Chapter 7 Vibration Measurement and Applications Dr. Tan Wei Hong School of Mechatronic Engineering Universiti Malaysia Perlis (UniMAP) Pauh Putra Campus ENT 346 Vibration Mechanics
Chapter Outline 7.1 Introduction 7. Transducers 7.3 Vibration Pickups 7.4 Frequency-Measuring Instruments 7.5 Vibration Exciters 7.6 Signal Analysis 7.7 Dynamic Testing of Machines and Structure 7.8 Experimental Modal Analysis ENT 346 Vibration Mechanics
7.1 Introduction Why we need to measure vibrations: To detect shifts in ω n which indicates possible failure To select operational speeds to avoid resonance Measured values may be different from theoretical values To design active vibration isolation systems To identify mass, stiffness and damping of a system To verify the approximated model 3 ENT 346 Vibration Mechanics
7.1 Introduction Type of vibration measuring instrument used will depend on: Expected range of frequencies and amplitudes Size of machine/structure involved Conditions of operation of the machine/structure Type of data processing used 4 ENT 346 Vibration Mechanics
7. Transducers A device that transforms values of physical variables into electrical signals Following slides show some common transducers for measuring vibration 5 ENT 346 Vibration Mechanics
7. Transducers Variable Resistance Transducers Mechanical motion changes electrical resistance, which cause a change in voltage or current Strain gage is a fine wire bonded to surface where strain is to be measured. 6 ENT 346 Vibration Mechanics
7. Transducers Variable Resistance Transducers Surface and wire both undergo same strain. Resulting change in wire resistance: R / R r L K 1 v 1 v L / L r L where K = Gage factor of the wire R = Initial resistance ΔR = Change in resistance L = Initial length of wire ΔL = Change in length of wire v = Poisson s ratio of the wire r = Resistivity of the wire Δr = Change in resistivity of the wire 0 for Advance 7 ENT 346 Vibration Mechanics
7. Transducers Variable Resistance Transducers Strain: L L R RK The following figure shows a vibration pickup: 8 ENT 346 Vibration Mechanics
7. Transducers Variable Resistance Transducers ΔR can be measured using a Wheatstone bridge as shown: E R R R R 1 3 4 R R R R V 1 3 4 9 ENT 346 Vibration Mechanics
10 7. Transducers Variable Resistance Transducers Initially, resistances are adjusted so that E=0 R 1 R 3 = R R 4 When R i change by ΔR i, 4 3 4 3 1 1 0 4 4 3 3 1 1 0 where R R R R R R R R r R R R R R R R R Vr E 10 ENT 346 Vibration Mechanics
7. Transducers Variable Resistance Transducers If the leads are connected between pts a and b, R 1 =R g, ΔR 1,= ΔR g, ΔR = ΔR 3 = ΔR 4 =0 R R g g E Vr 0 K or E KVr0 where Rg is the initial resistance of the gauge. Hence E can be calibrated to read ε directly. 11 ENT 346 Vibration Mechanics
7. Transducers Piezoelectric Transducers Certain materials generate electrical charge when subjected to deformation or stress. Charge generated due to force: Q x kf x kap x where k =piezoelectric constant A =area on which F x acts p x =pressure due to F x. 1 ENT 346 Vibration Mechanics
7. Transducers Piezoelectric Transducers E=vtp x v = voltage sensitivity t = thickness of crystal A piezoelectric accelerometer is shown. Output voltage proportional to acceleration 13 ENT 346 Vibration Mechanics
7. Transducers Example 10.1 Output Voltage of a Piezoelectric Transducer A quartz crystal having a thickness of.5mm is subjected to a pressure of 50psi. Find the output voltage if the voltage sensitivity is 0.055 V-m/N. 14 ENT 346 Vibration Mechanics
7. Transducers Example 10.1 Output Voltage of a Piezoelectric Transducer Solution E = vtp x =(0.055)(0.0054)(344738) = 47.4015V 15 ENT 346 Vibration Mechanics
7. Transducers Electrodynamic Transducers Voltage E is generated when the coil moves in a magnetic field as shown. E = Dlv Dl E v F I where D = magnetic flux density l = length of conductor v = velocity of conductor relative to magnetic field 16 ENT 346 Vibration Mechanics
7. Transducers Linear Variable Differential Transformer Transducer Output voltage depends on the axial displacement of the core. Insensitive to temp and high output. 17 ENT 346 Vibration Mechanics
7.3 Vibration Pickups Most common pickups are seismic instruments as shown Bottom ends of spring and dashpot have same motion as the cage Vibration will excite the suspended mass Displacement of mass relative to cage: z = x y 18 ENT 346 Vibration Mechanics
7.3 Vibration Pickups Y(t) = Ysinωt Equation of motion of mass m: mx c Steady-state solution: x y kx y 0 or mz cz kz my mz cz kz m Y sint z t Z sin t 19 ENT 346 Vibration Mechanics
0 7.3 Vibration Pickups n n m c r r r m k c r r Y r c m k Y Z, 1 tan tan 1 1 1 0 ENT 346 Vibration Mechanics
7.3 Vibration Pickups Vibrometer Measures displacement of a vibrating body Z/Y 1 when ω/ω n 3 (range II) z t Y sint if r 1 r r 1 In practice Z may not be equal to Y as r may not be large, to prevent the equipment from getting bulky 1 ENT 346 Vibration Mechanics
7.3 Vibration Pickups Example 10. Amplitude by Vibrometer A vibrometer having a natural frequency of 4 rad/s and ζ = 0. is attached to a structure that performs a harmonic motion. If the difference between the mximum and the minimum recorded values is 8 mm, find the amplitude of motion of the vibrating structure when its frequency is 40 rad/s. ENT 346 Vibration Mechanics
7.3 Vibration Pickups Example 10. Amplitude by Vibrometer Solution Amplitude of recorded motion: Z Y 10 110 0. 10 1.0093Y 4 mm Amplitude of vibration of structure: Y = Z/1.0093 = 3.9631 mm 3 ENT 346 Vibration Mechanics
7.3 Vibration Pickups Accelerometer Measures acceleration of a vibrating body. z If t z n Y sin t 1 r r 1 r r 1, t Y sint n 1 4 ENT 346 Vibration Mechanics
7.3 Vibration Pickups Accelerometer If 0.65< ζ < 0.7, 0.96 1 1 r r 1.04 for 0 r 0.6 Accelerometers are preferred due their small size. 5 ENT 346 Vibration Mechanics
7.3 Vibration Pickups Example 10.3 Design of an Accelerometer An accelerometer has a suspended mass of 0.01 kg with a damped natural frequency of vibration of 150 Hz. When mounted on an engine undergoing an acceleration of 1 g at an operating speed of 6000 rpm, the acceleration is recorded as 9.5 m/s by the instrument. Find the damping constant and the spring stiffness of the accelerometer. 6 ENT 346 Vibration Mechanics
7.3 Vibration Pickups Example 10.3 Design of an Accelerometer Solution 1 M easured value 1 r r True value or 7 ENT 346 Vibration Mechanics Thus 1 r r 1/ 0.9684 Operatingspeed d r 1 d 0.6667 150 n 1 1 n or r 6000 60 94.48 rad/s 1 r 0.4444 1 9.5 9.81 1.0663 68.3 rad/s 68.3 94.48 0.9684 (E.1) 0.6667 (E.)
7.3 Vibration Pickups Example 10.3 Design of an Accelerometer Solution Substitute (E.) into (E.1): 1.5801ζ 4.714ζ + 0.7576 = 0 Solution gives ζ = 0.753, 0.9547 Choosing ζ= 0.753 arbitrarily, n 1 n Damping constant 19.8571 N -s/m d k m c m n 94.48 1 0.753 0.011368.8889 1368.8889 rad/s 18738.568 N/m 0.011368.88890.753 8 ENT 346 Vibration Mechanics
7.3 Vibration Pickups Example 10.3 Design of an Accelerometer Solution Measures velocity of vibrating body: y t Y cos t Velocity: z If z t Y 1 r r 1 r r t Y cost r r cos t 1, then 9 ENT 346 Vibration Mechanics
7.3 Vibration Pickups Example 10.4 Design of a Velometer Design a velometer if the maximum error is to be limited to 1% of the true velocity. The natural frequency of the velometer is to be 80Hz and the suspended mass is to be 0.05 kg. 30 ENT 346 Vibration Mechanics
7.3 Vibration Pickups Example 10.4 Design of a Velometer Solution We have z t r Y 1 r r cos t R r 1 r r Recorded velocity True velocity (E.1) Maximum r r 1 1 (E.) 31 ENT 346 Vibration Mechanics
3 7.3 Vibration Pickups Example 10.4 Design of a Velometer Solution Substitute (E.) into (E.1), R R 4 4 4 1 1 1 4 1 1 1 1 1 3 ENT 346 Vibration Mechanics
7.3 Vibration Pickups Example 10.4 Design of a Velometer Solution R = 1.01 or 0.99 for 1% error ζ 4 ζ + 0.45075 = 0 and ζ 4 ζ + 0.55075=0 ζ = 0.570178, 0.4981 or ζ = 0.755101, 0.655607 33 ENT 346 Vibration Mechanics
7.3 Vibration Pickups Example 10.4 Design of a Velometer Solution Choosing ζ = 0.755101 arbitrarily, c 80 k n m n m n 37.9556 N -s/m 50.656 rad/s 0.0550.656 0.755101 50.6560.05 1633.157 N/m 34 ENT 346 Vibration Mechanics
7.3 Vibration Pickups Phase Distortion All vibrating-measuring instruments have phase lag. If the vibration consists of or more harmonic components, the recorded graph will not give an accurate picture phase distortion Consider vibration signal of the form as shown: 35 ENT 346 Vibration Mechanics
7.3 Vibration Pickups Phase Distortion Let phase shift = 90 for first harmonic Let phase shift = 180 for third harmonic Corresponding time lags: t 1 = 90 /ω, t = 180 /ω Output signal is as shown: 36 ENT 346 Vibration Mechanics
7.3 Vibration Pickups Phase Distortion In general, let the complex wave be y(t) = a 1 sinωt + a sinωt + Output of vibrometer becomes: z(t) = a 1 sin(ωt Φ 1 ) + a sin(ωt Φ ) + where j tan n j, j 1,,... 1 j n 37 ENT 346 Vibration Mechanics
7.3 Vibration Pickups Phase Distortion Φj π since ω/ω n is large. z(t) [a 1 sinωt + a sinωt + ] -y(t) Thus the output record can be easily corrected. Similarly we can show that output of velometer is z t y t Accelerometer: Let the acceleration curve be t a sin t a sin t y 1 Output of accelerometer: 38 ENT 346 Vibration Mechanics t a sin t a sin t z 1 1
7.3 Vibration Pickups Phase Distortion Since Φ varies almost linearly from 0 to 90 for ζ = 0.7, Φ αr = α(ω/ω n ) = βω where α and β are constants. Time lag t z is independent of frequency. t a1 sint a sint a sin a sin where t 1 Thus output of accelerometer represents the true acceleration being measured. 39 ENT 346 Vibration Mechanics
7.4 Frequency-Measuring Instruments Single-reed instrument or Fullarton Tachometer Clamped end pressed against vibrating body Adjust l until free end shows largest amplitude of vibration Read off frequency 40 ENT 346 Vibration Mechanics
7.4 Frequency-Measuring Instruments Multi-reed Instrument or Frahm Tachometer Clamped end pressed against vibrating body Frequency read directly off strip whose free end shows largest amplitude of vibration 41 ENT 346 Vibration Mechanics
7.4 Frequency-Measuring Instruments Stroboscope Produces light pulses A vibrating object viewed with it will appear stationary when frequency of pulse is equal to vibration frequency 4 ENT 346 Vibration Mechanics
7.5 Vibration Exciters Used to determine dynamic characteristics of machines and structures and fatigue testing of materials Can be mechanical, electromagnetic, electrodynamic or hydraulic type 43 ENT 346 Vibration Mechanics
7.5 Vibration Exciters Mechanical Exciters Force can be applied as an inertia force Force can be applied as an elastic spring force for frequency <30 Hz and loads <700N 44 ENT 346 Vibration Mechanics
7.5 Vibration Exciters Mechanical Exciters The unbalance created by two masses rotating at the same speed in opposite directions can be used as a mechanical exciter. 45 ENT 346 Vibration Mechanics
7.5 Vibration Exciters Electrodynamic Shaker The electrodynamic shaker can be considered as the reverse of an electrodynamic transducer. resonant frequencies are shown below. F DIl 46 ENT 346 Vibration Mechanics
7.6 Signal Analysis Acceleration-time history of a frame subjected to excessive vibration: Transformed to frequency domain: 47 ENT 346 Vibration Mechanics
7.6 Signal Analysis Spectrum Analyzers Separates energy of signal into various frequency bands Real-time analyzers useful for machine health monitoring types of real-time analysis procedures: digital filtering method and Fast Fourier Transform method Basic component of spectrum analyzer: Bandpass filter 48 ENT 346 Vibration Mechanics
7.6 Signal Analysis Bandpass Filter Permits passage of frequencies over a band and rejects all other frequency components Response of a filter: 49 ENT 346 Vibration Mechanics
7.6 Signal Analysis Bandpass Filter f u and f l are upper and lower cutoff frequencies respectively f c is centre (tuned) frequency Ripples within band is minimum for a good bandpass filter types of bandpass filters: constant percent bandwidth filters and constant bandwidth filters Constant percent: (f u f l )/f c is a constant E.g. octave, one-half-octave filters Constant bandwidth: f u f l is independent of f c 50 ENT 346 Vibration Mechanics
7.6 Signal Analysis Constant Percent Bandwidth and Constant Bandwidth Analyzers Spectrum analyzer with a set of octave and 1/3-octave band filters can be use for signal analysis Lower cutoff freq of a filter = upper cutoff freq of previous filter. Filter characteristics as shown 51 ENT 346 Vibration Mechanics
7.6 Signal Analysis Constant Percent Bandwidth and Constant Bandwidth Analyzers Constant bandwidth analyzer used to obtain more detailed analysis than constant percent bandwidth analyzer Wave or heterodyne analyzer is a constant bandwidth analyzer with a continuously varying centre frequency 5 ENT 346 Vibration Mechanics
7.7 Dynamic Testing of Machines and Structures Involves finding the deformation of machines/structures at a critical frequency Approaches: Operational Deflection Shape measurements Modal Testing 53 ENT 346 Vibration Mechanics
7.7 Dynamic Testing of Machines and Structures Using Operational Deflection Shape Measurements Forced dynamic deflection shape measured under steady-state frequency of system. Valid only for forces/frequency associated with operating conditions. If a particular location has excessive deflection, we can stiffen that location. 54 ENT 346 Vibration Mechanics
7.7 Dynamic Testing of Machines and Structures Modal Testing Any dynamic response of a machine/structure can be obtained as a combination of its modes. Knowledge of the mode shapes, modal frequencies and modal damping ratio will describe completely the machine dynamics. 55 ENT 346 Vibration Mechanics
7.8 Experimental Modal Analysis When a system is excited, its response exhibits a sharp peak at resonance Phase of response changes by 180 as forcing frequency crosses the natural frequency Equipment needed: Exciter to apply known input force Transducer to convert physical motion into electrical signal Signal conditioning amplifier Analyzer with suitable software 56 ENT 346 Vibration Mechanics
7.8 Experimental Modal Analysis Necessary Equipment Exciter Can be an electromagnetic shaker or impact hammer Shaker is attached to the structure through a stringer, to control the direction of the force Impact hammer is a hammer with built-in force transducer in its head Portable, inexpensive and much faster to use than a shaker But often cannot impart sufficient energy and difficult to control direction of applied force 57 ENT 346 Vibration Mechanics
7.8 Experimental Modal Analysis Necessary Equipment Transducer Piezoelectric transducers most popular Strain gauges can also be used Signal conditioner Outgoing impedance of tranducers not suitable for direct input into analyzers. Charge or voltage amplifiers are used to match and amplify the signals before analysis 58 ENT 346 Vibration Mechanics
7.8 Experimental Modal Analysis Necessary Equipment Analyzer FFT analyzer commonly used Analyzed signals used to find natural frequencies, damping ratios and mode shapes 59 ENT 346 Vibration Mechanics
7.8 Experimental Modal Analysis Necessary Equipment General arrangement for experimental modal analysis: 60 ENT 346 Vibration Mechanics
7.8 Experimental Modal Analysis Digital Signal Processing x(t) represents analog signal, x i = x(t i ) represents corresponding digital record. 61 ENT 346 Vibration Mechanics
7.8 Experimental Modal Analysis Determination of Modal Data from Observed Peaks Let the graph of H(iω) be as shown below. 4 peaks suggesting a 4-DOF system. 6 ENT 346 Vibration Mechanics
7.8 Experimental Modal Analysis Determination of Modal Data from Observed Peaks Partition into several frequency ranges. Each range is consider a 1-DOF system Damping ratio corresponding to peak j: j H j 1 1 i H i j j j where j 1 j H and i j j satisfy When damping is small, ωj ωn 63 ENT 346 Vibration Mechanics
7.8 Experimental Modal Analysis Example 10.5 Determination of Damping Ratio from Bode Diagram The graphs showing the variations of the magnitude of the response and its phase angle with the frequency of a single DOF system provides the frequency response of the system. Instead of dealing with the magnitude curves directly, if the logarithms of the magnitude ratios (in decibels) are used, the resulting plots are called Bode diagrams. Find the natural frequency and damping ratio of a system whose Bode diagram is as shown. 64 ENT 346 Vibration Mechanics
7.8 Experimental Modal Analysis Example 10.5 Determination of Damping Ratio from Bode Diagram 65 ENT 346 Vibration Mechanics
7.8 Experimental Modal Analysis Example 10.5 Determination of Damping Ratio from Bode Diagram Solution ω n = 10Hz, ω 1 = 9.6 Hz, ω = 10.5 Hz, Peak response = -35 db Damping ratio: 1 10.5 9.6 10.0 n 0.045 66 ENT 346 Vibration Mechanics
7.8 Experimental Modal Analysis Measurement of Mode Shapes Undamped multi-dof system: mx k x f Free harmonic vibration: k m y 0 i i Orthogonal relations for mode shapes: T Y m Y diagm T Y k Y diagk i K M i i 67 ENT 346 Vibration Mechanics
7.8 Experimental Modal Analysis Measurement of Mode Shapes Damped multi-dof system: mx cx k x f Assume proportional damping: c ak bm Undamped mode shapes of the system will diagonalize the damping matrix: Y T c Y diag k 68 ENT 346 Vibration Mechanics