A systematic investigation of the sound radiation of orthogonally stiffened plates is presented using a numerical procedure that combines the finite element method with the Rayleigh integral. Results are computed for plates with different numbers of stiffeners, stiffener depth, and plate thickness to investigate the dependence on the most important parameters. Differences are seen in the radiation efficiency of stiffened plates compared with unstiffened panels. In the monopole region, the result depends on the type of mode that dominates the response. For excitation within a bay, the radiation efficiency is reduced to that of a single bay if the stiffeners are stiff enough. If excited on a stiffener, the plate tends to radiate sound over its full surface area. In the short-circuiting region, on average, the radiation efficiency is equal to that of a smaller bay-sized panel with clamped edges, regardless of the excitation position. Results from the systematic study of 120 numerical cases are used to develop asymptotic formulae for the radiation efficiency of stiffened plates based on existing formulae for unstiffened panels. For all tested configurations, the average difference between the formulae and the numerical calculations was 0.3 dB over the whole frequency spectrum, with a standard deviation of ±1.5 dB. Between the frequency bands, the mean value varied between −2 and 3 dB, with a standard deviation of up to ±1.5 dB in the monopole region and a larger variation of up to ±5 dB in the short-circuiting region.
{"title":"An engineering approach for estimating the radiation efficiency of orthogonally stiffened plates","authors":"Christoph Knuth, G. Squicciarini, D. Thompson","doi":"10.1115/1.4056741","DOIUrl":"https://doi.org/10.1115/1.4056741","url":null,"abstract":"\u0000 A systematic investigation of the sound radiation of orthogonally stiffened plates is presented using a numerical procedure that combines the finite element method with the Rayleigh integral. Results are computed for plates with different numbers of stiffeners, stiffener depth, and plate thickness to investigate the dependence on the most important parameters. Differences are seen in the radiation efficiency of stiffened plates compared with unstiffened panels. In the monopole region, the result depends on the type of mode that dominates the response. For excitation within a bay, the radiation efficiency is reduced to that of a single bay if the stiffeners are stiff enough. If excited on a stiffener, the plate tends to radiate sound over its full surface area. In the short-circuiting region, on average, the radiation efficiency is equal to that of a smaller bay-sized panel with clamped edges, regardless of the excitation position. Results from the systematic study of 120 numerical cases are used to develop asymptotic formulae for the radiation efficiency of stiffened plates based on existing formulae for unstiffened panels. For all tested configurations, the average difference between the formulae and the numerical calculations was 0.3 dB over the whole frequency spectrum, with a standard deviation of ±1.5 dB. Between the frequency bands, the mean value varied between −2 and 3 dB, with a standard deviation of up to ±1.5 dB in the monopole region and a larger variation of up to ±5 dB in the short-circuiting region.","PeriodicalId":49957,"journal":{"name":"Journal of Vibration and Acoustics-Transactions of the Asme","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89613512","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The dynamic behaviors of a pinned-pinned spinning exponentially functionally graded shaft with unbalanced loads are investigated. The shaft is simulated in the Rayleigh beam model considering rotary inertia and gyroscopic effects. The governing equation for the flexural vibration of the shaft is derived via the Hamilton principle. Based on the boundary conditions, both the exact and approximate whirl frequency equations of the system are obtained analytically. Also, the validity of the proposed model is confirmed by comparing it with the results reported in the literature. Finally, a numerical study on the basis of the analytical solutions is performed to evaluate the main parameters, including slenderness ratio (α), gradient index (β), the mass ratio (μ), and eccentric distance (γ) on the whirl frequency, critical spinning speed, mode shapes, and stability of the system. The results reveal that the vibration and instability of the spinning shaft are strongly dependent on the unbalanced load and material gradient.
{"title":"Dynamic behavior of a spinning exponentially functionally graded shaft with unbalanced load","authors":"Guangding Wang, Qing Zhao, Liqing Chen, Huiqun Yuan","doi":"10.1115/1.4056656","DOIUrl":"https://doi.org/10.1115/1.4056656","url":null,"abstract":"\u0000 The dynamic behaviors of a pinned-pinned spinning exponentially functionally graded shaft with unbalanced loads are investigated. The shaft is simulated in the Rayleigh beam model considering rotary inertia and gyroscopic effects. The governing equation for the flexural vibration of the shaft is derived via the Hamilton principle. Based on the boundary conditions, both the exact and approximate whirl frequency equations of the system are obtained analytically. Also, the validity of the proposed model is confirmed by comparing it with the results reported in the literature. Finally, a numerical study on the basis of the analytical solutions is performed to evaluate the main parameters, including slenderness ratio (α), gradient index (β), the mass ratio (μ), and eccentric distance (γ) on the whirl frequency, critical spinning speed, mode shapes, and stability of the system. The results reveal that the vibration and instability of the spinning shaft are strongly dependent on the unbalanced load and material gradient.","PeriodicalId":49957,"journal":{"name":"Journal of Vibration and Acoustics-Transactions of the Asme","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89627094","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Acoustic radiation from stiffened double concentric large cylindrical shells with periodic cavities is analytically examined in the medium and high frequency range using the Sommerfeld-Watson transform. Creeping wave acoustic model of the stiffened double cylindrical shells in the shadow, penumbra and illuminated regions is established by the residue theorem. An asymptotic expression of far-field acoustic pressure is derived in the geometrical acoustic zone according to the stationary phase method. Sound field of the bare or stiffened double cylindrical shells with annular fluid is determined by the creeping wave poles in the broad circumferential region. Mechanisms of creeping wave propagation through the stiffened double cylindrical shells are shown. There are a great many elastic and acoustic waves that can't transmit through the annular fluid, annular bulkheads and periodic cavities with a moderate gap between the double circular cylindrical shells. Acoustic boundary layer theories are proposed to describe sound propagation through the annular fluid and bulkheads for the creeping waves.
{"title":"Acoustic radiation from stiffened double concentric large cylindrical shells: Part II Creeping waves","authors":"Xiongtao Cao","doi":"10.1115/1.4056634","DOIUrl":"https://doi.org/10.1115/1.4056634","url":null,"abstract":"\u0000 Acoustic radiation from stiffened double concentric large cylindrical shells with periodic cavities is analytically examined in the medium and high frequency range using the Sommerfeld-Watson transform. Creeping wave acoustic model of the stiffened double cylindrical shells in the shadow, penumbra and illuminated regions is established by the residue theorem. An asymptotic expression of far-field acoustic pressure is derived in the geometrical acoustic zone according to the stationary phase method. Sound field of the bare or stiffened double cylindrical shells with annular fluid is determined by the creeping wave poles in the broad circumferential region. Mechanisms of creeping wave propagation through the stiffened double cylindrical shells are shown. There are a great many elastic and acoustic waves that can't transmit through the annular fluid, annular bulkheads and periodic cavities with a moderate gap between the double circular cylindrical shells. Acoustic boundary layer theories are proposed to describe sound propagation through the annular fluid and bulkheads for the creeping waves.","PeriodicalId":49957,"journal":{"name":"Journal of Vibration and Acoustics-Transactions of the Asme","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90632076","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Acoustic radiation from stiffened double concentric large cylindrical shells with periodic cavities is analytically investigated via circumferential harmonic waves driven by a point force. The vibration of double isotropic circular cylindrical shells is described by the first-order shear deformation shell theory. One set of uniformly spaced annular bulkheads connects the inner and outer cylindrical shells. In-plane motion equations of the annular bulkheads are expressed by two displacement potential functions. Sound pressure loadings of periodic cavities exerting on the inner and outer cylindrical shells are derived according to the Fourier transform and Poisson summation formula. Far-field sound pressure of the stiffened double cylindrical shells is obtained using the stationary phase method and acoustic radiation features of stiffened double concentric large cylindrical shells with periodic cavities are analyzed in terms of sound pressure power spectra and sound pressure level. Acoustic propagation features of stiffened double cylindrical shells with or without acoustic cavities are shown.
{"title":"Acoustic radiation from stiffened double concentric large cylindrical shells: Part I Circumferential harmonic waves","authors":"Xiongtao Cao","doi":"10.1115/1.4056633","DOIUrl":"https://doi.org/10.1115/1.4056633","url":null,"abstract":"\u0000 Acoustic radiation from stiffened double concentric large cylindrical shells with periodic cavities is analytically investigated via circumferential harmonic waves driven by a point force. The vibration of double isotropic circular cylindrical shells is described by the first-order shear deformation shell theory. One set of uniformly spaced annular bulkheads connects the inner and outer cylindrical shells. In-plane motion equations of the annular bulkheads are expressed by two displacement potential functions. Sound pressure loadings of periodic cavities exerting on the inner and outer cylindrical shells are derived according to the Fourier transform and Poisson summation formula. Far-field sound pressure of the stiffened double cylindrical shells is obtained using the stationary phase method and acoustic radiation features of stiffened double concentric large cylindrical shells with periodic cavities are analyzed in terms of sound pressure power spectra and sound pressure level. Acoustic propagation features of stiffened double cylindrical shells with or without acoustic cavities are shown.","PeriodicalId":49957,"journal":{"name":"Journal of Vibration and Acoustics-Transactions of the Asme","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88256502","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A nonlinear dynamic model of a multi-mesh involute spur pinion-driven face-gear split-torque drive system is developed. The lumped mass model consists of five pinions and two face-gears with seven rotational degrees-of-freedom. Model includes two inputs, two outputs, and three idler gears meshing with two identical face gears at the bottom and top of the assembly. Face-gear tooth form and corresponding spur gear teeth are established by differential geometry and the theory of gearing. Face-gear drive system mesh stiffnesses are established by utilizing Finite Strip Method for the first time in the literature. The mesh stiffness and damping are time-variant. The model includes clearance type nonlinearity. The Harmonic Balance Method (HBM) is utilized to solve the nonlinear differential equations of motion. The results are compared with the time simulation results. The stability is checked with Floquet Theory and bifurcation diagrams from Poincare Sections. The model is fully capable of generating the tooth geometries, mesh stiffnesses, and gear train dynamics without reliance on any package programs. Hence, the model provides flexibility and fast computation for parametric studies compared to existing literature. The effect of the pinions' orientation angle patterns on the face-gear are investigated to evaluate the mesh phasing effects among system dynamic response. The effect of subharmonic motion on the dynamic response of the split-torque face-gear system is also demonstrated for the first time in the literature. The case studies demonstrate that subharmonic resonance peaks are avoided.
{"title":"Non-linear Dynamic Modelling and Analysis of Split-Torque Face-Gear Drive Systems","authors":"Mustafa Ozgur Aydogan, Z. Saribay, H. Ozguven","doi":"10.1115/1.4056632","DOIUrl":"https://doi.org/10.1115/1.4056632","url":null,"abstract":"\u0000 A nonlinear dynamic model of a multi-mesh involute spur pinion-driven face-gear split-torque drive system is developed. The lumped mass model consists of five pinions and two face-gears with seven rotational degrees-of-freedom. Model includes two inputs, two outputs, and three idler gears meshing with two identical face gears at the bottom and top of the assembly. Face-gear tooth form and corresponding spur gear teeth are established by differential geometry and the theory of gearing. Face-gear drive system mesh stiffnesses are established by utilizing Finite Strip Method for the first time in the literature. The mesh stiffness and damping are time-variant. The model includes clearance type nonlinearity. The Harmonic Balance Method (HBM) is utilized to solve the nonlinear differential equations of motion. The results are compared with the time simulation results. The stability is checked with Floquet Theory and bifurcation diagrams from Poincare Sections. The model is fully capable of generating the tooth geometries, mesh stiffnesses, and gear train dynamics without reliance on any package programs. Hence, the model provides flexibility and fast computation for parametric studies compared to existing literature. The effect of the pinions' orientation angle patterns on the face-gear are investigated to evaluate the mesh phasing effects among system dynamic response. The effect of subharmonic motion on the dynamic response of the split-torque face-gear system is also demonstrated for the first time in the literature. The case studies demonstrate that subharmonic resonance peaks are avoided.","PeriodicalId":49957,"journal":{"name":"Journal of Vibration and Acoustics-Transactions of the Asme","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89399937","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Time-domain equivalent source method (TDESM) has been employed for indoor simulation of pass-by noise (PBN). Despite removing the requirement of dedicated large semi-anechoic chambers for the microphone array technique, it is still difficult to obtain high simulation accuracy due to the existence of signal splicing error and ill-consideration of Doppler effect induced by the vehicle motion. In this paper, the TDESM is developed to improve the simulation accuracy of PBN. Unlike the previous TDESM, the improved TDESM (I-TDESM) models the PBN by superposing the contributions of an array of moving equivalent sources, which can take the influence of vehicle motion into account. The PBN at the receivers can be directly reconstructed via the sound field calculation of moving sources. Numerical studies and real vehicle experiments are implemented to demonstrate the feasibility of the I-TDESM. Simulation results show that the I-TDESM can naturally incorporate the Doppler effect and effectively eliminate the signal splicing error, thus improving the simulation accuracy. The experiment results of real vehicle further validate the I-TDESM.
{"title":"An improved time-domain equivalent source method for accurate indoor simulation of pass-by noise","authors":"Ce Liu, Xiao-Zheng Zhang, Yong-Bin Zhang","doi":"10.1115/1.4056195","DOIUrl":"https://doi.org/10.1115/1.4056195","url":null,"abstract":"\u0000 Time-domain equivalent source method (TDESM) has been employed for indoor simulation of pass-by noise (PBN). Despite removing the requirement of dedicated large semi-anechoic chambers for the microphone array technique, it is still difficult to obtain high simulation accuracy due to the existence of signal splicing error and ill-consideration of Doppler effect induced by the vehicle motion. In this paper, the TDESM is developed to improve the simulation accuracy of PBN. Unlike the previous TDESM, the improved TDESM (I-TDESM) models the PBN by superposing the contributions of an array of moving equivalent sources, which can take the influence of vehicle motion into account. The PBN at the receivers can be directly reconstructed via the sound field calculation of moving sources. Numerical studies and real vehicle experiments are implemented to demonstrate the feasibility of the I-TDESM. Simulation results show that the I-TDESM can naturally incorporate the Doppler effect and effectively eliminate the signal splicing error, thus improving the simulation accuracy. The experiment results of real vehicle further validate the I-TDESM.","PeriodicalId":49957,"journal":{"name":"Journal of Vibration and Acoustics-Transactions of the Asme","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2022-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89195615","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Mechanical face seals are wide-spread in many applications of powered equipment and turbomachinery. Often machine vibration and noise are unavoidable because of changing conditions which can be persistent and forceful. In critical applications when seals fail, they may have significant or even catastrophic consequences. To ensure the safety of such machinery and its associated mechanical components, machine vibration and noise must be diagnosed and quantified to keep the system's response within certain limits. This work focuses on the dynamics of a flexibly mounted stator mechanical face seal that is subjected to combinations of broad-band noisy vibrations of the shaft and the housing. In all previous work, the positions of the housing and the shaft have been considered fixed. The current work relaxes that condition, augmenting the equations of motion to incorporate equipment's noisy vibrations. Noises are expediently produced by the Weierstrass-Mandelbrot (WM) fractal function. A numerical simulation ensues, and the time-domain responses are subject to spectral analyses. Results show that under some design conditions, the seal is largely insensitive to machine vibrations. However, under other conditions, the seal response to exterior machine noise exhibits a rich spectral content that stems from various transient phenomena that include intensified half-frequency whirl, near synchronous response at steady-state, and super-synchronous higher harmonic oscillations caused by face contact.
{"title":"DYNAMIC SENSITIVITIES TO FRACTAL MACHINE NOISES IN A MECHANICAL FACE SEAL","authors":"I. Green","doi":"10.1115/1.4056194","DOIUrl":"https://doi.org/10.1115/1.4056194","url":null,"abstract":"\u0000 Mechanical face seals are wide-spread in many applications of powered equipment and turbomachinery. Often machine vibration and noise are unavoidable because of changing conditions which can be persistent and forceful. In critical applications when seals fail, they may have significant or even catastrophic consequences. To ensure the safety of such machinery and its associated mechanical components, machine vibration and noise must be diagnosed and quantified to keep the system's response within certain limits. This work focuses on the dynamics of a flexibly mounted stator mechanical face seal that is subjected to combinations of broad-band noisy vibrations of the shaft and the housing. In all previous work, the positions of the housing and the shaft have been considered fixed. The current work relaxes that condition, augmenting the equations of motion to incorporate equipment's noisy vibrations. Noises are expediently produced by the Weierstrass-Mandelbrot (WM) fractal function. A numerical simulation ensues, and the time-domain responses are subject to spectral analyses. Results show that under some design conditions, the seal is largely insensitive to machine vibrations. However, under other conditions, the seal response to exterior machine noise exhibits a rich spectral content that stems from various transient phenomena that include intensified half-frequency whirl, near synchronous response at steady-state, and super-synchronous higher harmonic oscillations caused by face contact.","PeriodicalId":49957,"journal":{"name":"Journal of Vibration and Acoustics-Transactions of the Asme","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2022-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90928139","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Recent results in the literature highlight the impact of nonlinear inertial forces on the post-flutter Limit Cycle Oscillation (LCO) characteristics of highly deflected structures in supersonic axial flow. The current investigation examines how the ability to passively modulate nonlinear inertial forces may alter the overall aeroelastic response. The structural model is a one-dimensional nonlinear inextensible plate subject to nonlinear aerodynamic forces in accordance with a new, geometrically modified third-order Piston Theory. For the linear aeroelastic case, we find that non-homogeneous mass distribution elicits discontinuous increases in the critical Mach number for flutter and several flutter mode-switching phenomena that are not observed when mass is added homogeneously. The existence of several different flutter mode mechanisms as a function of a concentrated mass location leads to different post-flutter LCO amplitude behavior. This is found to transition the underlying nonlinear structural dynamics to either be stiffening (when lower-order modes merge) or softening (when higher-order modes merge), which in turn alter the influence of nonlinear aerodynamic forces. We also address discrepancies in LCO amplitude trends due the nonlinear inertial forces previously reported in the literature.
{"title":"On the Influence of Nonlinear Inertial Forces on the Limit Cycle Oscillations of an Inextensible Plate in a Supersonic Axial Flow","authors":"S. Stanton, S. Choi, Kevin A. McHugh","doi":"10.1115/1.4056127","DOIUrl":"https://doi.org/10.1115/1.4056127","url":null,"abstract":"\u0000 Recent results in the literature highlight the impact of nonlinear inertial forces on the post-flutter Limit Cycle Oscillation (LCO) characteristics of highly deflected structures in supersonic axial flow. The current investigation examines how the ability to passively modulate nonlinear inertial forces may alter the overall aeroelastic response. The structural model is a one-dimensional nonlinear inextensible plate subject to nonlinear aerodynamic forces in accordance with a new, geometrically modified third-order Piston Theory. For the linear aeroelastic case, we find that non-homogeneous mass distribution elicits discontinuous increases in the critical Mach number for flutter and several flutter mode-switching phenomena that are not observed when mass is added homogeneously. The existence of several different flutter mode mechanisms as a function of a concentrated mass location leads to different post-flutter LCO amplitude behavior. This is found to transition the underlying nonlinear structural dynamics to either be stiffening (when lower-order modes merge) or softening (when higher-order modes merge), which in turn alter the influence of nonlinear aerodynamic forces. We also address discrepancies in LCO amplitude trends due the nonlinear inertial forces previously reported in the literature.","PeriodicalId":49957,"journal":{"name":"Journal of Vibration and Acoustics-Transactions of the Asme","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2022-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89882112","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this study, we investigate the importance of the fourth-order time derivative that appears in the equations derived by Jacques Antoine Charles Bresse in 1859, as well as in equations that were derived by Stephen Prokofievich Timoshenko and Paul Ehrenfest during years 1912-1913 and reported by Timoshenko in the 1916 book on theory of elasticity in the Russian language and then in two papers dated 1920 and 1921, in English. The first part of the study demonstrates that Timoshenko and Ehrenfest did not overestimate the importance of the fourth-order derivative term in their equations. The second part deals with the debate on the so called “second spectrum” attendant in original set of equations. It is shown that in the truncated Timoshenko-Ehrenfest equations--which is asymptotically consistent with elasticity theory—“the second spectrum” issue does not arise. Thus, two parts of this study are intricately interrelated with each other.
在本研究中，我们研究了出现在Jacques Antoine Charles Bresse于1859年推导的方程中的四阶时间导数的重要性，以及Stephen Prokofievich Timoshenko和Paul Ehrenfest在1912-1913年间推导的方程，这些方程由Timoshenko在1916年的俄语弹性理论书中报告，然后在1920年和1921年的两篇英文论文中报告。研究的第一部分表明Timoshenko和Ehrenfest并没有高估四阶导数项在他们的方程中的重要性。第二部分讨论了关于原方程组中所谓“第二谱”的争论。结果表明，在截断的Timoshenko-Ehrenfest方程中(与弹性理论渐近一致)，不会出现“第二谱”问题。因此，本研究的两个部分是错综复杂地相互关联的。
{"title":"Did S. P. Timoshenko and P. Ehrenfest Overestimate the Importance of the Fourth-Order Time Derivative in Their Beam Theory?","authors":"I. Elishakoff","doi":"10.1115/1.4055975","DOIUrl":"https://doi.org/10.1115/1.4055975","url":null,"abstract":"\u0000 In this study, we investigate the importance of the fourth-order time derivative that appears in the equations derived by Jacques Antoine Charles Bresse in 1859, as well as in equations that were derived by Stephen Prokofievich Timoshenko and Paul Ehrenfest during years 1912-1913 and reported by Timoshenko in the 1916 book on theory of elasticity in the Russian language and then in two papers dated 1920 and 1921, in English. The first part of the study demonstrates that Timoshenko and Ehrenfest did not overestimate the importance of the fourth-order derivative term in their equations. The second part deals with the debate on the so called “second spectrum” attendant in original set of equations. It is shown that in the truncated Timoshenko-Ehrenfest equations--which is asymptotically consistent with elasticity theory—“the second spectrum” issue does not arise. Thus, two parts of this study are intricately interrelated with each other.","PeriodicalId":49957,"journal":{"name":"Journal of Vibration and Acoustics-Transactions of the Asme","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2022-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83557368","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, the theory of a Timoshenko-Ehrenfest beam is revisited and given a new perspective with particular emphasis on the relative significances of the parameters underlying the theory. The investigation is intended to broaden the scope and applicability of the theory. It has been shown that the two parameters that characterise the Timoshenko-Ehrenfest beam theory, namely the rotary inertia and the shear deformation, can be related and hence they can be combined into one parameter when predicting the beam's free vibration behaviour. A theoretical proof is given that explains why the effect of the shear deformation on the free vibration behaviour of a Timoshenko-Ehrenfest beam for any boundary condition will be always more pronounced than that of the rotary inertia. The range of applicability of the Timoshenko-Ehrenfest beam theory for realistic problems is demonstrated by a set of new curves, which provide considerable insights into the theory.
{"title":"FURTHER INSIGHTS INTO THE TIMOSHENKO-EHRENFEST BEAM THEORY","authors":"J. Banerjee, D. Kennedy, I. Elishakoff","doi":"10.1115/1.4055974","DOIUrl":"https://doi.org/10.1115/1.4055974","url":null,"abstract":"\u0000 In this paper, the theory of a Timoshenko-Ehrenfest beam is revisited and given a new perspective with particular emphasis on the relative significances of the parameters underlying the theory. The investigation is intended to broaden the scope and applicability of the theory. It has been shown that the two parameters that characterise the Timoshenko-Ehrenfest beam theory, namely the rotary inertia and the shear deformation, can be related and hence they can be combined into one parameter when predicting the beam's free vibration behaviour. A theoretical proof is given that explains why the effect of the shear deformation on the free vibration behaviour of a Timoshenko-Ehrenfest beam for any boundary condition will be always more pronounced than that of the rotary inertia. The range of applicability of the Timoshenko-Ehrenfest beam theory for realistic problems is demonstrated by a set of new curves, which provide considerable insights into the theory.","PeriodicalId":49957,"journal":{"name":"Journal of Vibration and Acoustics-Transactions of the Asme","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2022-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84641829","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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