River Rapids Conference Proceedings of the Society for Experimental Mechanics Series Special Topics in Structural Dynamics, Volume 6 Gary Foss Christopher Niezrecki Proceedings of the 32nd IMAC, A Conference and Exposition on Structural Dynamics, 2014 River Publishers
Conference Proceedings of the Society for Experimental Mechanics Series Series Editor TomProulx Society for Experimental Mechanics, Inc., Bethel, CT, USA
River Publishers Gary Foss • Christopher Niezrecki Editors Special Topics in Structural Dynamics, Volume 6 Proceedings of the 32nd IMAC, A Conference and Exposition on Structural Dynamics, 2014
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Preface v Special Topics in Structural Dynamics, Volume 6represents one of the eight volumes of technical papers presented at the 32nd IMAC, A Conference and Exposition on Structural Dynamics, 2014 organized by the Society for Experimental Mechanics, and held in Orlando, Florida, February 3–6, 2014. The full proceedings also include volumes on Dynamics of Coupled Structures; Nonlinear Dynamics; Model Validation and Uncertainty Quantification; Dynamics of Civil Structures; Structural Health Monitoring; Topics in Modal Analysis I; and Topics in Modal Analysis II. Each collection presents early findings from experimental and computational investigations on an important area within structural dynamics. Special Topics in Structural Dynamics represents papers on enabling technologies for modal analysis measurements such as sensors and instrumentation, and applications of modal analysis in specific application areas. Topics in this volume include: Aircraft/aerospace Active control Analytical methods System identification Sensors and instrumentation The organizers would like to thank the authors, presenters, session organizers, and session chairs for their participation in this track. Seattle, WA, USA GaryFoss Lowell, MA, USA Christopher Niezrecki
Contents 1 Vibration Class at GIST, Korea........................................................................................ 1 Semyung Wang, Jongsuh Lee, Youngeun Cho, Homin Ryu, and Kihwan Park 2 Lab Exercises for a Course on Mechanical Vibrations.............................................................. 15 Anders Brandt 3 Variational Foundations of Modern Structural Dynamics.......................................................... 21 Robert N. Coppolino 4 Some Cornerstones of Signal Analysis History....................................................................... 33 Anders Brandt 5 Structural Dynamic Test-Analysis Correlation ...................................................................... 37 Robert N. Coppolino 6 A Brief History of 30 Years of Model Updating in Structural Dynamics.......................................... 53 François M. Hemez and Charles R. Farrar 7 Techniques for Synthesizing FRFs from Analytical Models ........................................................ 73 Hasan G. Pasha, Randall J. Allemang, and Allyn W. Phillips 8 An Analytical Method and Its Extension for Linear Modal Analysis of Beam-Type Systems Carrying Various Substructures ....................................................................................... 81 Zhenguo Zhang, Xiuchang Huang, Zhiyi Zhang, and Hongxing Hua 9 Computationally Efficient Nonlinear Dynamic Analysis for Stress/Strain Applications .......................................................................................... 89 Julie Harvie and Peter Avitabile 10 An Improved Expansion Process for Guyan Reduced Models: Technique for Improved Guyan Expansion Reconstruction (TIGER) .................................................................................. 109 Julie Harvie and Peter Avitabile 11 Towards a Technique for Nonlinear Modal Reduction.............................................................. 121 T.L. Hill, A. Cammarano, S.A. Neild, and D.J. Wagg 12 Identification of Independent Inputs and Their Spatial Positions ................................................. 129 D. Bernal and A. Ussia 13 Shock Response Fixture Developed from Analytical and Experimental Data and Customized Using Structural Dynamics Modification Techniques ............................................................... 135 Kai Aizawa and Peter Avitabile 14 Parameter Identification for Nonlinear Dynamic Systems via Multilinear Least Square Estimation ......... 169 Sushil Doranga and Christine Q. Wu 15 Support Systems for Developing System Models..................................................................... 183 Hasan G. Pasha, Karan Kohli, Randall J. Allemang, David L. Brown, and Allyn W. Phillips vii
viii Contents 16 Nonlinear Modeling for Adaptive Suppression of Axial Drilling Vibration ...................................... 195 Benjamin Winter, Garrison Stevens, Rex Lu, Eric Flynn, and Eric Schmierer 17 A Regenerative Approach to Energy Efficient Hydraulic Vibration Control..................................... 211 Jonathan L. du Bois 18 Virtual Sensing of Acoustic Potential Energy Through a Kalman Filter for Active Control of Interior Sound ........................................................................................................ 221 A. Kumar and S.V. Modak 19 Wavenumber Decomposition Applied to a Negative Impedance Shunts for Vibration Suppression onaPlate ................................................................................................................. 243 F. Tateo, K.A. Cunefare, M. Collet, and M. Ouisse 20 Modal Parameter Estimation of a Two-Disk- Shaft System by the Unified Matrix Polynomial Approach ... 251 Naim Khader 21 System Identification of an Isolated Structure Using Earthquake Records....................................... 265 Ruben L. Boroschek, Antonio A. Aguilar, and Fernando Elorza 22 Design of an Inertial Measurement Unit for Enhanced Training .................................................. 275 M. Bassetti, F. Braghin, F. Castelli Dezza, and F. Ripamonti 23 A Parameter Optimization for Mode Shapes Estimation Using Kriging Interpolation ......................... 287 Minwoo Chang and Shamim N. Pakzad 24 Determination of Principal Axes of a Wineglass Using Acoustic Testing ......................................... 295 Huinam Rhee, Sangjin Park, Junsung Park, Jongchan Lee, and Sergii A. Sarapuloff 25 Remote Placement of Magnetically Coupled Ultrasonic Sensors for Structural Health Monitoring .......... 301 Nipun Gunawardena, John Heit, George Lederman, Amy Galbraith, and David Mascareñas 26 Modular System for High-Speed 24-Bit Data Acquisition of Triaxial MEMS Accelerometers for Structural Health Monitoring Research .............................................................................. 313 Brianna Klingensmith, Stephen R. Burgess, Thomas A. Campbell, Peter G. Sherman, Michael Y. Feng, Justin G. Chen, and Oral Buyukozturk 27 Mode Shape Comparison Using Continuous-Scan Laser Doppler Vibrometry and High Speed 3D Digital Image Correlation .......................................................................................... 321 David A. Ehrhardt, Shifei Yang, Timothy J. Beberniss, and Matthew S. Allen 28 Triaxial Accelerometer, High Frequency Measurement and Temperature Stability Considerations .......... 333 Thomas Petzsche, Andy Cook, and Marine Dumont 29 Laser Speckle in Dynamic Sensing Applications..................................................................... 341 Will Warren, Logan Ott, Erynn Elmore, Erik Moro, and Matt Briggs 30 Sensor Placements for Damage Localization with the SDLV Approach .......................................... 347 D. Bernal, Q. Ma, R. Castro-Triguero, and R. Gallego 31 Diaphragm Flexibility in Floor Spectra............................................................................... 355 D. Bernal, E. Cabrera, and E. Rodríguez 32 Use of Zernike Polynomials for Modal Vector Correlation of Small Turbine Blades ............................ 361 Jonathan Salerno and Peter Avitabile 33 Modeling of Flexible Tactical Aerospace Vehicle for Hardware-in-Loop Simulations........................... 379 Yadunath Gupta, Pulak Halder, and Siddhartha Mukhopadhyay 34 Modal Test of Six-Meter Hypersonic Inflatable Aerodynamic Decelerator....................................... 401 Nijo Abraham, Ralph Buehrle, Justin Templeton, Mike Lindell, and Sean Hancock 35 Modal Testing of Space Exploration Rover Prototypes ............................................................. 415 Yvan Soucy and Frédérik Brassard
Chapter1 Vibration Class at GIST, Korea Semyung Wang, Jongsuh Lee, Youngeun Cho, Homin Ryu, and Kihwan Park Abstract Vibration class covers vibration phenomena of mechanical systems due to dynamic load is studied. It covers from single DOF to multi DOF and theory as well as numerical and experimental methods. It deals various subjects: Lagrange equation, Laplace transformation, Fourier transformation, mode superposition, finite element method, experimental modal analysis, random vibration, vibro-acoustics and model validation. Keywords Laboratory • Structural dynamics • Numerical analysis • Experimental modal analysis • Model updating 1.1 Introduction Advanced vibration course is intended for graduated students; its goal includes deep understanding and estimating the mechanism of how vibration takes place in a mechanical system as well as understanding the vibration theory [1–4]. The class begins with a detailed description of single degree of freedom (DOF) and two DOF systems for the underlying understanding of vibration. In these parts, the system responses, which are governed by differential equation, are investigated in time and frequency domain. These equations are used to explain the characteristics of response with respect to different damping ratio values (under, over and critical damping) and different kinds of damping (viscous, structural). For the case of multi DOF system which is represented by matrix, the system is analyzed based on the eigenvalue problem. Especially, mode shape of the system which is represented by eigenvector is introduced, and it is described that the several characteristics (reciprocal theorem, modal matrix) caused by the orthogonal property between this vector and system matrix. In addition, it is introduced that the frequency response function (FRF), which is the response of the applied force in the frequency domain, can be described by mode summation approach. For continuous system, modal parameters of the system are investigated through the governing equation, and the form of FRF is examined in the same way to the previous one by mode summation approach. In order to carry out experiments for vibration analysis, covers following contents are covered in the class [5]. 1. Different signals of force and types of sensors 2. Fundamental understanding of digital signal (leakage, window, power spectra, coherence and etc.) 3. Modal parameter estimation methods (peak picking, circle fitting, etc.) To relate the theory with the experiments and for students’ better understanding of the knowledge delivered in the class, two projects are assigned. One, as a common project to every student, is for verifying the theory, which they have learned in the class, by implementing the simple model (beam, plate). The basic purpose is to obtain the dynamic characteristics (natural frequency, damping, mode shape and FRF) of the system through analytic approach and these obtained results are verified from numerical analysis and experimental results. The other project is chosen as an individual topic that should be related to individual research area. S.Wang ( ) • J. Lee • Y. Cho • H. Ryu • K. Park Gwangju Institute of Science and Technology (GIST), Gwangju 500-712, Republic of Korea e-mail: smwang@gist.ac.kr G. Foss and C. Niezrecki (eds.), Special Topics in Structural Dynamics, Volume 6: Proceedings of the 32nd IMAC, A Conference and Exposition on Structural Dynamics, 2014, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-04729-4__1, © The Society for Experimental Mechanics, Inc. 2014 1
2 S. Wang et al. This paper is organized as follows. In Sect. 1.2, a brief description of the contents covered in the class is introduced, and in Sect. 1.3, the conducted projects are introduced as divided into common and individual topics. This paper is concluded in the Sect. 1.4. In addition, the detailed description of the commercialized non-contact sensor, i.e. laser scanning vibrometer (LSV), which is developed by this laboratory and is used in the common project, is attached as an Appendix. 1.2 Class Contents 1.2.1 Single Degree of Freedom In this chapter, a single DOF system as shown in Fig. 1.1 is introduced. It begins with derivation of the equation of motion shown in Eq. (1.1) for the system using Newton’s second law. mRxCcPx Ckx DF.t/ (1.1) Next, the explanation of the resonance and resonance frequency is introduced from the undamped free vibration case. The solution of the homogeneous differential equation is derived. Next, damping ratio is introduced followed by vibration characteristics according to the damping (underdamped, over damped, critical) and the types of damping (structural, coulomb, viscous damping) are presented to the students. In the next problem, forced vibration is examined and shown through an example of vibration due to rotating unbalance under the harmonic excitation. The solution of forced vibration equation of motion composed of homogeneous solution and particular solution is shown, and the concept of frequency response function as shown in Eq. (1.2) is introduced. X F D 1 q.k m!/2 C.c!/2 (1.2) Students can understand that the vibration frequency response is dominated by the stiffness term at lower frequency than resonance frequency, and by the mass term at frequencies higher than the resonance frequency, also by the damping term at near resonance frequency as shown in Eqs. (1.3)–(1.5). Then, structure modification using vibration frequency response characteristics and estimation of mass and stiffness based on an experiment are introduced. ! !n W X F 1 k (1.3) ! !n W X F 1 m!2 (1.4) ! !n W X F 1 jc! (1.5) Fig. 1.1 Damped single degree of freedom system
1 Vibration Class at GIST, Korea 3 Fig. 1.2 Half power points and Q-factor Fig. 1.3 (a) System excited by motion of support point (b) Transmissibility curve The estimation of the damping ratio using Q-factor in FRF is offered to the students. Specifically, the process of calculating the Q-factor using half power points is explained and the equivalent viscous damping can be estimated from this process as shown in Eq. (1.6) and Fig. 1.2. Q 1 2 !n !2 !1 (1.6) Next, support motion is introduced, and it leads to the explanation of transmissibility as shown in Fig. 1.3. It is shown that the principle of vibration isolation, and the transmissibility becomes smaller than 1 at the frequency range where the frequency ratio is larger thanp2. In addition, method of obtaining the transient response under non-periodic excitation is introduced. Transient vibration problem under impulse excitation is firstly introduced in order to convey the content to students successfully. Then, it is shown that vibration response under arbitrary excitation can be obtained using the combination of the impulse responses. 1.2.2 Multi Degree of Freedoms Even though an actual structure is a continuous system, the continuous system is discretized and modeled as a multi degree of freedoms (MDOF) system (Fig. 1.4) for a practical sense. Finite element method, boundary element method, and transfer
4 S. Wang et al. Fig. 1.4 Damped multi degree of freedoms system matrix method are introduced as general discretization methods, and the discretization process of the vibration system is explained. The equation of motion in matrix form is derived as shown in Eq. (1.7). MRx N C CPx N C Kx N DF N .t/ (1.7) The MDOF undamped free-vibration problem is switched to an eigenvalue problem, and the result shows that eigenvalues and eigenvectors can be equally considered as natural frequencies and mode shapes respectively. Also, the relationship between the number of the degree of freedom and the number of the natural frequencies and mode shapes can be explained. The eigenvectors of the system are shown to be orthogonal with respect to both mass and stiffness matrices. Modal matrix which assembles eigenvectors into a square matrix is introduced. By using the modal matrix, decoupling of the forced vibration terms and modal damping concept can be explained. In forced vibration case, FRF as shown in Eq. (1.8) can be obtained using the orthogonality of eigenvectors. Additionally, Maxwell’s reciprocity theorem states that Hik DHki for the linear system, Hik .!/ D N X rD1 i r k r .kr !2mr/ Cj .!cr/ (1.8) When the system becomes larger and more complex, DOF is increased. This leads to the difficulty of calculating the exact solution. To solve this kind of problem, an approximate solution is introduced. Mainly, superposition methods such as mode displacement method (MDM), mode acceleration method (MAM), load dependent Ritz vectors (LDRV) method, Krylov sequence, and Lanczos algorithm are explained, and the advantages and disadvantages of the each method are explained as well. Newton’s second law, energy method, and virtual work method are compared with each other so that the equation of motion is formulated. Consequently, Lagrange’s equation as shown in Eq. (1.9) is introduced to formulate the large and complex system. d dt @T @Pqi @T @qi C @U @qi DQi (1.9) 1.2.3 Experimental Modal Analysis Experimental modal analysis (EMA) contains experimental measurement process for the FRF of the system; signal processing, and extracting the modal parameters (natural frequencies, mode shapes, and damping ratios) from measured FRF. Verifying a numerical model, determining dynamic durability by experiments, and machinery diagnostics for maintenance are possible by using the modal parameters. The techniques needed to experimentally determine the FRF showing the relationship of response and excitation force are introduced. There are two kinds of methods to excite a structure. First case is supplying excitation force by attaching a vibration exciter as shown in Fig. 1.5a. In this case, in order to reduce the mass loading due to an attached vibration exciter, a stinger should be used. The type of signals that is applied to the vibration exciter is introduced,and the characteristics in the
1 Vibration Class at GIST, Korea 5 Fig. 1.5 Vibration excitation methods (a) Vibration exciter with a stinger (b) Impact hammer frequency domain corresponding to each signal are described. Second case is to use an impact hammer as shown in Fig. 1.5b. In this case, the characteristics of excited force in the frequency domain are explained. Also, effects of the header and tip of the hammer corresponding to the frequency range of interest are discussed. Lastly, the advantages and disadvantages of both methods are compared. The sensors for detecting a response and excitation force applied to the structure are explained. There are two kinds of sensors to measure vibration signal, which are contact and non-contact. The contact sensors are an accelerometer and a strain gauge. Because they are attached to the structure, there is a mass added effect. The non-contact sensors are a position sensitive detector (PSD) to measure displacement and a LSV to measure velocity. Because LSV can rapidly measure the vibration of several positions, it makes EMA easier. LSV is a device that was developed in this laboratory; a detailed description is attached in the Appendix. Digital signal processing theory for obtaining FRF from measured data is introduced. Because EMA contains the sampling process which is converting an analog signal into a digital signal, Nyquist sampling theorem as shown in Eq. (1.10) should be described. fs 2f0 (1.10) The aliasing phenomenon in which high-frequency component is detected in the low-frequency component occurs when the sampling rate is not enough. Hence, antialiasing filter is used to prevent the aliasing phenomenon. Fourier transform is explained because frequency conversion is needed to obtain FRF from the measured signal. Also, the leakage in which the frequency component power leaks to adjacent frequency component is explained. The window function which can reduce the leakage is shown and, its principle is explained. In processing techniques of experimental measurement signal, correlation function and spectral density function can obtain the correlation between the two signals in time domain and frequency domain. The linear relationship between the input and output signals (FRF) can be expressed in the correlation function and the spectral density function. And the coherence function which may indicate the degree of noise mixed in the signal via the spectral density function is introduced. Therefore, a coherence function can show whether the characteristic values measured in the experiment are being measured correctly. The peak picking and circle fitting which extracts the modal parameters from experimentally measured FRF are explained. And it is possible to observe the mode shapes using predicted values at some points, i.e. by the mode analysis method introduced.
6 S. Wang et al. Fig. 1.6 Model updating process 1.2.4 Model Updating Process It is possible to verify the numerical model using the modal parameters obtained by the EMA. However, since the error exists between FE model and EMA results, the need for reliable FE model should be explained. A model updating for matching between the dynamic characteristics of numerical model and modal parameters obtained via the EMA by changing the parameters of the numerical model is introduced. It is the required process to construct the FE model for the optimal design. Figure 1.6 is a flow chart showing the model updating process of a square steel plate. 1.3 Class Project 1.3.1 Midterm Project Midterm project is assigned to offer an opportunity to recognize the relationship between vibration theory and experiment based on the class contents introduced in Sect. 1.2. Specifically, student should obtain modal parameters using the governing equation of the system through analytical, numerical and experimental approach. In this way, students can understand each process and the relationship between them. Finally, numerical model updating process based on the experimental results is conducted repeatedly for the numerical model validation. The following two application systems are considered.
1 Vibration Class at GIST, Korea 7 Fig. 1.7 Midterm project of a cantilever beam using analytical, numerical, experimental approach 1. Cantilever beam: Accelerometer and Impact hammer 2. Plate and Brake-disk: LSV and Vibration exciter 1.3.1.1 Cantilever Beam Cantilever beam is given to the students for the experimental modal test using an impact hammer and an accelerometer. As mentioned previously, modal parameters using analytical and numerical approach are obtained using the governing equation and boundary condition of the system. Meanwhile, modal parameter using experimental approach is obtained through roving impact hammer test as shown in the experimental setup in Fig. 1.7. Student can verify the vibration theory of the cantilever beam by comparing the results from three different approaches. 1.3.1.2 Plate and Brake-Disk Next, plate and brake-disk are given to the student for the experimental modal test using a laser scanning vibration rather than an impact hammer and an accelerometer. Analytical approach is based on a related reference paper [6]. COMSOL (commercial numerical analysis program) offered to students for finding the modal parameters of the structure numerically; material properties of the structure are given in Table 1.1. A laser scanning vibrometer is offered in experimental approach (non-contact sensor) for measuring the vibration response of the system under the excitation condition using a vibration exciter. Tables 1.2 and 1.3 represent modal parameters of plate and brake-disk respectively obtained from student’s midterm report. Students can understand different facts from Tables 1.2 and 1.3. Firstly, modal parameter results of the plate using analytical approach and numerical approach are similar. On the other hand, modal parameter results using experimental approach has a little difference as compared with that of other two approaches. This result tells that the numerical model used in numerical approach has a little difference as compared to actual system. Finally, modal validation process is conducted while repeating the numerical model updating in order to reduce this difference.
8 S. Wang et al. Table 1.1 Description of plate and brake-disk and experimental setup Objectives Material properties Plate 1. Dimension: 0.119 0.119 0.0006 (m3) 2. Density: 2,700 (kg/m3) 3. Young’s Modulus: 75 (GPa) 4. Poisson’s ratio: 0.3 Brake-disk 1. Density: 7,450 (kg/m3) 2. Young’s Modulus: 115 (GPa) 3. Poisson’s ratio: 0.33 Boundary condition Free–free condition Experimental setup Table 1.2 Modal parameter results of a plate 1 2 4 5 6 Analytical approach (Hz) 145.076 212.833 376.687 376.687 661.719 Numerical approach (Hz) 144.515 210.729 373.430 374.450 656.764 Experimental approach (Hz) 144 225 336 343 536 1.3.1.3 Conclusion of Midterm Project Students have an opportunity to apply the vibration theory into the real system through a modal analysis using analytical, numerical and experimental approach. Moreover, students can understand the features and pros and cons of the each modal analysis approach. Lastly, they can learn the process of numerical model updating and validation based on the experimental result. 1.3.2 Individual Project Individual project proceeds with proposal and presentation. First of all, the description of the project and the information regarding student final goal are presented in the proposal, the students have to show detail project overflow, as develop through discussion and brainstorming. The evaluation on the conducted individual project is carried out depending on the
1 Vibration Class at GIST, Korea 9 Table 1.3 Modal parameter results of a brake-disk 1 2 3 4 Numerical approach (Hz) 1,358.199 2,555.593 2,578.565 3,050.416 Experimental approach (Hz) 1,434 2,616 2,912 3,370 Fig. 1.8 (a) Experimental configuration and measured nodes using LSV (b) Attached vibration exciter and accelerometer on the backside of the plate as an operator and a reference novelty, difficulty, degree of performance compared to the proposal, presentation material, etc. In this paper, a research on the damage detection by using operating deflection shape (ODS) is introduced, which is selected from the presented researches. Its purpose is to find out the location of damage or failure of a mechanical system by comparing [through Eq. (1.11)] the deflection shapes between un-damaged and damaged one. This project is briefly introduced in the following Figs. 1.8 and 1.9. This research performed in the class was published in the journal paper [7]. Damage detection D.'hi ui / 2 (1.11) 1.4 Conclusion This paper is an introduction given to graduate students in vibration class at GIST, Korea. Vibration class contents and individual projects are introduced. These individual projects are assigned to the students for better understanding of vibration by applying the vibration theory that they learned in the class to the real mechanical system. Midterm project is used to familiarize the students with the vibration system using analytical numerical and experimental approach. Finally, validation of the numerical model is conducted using model updating procedure. Another individual project is assigned to combine the theory with the individual research topic of the respective student. Moreover, Appendix covers the syllabus of the vibration class and the explanation of the commercial non-contact sensor developed and used by this laboratory.
10 S. Wang et al. Fig. 1.9 (a) Locations of the damages (b) Comparison result between damaged ODS and un-damaged ODS bar and contour plot A.1 Appendix A.1.1 Syllabus 5603 Advanced Vibration Summary: In this course, vibration phenomena of electro-mechanical systems due to dynamic load is studied. It covers from single DOF to multi DOF and theory as well as numerical and experimental methods. It deals various subjects: Lagrange equation, Laplace transformation, Fourier transformation, mode superposition, finite element method, experimental modal analysis, random vibration, mode component synthesis, rotor dynamics, vibro-acoustics. Text: 1. Theory of Vibration with Applications, 5th ed., W. T. Thomson and M. D. Dahleh, Prentice Hall, 1998. References: 2. Vibration with Control, D. J. Inman, Wiley, 2006. 3. Structural Dynamics: An Introduction to Computer Methods, R. R. Craig, John Wiley & Sons, 1981. 4. Finite Element Procedures, K.J.Bathe, Prentice Hall, 1996. Prerequisites by Topics: Engineering Mathematics Fundamental Vibration Tools Used: Experimental Modal Analysis: Laser Scanning Vibrometer, Pulse Modal Test, LMS CADA-X FEA codes: MSC/NASTRAN, ANSYS, COMSOL, SYSNOISE Math tools: Matlab Topics: SDOF [1] Free Vibration (damp free, damped) Forced Vibration (unbalance, vibration isolation, damping) Transient Vibration (impulse, arbitrary, shock) MDOF [1] (continued)
1 Vibration Class at GIST, Korea 11 (continued) MDOF (mode, forced harmonic vibration) Properties of Vibration Systems (flexibility influence, stiffness influence, Castigliano’s theorem, modal matrix) Lagrange’s Equation (virtual work, Hamilton’s Equation) Vibration Test [2] Measurement Hardware Digital Signal Processing Random Signal ModeShape Computational Methods [3] Finite Element Method Static Problem (Gaussian elimination, Cholesky decomposition) Eigenvalue Problem (Rayleigh method, Lanczos) Harmonic Problem (direct frequency, modal frequency) Acoustics Transient Problem (direct integration, mode superposition) Component Mode Synthesis [2] Static Condensation/Super Element Component Mode Synthesis Design Sensitivity Analysis DSA of Static Problem DSA of Eigenvalue Problem DSA of Noise and Vibration Random Vibration [1] Vibro-acoustics Rotor Dynamics Exams: Midterm (35) Projects: 2 (15, 30) Homeworks: (20) A.1.2 Introduction to the Development of Laser Scanning Vibrometer A.1.2.1 Motivation of LSV Development In general, vibration measurement equipment is essential in the development process of the structure system, which is used to identify the vibration characteristics of the structures including electric motors, automobiles, aircraft structures, nuclear reactors, towers, etc. In the past, the contact sensor is mostly used in identifying vibration characteristics. However, the contact sensor can induce changes in the dynamics characteristics of the structure and has several limitations in vibration measurement due to its own features. For these reasons, the demand for the laser vibrometer is on the increase globally [8]. Currently, the most famous and expensive laser vibrometer has been developed by Polytec. However, as it is too expensive, intelligent system design laboratory (ISD) and venture company “EM4SYS” (www.em4sys.com) are collaborating for the development of the LSV (Fig. 1.10) in the School of Mechatronics of Gwangju Institute of Science and Technology. A.1.2.2 Specification of LSV Next is the specification of LSV as shown in Table 1.4 [9].
12 S. Wang et al. Fig. 1.10 Image of Laser Scanning Vibrometer (LSV) Table 1.4 Specification of laser scanning vibrometer ESV-200 Laser scanning vibrometer Laser wavelength He–Ne laser 632.8 nm, Class II <1 mW, eye-safe Laser beam size 100 m at 1 m distance Working distance 0.4–40 m (depending on reflection condition) Sample size Several mm2 up tom2 range (depending on distance) Maximum velocity 2,530mm/s Velocity output ˙10 V (analog) Velocity gain 3 mm/s/V 15 mm/s/V 30 mm/s/V 120 mm/s/V 240 mm/s/V Displacement gain 4 m 20 m 40 m 160 m 320 m Velocity resolution 0.5 m/s Frequency range 2 Hz to 40 kHz Low pass filter 1 kHz 5 kHz 10 kHz 20 kHz 50 kHz Scan grid Multiple grid densities and coordinate systems Scan angle 25ı 25ı scanning range Angular resolution 0.01ı Scan speed Up to 10 pts/s Power 220 V AC (50/60 Hz), 70 W Operating temperature C5 ıC to C40 ıC Storage temperature 10 ıC toC65 ıC Relative humidity Max. 80 %, non-condensing Dimension (mm) Head Controller 305 200 160 (L W H) 360 435 132 (L W H) Weight Head Controller 10 kgf 11 kg A.1.2.3 Features of LSV 1. Noncontact vibration and displacement measurement 2. Quick and easy to setup 3. Time and cost saving with no transducer mounting, wiring, and signal conditioning 4. Modal analysis by measuring the complete spectra of all scanned points 5. Operational deflection shape (ODS) analysis by measuring the operating structural vibration 6. Dynamic displacement visualization of a vibrating object with an arbitrary shape 7. 3D vibration measurement using a single laser scanning vibrometer by moving to three different locations
1 Vibration Class at GIST, Korea 13 Table 1.5 Software features of laser scanning vibrometer Analysis Modal Analysis (Peak-picking method) and operational deflection shape (ODS) Analysis Versatile mesh generation Line, rectangle, circle, polygon, copy and delete and rotate and resize Excitation signal Random, sine-sweep, hammering Data analysis Data export to universal file format (UFF) for Me’scopeImport to MATLAB/ MS EXCEL Time data Measured time data is saved automatically in the setup folder Fig. 1.11 Applications of Laser Scanning Vibrometer (a) automobile (b) aircraft (c) brake-disk (d) micro-speaker (e) hard disk (f) small scale and heated object A.1.2.4 Software Features of LSV Next are software features of LSV as shown in Table 1.5. A.1.2.5 Applications of LSV There are a lot of applications of laser scanning vibrometer as shown in Fig. 1.11.
14 S. Wang et al. References 1. Thomson WT, Dahleh MD (1998) Theory of vibration with applications, 5th edn. Prentice Hall, Englewood Cliffs, NJ 2. Inman DJ (2007) Engineering vibration, 3rd edn. Prentice Hall, Englewood Cliffs, NJ 3. Craig RR (1981) Structural dynamics: an introduction to computer methods. Wiley, New York 4. Bathe KJ (1996) Finite element procedures. Prentice Hall, Englewood Cliffs, NJ 5. Ewins DJ (2000) Modal testing: theory, practice and application, 2nd edn. Research Studies Press, Baldock, Hertfordshire, England 6. Leissa AW (1973) The free vibration of rectangular plates. J Sound Vib 31(3):257–293 7. Bae WK, Kyong YS, Park KH, Dayou J, Wang SM (2011) Scaling the operating deflection shapes obtained from scanning laser Doppler Vibrometer. J Nondestruc Eval 30(2):91–98 8. SPIE Annual Report, 2000 9. Catalog of ‘Laser Scanning Vibrometer, VIXCEL Series’, EM4SYS, 2013
Chapter2 Lab Exercises for a Course on Mechanical Vibrations Anders Brandt Abstract This paper presents some exercises designed to teach fundamental aspects of mechanical vibrations in general, and experimental techniques for vibration measurements in particular. Teaching students to become good experimentalists is a very difficult task, and is perhaps not even possible inside standard curricula. However, some fundamental aspects of experimental work must be taught, and can be included in a course on vibrations as well as in other courses. The first exercise is designed to teach the student how careful one has to be when applying vibration sensors to a structure, and is based on the repeatability of mass calibration measurements. This makes it a good exercise to base a discussion on experiment setup and repeatability issues etc. The second exercise is an exercise where an approximate single-degree-of-freedom (SDOF) system is investigated by some simple analytical calculations as well as by an experimental measurement. This exercise serves to demonstrate the rather abstract notion of SDOF, and also illustrate the applicability of a simplified model in a limited frequency range. Finally, a third exercise is made where modal analysis of a slalom ski using impact testing is used as a demonstration of more advanced vibration analysis. Keywords Vibration measurement • Teaching vibration • Mass calibration • Accelerometer mounting • Experimental modal analysis 2.1 Introduction Making accurate vibration measurements is known to be rather difficult due to the many challenges of sensor choice, sensor mounting, and, in the case of investigation of eigenfrequencies (modal analysis), for example, the problems of suspending the structure properly. As this paper describes some exercises designed to teach students some good practice for vibration measurements, additional difficulties for students are often that they are not yet mature when it comes to perform good measurements; i.e. they have not yet learnt to be patient, to double check everything, etc. The course, for which the exercises described in the present paper are used, is a graduate (master) level course of a third semester length. The students have had a course on general signal processing, but are unfamiliar with the particular measurement and analysis techniques for vibration analysis. In the course, both wave theory of continuous structures and discrete mechanical systems are taught; the exercises covered in the present paper, however, focuses on the discrete system description of mechanical systems. The particular difficulties of vibration measurements addressed by the exercises described here are • to learn some experimental methodology to ensure good experimental results, such as checking repeatability, double checking everything that could potentially affect the measurement, etc., • to yield respect for the particular difficulties of applying accelerometers correctly, and • to correctly suspend a structure under free–free conditions for experimental modal analysis An additional point which is very important for these exercises is, of course, to tie the theory taught in the course to experimental results. A. Brandt ( ) Department of Technology and Innovation, University of Southern Denmark, Campusvej 55, 5230 Odense M, Denmark e-mail: abra@iti.sdu.dk G. Foss and C. Niezrecki (eds.), Special Topics in Structural Dynamics, Volume 6: Proceedings of the 32nd IMAC, A Conference and Exposition on Structural Dynamics, 2014, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-04729-4__2, © The Society for Experimental Mechanics, Inc. 2014 15
16 A. Brandt Perhaps a good way to summarize the spirit of the exercises described here is to use the famous quote by Albert Einstein; “A theory is something nobody believes, except the person who made it. An experiment is something everybody believes, except the person who made it.” The exercises described in this paper, addresses first and foremost the second part of this quote; the importance of being in constant doubt over ones experimental results. The exercises are, however, also touching on the essence of also questioningtheoretical results, and the importance of verifying ones theories (which in our engineering terminology are, of course, usually calledmodels). 2.2 Exercises The exercises we are about to describe here, require some basic vibration measurement equipment and sensors. The measurement hardware and software can be essentially any vibration measurement system. We are using a National Instruments 9234 USB box, driven by homemade MATLAB software using the Data Acquisition Toolbox, and the free ABRAVIBE [1] toolbox for the analysis. This means that the students record time data, for subsequent analysis in MATLAB. We have found that this ensures that the students understand every step in the processing of the data; something more “automatic” commercial systems often make more difficult. On the sensor side, in the exercises we use two accelerometers, a force sensor, an impact hammer, and a shaker with amplifier and random noise generator. Which particular sensors used are not particularly important, although the accelerometers should weigh less than 10 g, the force sensor should be of suitable sensitivity, and the same is true for the impact hammer. We currently use Dytran 3097A2T accelerometers with a sensitivity of 100 mV/g and mass of 4.3 g. The force sensor is a Kistler 9712B50 with a sensitivity of approximately 20 mV/N, and the impulse hammer is a Dytran 5800B3T with a sensitivity of approximately 10 mV/N. In addition to this, some relatively inexpensive measurement objects are needed, which will be described in conjunction with each exercise. The first three exercises described in Sects. 2.2.1–2.2.3 are made at one instance in the laboratory, in approximately 2 h, followed by 4–6 h of analysis and report writing. The last exercise, described in Sect. 2.2.4, is accomplished in a second laboratory session, in approximately 4 h, followed by 6–8 h of analysis and report writing. 2.2.1 Mass Calibration The first exercise we present, is based on calibration of an accelerometer (or a force sensor in an impact hammer) using a simple mass, as depicted in Fig. 2.1. This well-known technique, see for example [2], should be familiar to everyone making vibration measurements. It is a good technique not only for calibration, but also for checking that accelerometers are functional through the full frequency range, and for checking which frequency range a particular accelerometer can be used for (see Sect. 2.2.2), etc. The calibration procedure is simple; the accelerometer is attached to a mass, typically a piece of round rod of steel. The weight of the mass is accurately measured, and should be different depending on the impact hammer used, so that a soft hit with the hammer produces a suitable acceleration level. For this exercise we use a weight of approximately 1 kg, made of a piece of rod with approximately 40 mm diameter. The mass is hanging in two strings from a supporting rig, so that it can Mass, M Accelerometer Impact Hammer Fig. 2.1 Setup for mass calibration for the exercise described in Sect. 2.2.1
2 Lab Exercises for a Course on Mechanical Vibrations 17 move as a pendulum in the direction of the accelerometer as illustrated in Fig. 2.1, or alternatively placed on a soft foam pad. The mass is then excited by the impact hammer, while the force and accelerometer signals are recorded. The hardest tip for the hammer is chosen, to yield a frequency range as high as possible, typically up to 10 kHz. To give better accuracy, several impacts can be averaged, although this is generally not necessary if the sensitivities of the force sensor in the impact hammer and the accelerometer are appropriate, so that the measurement noise is minimal. After the time signals with the impact force and resulting acceleration are recorded, the students calculate the frequency response (FRF) of acceleration with force. Since Newton’s well-known formula for a mass is FDMa, this FRF should be a constant HDA/FD1/M, i.e. the FRF forms a straight line, independent of frequency. The exercise is made so that the students are given the sensitivity of the force sensor in the impact hammer, but not the sensitivity of the accelerometer: They are then asked to calculate the sensitivity of the accelerometer, given the measured frequency response at, for example, 159.2 Hz, which is equal to 1,000 rad/s, a common frequency for this purpose. 2.2.2 Accelerometer Mounting The next exercise is using the mass calibration method described in Sect. 2.2.2 to investigate the effects of different mounting techniques for mounting accelerometers. This exercise has several important objectives; first of all it obviously discusses different means of attaching an accelerometer to the test structure and, as we will see, what performance these mounting techniques result in. Second of all, it demonstrates the difficulty of getting repeatable measurements, as in most cases the students do not get the same result even if they use the same mounting technique twice. This also makes a good point of discussing the concept of repeatability, and the importance of this concept in engineering (or science in general). Third, I am using this exercise to teach the students not to trust their measurements, until they have investigated that the accelerometer they use with a particular mounting technique, actually has a frequency range high enough for the measurements they want to make; this is very easily investigated by using the mass calibration method. Fourth, this exercise also illustrates that accelerometers, like all measurement sensors, are not perfect, but vary rather much with frequency. In this exercise, the students use some different techniques to mount an accelerometer on the calibration mass, and perform measurements as described in Sect. 2.2.2. For each measurement the FRF is calculated and stored. In our case we mount the accelerometer with the following techniques: • a thin layer of wax, • a thick layer of wax, • a thin layer of hot glue (hot melt adhesive, using a “hot glue gun”), and • super glue (cyanoacrylate adhesive) Other techniques such as screw mount and magnetic base could also be used. They are, however, somewhat difficult to make identical to the techniques above, as they change the mass of the accelerometer unit. Since the students already have made a measurement with a thin layer of wax in the first exercise, this means that they obtain a total of five different FRFs, of which the two first should be identical (or very similar). These two FRFs based on a thin layer of wax are first compared. Regardless of whether they are identical or not (often they are not; usually because of too much wax) a fruitful discussion on repeatability issues is held, and the students who gets almost identical results are told that they have applied a sufficiently thin layer of wax. Many students are surprised how thin this layer has to be to yield identical FRFs. As the next step in this exercise, the students are asked to plot all five FRFs in one plot, and determine which of the mounting techniques works best. This produces a plot similar to Fig. 2.2, where the FRFs have first been normalized to have the same value at 159.2 Hz, and limits of ˙5 % around this value are plotted, to indicate the accuracy limits specified by the sensor manufacturer. Note that only one of the two thin wax measurements is included in the figure, which is for clarity only. The students are finally asked to find the frequency where the error reaches 5 % for all mounting techniques, and find the technique giving the highest frequency limit. This often results in a dead-end between superglue and wax. It should be stressed, and we do this during the lab exercise, that the frequency range obtained by the method described here, is not necessarily the frequency range obtained when the accelerometer is mounted in a point with less stiffness, such as on a lightly damped structure. It is therefore necessary to consider the obtained frequency range found on the calibration mass as a maximum frequency range, and to use some margin when using the accelerometer on a real structure.
18 A. Brandt 100 200 500 1000 5000 0.8 0.95 1 1.05 1.2 Frequency [Hz] Acceleration/Force [1/kg] Thin Wax Thich Wax Hot Glue Super Glue Fig. 2.2 Comparison of the FRFs from four measurements on a mass, using four different mounting techniques as described in Sect. 2.2.2. The FRFs are normalized to the same value at 159.2 Hz (1,000 rad/s) (from [2], Copyright 2011, John Wiley and Sons; reprinted with permission) Shaker M4 “spring” Force transducer Cubic mass Accelerometer Base plate Stinger Shaker support Fig. 2.3 Schematic illustration of the approximate SDOF system used for the exercise described in Sect. 2.2.3 2.2.3 SDOF Measurement and Analysis The single degree of freedom (SDOF) system is a key component in vibrations and structural dynamics. In this exercise, a simple system behaving as an approximate SDOF system is investigated, and used to illustrate the connection between theory and real world to the students. The system used is shown schematically in Fig. 2.3, and consists of a base plate of steel, approximately 100 300 10 mm; a steel cube with 30 mm side length; and a M4 bolt, approximately 50 mm long. There are two locking nuts locking the M4 bolt against the base plate and the mass. The mass is excited by a random force applied by a shaker through a stinger and a force sensor, and an accelerometer is mounted on the opposite to the force sensor. The FRF between the force and the acceleration is measured and compared to analytical results. This exercise has several objectives; first it illustrates that the SDOF system used in theory can, at least in a limited frequency range, be found in “real life”. Second, it allows for discussion between model and reality, since the results obtained are rarely identical to the analytical results calculated by the students. The stiffness of the M4 bolt is readily calculated using known formulas for moment of inertia and stiffness of a beam, and is omitted here so that professors can include this step as an exercise for the students with having direct access to the answer. The students are asked to measure the various components, and estimate the mass of the cube, including the accelerometer, and the stiffness, and from this estimate the “SDOF” natural frequency. From the measured FRF, the students are asked to approximately estimate the natural frequency and damping (through the 3 dB bandwidth and natural frequency). Once these parameters are obtained, the students should calculate the mass, stiffness, and damping coefficients of the system. The mass and stiffness coefficients thus obtained are compared with the mass calculated from the measurements of the cube, and the stiffness calculated for the beam. Since the results rarely compare particularly closely due to the lack of “precision” in the setup, it forms a good discussion point for differences between model and reality.
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