibrations; in this instance, the exciting force is continuously supplying energy to the system. These types of vibrations may be either determin-istic or random (see Figure 6.1). Self-excited vibrations are periodic and deterministic oscillations. Under certain conditions, the equilibrium state becomes unstable and any disturbances will cause the perturbations to grow until some effect limits any further growth.
eBook - PDF- Frank Fahy, David Thompson, Frank Fahy, David Thompson(Authors)
- 2015(Publication Date)
- CRC Press(Publisher)
77 Chapter 3 Fundamentals of vibration Brian Mace 3.1 INTRODUCTION Vibration of engineering structures is commonplace: for example, cars vibrate due to forces produced by the engine, or by road roughness; air-craft vibrate due to aerodynamic loads and turbulent boundary layers; and buildings vibrate due to forces produced by wind loading and earthquakes. We need to be able to model, analyse and design structures so that their vibrational behaviour is acceptable, and this requires us to understand the fundamentals behind the physics of vibration. Vibration is usually, but not always, a small-amplitude oscillatory motion about a static equilibrium position. Its effects are often unwanted: large displacements and stresses may occur, especially due to resonance, and these may lead to fatigue, breakage, wear or improper operation; structural vibration produces sound which radiates into the surrounding air (indeed, sound is also a form of vibration); vibration can produce physical discom-fort, motion sickness or physiological effects such as vibration white finger; instabilities can occur that lead to growing vibrations that often end in failure, such as flutter and galloping. However, not all vibration is undesir-able: music is one example. This chapter concerns the fundamental aspects of vibration. Emphasis will be placed on relatively simple systems (with one or two degrees of free-dom – see next section) and the fundamentals of their physical behaviour. In practice, vibration analysis often involves a finite element (FE) model of which the number of elements can be very large. Although this might be complicated, the underlying physics is the same – natural frequencies, reso-nance, modes and so on – and hence, understanding of the fundamentals is essential. 3.1.1 Some terminology and definitions The basic properties of a vibrating system are inertia (mass) and stiffness, which store kinetic and potential energy respectively, damping, which
eBook - PDFMechanical Vibrations
Active and Passive Control
- Tomasz Krysinski, François Malburet(Authors)
- 2010(Publication Date)
- Wiley-ISTE(Publisher)
xxii Mechanical Vibrations sufficiently important to cause destruction. When mechanical systems are exposed to increasing oscillations, they are referred to as unstable. Hence, it is important to study the stability of mechanical systems. The utilization field of modern mechanical systems is one in which the necessary stability margins are not enough and therefore active controls are needed. Among industrial examples, the helicopter represents one of the most complex systems in terms of sources of vibrations. This fact is the consequence of its architecture and operating mode. This system comprises many swiveling systems with very different speeds of rotation, hence the problems related to unbalance, connections, rotors, aerodynamic excitations, etc. On this type of structure, the excitations stresses are relatively important in relation to the mass of the structure (fuselage). Aeronautical structures are light and therefore flexible. Natural frequencies can be close to excitation frequencies, which may entail problems of vibration comfort and alternate constraints in the mechanical parts. The problems of dynamic optimization of the rotor and structure are very important. This optimization may require the introduction of insulating elements, such as suspensions, anti-vibrators or vibration control systems for the blades. These systems can be passive, self-adaptive or active. Some examples will be developed here. The authors wish to thank: – Eurocopter for being kind enough to allow them to use in this book the knowledge, experience and know-how developed by its employees, – the management of l’Ecole nationale superieure d’arts et metiers and la Societe d’etudes et recherches de l’Ecole nationale superieure d’arts et metiers for their help, – the teachers and students of l’Ecole nationale superieure d’arts et metiers of Aix-en-Provence, who were able to take part in some of these studies. PART I Sources of Vibrations This page intentionally left blank
eBook - ePubStructural Dynamics of Earthquake Engineering
Theory and Application Using Mathematica and Matlab
- S Rajasekaran(Author)
- 2009(Publication Date)
- Woodhead Publishing(Publisher)
4 Forced Vibration (harmonic force) of single-degree-of-freedom systems in relation to structural dynamics during earthquakes Abstract In this chapter, Forced Vibration of single-degree-of-freedom (SDOF) systems (both undamped and under-damped) due to harmonic force is considered. Governing equations are derived and the displacement response is determined using Wilson’s recurrence formula. Vibration excitation due to imbalance in rotating machines is discussed. Equations for transmissibility are derived for force and displacement isolation. The underlying principle of vibration-measuring instruments is illustrated. Key words resonance transient steady state magnification factor beating transmissibility seismometer accelerometer 4.1 Forced Vibration without damping In many important vibration problems encountered in engineering work, the exciting force is applied periodically during the motion. These are called Forced Vibrations. The most common periodic force is a harmonic force of time such as P = P 0 sin ω t 4.1 where P 0 is a constant, ω is the forcing frequency and t is the time. The motion is analysed using Fig. 4.1. m x ¨ + kx = P 0 sin ωt 4.2 4.1 Spring-mass system subjected to harmonic force. The general solution of Eq. 4.2 (non-homogeneous second order differential equation) consists of two parts x = x c + x p where x c = complementary solution, and x p = particular solution. The complementary solution is obtained by setting right hand side as zero. m x ¨ c + k x c = 0 4.3 x c = c 1 sin ω n t + c 2 cos ω n t 4.4 where ω n = k / m and c 1 and c 2 are arbitrary constants. Assume x p = A sin wt and. substituting m x ¨ p + k x p = P 0 sin ωt 4.5 − ω 2 mA + kA sin ωt = P 0 sin ωt 4.6 A = P 0 k 1 − ω 2 m / k = P 0 k 1 − ω 2 / ω n 2 4.7 Since β = ω / ω n, A = P 0 k 1 − β 2 = P 0 / k 1 − β 2 4.8 If[
eBook - PDFVibrational Mechanics: Nonlinear Dynamic Effects, General Approach, Applications
Nonlinear Dynamic Effects, General Approach, Applications
- Iliya I Blekhman(Author)
- 2000(Publication Date)
- World Scientific(Publisher)
It is only since the beginning of this century that a period of a rapid development of vibrational technology has started. °By vibration we mean here mechanical oscillations whose period is much shorter than that at which the motion of the system is being considered, and whose swing is far smaller than the characteristic size of the system. 4 Chapter 1 Introduction Without that technology, a number of most important industries, such as min-ing, the processing of natural resources, chemical technology, metallurgy, the manufacture of building materials, and the erecting of various constructions would have been absolutely unthinkable. 6 The diversity of trends in using vibration is reflected in the titles of sec-tions of the book [114]: Vibration shifts, Vibration transforms (in Vibrorhe-ology), Vibration separates and classifies, Vibration intensifies the pro-cesses and the treatment of workpieces, Vibration consolidates - vibration destroys, Vibration unites (self-synchronization of unbalanced rotors), Vi-bration maintains rotation - vibration retards rotation, Vibration cancels vibration - vibration intensifies vibration (the generalized principle of auto-balancing), Vibration helps in measuring - vibration hinders measuring, Vibration cures - vibration causes diseases. The title of a popular book by Goncharevich is also remarkable: Vibration as a nonstandard way [209]. The use of vibration made it possible to literally revolutionize many indus-tries, providing a great technical and economical effect. Potentialities, however, have not yet been exhausted. The application of vibrational technology seems to be most promising in the future. Apart from the book [114], there is an extensive literature in Russian (mono-graphs and reference books) devoted to general and special problems of the use of vibration in technology (see, e.g., [46, 84, 145, 192, 207, 208, 209, 215, 226, 259, 301, 302, 306, 315, 323, 364, 400, 402, 420, 417, 440, 443, 445, 454, 475, 550, 159, 227]).
- Robert D. Blevins(Author)
- 2016(Publication Date)
- Wiley(Publisher)
7 Forced Vibration This chapter provides the dynamic response to Forced Vibration. Steady-state response to sinusoidal excitation, generated by motors, pumps, and rotating machinery, is discussed in Section 7.1. Section 7.2 shows formulas for shock and transient response to time history loading. Vibration isolation is discussed in Section 7.3. Section 7.4 has random response to spectral loads. Section 7.5 has simplified methods for quick estimation of dynamic response. The natural frequencies and mode shapes of elastic structures that support these formulas are in Chapters 3, 4, and 5. 7.1 Steady-State Forced Vibration 7.1.1 Single-Degree-of-Freedom Spring–Mass Response Many practical systems consist of a massive component on an elastic support as shown in case 1 of Table 7.1 and Figure 7.1. The displacement of mass M at time t is x ( t ) . Newton’s second law (Eq. 2.4) applied to the spring–mass systems gives the equation of motion: M ̈ x + 2 M 𝜁𝜔 n ̇ x + kx = F ( t ) = F o cos 𝜔 t (7.1) F o cos ( 𝜔 t ) = external harmonic force with amplitude F o . 𝜔 = 2 𝜋 f = frequency in radians per second where f = forcing frequency f in Hertz, k = the spring stiffness (Table 3.2), and 𝜔 n = 2 𝜋 f n = ( k ∕ M ) 1 ∕ 2 = the circular natural frequency in radians per second (case 1 of Table 3.3). Overdot denotes derivative with respect to time. The middle term 2 M 𝜁𝜔 n dx ∕ dt on the left-hand side of Equation 7.1 is a linear viscous damping force that is proportional to velocity times the dimensionless damping factor 𝜁 . Steady-state response is at the forcing frequency. Displacement x ( t ) has amplitude X o and phase 𝜑 , relative to the harmonic force: x ( t ) = X o cos ( 𝜔 t − 𝜑 ) (7.2) Formulas for Dynamics, Acoustics and Vibration , First Edition. Robert D. Blevins. © 2016 John Wiley & Sons, Ltd. Published 2016 by John Wiley & Sons, Ltd.
- Giora Maymon(Author)
- 2008(Publication Date)
- Butterworth-Heinemann(Publisher)
C h a p t e r 1 / Some Basics of the Theory of Vibrations 1.1 A S INGLE D EGREE OF F REEDOM S YSTEM A good physical understanding of the behavior of vibrating structures can be achieved by analyzing the behavior of a single degree of freedom (SDOF) oscillator. The SDOF system is covered by an extremely large number of textbooks (e.g., refs. [1–12]), and is the basis for every academic course in vibration analysis in aerospace, mechanical, and civil engineering schools. It will be discussed briefly in the first chapters to create a common baseline for the analysis of the behavior of the cantilever beam described in most of the examples, and for any other continuous structure. The SDOF system can be excited either by a force (which is a function of time) acting on the mass, or by a forced movement of the support. The first type of excitation is usually called “force excitation” and the latter is called “base excitation.” These two major types of excitations (loadings) are basic to the structural response analyses of both the SDOF and the continuous elastic systems. The classical oscillator contains a point mass m (i.e., all the mass is con-centrated in one point), which is connected to a rigid support through two elements: a linear massless spring with a stiffness k and a viscous damper c (which creates a force proportional to the velocity), or a structural damper h (which creates a force proportional to the displacement and in 90 degrees phase lag behind it). The system can be excited either by a force f acting on the mass or by a base movement x s . The force-excited system is described in Figure 1.1. Note in the figure that two “elements” connect the mass to the 1 2 • Chapter 1 / Some Basics of the Theory of Vibrations k m f ( t ) x ( t ) c F IGURE 1.1 A force-excited SDOF.
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