Vibration Of Single Degree Of
Vibration of single degree offreedom systemsyAssoc. Prof. Dr. Pelin Gundes Bakirgundesbakir@yahoo.com
Course Schedule A shorth reviewi on theh ddynamici bbehaviourh ioff theh singlei l ddegree off ffreedomdsystems A short review on the dynamic behaviour of multi‐degree of freedom structures Objectives for vibration monitoring Fourier Series Expansion, Fourier Transforms, Discrete Fourier Transform Digital signal processing, problems associated with analog‐to‐digital conversion, sampling,aliasing,lleakage,l kwindowing,dfiltersfl Steps in instrumenting a structure, selection and installation of instruments,maintenance,vibrationib ti iinstrumentation,tt on,ifi tiddatat acquisitioni itisystems, strong‐motion data processing
Course Schedule RandomRdvariables,i bl stochastich i processes, statisticali i l analysis,l i correlationl i andd convolution,l icoherence, time and frequency domain representation of random dynamic loads Dynamic response of single and multi degree of freedom systems to random loads Modal analysis Applications in bridges, buildings, mechanical engineering and aerospace structures MATLAB exercises Term Projects
ReferencesModalMd l analysis:l i Heylen W., Lammens S. And Sas P., ‘Modal Analysis Theory and Testing’, KatholiekeUniversiteit Leuven, 1997. Ewins DD.J.,J ‘Modal TestingTesting, TheoryTheory, PracticePractice, and Application’ (Mechanical EngineeringResearch Studies Engineering Design Series), Research Studies Pre; 2 edition (August 2001)ISBN‐13: 978‐0863802188 Maia, N. M. M. and Silva, J. M. M.Theoretical and Experimental Modal Analysis, ResearchStudies Press Ltd,, Hertfordshire, 1997, 488 pp.,ISBN 0863802087Signal processing: Blackburn,, James A,, Modern instrumentation ffor scientists and engineers,g, New York :Springer, 2001 Stearns S. D. and David, R. A., Signal Processing Algorithms in Matlab, Prentice‐Hall Inc, 1996 Mitra S.K., ‘Digital Signal Processing’, A Computer based approach, Mc‐Graw Hill, 3rd Edition,2006. Heylen W., Lammens S. And Sas P., ‘Modal Analysis Theory and Testing’, KatholiekeUniversiteit Leuven, 1997. Keith Worden ‘Signal Processing and Instrumentation’, Lecture keith/mec409.htm
ReferencesSignal processing: Lynn, P. A. Introductory Digital Signal processing With Computer Applications. John Wiley &Sons, 1994. Stearns D. D. and David, R. A., Signal Processing Algorithms in Matlab, Prentice‐Hall Inc, 1996 Ifeachor E.C. and Jervis B.W. Digital Signal Processing: A Practical Approach, Addison‐Wesley,1997General vibration theory Rao S.S., ‘Mechanical vibrations’, Pearson, Prentice Hall, 2004. Inman D.J., ‘Engineering Vibration’, Prentice Hall, 1994. Meirovitch L., ‘Fundamentals of vibrations’, McGrawHill, 2001.
ReferencesR dRandomvibrations:ib i Bendat J.S. and Piersol A.G., ‘Random data analysis and measurement procedures’, Wiley Series inProbabilityy and Statistics,, 3rd Edition,, 2004. Lutes L.D. and Sarkani S., ‘Random Vibrations: Analysis of structural and mechanical systems’, Elsevier, 631pp, 2004. Newland D.E., ’An introduction to random vibrations, spectral and wavelet analysis’, Longman,1975/1984/1993. Soong T.T. and Grigoriu, ‘Random vibration of mechanical and structural systems’, Prentice Hall, 1993. Wirsching P.H. and Paez T.L. and Ortiz K. ‘Random vibrations: Theory and Practice’, John Wiley and sons,1995.199 Bendat J.S. and Piersol A.G. ‘Engineering applications of correlation and spectral analysis’, John Wiley andSons,, 2nd Edition,, 1993.
ReferencesVib i Instrumentation:VibrationIi Vibration, monitoring, testing and instrumentation handbook, CRC Press, Taylor and Francis,Edited byby: Clarence WW. De SilvaSilva. Aszkler C., ‘Acceleration, shock and vibration sensors’, Sensors handbook, Chapter 5, pages137 159137‐159. McConnell K.G., ‘Vibration testing, theory and practice’, John Wiley and Sons, 1995. Prerequisites: Basic knowledge on structural analysis.
Vibration based health monitoringB i Information:BasicI fti Instructor: Assoc. Prof. Dr. Pelin Gundes Bakir(http://atlas.cc.itu.edu.tr/ gundes Email: gundesbakir@yahoo.com Office hours TBD by email appointment Website:W b ithttp://atlas.cc.itu.edu.tr/ gundes/lectures Lecture time: Wednesday 14.0014 00‐1717.0000 Lecture venue: NH 404
Vibration based health monitoring‘Vibration based strstructuralct ral health monitoring’ isi a multidisciplinaryltidi i liresearch topic. The course is suitable both for undergraduate andgraduate students as well as the following departments: Civil engineering Earthquake engineering Mechanical engineering Aerospace engineering Electrical and electronic engineering
Vibration based health monitoringBasic Information: 70 % attendance is required.Grading: Quiz homeworks: 35% Mid‐term project:25% Final project:40%
Introduction ConceptsConcepts from vibrations Degrees of freedom Classification of vibration
Concepts from vibrationsNEWTON’S’ LAWSFirst law:If there are no forces acting upon a particle, then the particle will move ina straight line with constant velocity.Second law:A particle acted upon by a force moves so that the force vector is equal tothe time rate of change of the linear momentum vector.Third law:When two particles exert forces upon one another, the forces lie along theline joining the particles and the corresponding force vectors are thenegative of each other.
Definition Theh minimum numberb off independentddcoordinatesdrequiredd to ddeterminecompletely the positions of all parts of a system at any instant of timedefines the degree of freedom of the system. A single degree of freedomsystem requires only one coordinate to describe its position at any instantof time.
Single degree of freedom system FFor theth simplei l pendulumd lini ththe fifigure, theth motionti can beb statedt t d eitherith ini termstoff θ or x andd y. If ththecoordinates x and y are used to describe the motion, it must be recognized that these coordinates are notindependent. They are related to each other through the relationx2 y2 l 2where l is the constant length of the pendulum. Thus any one coordinate can describe the motion of thependulum. In this example, we find that the choice of θ as the independent coordinate will be moreconvenient than the choice of x and y.
Two degree of freedom system SSome examplesl off ttwo ddegree off freedomf dsystemstare shownhiin ththe fifigure. ThThe fifirstt fifigure showsha ttwomass – two spring system that is described by two linear coordinates x1 and x2. The second figure denotesa two rotor system whose motion can be specified in terms of θ1 and θ2. The motion of the system in thethird figure can be described completely either by X and θ or by x,y and X.
Discrete and continuous systems A largelnumberb off practicalti l systemstcan beb ddescribedib d usingi a finitefi it numberb off ddegrees off ffreedom,dsuchh asthe simple system shown in the previous slides. Some systems, especially those involving continuous elastic members, have an infinite number of degreesof freedom as shown in the figure. Since the beam in the figure has an infinite number of mass points, weneed an infinite number of coordinates to specify its deflected configuration. The infinite number ofcoordinates defines its elastic deflection curve. Thus, the cantilever beam has infinite number of degreesof freedom.
Discrete and continuous systems SSystemstwithith a fifiniteit numberb off ddegrees off ffreedomdare calledll d discretedit orlumped parameter systems, and those with an infinite number of degreesof freedom are called continuous or distributed systems. Most of the time, continuous systems are approximated as discretesystems, and solutions are obtained in a simple manner. Althoughtreatment of a system as continuous gives exact results, the analyticalmethods available for dealing with continuous systems are limited to anarrow selection of problems, such as uniform beams, slender rods andthinhi plates.l Hence,, most of the ppractical systemsyare studied byy treatingg them as finitelumped masses, springs and dampers. In general, more accurate resultsare obtained by increasing the number of masses, springs and dampers‐that is by increasing the number of degrees of freedom.
Classification of vibration Free vibration:ibration If a system,tafterft an iinitialiti l disturbancedi t biis lleftft ttovibrate on its own, the ensuing vibration is known as free vibration.No external force acts on the system. The oscillation of a simplependulum is an example of free vibration.vibration Forced vibration: If a system is subjected to an external force (oftena repeatingti typetoff force),f) theth resultinglti vibrationib ti isi knownkas forcedfdvibration.– If the frequency of the external force coincides with one of the naturalfrequencies of the system, a condition known as resonance occurs,and the system undergoes dangerously large oscillations. Failures ofsuch structures as buildingsbuildings, bridgesbridges, turbinesturbines, and airplane wingshave been assoicated with then occurrence of resonance.
Classification of vibration Undampeddd vibration:ib iIff no energy is lostl or ddissipatedd in ffriction or otherhresistance during oscillation, the vibration is known as undamped vibration.If any energy is lost in this way however, it is called damped vibration.While the springp g forms a physicalp ymodel for storingg kinetic energygy and hencecausing vibration, the dashpot, or damper, forms the physical model fordissipating energy and damping the response of a mechanical system. Adashpotp consists of a pistonpfit into a cylinderyfilled with oil. This pistonpisperforated with holes so that motion of the piston in the oil is possible. Thelaminar flow of the oil through the perforations as the piston moves causes adampingp g force on the piston.p
Classification of vibration Linear vibration:Liib tiIf allll ththe bbasici componentstof a vibratory system‐the spring, the mass,and the damper, behave linearly, theresulting vibration is known as linearvibration. The differential equations thatgovern the behaviour of vibratory linearyare linear. Therefore,, the principlepp ofsystemssuperposition holds. Nonlinear vibration: If however, any of thebasic components behave nonlinearly, thevibration is called ‘nonlinear vibration’. Thedifferential equations that govern thebehaviour of vibratory non‐linear systemsare non‐linear. Therefore, the principle ofsuperposition does not hold.
Classification of vibrationLinear andLid nonlinearlivibrationsib ticontd:td The nature of the spring force can bededuced by performing a simple staticexperiment With no mass attachedexperiment.attached, thespring stretches to a position labeled as xo 0in the figure. As successively more mass is attached to thespring, the force of gravity causes the springto stretch further. If the value of the mass isrecorded, along with the value of thedisplacement of the end of the spring eachtime more mass is added, the plot of theforce (mass denoted by m, times theacceleration due to gravity, denoted by g),versus this displacement denoted by x, yieldsa curve similar to that shown in the figure.
Classification of vibrationLinear andd nonlinearlvibrationsbcontd:d Note that in the region of values for xbetween 0 and about 20 mm,, the curve isa straight line. This indicates that fordeflections less than 20 mm and forcesless than 1000 N,N the force that is appliedby the spring to the mass is proportionalto the stretch of the spring. The constant of proportionality is the slopeof the straightg line.
Classification of vibration Deterministici i i vibration:ib iIff theh valuel or magnitudei d off theh excitationi i (f(force ormotion) acting on a vibratory system is known at any given time, the excitation iscalled ‘deterministic’. The resulting vibration is known as ‘deterministic vibration’. Nondeterministic vibration: In some cases, the excitation is non‐deterministic orrandom; the value of excitation at a given time cannot be predicted. In thesecases, a large collection of records of the excitation may exhibit some statisticalregularity. It is possible to estimate averages such as the mean and mean squarevalues of the excitation.
Classification of vibration Examplesl off randomdexcitations are windd velocity,lroadd roughness,handdground motion during earthquakes.If the excitation is random,, the resultingg vibration is called randomvibration. In the case of random vibration, the vibratory response of thesystem is also random: it can be described only in terms of statisticalquantities.quantities
Mathematical background HomogeneousHomogeneous linear ODEs withconstant coefficients NonhomogeneousN hODEsODE
Introduction ThThe dynamicdi behaviourb h ioff mechanicalh i l systemstiis ddescribedib d bby whath twe call second order Ordinary Differential Equations. The input to the mechanical structure appears on the right handside of the equation and is the Force and the solution of theequation gives the output which is usually the displacement. In order to be able solve these equations, it is imperative to have asolid backgroundgon the solution of homogeneousgandnonhomogeneous Ordinary Differential Equations. Homogeneous Ordinary Differential Equations represent the ‘FreeFreeVibrations’ and the non‐homogeneous Ordinary DifferentialEquations represent ‘Forced Vibrations’.
Homogeneous linear ODEs withconstant coefficients We shallh ll now considerd second‐orderd d hhomogeneous llinear ODEs whosehcoefficients a and b are constant.y′′ ay′ byy 0 The solution of a first order linear ODE:y′ ky 0 By separating variables and integrating, we obtain:dy kdxyln y kdx c *Taking exponents on both sides:y ( x) ce kdx ce kxLet’s tryy the above solution in the first equation.qUsingg a constantcoefficient k: λxy eSubstitutiSubsu ngg itss dederivativev ves : y′ λe λx(λ2 aλ b)e λx 0andd y′′ λ2 e λx
Homogeneous linear ODEs withconstant coefficients Hence iff λ is a solutionloff theh important characteristichequation ((orauxiliary equation)λ2 aλ b 0 Then the exponential solution y e λx is a solution of they′′ ay′ byy 0 Now from elementary algebra we recall that the roots of this quadraticequation are:1λ ( a a 2 4b)121λ2 ( a a 2 4b)2 The functions below are solutions to y′′ ay′ by 0y1 e λ1x andy 2 e λ2 x
Homogeneous linear ODEs withconstant coefficients From algebral b we knowkthath theh quadraticdequation belowb lmay havehthreehkinds of roots:λ2 aλ b 0 Case I: Two real roots if a 2 4b 0Case II: A real double root if a 2 4b 0Case III: Complex conjugate roots if a 2 4b 0 CASE I: In this casecase, a basis of solutions of y′′ ay′ by 0λxλ xin any interval is: y1 e and y2 ebecause y1 and y2 are defined and real for all x and their qquotient is notconstant. The corresponding general solution is:12y c1e λ1x c2 e λ2 x
Homogeneous linear ODEs withconstant coefficientsCASE II: Reall doubled bl root λ ‐a/2λ / If the discriminant a 2 4b is zero, we see from121λ2 ( a a 2 4b)2λ1 ( a a 2 4b)that we get only one root:λ λ1 λ2 a / 2,hence only one solution :y1 e ( a / 2) xTTo obtainb i a secondd independenti ddsolutionl i y2 neededd d forf a basis,b i we use thehmethod of order of reduction. Settingy2 uy1 , Substituting this and its derivatives y 2 ' u ' y1 uy1 ' and y2 ' ' intoy′′ ay′ by 0
Homogeneous linear ODEs withconstant coefficientsCASE II: Reall doubled bl root λ ‐a/2λ / We have : (u′′y1 2u′y1′ uy1′′) a(u′y1 uy1′ ) buy1 0 Collecting terms u′′y u′(2 y′ ay ) u( y′′ ay′ by ) 0 This expression in the last paranthesis is zero, since y1 is a solution of111111y′′ ay′ by 0 2 y′ aeThe expression in the second paranthesis is zero too sinceWe are thus left with1u ′′y1 0Henceu ′′ 0By two integrationsu c1 x c2 ax / 2 ay1
Homogeneous linear ODEs withconstant coefficientsCASE II: Reall doubled bl root λ ‐a/2λ /To get a second independent solution y2 uy1, we can simply choose c1 1and c2 0 and take u x. Then y2 xyy1. Since these solutions are notproportional, they form a basis. Hence in the case of a double root ofλ2 aλ b 0a basisb i off solutionsl ioffon any interval is: y′′ ay′ byb 0e ax / 2 , xe ax / 2The corresponding general solution is: y (c1 c2 x)e ax / 2
Homogeneous linear ODEs withconstant coefficientsCASE III: Complex roots –a/2 iωa/2 iω and –a/2‐iωa/2 iω This case occurs if the discriminant of the characteristic equationλ2 aλ b 0is negative. In this case, the roots of the above equation and thus the solutions of the ODEcome at first out complex. However, we show that from them we can obtain a basis of real solutions:y′′ ay′ by 0y1 e ax / 2 cos ω x ,wherey 2 e ax / 2 sini ωx14ω 2 b a2 This is proved in the next slides. It can be verified by substitution that these are solutions in the presentcase. They form a basis on any interval since their quotient cotωx is not constant. Hence, a real generalsolutionl ti iin CCase III iis:y e ax / 2 ( A cos ω x B sin ω x )( A , B arbitrary )
Homogeneous linear ODEs withconstant coefficients: Proof Complexl numberb representation off hharmonic motion: SincerrX OPthis vector can be represented as a complex number:rX a ibrwhere i 1 and a and b denote x and y components of X . Componentsa and b are also called the real and the imaginary parts of the vector X. If Adenotes the modulus or the absolute value of the vector X, and θ denotesrthe argumentgof the angleg between the vector and the x‐axis,, then X canalso be expressed as:
Homogeneous linear ODEs with constant coefficients:P fProofComplex number representation of harmonic motionrrX OPrX a ibrX
Homogeneous linear ODEs withconstant coefficients: Proof A apparentt we haveAshtwot complexl roots.t TheseThare:1212λ1 a iω and λ1 a iω We know from basic mathematics that a complex exponential function canbe expressed as:r itr itre e e e (cos t i sin t )Thus the roots of the second order Ordinary Differential Equation can bee λ x e ( a / 2) x iωx e ( a / 2) x (cos ωx i sin ωx)expressed as:1e λ2 x e ( a / 2 ) x iωx e ( a / 2) x (cos ωx i sin ωx) We now add these two lines and multiply the result by ½. This gives:y1 e ax / 2 cos ω xThen we subtract the second line from the first and multiply the result by1/2i. This gives:y 2 e ax / 2 sin ω x
Homogeneous linear ODEs withconstant coefficientsCaseRootsIDistinct realBasisGeneral solutione λ1x , e λ2 xy c1e λ1x c2 e λ2 xλ1, λ2IIReal double rootλ ‐a/2IIIe ax / 2, xe ax / 2y (c1 c2 x)e ax / 2Complexconjugate121λ2 a i ω2λ1 a iωy1 e ax / 2 cos ωxy2 e ax / 2 sin ωxy e ax / 2 ( A cos ω x B sin ω x )
Nonhomogeneous ODEs In thish section, we proceedd fromfhomogeneoushto nonhomogeneoushODEs.′′′y p ( x) y q ( x) y r ( x) The general solution consists of two parts:y ( x) yh ( x) y p ( x)where yh c1 y1 c2 y2 is a general solution of the homogeneous ODE.ODETerm in r(x)Choice for yp(x)keγxCeγxkx n (n 0,1,2,.)K n x n K n 1 x n 1 . K1 x K 0k cos ωxk sin ωxkeαx cos ωxkeαx sin ωxK cos ω x M sin ω xeαx ( K cos ωx M sin ωx)
Nonhomogeneous ODEsChoice rules for the method of undetermined coefficientsa)Basic rule: If r(x) is one of the functions in the first column in the Table,choose yp in the same line and determine its undetermined coefficientsby substituting yp and its derivatives intoy′′ p ( x) y′ q ( x) y r ( x)b) Modification rule: If a term in your choice for yp happens to be a solutionof the homogeneous ODE corresponding to the above equation, multiplyyyour choice of yp byy x ((or x 2 if this solution corres
Digital signal processing, problems associated with analog‐to‐digital conversion, sampling, aliasing, lkleakage, windowing, flfilters Steps in instrumenting a structure, selection and installation of instruments,maintenance,
The vibration analysis in Section 5 provides the FE-based vibration analysis procedure based on first principles direct calculations. The FE-based vibration analysis is recommended to evaluate the design during the detail design stage. If found necessary, the local vibration is tobe addressed in the detail vibration analysis. The
THEORY OF SHELLS VIBRATION AND STABILITY Course Outline PART 1 Waves and vibration I. Introduction 1. Plane waves a. Vibration b. Wave propagation 2. Beam with and without an elastic foundation 3. Ring, extensional and inextensional vibration II. Circular Plate 1. Waves and vibration in rectangular coordinates 2.
vibration of single degree of freedom dynamic systems. For a more comprehensive study see Engineering Vibration Analysis with Application to Control Systems by C. F. Beards (Edward Arnold, 1995). It should be noted that many of the techniques developed in single degree of freedom
vibration. Today, random vibration is thought of as the random motion of a structure excited by a random input. The mathematical theory of random vibration is essential to the realistic modeling of structural dynamic systems. This article summarizes the work of some key contributors to the theory of random vibration from
PRÜFTECHNIK AG Vibration Handbook 5 Introduction to vibration measurement The fundamental physical source of this vibration and of the recorded signal could be thought of, for example, as follows: The vibration velocity v is the change in displacement x over a period of time. Its waveform is shifted by 90 from the position
The dynamic vibration tests on structures are generally subdivided into two groups: (a) forced vibration test and (b) ambient vibration test. In the forced vibration test, the structure is usually excited by artificial means. The excita
The ISO 2631 suggests vibration measure-ments in the three translational axes on the seat pan, but only the axis with the greatest vibration is used to estimate vibration severity.5 The current trend in vibration research is to use multi-axis val-ues.
1 Fluke Corporation 3562 Screening Vibration Sensor FAQ Fluke 3562 Screening Vibration Sensor Frequently asked questions Q: Why should I use a vibration screening sensor and what is the benefit? A: Vibration anomalies are some of the first indications of misalignment, looseness, bearing wear
