Modal Characterization of a Thin Flat Plate in the Free-Free Condition with Non-Contact Particle Velocity Measurements by Kevin Wilby A Project Submitted to the Graduate Faculty of Rensselaer Polytechnic Institute in Partial Fulfillment of the Requirements for the degree of MASTER OF ENGINEERING in MECHANICAL ENGINEERING Approved: _________________________________________ Ernesto Gutierrez-Miravete, Project Adviser Rensselaer Polytechnic Institute Hartford, Connecticut December 2012 i CONTENTS 1. Introduction.............................................................................................................. - 1 2. Background .............................................................................................................. - 2 3. Theory/Methodology ............................................................................................... - 5 3.1 Theoretical Prediction of Natural Frequencies .............................................. - 6 3.1.1 Analytical Prediction of Natural Frequencies .................................... - 6 - 3.1.2 Finite Element Prediction of Natural Frequencies ............................. - 8 - 4. Results and Discussion .......................................................................................... - 10 4.1 Test Setup ..................................................................................................... - 10 - 4.2 Test Results .................................................................................................. - 13 - 5. Conclusions............................................................................................................ - 16 6. References.............................................................................................................. - 17 Appendix A – Analytical Prediction Spreadsheet ....................................................... - 18 Appendix B – Microflown Scan and Paint Inputs ....................................................... - 19 - ii LIST OF TABLES Table 1 Input Parameters to Theoretical Predictions ..................................................... - 6 Table 2 Analytically Derived Natural Frequencies ....................................................... - 7 Table 3 Calculated Natural Frequencies from FEA ...................................................... - 8 Table 4 Measured Natural Frequencies from White Noise Excitation ........................ - 14 Table 5 Comparison of Calculated and Measured Natural Frequencies ..................... - 16 - iii LIST OF FIGURES Figure 1. Picture of the Two Platinum Wires in the Microflown from [3] ................... - 3 Figure 2. Directionality of the Microflown PU Probe from [7] .................................... - 3 Figure 3. Comparison of the Point response from an Accelerometer, Laser Vibrometer and a Microflown Sensor [3] ......................................................................................... - 5 Figure 4. Predicted Mode Shapes from [10] .................................................................. - 7 Figure 5. FEA Mode Shapes (1st to 6th Modes From Upper Left to Lower Right) ....... - 9 Figure 6. Picture Showing Test Setup ......................................................................... - 10 Figure 7. Shaker Test Line Diagram ............................................................................ - 11 Figure 8. Typical Spherical Piston Calibration Setup ([13]) ....................................... - 12 Figure 9. Measured Velocity Response from a White Noise Excitation ..................... - 13 Figure 10. PU Mode Shapes (1st to 6th Modes from Upper Left to Lower Right) ....... - 15 - iv LIST OF SYMBOLS i horizontal half waves j vertical half waves πππ natural frequency (i horizontal and j vertical half waves) [Hz] πππ dimensionless frequency parameter (i horizontal and j vertical half waves) π length of the plate (meters) πΈ modulus of elasticity (pascals) β thickness of the plate (meters) πΎ mass per unit area (kg/meter^2) π poisson’s ratio π₯ displacement (meters) π΄ scale factor (meters) π sqrt(-1) π radian frequency (radians) π‘ time (seconds) π₯Μ velocity (meters/second) π₯Μ acceleration (meters/second^2) π pi v LIST OF ACRONYMS PU Pressure/Velocity FEA Finite Element Analysis G Acceleration due to Gravity (meters/second^2) vi ABSTRACT Structural acoustics is the study of the vibrations, and their propagation through solid objects. There are many industries that use vibratory analysis to provide insight to the key strengths and weaknesses of structures. This project describes a novel type of measurement (the Microflown sensor) used to perform a structural characterization of a flat plate and applies both analytical and finite element tools to validate the test results. The Microflown is a sensor that directly measures the particle velocity of the air moving past it. The Microflown measurement system has been shown in this project to accurately capture the dynamic response of a structure. vii 1. Introduction Structural acoustics is the study of the vibrations, and their propagation through solid objects. Understanding the structural dynamics of a structure can provide valuable information on its strengths and weaknesses for any given application. Vibratory analysis is constantly being performed in industry from assessing structural integrity in the aerospace industry [1] to minimizing the acoustic response of a structure in the automotive industry [2]. [1] discusses a computational technique for assessing the integrity of structures. This is a very powerful diagnostic tool that can provide indications of machine and structure wear by monitoring the vibrations. Signals related to rotational harmonics for example can indicate bearing failure before more damage is done to a machine. [2] discusses the design of a car muffler that uses microperforations to increase damping and in turn better absorb the sound waves. Understanding the structural response of the many moving parts in a car is very important as the perceived sound of a car can be a major part of its value. Generally speaking all structural acoustic analyses can be broken down into three types: simplistic analytical, detailed finite element and full or scale model testing. Typically analytical models are used wherever possible because of their ease of use, however, most real-life problems are much more complex then these models can replicate. Finite element models give the ability to analyze a structure well before it’s actually built. However, these models take longer to build and analyze than the analytical models and need a lot of validation for the results to be used for complex structures. Scale model testing can be used as insight into the dynamics of the structure as well as input into the finite element models, however can prove to be costly and may not fully replicate the full scale structure. Lastly, full scale testing is typically the best to analyze a structure, however, does not have the ability to be analyzed until the system is built, and is usually the most costly solution. This project focuses on the testing aspect of structural acoustics and uses both analytical and finite element tools to validate the test results. -1- 2. Background Modal testing is a way to quantify the dynamic response of a structure. Typically, this is done by exciting the structure (possibly with a calibrated force hammer or a shaker) and then measuring its response (usually with accelerometers). The procedure for these types of tests can vary significantly based on the application and in some cases can involve long setup times and/or long test duration. Accelerometers can be mounted in many ways (e.g. threading them into a hole, glue, wax) which depending on the test can be time consuming and potentially affect the structural response of the system (with the added mass of the accelerometer itself). Non-contact vibration measurements have the advantage that there is no influence on the structural response. This type of measurement can be performed with a sensor called Microflown [3]. The Microflown system performs non-contact measurements that are relatively easy to generate for both quick look and more detailed analyses. The Microflown is a sensor that directly measures the particle velocity of the air moving past it by the use of two heated platinum wires (Figure 1). As outlined in [3], the Microflown simply measures the difference in voltage caused by the heat of each platinum wire that is lost due to convection. By using two wires instead of one, and looking at the difference in voltage between these two wires, a highly directive measurement of velocity can be performed. The PU probe is a sensor package that combines this Microflown technology with a small 1/8” pressure sensor. This allows for a measurement of intensity and acoustic impedance (beyond just the velocity and pressure response). Additionally, the highly directional nature of the velocity probe (Figure 2) provides the ability for sound source localization. This project focuses on the velocity measurements, however, intensity and impedance measurements are discussed in more detail in [4] and [5], respectively and sound source localization is discussed in [6] as well as [3]. -2- Figure 1. Picture of the Two Platinum Wires in the Microflown from [3] Figure 2. Directionality of the Microflown PU Probe from [7] The Microflown PU probe will be used in this project to measure the surface velocity of a thin flat plate in the free-free condition. The Microflown Scan and Paint system (which uses a camera in conjunction with the PU probe) was chosen for this -3- project because it provides a visualization of the velocity profile of the plate. These measurements will be compared to both analytical and finite element methods to validate the measurement technique. The free-free boundary condition was chosen because the analytical and finite element methods should be very accurate for such a well researched case. Specific applications to this project are any industries that perform modal analyses (such as the aerospace and automotive industries). -4- 3. Theory/Methodology The Microflown is a sensor that directly measures the particle velocity of the air moving past it and will be used in this project and compared to both analytical and finite element prediction methods. This comparison involves the assumption that the particle velocity, when measured in the very near field (where the air can be approximated as incompressible), coincides with the surface velocity of a structure. This assumption is explained in detail in [8]. Additionally, this technique has been validated in [3] by comparing the point response of an accelerometer, laser vibrometer and a Microflown probe. Figure 3 shows the test setup from [3] as well as the comparison of the three sensors. Figure 3. Comparison of the Point response from an Accelerometer, Laser Vibrometer and a Microflown Sensor [3] The structure in this report will be a thin plate in the free-free condition. As defined in [9], a thin plate is one with a thickness that is less than 1/10 the minimum lateral plate dimension. The plate used in this project is made of steel and has the dimensions of 8in by 12in by ½ in. The free-free condition of a thin flat plate was used to provide a condition that has been well investigated and the accuracy of the prediction methods is known. The test setup is a plate hung with rope, excited by two shakers, and measured with the Microflown Scan and Paint system. This method is validated using [9] and [10], using theory of vibrations of plates and by creating the finite element model and solving for the natural frequencies. -5- 3.1 Theoretical Prediction of Natural Frequencies In this section, the natural frequencies of the thin flat plate are calculated by a textbook analytical method and by a finite element method. The parameters (geometry and material properties) used in each prediction method are outlined in Table 1. Table 1 Input Parameters to Theoretical Predictions Parameter length width thickness Young's Modulus Poisson's Ratio Density 3.1.1 Value 0.3048 0.2032 0.0127 2.0684E+11 0.3 7854.17 Unit m m m Pa kg/m^3 Analytical Prediction of Natural Frequencies The vibration of a thin plate in the free-free condition has been well investigated. Some examples include [11] which calculates the mode shapes of a free-free plate that is antisymmetric with respect to the x and y axis but symmetric about the diagonals and [12] which experimentally measures the frequencies and mode shapes for square brass plates. [9] provides a method for determining the natural frequencies of the first six modes of rectangular plates for all 21 possible combinations of the three elementary boundary conditions on the four edges of the plates (free, simply supported and clamped). The general assumptions of this method are: ο· The plates are flat and have constant thickness ο· The plates are composed of a homogeneous, linear elastic, isotropic material ο· The plates are thin (defined as the thickness of the plate is less than 1/10 the minimal lateral plate dimension) ο· The plates deform through flexural deformation ο· The in-plane load on the plates is zero Equation (1) provides an equation for calculating the natural frequency of the mode sequence for a plate: \ π2 πΈβ3 ππ √ πππ = 2ππ ; i=1,2,3… ; j=1,2,3… 2 12πΎ(1−π 2 ) -6- (1) Where fij is the natural frequency (in hertz) of the mode sequence of i horizontal half waves and j vertical half waves, λij is a dimensionless frequency parameter, a is the length of the plate, E is the modulus of elasticity, h is the thickness of the plate, γ is the mass per unit area, and ν is Poisson’s ratio. The values for the square of λij were taken from [9] for a length of width ratio of 1.5. The Table 1 geometry and material properties were used in this analysis. Table 2 and Figure 4 show each calculated frequency and mode shape, respectively. Note the different naming conventions used in [9] and [10] (i and j vs. m and n) and that modes 1-6 using the [10] naming convention would be would be [1,1], [2,0], [2,1], [0,3], [3,0], and [1,2]. Table 2 Analytically Derived Natural Frequencies Mode [i,j] 1 [2,2] 2 [3,1] 3 [3,2] 4 [1,3] 5 [4,1] 6 [2,3] 0 0 1 2 3 m ο¨ ο§ n 1 Frequency (Hz) 680.1 729.8 1576.2 1699.2 1966.4 2280.3 2 3 Figure 4. Predicted Mode Shapes from [10] -7- 4 3.1.2 Finite Element Prediction of Natural Frequencies A finite element analysis (FEA) was performed to predict the undamped natural frequencies and mode shapes of the thin flat plate in the free-free conditions. The model was built in HYPERMESH v11.0 and the Table 1 geometry and material properties were used. Once the model was built, the natural frequencies were calculated using the mass and stiffness of each element by importing the model into VIBES-SABER. Table 3 shows the calculated natural frequencies for each mode and Figure 5 shows a snapshot of the mode shape animation. Notice how each of the mode shapes correspond with the analytical mode shapes identified in Figure 4. Table 3 Calculated Natural Frequencies from FEA Mode [i,j] 1 [2,2] 2 [3,1] 3 [3,2] 4 [1,3] 5 [4,1] 6 [2,3] Frequency (Hz) 678.1 722.3 1563.0 1680.8 1943.7 2258.7 -8- Figure 5. FEA Mode Shapes (1st to 6th Modes From Upper Left to Lower Right) -9- 4. Results and Discussion Section 3 provided a prediction of the natural frequencies of the plate by two different methods (analytical and finite element). The focus of this project is to use the Microflown particle velocity sensor to measure the natural frequencies of this plate. The noncontact measurements that are possible with the Microflown system provide an easy to use tool that can quickly provide a detailed analysis of the dynamics of a structure. 4.1 Test Setup A thin flat plate made of steel and with the geometry of 8in by 12in by1/2in was used in this project. The plate had two holes drilled in it to provide a means for mounting it. The plate was hung by a rope to approximate a free-free condition. Two shakers were attached to the structure and excited in parallel to reduce the risk of the shakers placement on an anti-node for any given mode shape. The locations of the shakers can be seen in Figure 6 and are generally indicated on each of the figures in Section 4.2 that show test results. Figure 7 shows a line diagram of the test setup, where the entire test was run using the Microflown Scan and Paint hardware and software. Shakers Figure 6. Picture Showing Test Setup - 10 - PU Pressure PU Velocity LEMO LEMO Signal Conditioner BNC BNC Camera USB Microflown Digitizer USB CPU BNC Microdot Shaker 1 Microdot Shaker 2 Figure 7. Shaker Test Line Diagram The Microflown Scan and Paint hardware includes a PU probe, a signal conditioner (MSFC-4) and a digitizer (Scout 422). The PU probe simultaneously measures pressure and velocity. The signal conditioner receives both the pressure and velocity measurements, and provides a calibrated correction to the signal. The digitizer has two functions: it receives the pressure and velocity signals and sends them to the Scan and Paint software and provides the shakers with the excitation voltage. A camera is also used to record a video of the PU probe measurement. The Microflown Scan and Paint software takes in the pressure and velocity measurements and the video feed from the camera. The camera is set up perpendicular to the surface of interest. The probe is then scanned by the user over the entire surface of interest while maintaining a consistent distance (roughly 1-2 inches from the surface) and aiming the probe perpendicular to the surface. Both the audio (pressure and velocity) and video signals are recorded while the probe is scanned. Although there is no actual painting occurring, the Microflown Scan and Paint software will follow the probe from the video signal (typically by following a brightly colored object attached to the probe) - 11 - and synchronize it with the audio signal. Once the signals are synchronized, the measurements (such as those in Section 4.2) can be processed and analyzed. The Microflown PU probe requires a special spherical piston calibrator to perform an official calibration of the sensor (this is performed once a year). The procedure for this test is generally to excite the calibrator with a known signal, and use a reference microphone to relate this signal to both pressure and particle velocity [13]. Figure 8 shows the spherical piston calibrator, the Microflown probe and the reference microphone in a typical setup. Spherical Piston Reference Microphone PU Probe Figure 8. Typical Spherical Piston Calibration Setup ([13]) It is also good practice to do a quick bench top calibration to confirm that the probe is measuring correctly and that the appropriate sensitivity is being applied. This is done by taking a standard accelerometer calibrator that provides a G of acceleration (120dB re: μG) on a flat vibrating surface at 100Hz. By using Equation (2), one can easily relate acceleration response and velocity measurements. π₯ = π΄π πππ‘ Since: (ππ‘) π₯ = π₯Μ π ο¨ π₯Μ = πππ΄π πππ‘ π ο¨ π₯Μ = −π2 π΄π πππ‘ = πππ₯Μ Since: (ππ‘) π₯Μ = π₯Μ - 12 - (2) Where x is displacement, A is a scale factor, j is the sqrt(-1), ω is the radian frequency, t is time, αΊ is velocity, and αΊ is acceleration. Therefore, by placing the PU probe directly above the calibrator, and converting the velocity reading by simply multiplying by ω or 2*π*100Hz, the probe can be quickly calibrated to the signal. 4.2 Test Results The two shakers that were mounted to the plate were first excited with white noise and scanned using the Scan and Paint system. Figure 9 shows the measured velocity response of the plate (an averaged linear spectrum of each location). Each of the peaks shown represents a natural frequency of the plate. Table 4 shows the measured natural frequency for each of the first 6 modes (i.e. the first 6 peak frequencies). Total Average Linear Spectrum of U PVL (re 50e-9 m/s) [dB] 70 60 50 40 30 20 10 2 10 3 10 [Hz] 4 10 Figure 9. Measured Velocity Response from a White Noise Excitation - 13 - Table 4 Measured Natural Frequencies from White Noise Excitation Mode [i,j] 1 [2,2] 2 [3,1] 3 [3,2] 4 [1,3] 5 [4,1] 6 [2,3] Frequency (Hz) 684.8 726.3 1505 1682 1913 2207 Once the natural frequencies were identified with the white noise, the shakers were then excited with a sine wave at each of the frequencies of the peaks shown in Figure 9. Figure 10 shows the measured velocity profile for each of the first 6 modes. The colors on the graph correspond to the areas where the velocity is highest (red) and lowest (blue). Note that these measurements are taken as a steady state (i.e. the velocity profile is the absolute value of the velocity) as opposed to Figure 5 which is the instantaneous modal distribution of the plate. With this in mind, Figure 10 can be directly compared to the modal patterns in Figure 5 (e.g. 1st mode in the upper left of each figure should show the same pattern). The expected mode shapes (from both the FEA and the analytical solutions outlined in Section 3) as well as the shaker locations are also shown on each graph in Figure 10. The first, second and third modes (upper left, upper right and center left of Figure 10) show very clean mode shapes that closely follow the expected patterns. The fourth, fifth and sixth modes (center right, lower left and lower right of Figure 10), resemble the expected patterns, however they are not as clean. This could potentially be due to many factors. The shakers were randomly placed in an attempt to reduce the risk that they will both be placed on a nodal pattern line. However, it can be seen in the 5th and 6th modes (lower left and lower right of Figure 10) that the shaker locations appear to be located very close to a nodal line. Using a larger plate or more shakers could potentially have further reduced the risk of placing a shaker on a nodal line. Additionally, as can be seen in the 4th mode (center right of Figure 10), while the center nodal line can be easily seen, the edges do not show the detailed character of the expected pattern. This could potentially be improved upon by using a larger plate and/or using a finer grid in the Scan and Paint software. - 14 - Shaker Location Shaker Location Nodal Lines Nodal Lines Shaker Location Shaker Location Nodal Lines Nodal Lines Shaker Location Shaker Location Nodal Lines Nodal Lines Figure 10. PU Mode Shapes (1st to 6th Modes from Upper Left to Lower Right) - 15 - 5. Conclusions The Microflown measurement system has shown the capability to accurately measure the dynamic response of a structure. Table 5 shows the comparison of each of the three methods outlined in this project. The test was shown to be within 5% of either prediction method for each of the first 6 modes. The Microflown measurements were also shown to generally follow the mode shapes that were predicted from both the analytical and finite element methods. Improvements upon the test setup could potentially have shown a cleaner measurement for the higher modes (as well as the modes above the 6th). Table 5 Comparison of Calculated and Measured Natural Frequencies Mode [i,j] 1 [2,2] 2 [3,1] 3 [3,2] 4 [1,3] 5 [4,1] 6 [2,3] Analytical Frequency (Hz) 680.1 729.8 1576.2 1699.2 1966.4 2280.3 FEA Frequency (Hz) 678.1 722.3 1563.0 1680.8 1943.7 2258.7 Test Frequency (Hz) 684.8 726.3 1505.0 1682.0 1913.0 2207.0 Three different types of modal analyses were used in this project: analytical, finite element and test methods. Depending on the application, anywhere from one to all three of these methods may need to be used to fully understand the structure. Each of the analyses has their own strengths and weaknesses. The Microflown particle velocity sensor has advanced measurement technology. The results of this project have implications in any of the many industries that currently perform vibratory analysis. Non contact velocity measurements can be used to define the dynamics of a structure, provide insight to proper accelerometer placement and/or provide important input into analytical or finite element models. - 16 - 6. References [1] Shepherd, M; Conlon, S. “Structural Intensity Modeling and Simulations for Damage Detection”. ASME Journal of Vibration and Acoustics. October 2012. Volume 134 Issue 5 [2] Allam, S; “A New Type of Muffler Based on Microperforated Tubes”. ASME Journal of Vibration and Acoustics. June 2011. Volume 133 Issue 3 [3] de Bree, “The H-E. Microflown, E-Book”. <http:/www.microflown.com/library/books>. Microflown Technologies. Arnhem, The Netherlands. 2012. [4] Tijs, E; Nejade, A; de Bree, H-E. “Verification of P-U Intensity Calculation”. Microflown Technologies [5] Tijs, E; de Bree, H-E. “Recent Developments Free Field PU Impedance Technique”. Microflown Technologies. [6] Wind, J; de Bree, H-E. “Sound Source Localization and Sound Mapping Using a PU Sensor Array”. Microflown Technologies. [7] “Microflown Technologies – PU Regular Probe Manual”. V1.0 2009-03 [8] de Bree, H-E. “The Very Near Field Theory, Simulations and Measurements of Sound Pressure and Particle Velocity in the Very Near Field”. University of Twente. International Conference on Sound and Vibration. June 2004 [9] Blevins, R. “Formulas for Natural Frequency and Mode Shape”. Krieger Publishing Company. Copyright 1979 [10] Waller, M. “Vibration of Free Rectangular Plates”. Proc. Phys. Soc. (London), ser. B, vol. 62, no. 353, 1949, pp. 277-285 [11] Bazley, N. W.; Fox, D. W.; Stadter, J. T. “Upper and Lower Bounds for the Frequencies of Rectangular Free Plates. Tech. Memo TG-707, Appl. Physics Lab., The Johns Hopkins University., August 1965 [12] Waller, M. “Vibrations of Free Square Plates. Proc. Phys. Soc. (London), Vol. 51, January 1939, pp. 831-844 [13] de Bree, H-E; Basten, T. “A Full Bandwidth Calibrator for a Sound Pressure and Particle Velocity Sensor”. German Annual Conference on Acoustics. March 2008 - 17 - Appendix A – Analytical Prediction Spreadsheet Table A1 Values of Lambda Squared Used in Analytical Analysis ([9]) lambda^2 1 20.13 2 21.6 3 46.65 - 18 - 4 50.29 5 58.2 6 67.49 Appendix B – Microflown Scan and Paint Inputs Figure B1 Parameters Used in Microflown Analysis Parameter Value Sample Rate (Samples/Second) 50000 Number of FFT Points 8192 Overlap (%) 50 Number of Averages 5 Window Type Hanning Number of Samples per Probe Position 24576 - 19 -