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Analysis of free and forced vibrations of ultrasonic vibrating tools, case study: ultrasonic assisted surface rolling process

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Abstract

In this study, the mechanism design and analysis of free and forced vibrations of vibrating tool in ultrasonic assisted surface rolling process (UASR) were taken into consideration. In this regard, three free vibrations problems are studied: first, a simple five-element horn with cylindrical-conical geometry; second, an UASR horn with complex geometrical shape; and third, an integrated UASR system, including transducer and booster (made earlier) associated with UASR horn. Moreover, their natural frequency, mode shape, and amplification factor were achieved individually. UASR system is experimentally manufactured and its oscillations amplitudes are extracted for different forced excitation frequencies. It was observed that the empirical resonance frequency was more consistent with the numerical results of UASR system than UASR horn, while the latter has been used for ultrasonic analyses and designing till now. Considering appropriate damping, numerical harmonic analysis of UASR horn was performed to extract the vibrational behavior of its end involved and not involved with the workpiece, whereas an external oscillatory force excites the primary section of the UASR horn. The loading history on the workpiece surface and fluctuations amplitude of the involved horn end were extracted, where their experimental measurement was impossible. In both cases, maximum mechanical oscillations of the horn end were observed at forced excitation with the same natural frequency obtained in numerical modal analysis, which is due to resonance occurrence and standing wave formation at the horn.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Golpayegan University of Technology, Golpayegan, Iran

    Hamed Razavi, Mohammad Keymanesh & Iman Fakhari Golpayegani

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  1. Hamed Razavi
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  2. Mohammad Keymanesh
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  3. Iman Fakhari Golpayegani
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Correspondence to Hamed Razavi.

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Appendix

Appendix

$$ {\displaystyle \begin{array}{l} At\kern1em x=-{L}_1\to {u}_1^{\prime }=0\to {a}_1 sink{L}_1+{b}_1 cosk{L}_1=0\\ {} At\kern1em x=0\to \left\{\begin{array}{l}{u}_1={u}_2\to {a}_1\mathit{\cos}0+{b}_1\mathit{\sin}0={F}_2\left({a}_2\mathit{\cos}0+{b}_2\sin 0\right)\to {a}_1={F}_2{a}_2\\ {}{u}_1^{\prime }={u}_2^{\prime}\to k{b}_1={F}_2^{\prime }{a}_2+{F}_2k{b}_2\end{array}\right\},\kern1em {F}_2=\frac{1}{L_2{d}_1},{F}_2^{\prime }=\frac{d_1-{d}_3}{l_2^2{d}_1^2}\end{array}} $$
$$ At\kern1em x={L}_2\to \left\{\begin{array}{l}{u}_2={u}_3\to {a}_3 cosk{L}_2+{b}_3 sink{L}_2={F}_2\left({a}_2\cos k{L}_2+{b}_2\sin k{L}_2\right)\\ {}{u}_2^{\prime }={u}_3^{\prime}\to k{b}_3\cos k{L}_2-k{a}_3\sin k{L}_2=\left({F}_2^{\prime }{a}_2+{F}_2k{b}_2\right)\cos k{L}_2+\left({F}_2^{\prime }{b}_2-{F}_2k{a}_2\right) sink{L}_2\end{array}\right\},{F}_2=\frac{1}{L_2{d}_3},{F}_2^{\prime }=\frac{d_1-{d}_3}{L_2^2{d}_3^2} $$
$$ {\displaystyle \begin{array}{l} At\kern1em x={L}_2+{L}_3\to \left\{\begin{array}{l}{u}_3={u}_4\to {a}_3\cos k\left({L}_2+{L}_3\right)+{b}_3\sin k\left({L}_2+{L}_3\right)={F}_4\left[{a}_4\cos k\left({L}_2+{L}_3\right)+{b}_4\sin k\left({L}_2+{L}_3\right)\right]\\ {}{u}_3^{\prime }={u}_4^{\prime}\to -k{a}_3\sin k\left({L}_2+{L}_3\right)+k{b}_3\cos k\left({L}_2+{L}_3\right)=\left({F}_4^{\prime }{a}_4+{F}_4k{b}_4\right)\cos k\left({L}_2+{L}_3\right)\\ {}\kern19.6em +\left({F}_4^{\prime }{b}_4-{F}_4k{a}_4\right)\sin k\left({L}_2+{L}_3\right)\end{array}\right\},\\ {}\kern2em {F}_4=\frac{1}{L_4{d}_3},{F}_4^{\prime }=\frac{d_3-{d}_5}{L_4^2{d}_3^2}\end{array}} $$
$$ {\displaystyle \begin{array}{l} At\kern1em x={L}_2+{L}_3+{L}_4\to \left\{\begin{array}{l}{u}_4={u}_5\to {a}_5\cos k\left({L}_2+{L}_3+{L}_4\right)+{b}_5\sin k\left({L}_2+{L}_3+{L}_4\right)={F}_4\left[\begin{array}{l}{a}_4\cos k\left({L}_2+{L}_3+{L}_4\right)\\ {}+{b}_4\sin k\left({L}_2+{L}_3+{L}_4\right)\end{array}\right]\\ {}\kern4.12em \\ {}{u}_4^{\prime }={u}_5^{\prime}\to -k{a}_5\sin k\left({L}_2+{L}_3+{L}_4\right)+k{b}_5\cos k\left({L}_2+{L}_3+{L}_4\right)=\left({F}_4^{\prime }{a}_4+{F}_4k{b}_4\right)\cos k\left({L}_2+{L}_3+{L}_4\right)\\ {}\kern22.84em +\left({F}_4^{\prime }{b}_4-{F}_4k{a}_4\right)\sin k\left({L}_2+{L}_3+{L}_4\right)\end{array}\right\},\\ {}\kern1.44em {F}_4=\frac{1}{L_4{d}_5}\to {F}_4^{\prime }=\frac{d_3-{d}_5}{L_4^2{d}_5^2}\end{array}} $$
$$ {\displaystyle \begin{array}{l} At\kern1em x={L}_2+{L}_3+{L}_4+{L}_5\to {u}_5^{\prime }=0\to -{a}_5\sin k\left({L}_2+{L}_3+{L}_4+{L}_5\right)+{b}_5\cos k\left({L}_2+{L}_3+{L}_4+{L}_5\right)=0\\ {}N=\mid {u}_{out}/{u}_{in}\mid =\mid \left\{{a}_5\cos k\left({L}_2+{L}_3+{L}_4+{L}_5\right)+{b}_5\sin k\left({L}_2+{L}_3+{L}_4+{L}_5\right)\right\}/\left\{{a}_1\cos k{L}_1-{b}_1\sin k{L}_1\right\}\mid \end{array}} $$

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Razavi, H., Keymanesh, M. & Golpayegani, I.F. Analysis of free and forced vibrations of ultrasonic vibrating tools, case study: ultrasonic assisted surface rolling process. Int J Adv Manuf Technol 103, 2725–2737 (2019). https://doi.org/10.1007/s00170-019-03718-x

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