When a ruler is clamped at one end and struck, it oscillates and emits a characteristic sound. Investigate how the sound depends on relevant parameters.
*The singing ruler (YouTube, Никита Черников, 08.07.2025), https://youtu.be/v0dLL8ybKLw *PET Ruler Vibrations (YouTube, PET Physics and Everyday Thinking - HS, 23.07.2015), https://youtu.be/4SpSwTvbZI4
*F. Norton. CHAPTER 16: Oscillatory Motion and Waves (theexpertta.com), https://theexpertta.com/book-files/OpenStaxCollegePhysics/CP_Ch16. Oscillatory Motion and Waves.pdf
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A even better resource, Unit2 of Vol 1 in University Physics, can be downloaded from Science – OpenStax. You need to know how we can determine the characteristic frequency from wave equations and the corresponding boundary conditions.
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*M. Aramaki, H. Baillères, L. Brancheriau, R. Kronland-Martinet, and S. Ystad. Sound quality assessment of wood for xylophone bars. J. Acoust. Soc. Am. 121, 4, 2407-2420 (2007), https://doi.org/10.1121/1.2697154
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Sound quality assessment in Figs. 2 and 3.
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*R. Selfridge, M. Andreasson, L. Bengtsson, B. V. Kristjánsson, E. Lindborg, M. Rydén, H. E. Tez, and J. D. Reiss. Twang! A physically derived synthesis model for the sound of a vibrating bar (Audio Engineering Society 152nd Convention Paper 10553, May 2022), https://eecs.qmul.ac.uk/~josh/documents/2022/Selfridge AES152.pdf
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A physically derived synthesis model of the sound generated when a ruler is twanged while hanging over the edge of a solid surface is presented. A set of physical parameters to control ruler length as well as the material properties are provided.
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*J. P. Chopade and R. B. Barjibhe. Free vibration analysis of fixed free beam with theoretical and numerical approach method. Int. J. Innov. Engin. Techn. (IJIET) 2, 1, 352-356 (2013), https://ijiet.com/wp-content/uploads/2013/02/53.pdf
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The vibration problems of Euler-Bernoulli beams were solved analytically or approximately, but the print quality is not as good as [Coskun11].
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*C. Brum Medeiros and M. Wanderley. Evaluation of sensor technologies for the rulers, a kalimba-like digital musical instrument (McGill Univ., 2011), https://www.researchgate.net/publication/266608335_Evaluation_of_sensor_technologies_for_the_rulers_a_kalimba-like_digital_musical_instrument
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Evaluation of three sensor technologies for the rulers.
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*S. B. Coşkun, M. T. Atay, and B. Öztürk. Transverse vibration analysis of Euler-Bernoulli beams using analytical approximate techniques. In: Advances in Vibration Analysis Research (Ed. Farzad Ebrahimi, intechopen.com, 2011), https://doi.org/10.5772/15891
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The vibration problems of Euler-Bernoulli beams were solved analytically or approximately for various boundary conditions. The characteristic frequencies and vibration modes were derived.
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*Lord Rayleigh. The Theory of Sound. (London, Macmillan, 1877, Courier Dover Publications, 1945), https://books.google.com/books?id=v4NSAlsTwnQC, https://books.google.com/books?id=Frvgu1wSFfUC *N. H. Fletcher and T. D. Rossing. The Physics of Musical Instruments. (New York, Springer-Verlag, 1991), https://books.google.com/books?id=9CRSRYQlRLkC
Oscillations of a Ruler: Equation, Explanation & Answers (thebosonbreaker, physicsforums.com, Feb 28, 2016), https://www.physicsforums.com/threads/oscillations-of-a-ruler-equationexplanation-answers.859699/
*T. Beleândez, C. Neipp, and A. Belea Ndez. Numerical and experimental analysis of a cantilever beam: A laboratory project to introduce geometric nonlinearity in mechanics of materials. Int. J. Engng Ed. 19, 6, 885-892 (2003), https://www.ijee.ie/articles/Vol19-6/IJEE1457.pdf