Experimental Modal Analysis of Rectangular and Circular Beams B
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Experimental Modal Analysis of Rectangular and Circular Beams B. H. Emory and W. D. Zhu University of Maryland, Baltimore County of boundary conditions [6-9]. The effect of the 1. INTRODUCTION sensor location is demonstrated along with Abstract The main focus of this work is to develop that of the torque in the bolts used to secure two test stands and accompanying activity to the rectangular beam at the clamped and Analytical and experimental meth- teach undergraduate and graduate students pinned boundaries. In many cases there is ods are used to determine the about vibrations and experimental modal no discussion of accounting for mass loading natural frequencies and mode shapes of Aluminum 6061-T651 analysis. Currently our department does not by the sensor and determining its placement beams with rectangular and cir- have an instructional module that can be before testing [6-8]. Proper selection of the cular cross-sections. A unique used to educate students on how to perform exciter tip is also addressed by examining the test stand is developed to pro- modal testing and use modern vibrations and measured frequency response function (FRF) vide the rectangular beam with modal testing equipment and software. Within and coherence. different boundary conditions including clamped-free, clamped- modal testing there are many mistakes that Knowledge of material properties, such clamped, clamped-pinned, and can be made, which are not noticed when as the elastic modulus, shear modulus, and pinned-pinned. The first 10 bend- doing computer simulation using finite element Poisson’s ratio, are important in mechanical ing natural frequencies and mode software. A few simple mistakes include choice design and research. Determination of the shapes for each set of boundary of the wrong accelerometer location, improper material properties of a specimen is important conditions are measured. The effect of the bolt torque on the accelerometer mounting, choice of the wrong since they vary from one to another. Vibration measured natural frequencies is exciter tip, impacting the structure with too testing is preferred over destructive testing examined. A new technique is much force that would overload the instruments, because the specimen can be used in further used to mount an accelerometer double impacts, and improper setup of the data experimentation. Vibration testing is also on the circular beam to measure collection equipment and software. Laboratory- advantageous for material property testing its torsional modes; its first 20 natural frequencies and first 10 based engineering education is important because it is less costly than using a tensile mode shapes are measured. The because “engineering students see themselves or torsion test. A circular beam test stand is measured natural frequencies as essentially practical” [1]. Laboratory experience presented for material property testing because and mode shapes of both beams is critical to a student’s development because unlike the torsional equation of motion is exact for the are compared with their theoreti- computer simulation, laboratory work teaches circular beam. A new apparatus is used for cal predictions. The Timoshenko beam theory is shown to provide students that attention to detail is paramount. It measuring the torsional vibration of the circular better predictions of the higher is also important to remember that engineers will beam with an accelerometer [10]. Free-free bending natural frequencies of the always be responsible for their actions and that an boundary conditions are used because they circular beam than the Bernoulli- education, which separates them in any way from are more easily verifiable than other boundary Euler beam theory. The material reality, introduces danger [2]. conditions. The linear curve-fit method [11] properties of the circular beam, in- cluding the elastic modulus, shear The test stand developed for a rectangular is used to determine the material properties. modulus, and Poisson’s ratio, are beam provides different sets of boundary The Timoshenko and Euler-Bernoulli beam determined accurately. The use conditions. It can be used for undergraduate and theories for the bending modes are compared of the rectangular and circular graduate education as well as other experiments to determine which theory gives more accurate beam test stands as a teaching in the laboratory. For example, it can be used to results. Students can gain a wide range of tool for undergraduate and gradu- ate students is discussed. The validate a robust, iterative algorithm for structural experience in modal testing by performing laboratory demonstration using damage detection [3] and the optimal damper similar experiments with the test stands. the test stands was well received location that can dissipate the largest number by students in the undergraduate of modes of a beam [4]. The test stand could vibrations class. also be used to complement computational 2. EXPERIMENTAL SETUP KEYWORDS: modal testing, Ti- simulations like the one presented by Pota and To perform modal testing a personal moshenko beam theory, material Alberts [5]. The boundary conditions are studied computer is used to control and collect data property measurement, educa- by comparing the natural frequencies and mode from a dynamic signal analyzer (Siglab Model tional laboratories shapes from experimental measurements 20-42). Inputs from the accelerometer (PCB with those predicted using the Euler-Bernoulli U352C66) and impact hammer (PCB Model beam theory. In many vibrating beam or plate 086C01) are conditioned using a signal experiments there is no discussion of validity conditioner (PCB Model 482A17). In order Journal of STEM Education Volume 7 • Issue 3 & 4 July–December 2006 80 to measure all the frequencies in a given accommodate various fixtures, some of which bandwidth, the sampling frequency ws needs are not utilized in this work. The steel-threaded to be at least twice the maximum frequency base has two sets of threads to receive the to be measured. Otherwise aliasing will occur. 190.5 mm bolts. Steel is used because torque In Siglab the sampling frequency is 2.56 times is applied to the bolts; boundary conditions the bandwidth. In all experiments in this work made of aluminum can be ruined after torquing the bandwidth starts at the zero frequency and a few times. The clamped or pinned boundaries aliasing will not occur. To obtain the desired are secured with two 190.5 mm bolts, the steel- frequency resolution dF, the record length RL threaded base is fastened to the aluminum base Fig. 1 Clamped boundary in the time domain is chosen to be RL = —— . with 95.25 mm bolts, and the aluminum base condition The bandwidth is adjusted to give the desired is secured to an optical table using five 95.25 accuracy. mm bolts. Once the boundary condition type is Care should be taken in selecting the chosen, a torque wrench is used to ensure that record length. When the record length is too both bolts have nearly the same torque and that short, the vibration of the structure does not the boundary does not impose a moment on the dissipate fully, which causes leakage. Applying beam; a level can also be used on top of the a windowing function to the response signal boundary for this purpose. A level is important helps to force the signal to zero near the end when setting up a pinned boundary as very little of the measurement, which reduces leakage. torque is needed. Another advantage of this windowing function The first experiment conducted on the beam is that it places more weight on the initial part is to determine the amount of torque at which of the signal than the latter. While the use of the first natural frequency of the clamped-free Fig. 2 Pinned boundary the exponential window in Siglab can affect rectangular beam becomes constant. The condition the measurement of damping ratios of the impact and accelerometer locations remain structure, it does not affect the measurement fixed in this experiment, and the extender is of its natural frequencies and mode shapes. A used for the hammer as the bandwidth is from Hole for Mounting Steel Threaded Base boxcar window is used for the input. 0 to 10 Hz. For the first test the nuts are only While testing over different frequency ranges screwed down so that the bottom of the nut the hammer tip is changed accordingly. At low touches the top steel bar (hand-tight). The frequencies an extender is used for the hammer. torque is increased in increments of 6.8 Nm and For all experiments, only one accelerometer with an impact test is performed accordingly. 2 gram mass is used to keep mass loading to a When using a clamped boundary condition, minimum. For every impact test three impacts the bolts are torqued to 40.7 Nm. For a pinned are averaged to ensure repeatable results and boundary condition the bolts are only hand- Hole for Mounting Aluminum Base to Optical Table to obtain a coherence measurement. The tightened so that the top bar does not rotate but Fig. 3 Bottom view of the accelerometer is affixed to the specimen using the beam can still rotate. Tightening the bolts aluminum base Petro wax. more than hand-tight can make the pin cut into the beam and change the boundary from pinned to clamped. The pinned boundary condition is Slot to Receive 3. RECTANGULAR BEAM also modeled physically by allowing as little of Bolt Hole Pinned Side The rectangular beam has a cross-section of the beam as possible to extend past the pinned .0761 m by .0065 m and a total length of .916 m. end because the weight of the extended portion Its mass density is 2715 kg/m3. The boundary of the beam creates a moment at the pinned conditions are composed of steel clamps with end. Fig.