Survey

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Transcript

AE 568 - Experimental Analysis of Vibrating Structures “Theory Guides; Experiment Decides.” AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring AE 568 - Experimental Analysis of Vibrating Structures Experimental vibration analysis of engineering structures is a field of increasing importance and popularity for researchers as consequence both of significant technological improvement of measurement equipments and theoretical formulations and of the extreme importance on the structural safety, serviceability conditions and durability of vibrating structures. AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring AE 568 - Experimental Analysis of Vibrating Structures Experimental vibration analysis is one of the most important tools for analysing dynamic properties of mechanical structures as the information obtained is used in the development or modification of structures to obtain a desired dynamic behaviour. AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring AE 568 - Experimental Analysis of Vibrating Structures This course is designed to use the experimental techniques in vibration measurements and thus to provide the students especially for the ones working on structural dynamics, mechanical vibrations and modal testing areas by providing unique inside on the general understanding of vibration test planning, selection and use of exciters, transducers and sensors, data collection, processing and assessment in particular with hands on environment for modal analysis and testing. AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring AE 568 - Experimental Analysis of Vibrating Structures Therefore, the course mainly focuses on - investigating structural vibrations by putting particular emphasize on the real application of experimental techniques in vibration measurements by maintaining the balance between theory and practical training. AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring AE 568 - Experimental Analysis of Vibrating Structures Structures Lab Capabilities AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring AE 568 - Experimental Analysis of Vibrating Structures Structures Lab Capabilities SOFTWARE MATLAB 2009a ANSYS 11.0 MSC PATRAN/ NASTRAN 2007r1 NI LabVIEW 8.6 HARDWARE B&K 6 channel Pulse portable data acquisition unit with special software of FFT Analysis, Time Data Record, Modal Test Consultant, Operational Modal Analysis, Reflex Modal Analysis Software B&K Modal Vibration Exciter (200N) B&K Impact Hammer AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring AE 568 - Experimental Analysis of Vibrating Structures Structures Lab Capabilities HARDWARE - Cont’ Various B&K Single-axis, Triaxial and miniature accelorometers, Empedance head. Keyence Laser Displacement Sensor Polytec Scanning Laser Vibrometer (New! – METUWIND Structural Dynamics LAB) Agilent Signal Generator Hameg Oscilloscope Additionally, Various Uni-axial Strain Gauges and Installation Kits Dedicated equipment for smart structure applications comprising programmable controller (SS10), high voltage power amplifiers, high voltage power supplies, preamplifiers and piezoelectric (PZT) patches in various size and shape. AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring Modal Analysis Theory Lecture # 1 1. Modal Analysis Theory 1.1 Theoretical Basis and Terminology 1.2 Modal Analysis of SDoF Dynamic Systems 1.3 Modal Analysis of MDoF Dynamic Systems AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring Modal Analysis Theory Theoretical Basis and Terminology Modal Analysis Theory Theoretical (Analytical) Experimental (Modal Testing) Aim is to develop “Reliable Dynamic Models”!! AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring Modal Analysis Theory Application of Modal Analysis: • Identification and evaluation of vibration phenomena • Validation, correction and updating of analytical dynamic models • Development of experimentally based dynamic models • Structural integrity assessment • Structural modification and damage detection • Reduction of mathematical models • Determining, improving and optimising dynamic characteristics of engineering structures. AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring Modal Analysis Theory Wind induced Vibration Tacoma Narrows Bridge (Washington State), 1940 AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring Modal Analysis Theory Forced Vibration - Resonance London Millenium Bridge Opening,2000 AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring Modal Analysis Theory Ground Resonance A Chinook Helicopter AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring Modal Analysis Theory Flutter Piper PA-30 Twin Comanche Aircraft Tail Flutter Test,1966 AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring Modal Analysis Theory Flutter AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring Modal Analysis Theory Assumptions: • Structure is linear and time-invariant • Structure obeys Maxwell’s Reciprocity Theorem. • About FRFs (Frequency Response Functions) • The positions of the shaker and accelerometer are reversed in multiple single-input RECIPROCITY checks! i.e. Various seperate single-input tests with the shaker located at different position for each test. AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring Modal Analysis Theory Assumptions: • Structure obeys Maxwell’s Reciprocity Theorem. The measured FRF for a force at location j and response at location i should correspond directly with the measured FRF for a force at location i and response at location j. The FRF matrix is symmetric and this property can be used as a check on the quality of the measured data. MODAL ANALYSIS is the process of determining the inherent dynamic characteristics of a system in the forms of Natural Frequencies, Damping Factors and Mode Shapes. AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring Modal Analysis Theory The three main phases of modal testing • The theoretical basis of vibration, • Accurate measurement of vibration (Controlled testing conditions), • Realistic and detailed data analysis (Signal processing, Range of curve fitting procedures in an attempt to find the mathematical model which provides the closest description of the actually observed behaviour). AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring Modal Analysis Theory Theoretical Route to Vibration Analysis Structure will vibrate under Structure's physical characteristics given excitation condition Description of Vibration Response Structure Modes Level SPATIAL MODEL MODAL MODEL RESPONSE MODEL • Mass • Natural Frequencies • Set of FRFs • Stiffness • Mode Shapes • Impulse Responses • Damping • Modal Damping Factors Structure's behaviour as a set of vibration modes AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring Modal Analysis Theory Experimental Route to Vibration Analysis Experimental Modal Analysis Response Vibration Structural Properties Modes Model RESPONSE MODEL MODAL MODEL SPATIAL MODEL • Set of FRFs • Natural Frequencies • Mass • Impulse Responses • Mode Shapes • Stiffness • Modal Damping Factors • Damping AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring Modal Analysis Theory Basic Vibration Theory SDOF Systems MDOF Systems Continuous Systems (Single Degree of Freedom) (Multi Degree of Freedom) (Infinitely many number of DOF) DOF: The minimum number of independent coordinates required to determine completely the motion of all parts of the system at any instant of time. A different selection of coordinates will lead to different equations of motion but end up with same natural frequencies regardless of the choice of coordinates (i.e. same system!) AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring Modal Analysis Theory Basic Vibration Theory Terms Vibration Type Description External Excitation • Free • Forced • Vibration induced by initial inputs only • Vibration subjected to one or more continuous external inputs Presence of Damping • Undamped • Damped • Vibration with no energy loss or dissipation • Vibration with energy loss Linearity of Vibration • Vibration for which superposition principle holds • Linear Vibration • Non-linear Vibration • Vibration that violates superposition principle Predictability • Deterministic • Random AE 568 Experimental Analysis of Vibrating Structures • The value of vibration is known at any given time • The value of vibration is not known at any given time but the statistical properties of vibration are known Dr. M. ŞAHİN – 2015/16 Spring Modal Analysis Theory The Phasor Phasor A is a vector that rotates in a counterclockwise direction with Angular velocity ω in the complex plane. A = A{Cos (ωt ) + jSin(ωt )} = A ⋅ e jωt where j = − 1 e ± jθ = Cos (θ ) ± jSin(θ ) { } A1 = A1 ⋅ e j (ωt +φ ) = A1 ⋅ e jφ ⋅ e jωt AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring Modal Analysis Theory The Phasor Imaginary Real AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring Modal Analysis Theory The Phasor dA j (ωt + π2 ) j ωt = jωAe = ωAe dt d2A j ωt 2 2 j Ae = ω dt 2 = −ω 2 Ae j (ωt ) = ω 2 Ae j (ωt +π ) where AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring Modal Analysis Theory Modal Analysis of SDOF Dynamic Systems Although very few practical structures could realistically be modeled by SDOF System, a mode complex multi-degree-of-freedom (MDOF) system can always be represented as the linear superposition of a number of SDOF systems. (a) Undamped (b) Viscously Damped (c) (c) Hysterically (or structurally) Damped (d or h) AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring Modal Analysis Theory Modal Analysis of SDOF Dynamic Systems Dynamic Properties of Mechanical Systems: • • • Mass (Responsible for Inertia), Stiffness (Responsible for Elastic Forces), Damping (Responsible for Dissipative Forces) (a) Undamped Spatial Model: m, k (Simple Harmonic Oscillator = Spring&Mass System) If no forcing; f (t ) = 0 mx + kx = 0 x(t ) = xeiωt , ω0 = AE 568 Experimental Analysis of Vibrating Structures k m (or ωn ) (Modal Model) Dr. M. ŞAHİN – 2015/16 Spring Modal Analysis Theory Modal Analysis of SDOF Dynamic Systems (a) Undamped Response Model f (t ) = feiωt , x(t ) = xeiωt (k − ω 2 m) xeiωt = feiωt H (ω ) = x 1 = = α (ω ) 2 f k −ω m (In the form of FRF) x and f are complex to accommodate both the Amplitude and Phase information Harmonic Displacement Response H (ω ) = Harmonic Force AE 568 Experimental Analysis of Vibrating Structures Complex if damping is not zero! Real if damping is zero! Dr. M. ŞAHİN – 2015/16 Spring Modal Analysis Theory Modal Analysis of SDOF Dynamic Systems (a) Undamped Circular Frequency ωn fn = [rad / s ] (repetitiveness of the oscillation) ωn ωn [rad / s ] ω [cycle] ωn = = n = [ Hz ] 2π 2π 2π [rad / cycle] 2π [ s ] 1 Hz = 1 [cycle / s ] AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring Modal Analysis Theory Modal Analysis of SDOF Dynamic Systems (a) Undamped Simple Harmonic Motion: AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring Modal Analysis Theory Modal Analysis of SDOF Dynamic Systems (a) Undamped Recall Phasor! AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring Modal Analysis Theory Modal Analysis of SDOF Dynamic Systems Useful quantities describing the vibration; Average Value: Mean Square Value: Square of displacement is associated with a system’s potential energy. Average of the displecement squared is also a useful vibration property. Root Mean Square (RMS) Value: Square root of the Mean Square value is commonly used in specifying vibration. AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring Modal Analysis Theory Modal Analysis of SDOF Dynamic Systems (b) Viscous Damping (c) Viscous dashpot c or damper, physical model for dissipating energy Equation of motion for free vibration case; mx + cx + kx = 0 m → [kg ], k → [ N / m], c → [ Ns / m], [kg / s ] In Laplace Domain, 2 s1, 2 c k c − =− ± 2m 2m m AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring Modal Analysis Theory Modal Analysis of SDOF Dynamic Systems (b) Viscous Damping 2 c k Overdamped: Both roots are real. 2m 〉 m 2 c k 〈 Underdamped: Two roots are complex conjugate. 2m m 2 Critically damped: Two equal real roots. c = k m 2m cc → Critical Damping Coefficient = 2 km = 2m k = 2mωn m ωn → Undamped natural frequency ξ → Damping Ratio = Damping Constant Damping Constant for Critically Damped Condition AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring Modal Analysis Theory Modal Analysis of SDOF Dynamic Systems (b) Viscous Damping Overdamped: ξ= ξ 〉1 Underdamped: ξ 〈1 Critically damped: ξ =1 c → Dimensionless quantity cc s1,2 = −ωnξ ± ωn ξ 2 − 1 ωd ωd → Damped natural frequency Damping ratio for critically damped systems seperates oscillatory motion from nonoscillatory motion and critical damping is the value of damping that provides the fastest return to zero without oscillation. AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring Modal Analysis Theory Critical Damping Army gun firing – Explosion and recoil take a few miliseconds With recovery of less than 1 second. In SDOF systems, it all about using all the spring’s potential energy! AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring Modal Analysis Theory Modal Analysis of SDOF Dynamic Systems (c) Structural (Hysteretic) Damping (d or h) Describes more closely the energy dissipation mechanism. By making viscous damping rate vary inversly with the frequency. c= d ω Provides much simpler analysis for MDOF systems! η → Structural Damping Loss Factor = ξ= ηk η 1 c c = = = cc 2 km 2 kmω 2 ω AE 568 Experimental Analysis of Vibrating Structures c d ω= k k k at ω = ω n → η = 2ξ m Dr. M. ŞAHİN – 2015/16 Spring Modal Analysis Theory Modal Analysis of SDOF Dynamic Systems A common unit of measurement for vibration amplitudes and RMS values is the decibel (dB). The decibel is defined in terms of the base 10 logarithm of the power ratio of two electrical signals (or as the ratio of the square of the amplitudes of two signals) 2 x x dB → 10 log10 1 = 20 log10 1 x2 x2 Voltage ratios in dB are calculated by V dB → 20 log10 1 V2 dB Scale expands of compresses vibration response information AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring Modal Analysis Theory Modal Analysis of SDOF Dynamic Systems Decade: A 1:10 increase or decrease of a variable, usually in frequency. Example: A 20 dB/decade gain: A gain change of 20 dB for each 10 fold increase or decrease in frequency. Numerically: ω = 10 rad / s 20 log10 (10 ) = 20 dB 10 = 1 rad / s and 10 ×10 = 100 rad / s 10 For ω1 ⇒ 20 log10 (1) = 0 dB / decade For ω2 ⇒ 20 log10 (100 ) = 40 dB / decade AE 568 Experimental Analysis of Vibrating Structures 20 dB decrease! 20 dB increase! Dr. M. ŞAHİN – 2015/16 Spring Modal Analysis Theory Modal Analysis of SDOF Dynamic Systems Octave: A doubling or halving, usually applied to frequency. Example: A 6 dB/octave gain: A gain change of 6 dB for each doubling or halving of frequency. Numerically: ω = 10 rad / s 20 log10 (10) = 20 dB 10 = 5 rad / s and 10 × 2 = 20 rad / s 2 For ω1 ⇒ 20 log10 (5) ≅ 14 dB / octave For ω2 ⇒ 20 log10 (20 ) ≅ 26 dB / octave 6 dB decrease! 6 dB increase! Slopes can be defined as either dB/octave or dB/decade. ∴ 20 dB / decade AE 568 Experimental Analysis of Vibrating Structures ≅ 6 dB / octave Dr. M. ŞAHİN – 2015/16 Spring Modal Analysis Theory Modal Analysis of SDOF Dynamic Systems Definitions of FRFs; AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring Modal Analysis Theory Modal Analysis of SDOF Dynamic Systems A most effective way of investigation for modal analysis is using the Frequency Response Function (FRF) RECEPTANCE: (recall) f ( t ) = fe iωt , x( t ) = xeiωt ( k − ω 2 m )xeiωt = fe iωt xeiωt x H ( ω ) = i ωt = = α ( ω ) fe f Harmonic Displacement Response H (ω ) = Harmonic Force AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring Modal Analysis Theory Modal Analysis of SDOF Dynamic Systems Alternative forms of FRF; f (t ) = fe iωt , x(t ) = xeiωt MOBILITY: (k − ω 2 m) xeiωt = fe iωt x iωxeiωt Y (ω ) = = = iωα (ω ) i ωt f fe Magnitude : Y (ω ) = ω α (ω ) Phase AE 568 Experimental Analysis of Vibrating Structures : θ Y = θα + π 2 Dr. M. ŞAHİN – 2015/16 Spring Modal Analysis Theory Modal Analysis of SDOF Dynamic Systems Summary; Magnitude : A( ω ) = ω Y ( ω ) = ω 2 α ( ω ) Phase : θ A = θY + π 2 = θα + π The reciprocals of the three FRFs of an SDOF system. Dynamic Stiffness = Force 1 = Displacement Response α ( ω ) Mechanical Inpedance = Apparent Mass = AE 568 Experimental Analysis of Vibrating Structures Force 1 = Velocity Response Y ( ω ) Force 1 = Acceleration Response A( ω ) Dr. M. ŞAHİN – 2015/16 Spring Modal Analysis Theory Modal Analysis of SDOF Dynamic Systems (b) Viscous Damping, c Response Model (In the form of FRF) Receptance : Mobility : H( ω ) = α( ω ) = 1 X(ω ) = F ( ω ) k − ω 2 m + i( ωc ) ( ) X ( ω ) iω = Y(ω ) = F ( ω ) k − ω 2 m + i( ωc ) Accelerance : ( ) −ω2 X ( ω ) = A( ω ) = F ( ω ) k − ω 2 m + i( ωc ) AE 568 Experimental Analysis of Vibrating Structures ( ) Dr. M. ŞAHİN – 2015/16 Spring Modal Analysis Theory Modal Analysis of SDOF Dynamic Systems (c) Structural (Hysteretic) Damping, d or h Response Model (In the form of FRF) Receptance : Mobility : H( ω ) = α( ω ) = 1 X(ω ) = F ( ω ) k − ω 2 m + ih ( ) X ( ω ) iω = Y(ω ) = F ( ω ) k − ω 2 m + ih Accelerance : ( ) −ω2 X ( ω ) = A( ω ) = F ( ω ) k − ω 2 m + ih AE 568 Experimental Analysis of Vibrating Structures ( ) Dr. M. ŞAHİN – 2015/16 Spring Modal Analysis Theory 1. Modal Analysis Theory 1.1 Theoretical Basis and Terminology 1.2 Modal Analysis of SDoF Dynamic Systems (cont’) 1.3 Modal Analysis of MDoF Dynamic Systems AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring

Similar