# Estimation of Fundamental Natural Frequency and Damping Ratio

Modal Testing 421L/521L (Lab 9) 10/21/2016 Frequency Response Frequency Response Function System characteristics in frequency domain F(s) Input G(s) System Output How to find FRF Mathematical modeling based on known parameter System identification through experimental Apply known input to your system Example of known input: impulse (impact hammer), sine sweep (shaker), Pseudo random (Function Generator), operational condition, etc

Measure the output Example of measured output: Accelerometer, Displacement sensors, Strain gage, load cell, LVDT, etc Find G(jw) = FFT(x(t))/FFT(f(t)) = x(jw)/f(jw) Where G(jw) = FRF, X(t) = output and f(t) = input X(s) Signal Processing and Window Analog Signal Input Ch AntiAliasing Filter Window

ADC FFT Averaging Visualization Signal Processing and Window FFT based signal processing involves ADC. Analog to Digital Conversion Sampling, Nyquist frequency and frequency folding Aliasing (or Anti-aliasing: make 0 if higher than Nyquist freq.) Frequency folding fs Nyquist frequency Signal Processing and Window Finite sampling which does not match exact period creates leakage

10Hz sine Signal FFT 9.5Hz sine Signal Processing and Window Window tailors the finite signal such that the start and end matches to 0. By applying window, spectral leakage could be improved. There are multiple shapes of Windows Signal Processing and Window Proc. of SEM, H. Gaberson, 2002

Frequency Response of 1-DOF System c k M , Substitute, , Where, n=sqrt(k/m), undamped natural frequency, rad/s =c/sqrt(2mk), damping ratio = excitation/Input frequency x,f k, stiffness, N/m m, mass, kg c, damping coefficient, N/(m/s) Frequency Response of 1-DOF System

0.04 0.035 m a g n it u d e 0.03 0.025 0.02 0.015 0.01 0.005 0 0 10 20 30

40 50 60 frequency (rad/s) 70 80 90 100 0 10 20 30

40 50 60 frequency (rad/s) 70 80 90 100 0 -20 p h a s e (d e g ) -40 -60 -80

-100 -120 -140 -160 -180 Frequency Response of Multi DOF System c1 k1 m1 c2 k2 , X1,f1

m2 X2,f2 c3 k3 m3 k, stiffness, N/m m, mass, kg c, damping coefficient, N/(m/s) Mode freq = det. of [K-w2M]=0, mode shape = eigen vector X3,f3 Frequency Response of Cantilever Beam Y(x) E, I, L, x E: Youngs modulus

I: Moment of inertia L: length : mass per unit length See Handout Experiment Identify mode shape and corresponding frequencies Mount Accelerometer onto beam End for cantilever beam Mark excitation points Excite beam by applying impulse using impact hammer at the marked points Observe input, time response and frequency response Collect Frequency response (5 sets then average) Create waterfall chart Find resonant frequency and corresponding mode shape

Experimental setup: Cantilever Beam Aluminum Beam Thickness = 4.84mm Width = 19.09mm Length = 640mm Accelerometer is mounted at the end of the beam Mass of accelerometer = 7.83 gram 1 2 3 4

5 6 7 8 Example 3rd mode 2nd mode magnitude (dB) 100 1st mode 50 0 -50 500

8 6 400 300 4 200 2 100 Frequency (Hz) 0 0 Excitation Position Example of FRF

60 50 X: 52 Y: 52.22 X: 8 Y: 36.39 40 magnitude (dB) X: 148 Y: 42.81 30 20 10 0 -10

-20 0 50 100 150 200 250 300 Frequency (Hz) 350 400 450

500 Experiment Install test setup using 4 stainless steel rods and one AL plate as in the direction Mount 2 tri-axial Accelerometers onto structure Mark excitation points Excite Structure by applying impulse using impact hammer at the marked points Observe input, time response and frequency response Collect Frequency response (5 sets then average) Create waterfall chart Find resonant frequency and corresponding mode shape Identify mode shape and corresponding frequencies from FEA

Experimental setup Stainless Steel rod Accelerometer AL plate Diameter = 0.5in Length = 10in Aluminum Plate Length = 12in Width = 12in Thickness = 0.5in 2 Tri-axial accelerometer is mounted at the top plate d2 d1

Stainless steel rod Experiment: Configuration #1 Install 12x12in AL plate using 4 stainless steel rods on optical table Set the spacing between the stainless steel rods, *d1 = d2 = 9in Attach 2 tri-axial accelerometer on the top surface and apply impulse each point marked A, B, C and D Apply impulse to each marked point. Log data and find FRF with window Compare experimental results from A, B, C and D Analyze static deformation from 3 load cases under unit force at A, B, C and D Compare analysis results from each load case A, B, C and D Perform modal analysis and compare the results

to the experiment A B D C Experiment: Configuration #2 Install 12x12in AL plate using 4 stainless steel rods on optical table Set the spacing between the stainless steel rods, *d1 = 9in *d2 = 3in Attach 2 tri-axial accelerometer on the top surface and apply impulse each point marked A, B, C and D Apply impulse to each marked point. Log data and find FRF with window

Compare experimental results from A, B, C and D Analyze static deformation from 3 load cases under unit force at A, B, C and D Compare analysis results from each load case A, B, C and D Perform modal analysis and compare the results to the experiment A B D C ? Does your measurement match to your estimation? Show your measurement and measured value

How does the geometry affect to the result? Translation? Rotation? Have you observed any higher mode? ANSYS Install ANSYS/student http://www.ansys.com/Student Free to use for educational purpose General Steps of FEA Review your system Geometric Modeling Direct Modeling Import 3D(or CAD) model and Finite Element Modeling Define Material Properties and Real Constants Define Nodes and Elements Apply Boundary Conditions

Applying Loading Conditions Solve Configure Solver and Solve Post Processing Visualize Result and/or Export Data Real Hardware Example HET Wide Field Corrector 8/14/09 McDonald Observatory Top Level Mechanical Design Requirement WFC subsystem should be designed for: 35 nominal zenith angle +/- 8.5 Operational temperature of 10 C +/- 20 C 20 Hz minimum fundamental frequency

Meet or exceed the defined mirror positioning requirements including gravity, thermal and initial alignment Meet or exceed the required mirror adjustment resolution Serviceability and maintainability 8/14/09 McDonald Observatory Mechanical Design Overview System Weights (1802 lbs total)

8/14/09 950 lbs glass 550 lbs steel and Invar 120 lbs aluminum baffles and Cover 182 lbs lower strongback mounted instruments System interface to PFIP 3 point kinematic interface on 1400mm in diameter and 195mm from the vertex of M2 Solid mirrors (ClearceramZ-HS by OHARA), undercut for lightweighting Stainless Steel and Invar truss tubes M4 support includes truss head ring and pre-tensioned spider vanes that are aligned to the diffraction pattern of the HET primary mirror segments

M2/M5 and M3 supported by steel strongback McDonald Observatory FEA System Model 8/14/09 McDonald Observatory PDR CDR FEA Iterations PDR CDR Structural Updates

8/14/09 Updated interface points (optical axis) Updated both strongbacks to 1018 steel Updated cross sections Updated all material properties Added lumped masses for UT instruments Updated head ring and added head ring compliance McDonald Observatory Frequency Response Fundamental

Frequency 25.47 Hz 8/14/09 2nd Frequency 29.53 Hz McDonald Observatory Frequency Response 3rd Frequency = 29.86 Hz 4th Frequency = 30.08 Hz 5th Frequency = 30.69 Hz 6th Frequency = 31.19 Hz Movement of lower instrument package masses coupled with slight movement of the M2/M5 strongback. 8/14/09

3rd frequency shown McDonald Observatory Frequency Response 7th Frequency 38.61 Hz 8/14/09 8th Frequency 43.68 Hz McDonald Observatory WFC Structure Modal Test Straps to hang Accelerometer

DAQ device McDonald Observatory Data Acquisition and Analysis 50 20 40 40 30 60 20 10 80

0 100 -10 120 -20 140 1 2 3 4 5 6

7 Dominant mode: headring movement in z direction at 43Hz (FEA = 49Hz)

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