The velocity and flux of spherical glass beads with nominal diameters of 200, 80 and 40 μm have been obtained by phase-Doppler anemometry in a round unconfined air jet over the first 28 diameters. The jet diameter was 15 mm and the exit velocity was 13 ms-1 giving a Reynolds number of 13 000 and a timescale of 1.15 ms, which increased quadratically with axial distance: the bead inertial time constants were 298, 48 and 12 ms. The purposes of the experiments were to quantify the velocity and flux characteristics of the beads and of the gas phase in the presence of the beads as a function of bead diameter and of the mass loading in the jet nozzle. Due to the large inertia of the 200 μm beads, the mean bead velocity downstream of the exit of the jet was constant and independent of mass loading up to 0.37 and the axial root mean square (r.m.s.) bead velocity decayed by about one-fifth : at the exit of the jet, the axial r.m.s. bead velocity was higher than that of the corresponding clean jet. The mean centreline velocity of the 80 μm beads decayed to about one-half of the bead exit velocity by 28 diameters downstream and was independent of mass loading up to 0.86. The decay rate of the mean gas centreline velocity in the presence of the beads reduced as the loading increased because of momentum transfer from the discrete to the gaseous phase. The axial r.m.s. velocity of the beads was comparable to that of the gas phase and both decreased with increasing loading and the rate of spread of the half width of the jet increased with increasing loading. For the 40 μm beads, the decay rate of the mean centreline velocity of the beads decreased with increasing loading and, in contrast to the 80 μm beads, the rate of spread decreased with increasing loading up to 0.80. The axial r.m.s. velocity of the beads became largest at a position downstream of the nozzle exit, which moved downstream with increasing loading and was larger than the axial r.m.s. velocity of the clean jet, although the beads were not expected to be responsive to the frequencies of the energy-containing eddies. The bead axial r.m.s. velocity was more than twice as large as the radial r.m.s. velocity and the correlation coefficient of the cross correlation was larger than that of the clean jet. The large bead turbulence, anisotropy and strong correlation coefficient are explained by the superposition of bead trajectories from regions of different bead mean velocity and are not because of acquisition of axial turbulent motion from the gaseous phase.
{"title":"Velocity and particle-flux characteristics of turbulent particle-laden jets","authors":"Y. Hardalupas, A. M. Taylor, J. Whitelaw","doi":"10.1098/rspa.1989.0117","DOIUrl":"https://doi.org/10.1098/rspa.1989.0117","url":null,"abstract":"The velocity and flux of spherical glass beads with nominal diameters of 200, 80 and 40 μm have been obtained by phase-Doppler anemometry in a round unconfined air jet over the first 28 diameters. The jet diameter was 15 mm and the exit velocity was 13 ms-1 giving a Reynolds number of 13 000 and a timescale of 1.15 ms, which increased quadratically with axial distance: the bead inertial time constants were 298, 48 and 12 ms. The purposes of the experiments were to quantify the velocity and flux characteristics of the beads and of the gas phase in the presence of the beads as a function of bead diameter and of the mass loading in the jet nozzle. Due to the large inertia of the 200 μm beads, the mean bead velocity downstream of the exit of the jet was constant and independent of mass loading up to 0.37 and the axial root mean square (r.m.s.) bead velocity decayed by about one-fifth : at the exit of the jet, the axial r.m.s. bead velocity was higher than that of the corresponding clean jet. The mean centreline velocity of the 80 μm beads decayed to about one-half of the bead exit velocity by 28 diameters downstream and was independent of mass loading up to 0.86. The decay rate of the mean gas centreline velocity in the presence of the beads reduced as the loading increased because of momentum transfer from the discrete to the gaseous phase. The axial r.m.s. velocity of the beads was comparable to that of the gas phase and both decreased with increasing loading and the rate of spread of the half width of the jet increased with increasing loading. For the 40 μm beads, the decay rate of the mean centreline velocity of the beads decreased with increasing loading and, in contrast to the 80 μm beads, the rate of spread decreased with increasing loading up to 0.80. The axial r.m.s. velocity of the beads became largest at a position downstream of the nozzle exit, which moved downstream with increasing loading and was larger than the axial r.m.s. velocity of the clean jet, although the beads were not expected to be responsive to the frequencies of the energy-containing eddies. The bead axial r.m.s. velocity was more than twice as large as the radial r.m.s. velocity and the correlation coefficient of the cross correlation was larger than that of the clean jet. The large bead turbulence, anisotropy and strong correlation coefficient are explained by the superposition of bead trajectories from regions of different bead mean velocity and are not because of acquisition of axial turbulent motion from the gaseous phase.","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":"411 1","pages":"31 - 78"},"PeriodicalIF":0.0,"publicationDate":"1989-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86838355","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A simple method is given to estimate the power, transmitted through a large elastic panel, when averaged over all directions of an incident plane sound wave, in the limit of short waves compared with plate dimensions. Details are given for the simple prototype example of a rectangular isotropic bending plate. The procedure provides a simple estimate of the forced non-resonant contribution to the power flow, in the short wave asymptotic limit, and previous results are recovered. The method is readily amenable to generalizations to deal with more complicated geometries and plate equations. As an illustration, results are presented for the case of a circular isotropic panel.
{"title":"The transmission of randomly incident sound through an elastic panel","authors":"F. G. Leppington","doi":"10.1098/rspa.1989.0121","DOIUrl":"https://doi.org/10.1098/rspa.1989.0121","url":null,"abstract":"A simple method is given to estimate the power, transmitted through a large elastic panel, when averaged over all directions of an incident plane sound wave, in the limit of short waves compared with plate dimensions. Details are given for the simple prototype example of a rectangular isotropic bending plate. The procedure provides a simple estimate of the forced non-resonant contribution to the power flow, in the short wave asymptotic limit, and previous results are recovered. The method is readily amenable to generalizations to deal with more complicated geometries and plate equations. As an illustration, results are presented for the case of a circular isotropic panel.","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":"7 1","pages":"153 - 165"},"PeriodicalIF":0.0,"publicationDate":"1989-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89025235","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The explicit form of the Green’s tensor in unbounded inhomogeneous dielectric, magnetic and partly conducting material is obtained. The tensor is given explicitly in terms of two scalar Green’s function potentials, each of which obeys its own wave equation. It is found that the gradients of the constitutive parameters at the source of the radiating currents may have a strong effect on the multipolar field. For example, an electric dipole embedded in a uniaxially inhomogeneous source-zone is found to produce additional fields with the characteristics of electric quadrupole (l = 2, m = ± 1) and a magnetic dipole (l = 1, m = ± 1). I present a mathematical formulation of Huygens’ principle for inhomogeneous media in terms of the Green’s tensor and its associated tensor potentials, thus extending the use of the Green’s tensor to scattering problems in inhomogeneous finite domains.
得到了无界非均匀介质、磁性和部分导电材料中格林张量的显式形式。张量被明确地用两个标量格林函数势给出，每一个都服从自己的波动方程。研究发现，辐射电流源处本构参数的梯度对多极场有很强的影响。例如，发现嵌入在单轴非均匀源区中的电偶极子会产生具有电四极子(l = 2, m =±1)和磁偶极子(l = 1, m =±1)特征的附加场。我提出了基于格林张量及其相关张量势的非均匀介质惠更斯原理的数学公式，从而将格林张量的使用扩展到非均匀有限域中的散射问题。
{"title":"Green’s tensors and associated potentials for electromagnetic waves in inhomogeneous material media","authors":"A. Ben-menahem","doi":"10.1098/rspa.1989.0118","DOIUrl":"https://doi.org/10.1098/rspa.1989.0118","url":null,"abstract":"The explicit form of the Green’s tensor in unbounded inhomogeneous dielectric, magnetic and partly conducting material is obtained. The tensor is given explicitly in terms of two scalar Green’s function potentials, each of which obeys its own wave equation. It is found that the gradients of the constitutive parameters at the source of the radiating currents may have a strong effect on the multipolar field. For example, an electric dipole embedded in a uniaxially inhomogeneous source-zone is found to produce additional fields with the characteristics of electric quadrupole (l = 2, m = ± 1) and a magnetic dipole (l = 1, m = ± 1). I present a mathematical formulation of Huygens’ principle for inhomogeneous media in terms of the Green’s tensor and its associated tensor potentials, thus extending the use of the Green’s tensor to scattering problems in inhomogeneous finite domains.","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":"2 1","pages":"106 - 79"},"PeriodicalIF":0.0,"publicationDate":"1989-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81272736","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
R. Dalitz, D. Davis, P. Fowler, A. Montwill, J. Pniewski, J. Zakrzewski
The existence of the H particle, the dihyperon predicted by Jaffe, would bring into question the existence of double hypernuclei. We review the two double hypernucleus events published in the literature. We include an independent report, hitherto unpublished, which was made on the ΛΛ10Be event in 1963 and clarifies the salient features of the event; this report reaffirms its published interpretation. We have made a simple calculation of the energy spectrum for Ξ- hyperons produced with K- beams in past emulsion experiments, with a result which accounts adequately for the paucity of reported double hypernucleus events. We outline a hybrid emulsion experiment that would locate Ξ- hyperon interactions efficiently and could thereby greatly improve our knowledge of double hypernuclei.
{"title":"The identified ΛΛ-hypernuclei and the predicted H-particle","authors":"R. Dalitz, D. Davis, P. Fowler, A. Montwill, J. Pniewski, J. Zakrzewski","doi":"10.1098/rspa.1989.0115","DOIUrl":"https://doi.org/10.1098/rspa.1989.0115","url":null,"abstract":"The existence of the H particle, the dihyperon predicted by Jaffe, would bring into question the existence of double hypernuclei. We review the two double hypernucleus events published in the literature. We include an independent report, hitherto unpublished, which was made on the ΛΛ10Be event in 1963 and clarifies the salient features of the event; this report reaffirms its published interpretation. We have made a simple calculation of the energy spectrum for Ξ- hyperons produced with K- beams in past emulsion experiments, with a result which accounts adequately for the paucity of reported double hypernucleus events. We outline a hybrid emulsion experiment that would locate Ξ- hyperon interactions efficiently and could thereby greatly improve our knowledge of double hypernuclei.","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":"33 1","pages":"1 - 17"},"PeriodicalIF":0.0,"publicationDate":"1989-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86975616","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Questions of belief are essential in analysing protocols for the authentication of principals in distributed computing systems. In this paper we motivate, set out, and exemplify a logic specifically designed for this analysis: we show how various protocols differ subtly with respect to the required initial assumptions of the participants and their final beliefs. Our formalism has enabled us to isolate and express these differences with a precision that was not previously possible. It has drawn attention to features of protocols of which we and their authors were previously unaware, and allowed us to suggest improvements to the protocols. The reasoning about some protocols has been mechanically verified. This paper starts with an informal account of the problem, goes on to explain the formalism to be used, and gives examples of its application to protocols from the literature, both with shared-key cryptography and with public-key cryptography. Some of the examples are chosen because of their practical importance, whereas others serve to illustrate subtle points of the logic and to explain how we use it. We discuss extensions of the logic motivated by actual practice; for example, to account for the use of hash functions in signatures. The final sections contain a formal semantics of the logic and some conclusions.
{"title":"A logic of authentication","authors":"M. Burrows, M. Abadi, R. Needham","doi":"10.1098/rspa.1989.0125","DOIUrl":"https://doi.org/10.1098/rspa.1989.0125","url":null,"abstract":"Questions of belief are essential in analysing protocols for the authentication of principals in distributed computing systems. In this paper we motivate, set out, and exemplify a logic specifically designed for this analysis: we show how various protocols differ subtly with respect to the required initial assumptions of the participants and their final beliefs. Our formalism has enabled us to isolate and express these differences with a precision that was not previously possible. It has drawn attention to features of protocols of which we and their authors were previously unaware, and allowed us to suggest improvements to the protocols. The reasoning about some protocols has been mechanically verified. This paper starts with an informal account of the problem, goes on to explain the formalism to be used, and gives examples of its application to protocols from the literature, both with shared-key cryptography and with public-key cryptography. Some of the examples are chosen because of their practical importance, whereas others serve to illustrate subtle points of the logic and to explain how we use it. We discuss extensions of the logic motivated by actual practice; for example, to account for the use of hash functions in signatures. The final sections contain a formal semantics of the logic and some conclusions.","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":"30 1","pages":"233 - 271"},"PeriodicalIF":0.0,"publicationDate":"1989-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75503251","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Thermal analysis in general, and differential scanning calorimetry in particular, offer rapid and automated techniques for studying exothermic decompositions. When reaction is not too complex, values for the exothermicities and Arrhenius parameters may be extracted. The basis of most mathematical treatments is not deep and quite commonly rests on analyses of ‘static’ situations, in which the responses to constant ambient temperatures are the starting point. In this paper we give an economical analysis of the dynamic process of heating a reactive sample continuously. The reduced variables of thermal explosion theory provide a compact representation of the standard case of a single reaction obeying a first-order rate law with an Arrhenius temperature dependence. An asymptotic treatment is used to refine the equations of mass and energy conservation to a practically important form. The natural variables that arise in this way lead to a model of the system in terms of the rate of the chemical reaction, rather than the more usual temperature of the sample. This new treatment leads to a clear and compact description of the behaviour of the reaction for all useful operating conditions. The well-known Kissinger relation normally derived from static treatments is seen to be relevant in this model, but care must be taken in choosing the precise physical quantities to which it relates.
{"title":"The effects of a linearly increasing ambient temperature on an exothermic reaction in a closed system","authors":"T. Boddington, P. Gray, S. R. Kay","doi":"10.1098/rspa.1989.0106","DOIUrl":"https://doi.org/10.1098/rspa.1989.0106","url":null,"abstract":"Thermal analysis in general, and differential scanning calorimetry in particular, offer rapid and automated techniques for studying exothermic decompositions. When reaction is not too complex, values for the exothermicities and Arrhenius parameters may be extracted. The basis of most mathematical treatments is not deep and quite commonly rests on analyses of ‘static’ situations, in which the responses to constant ambient temperatures are the starting point. In this paper we give an economical analysis of the dynamic process of heating a reactive sample continuously. The reduced variables of thermal explosion theory provide a compact representation of the standard case of a single reaction obeying a first-order rate law with an Arrhenius temperature dependence. An asymptotic treatment is used to refine the equations of mass and energy conservation to a practically important form. The natural variables that arise in this way lead to a model of the system in terms of the rate of the chemical reaction, rather than the more usual temperature of the sample. This new treatment leads to a clear and compact description of the behaviour of the reaction for all useful operating conditions. The well-known Kissinger relation normally derived from static treatments is seen to be relevant in this model, but care must be taken in choosing the precise physical quantities to which it relates.","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":"105 1","pages":"269 - 283"},"PeriodicalIF":0.0,"publicationDate":"1989-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79259922","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
I consider fluid motions in the neighbourhood of a cosmic string by using complex variable techniques. In particular I study the motion of line vortices and show that a single vortex cannot remain at rest but must orbit the cosmic string. I also discuss the motion of vortices in the flow of a wake induced by a moving cosmic string. Some possible connections with string theory are noted.
{"title":"Vortex motion in the neighbourhood of a cosmic string","authors":"G. Gibbons","doi":"10.1098/rspa.1989.0113","DOIUrl":"https://doi.org/10.1098/rspa.1989.0113","url":null,"abstract":"I consider fluid motions in the neighbourhood of a cosmic string by using complex variable techniques. In particular I study the motion of line vortices and show that a single vortex cannot remain at rest but must orbit the cosmic string. I also discuss the motion of vortices in the flow of a wake induced by a moving cosmic string. Some possible connections with string theory are noted.","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":"15 1","pages":"431 - 439"},"PeriodicalIF":0.0,"publicationDate":"1989-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83388215","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Following the accident at the Chernobyl 4 nuclear power reactor in the Ukraine, U. S. S. R., on 25 April 1986, computer models were used to analyse dispersal of the radioactive material across Europe according to the meteorological situation, and interpret the available radiological measurements. Subsequently, after Soviet scientists presented their accounts of the accident and its causes in Vienna in August 1986, and as further measurements became available from other countries, more detailed assessments were undertaken of the release pattern over the 10-day period after the initial explosions, and the resulting contamination outside the U. S. S. R.
{"title":"Analysis of the dispersal and deposition of radionuclides from Chernobyl across Europe","authors":"H. ApSimon, J. J. Wilson, K. Simms","doi":"10.1098/rspa.1989.0111","DOIUrl":"https://doi.org/10.1098/rspa.1989.0111","url":null,"abstract":"Following the accident at the Chernobyl 4 nuclear power reactor in the Ukraine, U. S. S. R., on 25 April 1986, computer models were used to analyse dispersal of the radioactive material across Europe according to the meteorological situation, and interpret the available radiological measurements. Subsequently, after Soviet scientists presented their accounts of the accident and its causes in Vienna in August 1986, and as further measurements became available from other countries, more detailed assessments were undertaken of the release pattern over the 10-day period after the initial explosions, and the resulting contamination outside the U. S. S. R.","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":"20 1","pages":"365 - 405"},"PeriodicalIF":0.0,"publicationDate":"1989-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78746328","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In thermal explosion theory it is usually impossible analytically, and sometimes a substantial task numerically, to locate the ambient temperature at which transition from discontinuous to continuous behaviour occurs. It is possible to establish analytically a lower bound that is remarkably close to the numerically computed value.
{"title":"Thermal explosions, criticality and the disappearance of criticality in systems with distributed temperatures IV. Rigorous bounds and their practical relevance","authors":"G. Wake, T. Boddington, P. Gray","doi":"10.1098/rspa.1989.0107","DOIUrl":"https://doi.org/10.1098/rspa.1989.0107","url":null,"abstract":"In thermal explosion theory it is usually impossible analytically, and sometimes a substantial task numerically, to locate the ambient temperature at which transition from discontinuous to continuous behaviour occurs. It is possible to establish analytically a lower bound that is remarkably close to the numerically computed value.","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":"26 1","pages":"285 - 289"},"PeriodicalIF":0.0,"publicationDate":"1989-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81728711","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper the simplest and most widely used model of a rigid block undergoing harmonic forcing is analysed in detail. The block is shown to possess extremely complicated dynamics, with many different types of response being revealed. Symmetric single-impact subharmonic orbits of all orders are found and regions of parameter space in which they occur are given. In particular, period-doubling cascades of asymmetric orbits are found, which ultimately produce an apparently non-periodic or chaotic response. Sensitivity to initial conditions is illustrated, which leads to uncertainty in the prediction of the asymptotic dynamics. Nevertheless, the transient response may be the most important in connection with real earthquakes. To this end, the concept of the domain of maximum transients is introduced. In this light the response is shown to be quite ordered and predictable, despite the chaotic nature of the asymptotic domain of attraction. It is shown that safety issues cannot be satisfactorily resolved until an agreed set of initial conditions is established. It appears that blocks may survive under very high accelerations and topple at very low accelerations provided the initial conditions are correct. Consideration is also given to the use of actual earthquake recordings in attempting to reproduce responses in given structures. If the present conclusions carry over to general excitations, then small errors in recordings may produce large differences in response. The present methods include orbital stability techniques together with detailed numerical computations. These results are backed up by encouraging qualitative agreement from an electronic analogue circuit.
{"title":"On the dynamics of rigid-block motion under harmonic forcing","authors":"S. Hogan","doi":"10.1098/rspa.1989.0114","DOIUrl":"https://doi.org/10.1098/rspa.1989.0114","url":null,"abstract":"In this paper the simplest and most widely used model of a rigid block undergoing harmonic forcing is analysed in detail. The block is shown to possess extremely complicated dynamics, with many different types of response being revealed. Symmetric single-impact subharmonic orbits of all orders are found and regions of parameter space in which they occur are given. In particular, period-doubling cascades of asymmetric orbits are found, which ultimately produce an apparently non-periodic or chaotic response. Sensitivity to initial conditions is illustrated, which leads to uncertainty in the prediction of the asymptotic dynamics. Nevertheless, the transient response may be the most important in connection with real earthquakes. To this end, the concept of the domain of maximum transients is introduced. In this light the response is shown to be quite ordered and predictable, despite the chaotic nature of the asymptotic domain of attraction. It is shown that safety issues cannot be satisfactorily resolved until an agreed set of initial conditions is established. It appears that blocks may survive under very high accelerations and topple at very low accelerations provided the initial conditions are correct. Consideration is also given to the use of actual earthquake recordings in attempting to reproduce responses in given structures. If the present conclusions carry over to general excitations, then small errors in recordings may produce large differences in response. The present methods include orbital stability techniques together with detailed numerical computations. These results are backed up by encouraging qualitative agreement from an electronic analogue circuit.","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":"66 1","pages":"441 - 476"},"PeriodicalIF":0.0,"publicationDate":"1989-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91302970","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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