1、机械振动 Fundamentals of Vibration Lecture 1 Ji Lin Section of Mechanical Design and Theory School of Mechanical Engineering Shandong University Vibration Problems 1. Vibrations are usually small, oscillatory motion about a static equilibrium position. 2. Most engineering structures vibrate. (Rotating m
2、achinery)Practical Examples Effects 3. Becoming lighter, faster, quieter, and more flexible are often more prone to vibrations. 4. Engineers need to be equipped with the knowledge required to tackle vibration problems encountered in industryto understand, model, analysis, design and treat General Ai
3、m To introduce students with little or no previous experience of mechanical vibrations, and with quite different backgrounds, to the basic concepts of vibrational behavior, to provide a general introduction to vibration modeling, analysis and control. Course Overview 1. Course Content 9 Sections 2.
4、Resources 1) Lecture notes (including examples and problems, mainly written by Prof. Mace, ISVR, Univ. of Soton) 2) Mechanical Vibrations by S.S. Rao, Addison Wesley Publishing. (Core text) 3)机械振动基础 , 胡海岩, 航空工业出版社 (Secondary text) 3. Credit Value: 2 points 4. Formal Contact Hours: 32 5. Assessment A
5、ssignments(80%) + Attendance(10%) + Others(10%) Introduction 1. Terminology Free/forced vibration; Damped/undamped system; Linear/non-linear system; Deterministic/random Vibration; Discrete/continuous system. 2. Basic Principles ( to find system equations) Newton Laws; Work-energy; Impulse momentum;
6、 Lagranges equation 3. Basic Concepts Degree of freedom; Simple harmonic motion; Complex exponential notation (C.E.N); Frequency response function (FRF) Fundamentals For vibration to occur we need ? mass ? stiffness k ? The other vibration quantity is damping c System vibrates about its equilibrium
7、position Ingredients of Vibration Mass store of kinetic energy Stiffness store of potential (strain) energy Damping: dissipates energy Force provide energy Vibro-acoustic Problems Interior Noise Effects of Vibration 1. Large displacements and stresses (esp. resonance) 2. Fatigue 3. Noise, sound 4. B
8、reakage, wear, improper operation 5. Physical discomfort, physiological effects 6. Instabilities (flutter, galloping) Free Vibration no external forces act System vibrates at its natural frequency Fundamentals -damping Mechanical Systems ? Systems maybe linear or nonlinear ? Linear Systems (idealiza
9、tion) 1 Output frequency = Input frequency 2 If the magnitude of the excitation is changed, the response will change by the same amount 3 Superposition applies (Non-linear systems are not considered in this course.) Mechanical Systems Linear system Mechanical Systems ? Linear system y = Ma + Mb = M(
10、a + b) Mechanical Systems Nonlinear system Contain nonlinear springs and dampers; Do not follow the principle of superposition output comprises frequencies other than the input frequency output not proportional to input Newton Laws Force = mass acceleration Moment = rotation inertia angular accelera
11、tion Work-energy = kinetic energy + potential strain ) energy Energy ( Work of external forces = change in energy Impulse-momentum theorem Impulse = change in momentum Lagranges equation Systematic method (see the last sec tion ) Degrees of Freedom (DOFs) Modelling Number of DOFs = number of indepen
12、dent coordinates we use to describe the motion Coordinates may be displacements of some points, rotation, relative displacement, other (modal amplitudes). Number depends on 1) how complex the system is; 2) how we choose to model it; 3) modelling simplifications and assumptions; 4) what we want from
13、the model. (FEA? SEA?) Harmonic motion 2 radians Solution can be written as any of xt() =A sin( t ) +B cos( t ) t +)xt() =C sin( (sinusoidal or t +) time harmonic)xt() =D cos( frequency :(rad /) f = (cycle / sec s ond )2 period :T (, )s time per cycle ( ): =A22amplitude magnitude CD +B mean value :
14、x =0 21 2 Cmean square value : x =C . rms r m s value : x = 22 dxvelocity x dt 2dxacceleration x dt2 Complex Exponential Notation b x = Acos + iAsin x = A(cos+ isin )+ real + imaginary i phaseSo x = Ae Eulers Equation ie = cos isin magnitude 22 magnitude x = A= a + b phase = tan 1 (ba) Complex Expon
15、ential Notation (C.E.N) Time harmonic quantity written as xt() = In the “real” world we see ReX(t) xiX=eit Time derivatives x 2 ei Xit differential equationalgebraic equation Make life easy but introduce complex numbers. it+( )xt()= Xemagnitude phase Deterministic vibration Force and response known
16、+ predictable (e.g. rotating machinery, impulse, ect.) Random vibration Force and response unknown/unpredictable e.g. uneven road, wind, turbulence boundary layers (TBL) Discrete Systems finite number of rigid masses + massless stiffness elements Multi-degree-of-freedom (lumped parameter systems) (N
17、 modes, N natural frequencies) x3x1 x2 x4 Continuous systems Systems having distributed mass and stiffness (Infinite number of degrees-of-freedom) e.g. beams, plates etc. Example -beam Frequency Response Functions (FRFs) Define the system in terms of response to sinusoidal inputs (e.g. harmonic forc
18、e excitations). FRF: The ratio output/input of a system in steady-state when time harmonic. it e.g. force (input) f =Fe itdisplacement (output) x =Xe it ratio of (complex)Xe XFRF it= amplitude, does notFe F depend on time Complex, (usually) frequency dependent; magnitude phase H ()H () Harmonic Forc
19、es We often deal with time harmonic behaviour. Main Reasons 1. often have harmonic forces, e.g. rotating machine; 2. often have periodic forces comprising harmonic components, e.g. Fourier series; 3. general forces transformed as a sum of harmonics by Fourier Transform. Harmonic Response Frequency R
20、esponse Function (FRF) The ratio output/input of a system in steady-state when time harmonic. Note that V = iX; A = iV Frequency Response Functions (FRFs) Acceleration Force Accelerance = Apparent Mass = Force Acceleration Displacement Force Receptance = Dynamic Stiffness = Force Displacement In vibrations, FRFs depend on what we are interested in.
