- Email: [email protected]
Experimental determination of mode shapes of beams by roving impact test
Materials Today: Proceedings xxx (xxxx) xxx
Contents lists available at ScienceDirect
Materials Today: Proceedings journal homepage: www.elsevier.com/locate/matpr
Experimental determination of mode shapes of beams by roving impact test C. Amara Chandra, Prasanta Kumar Samal ⇑ The National Institute of Engineering, Manandavadi Road, Mysuru 570008, India
a r t i c l e
i n f o
Article history: Received 11 November 2019 Received in revised form 13 December 2019 Accepted 6 January 2020 Available online xxxx Keywords: Mode shapes Natural frequency Roving impact test LabVIEW MATLAB
a b s t r a c t In this work, an attempt has been made to determine the mode shapes of a beam by experimental roving impact tests without using an FFT analyzer. The roving impact test was performed by acquiring the vibration using the National Instruments data acquisition system (NIDAQ) and LabVIEW. The acquired data was subjected to post-processing using MATLAB to determine the mode shapes. The mode shapes were determined using both roving accelerometer and roving hammer impact tests. The results were found to be in good agreement with each other. The numerical modal analysis of the beam was also carried out using the finite element package, ANSYS and compared with the experimental results. The maximum percentage error in natural frequency was found to be 4.26%. Ó 2020 Elsevier Ltd. All rights reserved. Selection and Peer-review under responsibility of the scientific committee of the International Mechanical Engineering Congress 2019: Materials Science.
1. Introduction Vibration is defined as a motion that repeats itself after an interval of time. The examples of vibration include swinging of a pendulum or the motion of a plucked string. The vibration theory deals with the study of oscillatory motions of bodies and the corresponding forces. Vibration is involved in most of the human activities. For example, the vibration of our eardrums to hear, the vibration of light waves to see, the vibration of lungs for breathing, etc. [1]. All bodies possessing mass and elasticity are capable of vibrations. Sometimes vibrations are functionally designed to fulfill the requirements, for example, guitar, vibration shaker, compactors, etc. Most engineering machines and structures experience vibrations to some degree and their design generally requires consideration of their oscillatory motions. When external excitation frequency coincides with the natural frequency of a machine or structure, it is called resonance, which leads to excessive deflections and failure [2]. Because of the devastating effects that vibrations can have on machines and structures, vibration testing has become a standard procedure in the design and development of most engineering systems. Vibration analysis can be done analytically, numerically and experimentally. But experimental analysis method (modal analy⇑ Corresponding author. E-mail address: [email protected] (P.K. Samal).
sis) has an advantage over the analytical and numerical method i.e. in case of experimental analysis modal characteristics are defined from actual measurements. To determine the dynamic characteristics of structures, experimental modal analysis is used. Structures vibrate with high amplitude at its resonant frequency. It is important to know the modal parameters such as natural frequency and mode shape of the structure to improve its strength and reliability while designing [3]. A natural frequency is a frequency at which the structure would oscillate if it were disturbed from its rest position and then allowed to vibrate freely [4]. And a mode shape is a specific pattern of vibration executed by a mechanical system at a specific frequency [5]. In this work, the vibration analysis is carried out on common structures like a beam. A beam can be defined as a slender horizontal structure that resists lateral loads [6]. Beams are important members of engineering structures used in various forms and comprise various artifacts, such as railway tracks, supporting members in buildings, long-span bridges, airplane wings, gun barrels, robot arms, etc. [7]. The experimental modal analysis developed steadily in popularity since the advent of the digital FFT spectrum analyzer in the early 19790 s. At present, impact testing has become a common practice to find modes of the vibrating structure of machines [8]. The literature is focused on free vibration analysis of beams and plates and their mode shapes. By reviewing the work done on this topic it is possible to understand the methodology and basic
https://doi.org/10.1016/j.matpr.2020.01.119 2214-7853/Ó 2020 Elsevier Ltd. All rights reserved. Selection and Peer-review under responsibility of the scientific committee of the International Mechanical Engineering Congress 2019: Materials Science.
Please cite this article as: Amara Chandra C and P. K. Samal, Experimental determination of mode shapes of beams by roving impact test, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2020.01.119
2
Amara Chandra C and P. K. Samal / Materials Today: Proceedings xxx (xxxx) xxx
terminologies. The free vibration analysis was performed for the beams by various researchers [6–19]. Mode shapes of plates are determined experimentally by roving impact hammer test using FFT analyzer [20]. The review indicated that, the vibration analysis carried out so far was majorly by Finite Element methods. Further, most of the research on experimental vibration analysis focused primarily on the determination of natural frequencies. The mode shapes have been determined by some of the researchers by using an FFT analyzer. However, since an FFT analyzer is highly expensive, it cannot
be afforded easily. Until now, there are no reported literatures on the experimental determination of mode shapes without the aid of FFT analyzer. Hence, in this work, an attempt has been made to determine the mode shapes of a beam solely by employing LabVIEW and MATLAB thereby eliminating the need of an FFT analyser. 2. Experimental details 2.1. Specimen details The beam specimen used was Aluminum whose material properties and dimensions are given in Table 1.
Table 1 Specimen Details. Details
Value
Young’s modulus (E) Poisson’s ratio (t) Density (q) Dimension
69.1 MPa 0.334 2720 kg/m3 350 21 3 mm
2.2. Experimental setup The aluminum cantilever beam was used. The entire beam length of 350 mm was divided into 14 nodes (excluding the free end of the beam) as shown in Fig. 1. Thus the distance between two adjacent nodes was found to be 25 mm. The experimental configuration consists of a beam fixed by nuts and bolts on a stand at one end as shown in Fig. 2. At one of the nodal points considered, the accelerometer (356A15) is mounted on the beam. Using cables, the accelerometer is connected to the DAQ system. As only the transverse amplitude is analyzed, the accelerometer is mounted with its vertical Z-axis. The impact hammer is connected to DAQ (NI-9234) module so that the excitation force can be measured. The DAQ (NI-9234) is inserted into the cDAQ chassis (NI CDAQ-9178), which is connected via the USB cable to the personnel computer having LabVIEW in it. 3. Results
Fig. 1. Cantilever beam divided into 14 nodes.
Fig. 2. Experimental setup.
3.1. Experimental modal analysis by roving accelerometer impact test The accelerometer was initially mounted on the first node and the beam was excited with a force of 17 N by the impact hammer at the 14th node. The vibration data were acquired using the NI9234 module and LabVIEW and was saved for further analysis in MATLAB. The experiment was repeated by positioning the accelerometer at the rest of the nodes one after the other by applying the same amount of force at 14th node. The frequency response curves were plotted using MATLAB for each nodal impact. Fig. 3 shows the frequency response curves for the 13th and 14th nodal position of the accelerometer. It can be observed that for 13th, 14th and all other nodes of the beam, the values of the first and second natural frequencies were the same when the accelerometer was fixed at each node, while the amplitude of acceleration values attained at those natural
Fig. 3. Frequency response curve for the nodal impact at: a) node 13 and (b) node 14.
Please cite this article as: Amara Chandra C and P. K. Samal, Experimental determination of mode shapes of beams by roving impact test, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2020.01.119
Amara Chandra C and P. K. Samal / Materials Today: Proceedings xxx (xxxx) xxx Table 2 Nodal Amplitude Values for 1st and 2nd modes by Roving Accelerometer Impact Test. Nodes
1st mode
2nd mode
1 2 3 4 5 6 7 8 9 10 11 12 13 14
0 1.21 3.28 4.17 4.58 5.32 6.48 6.98 8.75 8.91 10.75 11.26 12.2790 13.3830
0 1.61 2.47 2.98 3.74 4.98 4.56 0.23 0.98 1.62 2.54 4.56 9.02 8.21
3
By executing the code, the plot of amplitude of vibration against nodal distance was obtained, which represents the mode shapes. When the amplitude values corresponding to the first mode (second column of Table 2) were plotted against nodal distance, the first mode shape of cantilever beam was obtained as shown in Fig. 4(a). Whereas, the second mode shape of the cantilever beam was obtained by plotting the amplitude values corresponding to the second mode (third column of Table 2) against nodal distance, as shown in Fig. 4(b). It can be observed from Fig. 4 that the graphs are not smooth. The reason could be the consideration of only 14 nodes. Since the beam is a continuous system, infinity will be the number of nodes to be taken to get the exact mode shape. These mode shapes were therefore subjected to curve smoothing to obtain a smooth curve. The mode shapes obtained are as shown in Fig. 5 after curve fitting. 3.2. Application of Maxwell-Betti reciprocal theorem
frequencies are different. The amplitude corresponding to frequency values was then tabulated. Table 2 shows the amplitude values corresponding to first and second natural frequencies at all the 14 nodes. The tabulated acceleration values were plotted against nodal distance to get mode shapes using MATLAB. The code was written in such a way that distance between each node was 25 mm up to 350 mm (Since the beam is divided into 14 nodes) were taken as X-axis (Nodal distance) in the graph and the amplitude value in Table 2 was taken as Y-axis.
Maxwell-Betti reciprocal theorem, discovered by Enrico Betti in 1872, states that ‘‘for a linear elastic structure subject to two sets of forces {Ai} i = 1,. . .,m and {Bj}, j = 1,2,. . .,n, the work done by the set A through the displacements produced by the set B is equal to the work done by the set B through the displacements produced by the set A.” Since displacement at point B due to force applied at point A is the same as the displacement at point A due to force applied at point B. The amplitude of acceleration at node B due to impact load
Fig. 4. Mode shapes of the cantilever beam by roving accelerometer impact test. a) First mode shape; (b) second mode shape.
Fig. 5. Mode shapes of the cantilever beam by roving accelerometer impact test after curve smoothing a) first mode shape; (b) second mode shape.
Please cite this article as: Amara Chandra C and P. K. Samal, Experimental determination of mode shapes of beams by roving impact test, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2020.01.119
4
Amara Chandra C and P. K. Samal / Materials Today: Proceedings xxx (xxxx) xxx
at node A would the same as the amplitude of acceleration at node A due to impact load at node B. With the help of this theorem, it is evident that the results obtained by change of the position of accelerometer and impact at the same position (roving accelerometer impact test), and by impacting at different position and mounting the accelerometer at the same position (roving hammer impact test) will be same.
Table 3 Nodal amplitude values for 1st and 2nd modes by Roving hammer impact Test. Nodes
1st mode
2nd mode
1 2 3 4 5 6 7 8 9 10 11 12 13 14
0 11.46 12.74 12.45 12.6 13.04 13.29 13.96 14.56 16.31 17.03 20.2 22.48 24.48
0 1.46 2.28 2.32 3.81 4.74 4.51 0.285 1.428 2.02 2.95 4.51 7.086 8.45
3.3. Experimental modal analysis by roving accelerometer impact test The experimental procedure of obtaining the data from LabVIEW and post-processing the data in MATLAB remains the same. The only change that we have made in this experiment is, here we fix the accelerometer in one node and impact force is applied at each node to get the amplitude corresponds to its natural frequency values at each node. Table 3 shows the amplitude values corresponding to first and second natural frequencies when impacted at all the 14 nodes keeping the accelerometer at one position. The tabulated acceleration values were plotted against nodal distance to get mode shapes of the beam using MATLAB as shown in Fig. 6. And Fig. 7 shows the first and second mode shapes after curve smoothing. From Figs. 5 and 7 it is evident that mode shapes obtained by changing the position of accelerometer and without changing the position of the accelerometer are the same. Hence it is convenient to adopt the test by mounting the accelerometer at one position and applying impact at different positions (roving hammer impact test) as compared to the roving accelerometer impact test. This method overcomes the problem of removing and fixing the accelerometer again and again, and a considerable amount of time can be saved.
Fig. 6. Mode shapes of the cantilever beam by roving hammer impact test. a) first mode shape; (b) second mode shape.
Fig. 7. Mode shapes of the cantilever beam by roving hammer impact test after curve smoothing. a) First mode shape; (b) second mode shape.
Please cite this article as: Amara Chandra C and P. K. Samal, Experimental determination of mode shapes of beams by roving impact test, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2020.01.119
Amara Chandra C and P. K. Samal / Materials Today: Proceedings xxx (xxxx) xxx
5
Fig. 8. Mode shapes of the cantilever beam obtained from numerical analysis. a) First mode shape; (b) Second mode shape.
Mode no.
Experimental analysis
Numerical analysis
Difference (%)
The mode shapes and corresponding natural frequency were also determined by finite element analysis using ANSYS. The results are compared and observed to be in good agreement.
1 2
18 106
17.168 107.56
4.26 1.25
CRediT authorship contribution statement
Table 4 Comparison of natural frequencies (in Hz).
3.4. Numerical modal analysis of the cantilever beam The modal analysis was conducted in the ANSYS Modal Module. This has a better graphical user interface and makes the selection of elements more flexible. The 350 21 3 mm beam was modeled. In the Engineering data section, the material properties were added and meshed. For meshing, hexahedral elements with a size of 1 mm in the dimension of thickness and 3 mm in the other two dimensions were used. The beam was fixed at one end to make it as a cantilever. In the Analysis settings, the beam was solved for the first two modes. The results are shown in Fig. 8. The natural frequencies corresponding to the first two-mode shapes, determined from the experimental analysis and the numerical analysis, are compared and presented in Table 4. 4. Discussions From Figs. 5 and 7 it is observed that mode shapes obtained by changing the position of accelerometer and without changing its position are almost the same. Hence it is convenient to adopt the test by mounting the accelerometer at one position and applying impact at different positions (roving hammer impact test) as compared to the roving accelerometer impact test. This method overcomes the problem of removing and fixing the accelerometer again and again, and a considerable amount of time can be saved. Comparing Figs. 5, 6 and 8, it can be observed that the first twomode shapes determined by the numerical method and experimental method are almost the same. From Table 4, it is observed that the natural frequency for mode 1 and mode 2 determined by the experimental and the numerical analysis are in good agreement. 5. Conclusions This work is mainly aimed at the evaluation of mode shapes and the natural frequencies of the beam. The mode shapes are determined experimentally in two methods namely roving accelerometer and roving hammer impact test. It is observed that mode shapes obtained by both the methods are the same. Hence it is convenient to adopt the test by mounting the accelerometer at one position and applying impact at different positions (roving hammer impact test) as compared to the roving accelerometer impact test. With this, the problem of attaching the accelerometer at each node can be eliminated which in turn yields in a simple experimental procedure.
C. Amara Chandra: . Prasanta Kumar Samal: Conceptualization, Project administration. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. References [1] Singiresu S. Rao. ‘‘Mechanical Vibrations”, Fourth Edition2015. [2] M. L. Chandravanshi A. K. Mukhopadhyay ‘‘Modal Analysis of Structural Vibration”, Proceedings of the International Mechanical Engineering Congress and Exposition, November 15-21-2013. [3] Daniel J. Inman, ‘‘Engineering Vibration”, PEARSON publication. [4] William T. Thomson, ‘‘Theory of Vibrations with Applications”, Fifth Edition PEARSON publications. [5] Prashant S. Walunj, V.N. Chougule, Anirban C. Mitra, investigation on modal parameters of rectangular cantilever beam using experimental modal analysis, Mater. Today. Proceed. (2015) 2121–2130. [6] C. Srinivasan, S. Vijayakumar, K. Pasupathi, S. Sasidharan, ‘‘Investigation on Vibrational Characteristics of Jute Fiber Reinforced Composite Material”. [7] S.G. Patel, G.R. Vesmawala, Experimental and numerical study of a steel bridge model through vibration testing using sensor networks, J. Mech. Civil Eng. 12 (6) (2015) 83–89. [8] Itishree Mishra Shishir Kumar Sahu ‘‘An Experimental Approach to Free Vibration Response of Woven Fiber Composite Plates under Free-Free Boundary Condition”. ISSN 1 2 22315721, 2012. [9] Ashish R. Sonawane, Poonam S. Talmale, Modal analysis of single rectangular cantilever plate by mathematically, FEA and experimental, Inter. Res. J. Eng. Technol. 04 (08) (2017) 264–269. [10] G.A. Yashavantha, Kumar, K.M Sathish, Kumar, Free vibration analysis of smart composite beam, Mater. Today Proceed. 4 (2) (2017) 2487–2491. [11] P. Kumar, S. Bhaduri, A. Kumar, ‘‘Vibration Analysis of Cantilever Beam: An Experimental Study”, ISSN: 2P. Kumar 653. [12] M.S. Kotambkar, Effect of mass attachment on natural frequency of free-free beam: analytical, numerical and experimental investigation, Inter. J. Eng. Res. Studies (2014). [13] Daniel Ambrosiana, Experimental validation of free vibrations from nonsymmetrical thin walled beams, Eng. Struct. 32 (2010) 1324–1332. [14] Mehmet Avcar, ‘‘Free Vibration Analysis of Beams Considering Different Geometric Characteristics and Boundary Conditions”, DOI: 10.5923, mechanics, 2014. [15] M.N. Hamdan, B.A. Jubran, Free and Forced Vibrations of a Restrained Cantilever Beam Carrying a Concentrated Mass, JKAU, Eng. Sci. 3 (1991) 71–83. [16] H. Nahvi, M. Jabbari, Crack detection in beams using experimental modal data and finite element model, Inter. J. Mech. Sci. 47 (2005). [17] T. Nikhil, T. Chandrahas, C. Chaitanya, I. Sagar, G.R. Sabareesh, Design and development of a test-rig for determining vibration characteristics of a beam, Procedia Eng. (2016). [18] S.H. Gawande, R.R. More, Investigations on effect of notch on performance evaluation of cantilever beams, Inter. J. Acoust. Vibr. 22 (4) (2017) 493–500. [19] D.P. Kamble, Chandan Kumar, Shivprasad R. Sontakke, Ratnadip T. Gaikwad, Analytical and experimental analysis of cantilever beam under forced vibration, Inter. J. Eng. Sci. Comput. 6 (3) (2016) 2168–2171. [20] Brian J. Schwarz, Mark H. Richardson, Experimental modal analysis, Vibrant Technol. (1999).
Please cite this article as: Amara Chandra C and P. K. Samal, Experimental determination of mode shapes of beams by roving impact test, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2020.01.119
Contents lists available at ScienceDirect
Materials Today: Proceedings journal homepage: www.elsevier.com/locate/matpr
Experimental determination of mode shapes of beams by roving impact test C. Amara Chandra, Prasanta Kumar Samal ⇑ The National Institute of Engineering, Manandavadi Road, Mysuru 570008, India
a r t i c l e
i n f o
Article history: Received 11 November 2019 Received in revised form 13 December 2019 Accepted 6 January 2020 Available online xxxx Keywords: Mode shapes Natural frequency Roving impact test LabVIEW MATLAB
a b s t r a c t In this work, an attempt has been made to determine the mode shapes of a beam by experimental roving impact tests without using an FFT analyzer. The roving impact test was performed by acquiring the vibration using the National Instruments data acquisition system (NIDAQ) and LabVIEW. The acquired data was subjected to post-processing using MATLAB to determine the mode shapes. The mode shapes were determined using both roving accelerometer and roving hammer impact tests. The results were found to be in good agreement with each other. The numerical modal analysis of the beam was also carried out using the finite element package, ANSYS and compared with the experimental results. The maximum percentage error in natural frequency was found to be 4.26%. Ó 2020 Elsevier Ltd. All rights reserved. Selection and Peer-review under responsibility of the scientific committee of the International Mechanical Engineering Congress 2019: Materials Science.
1. Introduction Vibration is defined as a motion that repeats itself after an interval of time. The examples of vibration include swinging of a pendulum or the motion of a plucked string. The vibration theory deals with the study of oscillatory motions of bodies and the corresponding forces. Vibration is involved in most of the human activities. For example, the vibration of our eardrums to hear, the vibration of light waves to see, the vibration of lungs for breathing, etc. [1]. All bodies possessing mass and elasticity are capable of vibrations. Sometimes vibrations are functionally designed to fulfill the requirements, for example, guitar, vibration shaker, compactors, etc. Most engineering machines and structures experience vibrations to some degree and their design generally requires consideration of their oscillatory motions. When external excitation frequency coincides with the natural frequency of a machine or structure, it is called resonance, which leads to excessive deflections and failure [2]. Because of the devastating effects that vibrations can have on machines and structures, vibration testing has become a standard procedure in the design and development of most engineering systems. Vibration analysis can be done analytically, numerically and experimentally. But experimental analysis method (modal analy⇑ Corresponding author. E-mail address: [email protected] (P.K. Samal).
sis) has an advantage over the analytical and numerical method i.e. in case of experimental analysis modal characteristics are defined from actual measurements. To determine the dynamic characteristics of structures, experimental modal analysis is used. Structures vibrate with high amplitude at its resonant frequency. It is important to know the modal parameters such as natural frequency and mode shape of the structure to improve its strength and reliability while designing [3]. A natural frequency is a frequency at which the structure would oscillate if it were disturbed from its rest position and then allowed to vibrate freely [4]. And a mode shape is a specific pattern of vibration executed by a mechanical system at a specific frequency [5]. In this work, the vibration analysis is carried out on common structures like a beam. A beam can be defined as a slender horizontal structure that resists lateral loads [6]. Beams are important members of engineering structures used in various forms and comprise various artifacts, such as railway tracks, supporting members in buildings, long-span bridges, airplane wings, gun barrels, robot arms, etc. [7]. The experimental modal analysis developed steadily in popularity since the advent of the digital FFT spectrum analyzer in the early 19790 s. At present, impact testing has become a common practice to find modes of the vibrating structure of machines [8]. The literature is focused on free vibration analysis of beams and plates and their mode shapes. By reviewing the work done on this topic it is possible to understand the methodology and basic
https://doi.org/10.1016/j.matpr.2020.01.119 2214-7853/Ó 2020 Elsevier Ltd. All rights reserved. Selection and Peer-review under responsibility of the scientific committee of the International Mechanical Engineering Congress 2019: Materials Science.
Please cite this article as: Amara Chandra C and P. K. Samal, Experimental determination of mode shapes of beams by roving impact test, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2020.01.119
2
Amara Chandra C and P. K. Samal / Materials Today: Proceedings xxx (xxxx) xxx
terminologies. The free vibration analysis was performed for the beams by various researchers [6–19]. Mode shapes of plates are determined experimentally by roving impact hammer test using FFT analyzer [20]. The review indicated that, the vibration analysis carried out so far was majorly by Finite Element methods. Further, most of the research on experimental vibration analysis focused primarily on the determination of natural frequencies. The mode shapes have been determined by some of the researchers by using an FFT analyzer. However, since an FFT analyzer is highly expensive, it cannot
be afforded easily. Until now, there are no reported literatures on the experimental determination of mode shapes without the aid of FFT analyzer. Hence, in this work, an attempt has been made to determine the mode shapes of a beam solely by employing LabVIEW and MATLAB thereby eliminating the need of an FFT analyser. 2. Experimental details 2.1. Specimen details The beam specimen used was Aluminum whose material properties and dimensions are given in Table 1.
Table 1 Specimen Details. Details
Value
Young’s modulus (E) Poisson’s ratio (t) Density (q) Dimension
69.1 MPa 0.334 2720 kg/m3 350 21 3 mm
2.2. Experimental setup The aluminum cantilever beam was used. The entire beam length of 350 mm was divided into 14 nodes (excluding the free end of the beam) as shown in Fig. 1. Thus the distance between two adjacent nodes was found to be 25 mm. The experimental configuration consists of a beam fixed by nuts and bolts on a stand at one end as shown in Fig. 2. At one of the nodal points considered, the accelerometer (356A15) is mounted on the beam. Using cables, the accelerometer is connected to the DAQ system. As only the transverse amplitude is analyzed, the accelerometer is mounted with its vertical Z-axis. The impact hammer is connected to DAQ (NI-9234) module so that the excitation force can be measured. The DAQ (NI-9234) is inserted into the cDAQ chassis (NI CDAQ-9178), which is connected via the USB cable to the personnel computer having LabVIEW in it. 3. Results
Fig. 1. Cantilever beam divided into 14 nodes.
Fig. 2. Experimental setup.
3.1. Experimental modal analysis by roving accelerometer impact test The accelerometer was initially mounted on the first node and the beam was excited with a force of 17 N by the impact hammer at the 14th node. The vibration data were acquired using the NI9234 module and LabVIEW and was saved for further analysis in MATLAB. The experiment was repeated by positioning the accelerometer at the rest of the nodes one after the other by applying the same amount of force at 14th node. The frequency response curves were plotted using MATLAB for each nodal impact. Fig. 3 shows the frequency response curves for the 13th and 14th nodal position of the accelerometer. It can be observed that for 13th, 14th and all other nodes of the beam, the values of the first and second natural frequencies were the same when the accelerometer was fixed at each node, while the amplitude of acceleration values attained at those natural
Fig. 3. Frequency response curve for the nodal impact at: a) node 13 and (b) node 14.
Please cite this article as: Amara Chandra C and P. K. Samal, Experimental determination of mode shapes of beams by roving impact test, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2020.01.119
Amara Chandra C and P. K. Samal / Materials Today: Proceedings xxx (xxxx) xxx Table 2 Nodal Amplitude Values for 1st and 2nd modes by Roving Accelerometer Impact Test. Nodes
1st mode
2nd mode
1 2 3 4 5 6 7 8 9 10 11 12 13 14
0 1.21 3.28 4.17 4.58 5.32 6.48 6.98 8.75 8.91 10.75 11.26 12.2790 13.3830
0 1.61 2.47 2.98 3.74 4.98 4.56 0.23 0.98 1.62 2.54 4.56 9.02 8.21
3
By executing the code, the plot of amplitude of vibration against nodal distance was obtained, which represents the mode shapes. When the amplitude values corresponding to the first mode (second column of Table 2) were plotted against nodal distance, the first mode shape of cantilever beam was obtained as shown in Fig. 4(a). Whereas, the second mode shape of the cantilever beam was obtained by plotting the amplitude values corresponding to the second mode (third column of Table 2) against nodal distance, as shown in Fig. 4(b). It can be observed from Fig. 4 that the graphs are not smooth. The reason could be the consideration of only 14 nodes. Since the beam is a continuous system, infinity will be the number of nodes to be taken to get the exact mode shape. These mode shapes were therefore subjected to curve smoothing to obtain a smooth curve. The mode shapes obtained are as shown in Fig. 5 after curve fitting. 3.2. Application of Maxwell-Betti reciprocal theorem
frequencies are different. The amplitude corresponding to frequency values was then tabulated. Table 2 shows the amplitude values corresponding to first and second natural frequencies at all the 14 nodes. The tabulated acceleration values were plotted against nodal distance to get mode shapes using MATLAB. The code was written in such a way that distance between each node was 25 mm up to 350 mm (Since the beam is divided into 14 nodes) were taken as X-axis (Nodal distance) in the graph and the amplitude value in Table 2 was taken as Y-axis.
Maxwell-Betti reciprocal theorem, discovered by Enrico Betti in 1872, states that ‘‘for a linear elastic structure subject to two sets of forces {Ai} i = 1,. . .,m and {Bj}, j = 1,2,. . .,n, the work done by the set A through the displacements produced by the set B is equal to the work done by the set B through the displacements produced by the set A.” Since displacement at point B due to force applied at point A is the same as the displacement at point A due to force applied at point B. The amplitude of acceleration at node B due to impact load
Fig. 4. Mode shapes of the cantilever beam by roving accelerometer impact test. a) First mode shape; (b) second mode shape.
Fig. 5. Mode shapes of the cantilever beam by roving accelerometer impact test after curve smoothing a) first mode shape; (b) second mode shape.
Please cite this article as: Amara Chandra C and P. K. Samal, Experimental determination of mode shapes of beams by roving impact test, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2020.01.119
4
Amara Chandra C and P. K. Samal / Materials Today: Proceedings xxx (xxxx) xxx
at node A would the same as the amplitude of acceleration at node A due to impact load at node B. With the help of this theorem, it is evident that the results obtained by change of the position of accelerometer and impact at the same position (roving accelerometer impact test), and by impacting at different position and mounting the accelerometer at the same position (roving hammer impact test) will be same.
Table 3 Nodal amplitude values for 1st and 2nd modes by Roving hammer impact Test. Nodes
1st mode
2nd mode
1 2 3 4 5 6 7 8 9 10 11 12 13 14
0 11.46 12.74 12.45 12.6 13.04 13.29 13.96 14.56 16.31 17.03 20.2 22.48 24.48
0 1.46 2.28 2.32 3.81 4.74 4.51 0.285 1.428 2.02 2.95 4.51 7.086 8.45
3.3. Experimental modal analysis by roving accelerometer impact test The experimental procedure of obtaining the data from LabVIEW and post-processing the data in MATLAB remains the same. The only change that we have made in this experiment is, here we fix the accelerometer in one node and impact force is applied at each node to get the amplitude corresponds to its natural frequency values at each node. Table 3 shows the amplitude values corresponding to first and second natural frequencies when impacted at all the 14 nodes keeping the accelerometer at one position. The tabulated acceleration values were plotted against nodal distance to get mode shapes of the beam using MATLAB as shown in Fig. 6. And Fig. 7 shows the first and second mode shapes after curve smoothing. From Figs. 5 and 7 it is evident that mode shapes obtained by changing the position of accelerometer and without changing the position of the accelerometer are the same. Hence it is convenient to adopt the test by mounting the accelerometer at one position and applying impact at different positions (roving hammer impact test) as compared to the roving accelerometer impact test. This method overcomes the problem of removing and fixing the accelerometer again and again, and a considerable amount of time can be saved.
Fig. 6. Mode shapes of the cantilever beam by roving hammer impact test. a) first mode shape; (b) second mode shape.
Fig. 7. Mode shapes of the cantilever beam by roving hammer impact test after curve smoothing. a) First mode shape; (b) second mode shape.
Please cite this article as: Amara Chandra C and P. K. Samal, Experimental determination of mode shapes of beams by roving impact test, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2020.01.119
Amara Chandra C and P. K. Samal / Materials Today: Proceedings xxx (xxxx) xxx
5
Fig. 8. Mode shapes of the cantilever beam obtained from numerical analysis. a) First mode shape; (b) Second mode shape.
Mode no.
Experimental analysis
Numerical analysis
Difference (%)
The mode shapes and corresponding natural frequency were also determined by finite element analysis using ANSYS. The results are compared and observed to be in good agreement.
1 2
18 106
17.168 107.56
4.26 1.25
CRediT authorship contribution statement
Table 4 Comparison of natural frequencies (in Hz).
3.4. Numerical modal analysis of the cantilever beam The modal analysis was conducted in the ANSYS Modal Module. This has a better graphical user interface and makes the selection of elements more flexible. The 350 21 3 mm beam was modeled. In the Engineering data section, the material properties were added and meshed. For meshing, hexahedral elements with a size of 1 mm in the dimension of thickness and 3 mm in the other two dimensions were used. The beam was fixed at one end to make it as a cantilever. In the Analysis settings, the beam was solved for the first two modes. The results are shown in Fig. 8. The natural frequencies corresponding to the first two-mode shapes, determined from the experimental analysis and the numerical analysis, are compared and presented in Table 4. 4. Discussions From Figs. 5 and 7 it is observed that mode shapes obtained by changing the position of accelerometer and without changing its position are almost the same. Hence it is convenient to adopt the test by mounting the accelerometer at one position and applying impact at different positions (roving hammer impact test) as compared to the roving accelerometer impact test. This method overcomes the problem of removing and fixing the accelerometer again and again, and a considerable amount of time can be saved. Comparing Figs. 5, 6 and 8, it can be observed that the first twomode shapes determined by the numerical method and experimental method are almost the same. From Table 4, it is observed that the natural frequency for mode 1 and mode 2 determined by the experimental and the numerical analysis are in good agreement. 5. Conclusions This work is mainly aimed at the evaluation of mode shapes and the natural frequencies of the beam. The mode shapes are determined experimentally in two methods namely roving accelerometer and roving hammer impact test. It is observed that mode shapes obtained by both the methods are the same. Hence it is convenient to adopt the test by mounting the accelerometer at one position and applying impact at different positions (roving hammer impact test) as compared to the roving accelerometer impact test. With this, the problem of attaching the accelerometer at each node can be eliminated which in turn yields in a simple experimental procedure.
C. Amara Chandra: . Prasanta Kumar Samal: Conceptualization, Project administration. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. References [1] Singiresu S. Rao. ‘‘Mechanical Vibrations”, Fourth Edition2015. [2] M. L. Chandravanshi A. K. Mukhopadhyay ‘‘Modal Analysis of Structural Vibration”, Proceedings of the International Mechanical Engineering Congress and Exposition, November 15-21-2013. [3] Daniel J. Inman, ‘‘Engineering Vibration”, PEARSON publication. [4] William T. Thomson, ‘‘Theory of Vibrations with Applications”, Fifth Edition PEARSON publications. [5] Prashant S. Walunj, V.N. Chougule, Anirban C. Mitra, investigation on modal parameters of rectangular cantilever beam using experimental modal analysis, Mater. Today. Proceed. (2015) 2121–2130. [6] C. Srinivasan, S. Vijayakumar, K. Pasupathi, S. Sasidharan, ‘‘Investigation on Vibrational Characteristics of Jute Fiber Reinforced Composite Material”. [7] S.G. Patel, G.R. Vesmawala, Experimental and numerical study of a steel bridge model through vibration testing using sensor networks, J. Mech. Civil Eng. 12 (6) (2015) 83–89. [8] Itishree Mishra Shishir Kumar Sahu ‘‘An Experimental Approach to Free Vibration Response of Woven Fiber Composite Plates under Free-Free Boundary Condition”. ISSN 1 2 22315721, 2012. [9] Ashish R. Sonawane, Poonam S. Talmale, Modal analysis of single rectangular cantilever plate by mathematically, FEA and experimental, Inter. Res. J. Eng. Technol. 04 (08) (2017) 264–269. [10] G.A. Yashavantha, Kumar, K.M Sathish, Kumar, Free vibration analysis of smart composite beam, Mater. Today Proceed. 4 (2) (2017) 2487–2491. [11] P. Kumar, S. Bhaduri, A. Kumar, ‘‘Vibration Analysis of Cantilever Beam: An Experimental Study”, ISSN: 2P. Kumar 653. [12] M.S. Kotambkar, Effect of mass attachment on natural frequency of free-free beam: analytical, numerical and experimental investigation, Inter. J. Eng. Res. Studies (2014). [13] Daniel Ambrosiana, Experimental validation of free vibrations from nonsymmetrical thin walled beams, Eng. Struct. 32 (2010) 1324–1332. [14] Mehmet Avcar, ‘‘Free Vibration Analysis of Beams Considering Different Geometric Characteristics and Boundary Conditions”, DOI: 10.5923, mechanics, 2014. [15] M.N. Hamdan, B.A. Jubran, Free and Forced Vibrations of a Restrained Cantilever Beam Carrying a Concentrated Mass, JKAU, Eng. Sci. 3 (1991) 71–83. [16] H. Nahvi, M. Jabbari, Crack detection in beams using experimental modal data and finite element model, Inter. J. Mech. Sci. 47 (2005). [17] T. Nikhil, T. Chandrahas, C. Chaitanya, I. Sagar, G.R. Sabareesh, Design and development of a test-rig for determining vibration characteristics of a beam, Procedia Eng. (2016). [18] S.H. Gawande, R.R. More, Investigations on effect of notch on performance evaluation of cantilever beams, Inter. J. Acoust. Vibr. 22 (4) (2017) 493–500. [19] D.P. Kamble, Chandan Kumar, Shivprasad R. Sontakke, Ratnadip T. Gaikwad, Analytical and experimental analysis of cantilever beam under forced vibration, Inter. J. Eng. Sci. Comput. 6 (3) (2016) 2168–2171. [20] Brian J. Schwarz, Mark H. Richardson, Experimental modal analysis, Vibrant Technol. (1999).
Please cite this article as: Amara Chandra C and P. K. Samal, Experimental determination of mode shapes of beams by roving impact test, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2020.01.119