Abstract. In this paper a model of an –stepped bar with variable Cross-sections coupled with foun... more Abstract. In this paper a model of an –stepped bar with variable Cross-sections coupled with foundation by means of lumped masses and springs is studied. It is assumed that the process of vibrations in each section of the bar is described by a wave equation. The analytical tools of vibration analysis are based on finding eigenfunctions with piecewise continuous derivatives, which are orthogonal with respect to a generalized weight function. These eigenfunctions automatically satisfy the boundary conditions at the end points as well as the non-classical boundary conditions at the junctions. The solution of the problems is formulated in terms of Green function. By means of the proposed algorithm a problem of arbitrary complexity could be considered in the same terms as a single homogeneous bar. This algorithm is efficient in design of low frequency transducers. An example is given to show the practical application of the algorithm to a two-stepped transducer.
Advances in Geotechnical Earthquake Engineering - Soil Liquefaction and Seismic Safety of Dams and Monuments, 2012
This constant of proportionality BF has come to be known as "Bryan's factor". The authors ) inves... more This constant of proportionality BF has come to be known as "Bryan's factor". The authors ) investigated Bryan's effect for an elastic, isotropic, spherically symmetric body, rotating in three-dimensional space. Among other results, they demonstrated that Bryan's factor depends on the vibration mode. Their concise formulation is given in general terms without computational detail or assumptions on the magnitude of rotation or illustrative examples. In "slow rotation" (explained below) was assumed for spherical bodies consisting of concentric layers of elastic and/or acoustic media. In that paper, some detail was supplied for computations and an illustrative example was presented. They did not assume a "thin shell theory", as in
ABSTRACT For an asymmetric rotor vibratory gyroscope that is oscillating in an elastic suspension... more ABSTRACT For an asymmetric rotor vibratory gyroscope that is oscillating in an elastic suspension means, the equation of motion is derived from the Eurler–Lagrange equation and the exact solution is obtained as Heun functions (Hfs). A fast and effective method for calculating Heun functions by direct calculation of solutions of the Heun differential equation (Hde) using standard numerical integration methods is developed. Three methods of accuracy check are employed in this case. The accuracy of the numerical solutions deteriorated in the vicinity of the singularity. To overcome this difficulty, an optimised method for calculating the Hfs is developed, which give a uniform accuracy of the calculated values on the interval. The optimised method for calculating the numerical Hfs by means of the solution of the governing initial value problem gave acceptable accuracy in modelling the behaviour of an asymmetric rotor gyroscope.
2011 4th IEEE International Workshop on Advances in Sensors and Interfaces (IWASI), 2011
In a recent article in the Journal of Sound and Vibration (JSV) we discussed the inDlence of mass... more In a recent article in the Journal of Sound and Vibration (JSV) we discussed the inDlence of mass imperfections and isotropic (viscous) damping on the vibrating pattern of a slowly rotating spherical body. Using the mathematical tools developed in the JSV article, in ad dition to mass imperfections, we demonstrate how to introduce prestress using a non-linear theory of elasticity
2007 International Conference - Days on Diffraction, 2007
Bryan's effect"-that is, the effect of a vibrating pattern's precession in the direction of inert... more Bryan's effect"-that is, the effect of a vibrating pattern's precession in the direction of inertial rotation of a vibrating ring-was discovered by G. Bryan in 1890. This effect has several applications in navigational instruments, such as cylindrical, hemispherical and planar circular disc rotational sensors. The model of a thin circular disc vibrating in its plane and subjected to inertial rotation is considered. The dynamics of the disc gyroscope are considered in terms of linear elasticity. Two models are considered: solid discs and a composite disc consisting of concentric annular discs with various boundary conditions on the inner and outer circumferences. It is assumed that the angular rate of inertial rotation of the composite disc is constant and has axial orientation. It is also assumed that this angular rate is much smaller than the lowest eigenvalue of the composite disk. Hence any centrifugal effects and quantities that are proportional to the square of the angular rate are negligible. Our model is formulated in general terms and then compared to a formulation in terms of Novozhilov-Arnold-Warburton's theory of thin shells. The system of equations of motion of the disc is separated and transformed into a pair of wave equations in polar coordinates. A solution is obtained in terms of Bessel and Neumann functions. Various non-axisymmetric modes of the composite disc are considered and the dependence of Bryan's effect on eigenvalues, mass densities of the composite disc, its modulii of elasticity, Poisson ratios, outer and inner radii of the disc, and for various types of boundary conditions, are investigated.
Students of mathematical physics, engineering, natural and biological sciences sometimes need to ... more Students of mathematical physics, engineering, natural and biological sciences sometimes need to use special functions that are not found in ordinary mathematical software. In this paper a simple universal numerical algorithm is developed to compute the Legendre function values of the first kind using the Legendre differential equation. The computed function values are compared to built-in values in Mathcad14 and Derive6. Error analysis is performed to test the accuracy of the algorithm. Graphical residuals are found to be of order 10-12. Finally, some physical application is presented.
In 1890 Bryan observed that when a vibrating structure is rotated the vibrating pattern rotates a... more In 1890 Bryan observed that when a vibrating structure is rotated the vibrating pattern rotates at a rate proportional to the rate of rotation. During investigations of the effect in various solid and fluid-filled objects of various shapes, an interesting commonality was found in connection with the gyroscopic effects of the rotating object. The effect has also been discussed in connection with a rotating fluid-filled wineglass. A linear theory is developed, assuming that the rotation rate is constant and much smaller than the lowest eigenfrequency of the vibrating system. The associated physics and mathematics are easy enough for undergraduate students to understand.
Buffelspoort TIME2008 Peer-reviewed Conference Proceedings, 22 – 26 September In this paper a mat... more Buffelspoort TIME2008 Peer-reviewed Conference Proceedings, 22 – 26 September In this paper a mathematical model describing the interaction of a lion population with that of the zebra and wildebeest populations is considered. The traditional method uses a model with known coefficients and a CAS numerical routine to determine a numerical solution that can be compared to historical data about the populations. The numerical values of the coefficients involved are usually "educated guesses" made by the team consisting of, for example, biologists, game rangers and experienced applied mathematicians. The coefficients are usually described in terms of quantities such as "carrying capacity", "birth rate" et cetera, and might mean little to the mathematician. In this paper an "inverse method" is considered, that is, a method easy enough for senior undergraduate and graduate mathematics majors to understand and apply as part of a "biomechanics"...
Proceedings of the International Conference Days on Diffraction-2005, 2005
In this paper we consider a model of an N-stepped bar. It is assumed that the longitudinal vibrat... more In this paper we consider a model of an N-stepped bar. It is assumed that the longitudinal vibrations in each section of the thick bar is described by a fourth order Rayleigh equation. The analytical tools of vibration analysis are based on finding eigenfunctions with piecewise continuous derivatives, which are orthogonal with respect to a specially chosen weight function. The solution of the problem is formulated in terms of Green function.
Days on Diffraction, St. Petersburg, Russia 2006 It was found by G. Bryan in 1890 that vibrating ... more Days on Diffraction, St. Petersburg, Russia 2006 It was found by G. Bryan in 1890 that vibrating pattern of a rotating ring follows to a direction of the inertial rotation of this ring with an angular rate of the vibrating pattern smaller than the inertial rate. In 1979 E. Loper and D. Lynch proposed a hemispherical vibrating bell gyroscope utilising the Bryan’s effect, which can measure an inertial angular rate and angle of rotation about the symmetry axis of the hemispherical shell. All these works exploited the precession properties of thin vibrating shells subjected to an inertial rotation around their axes of symmetry. In 1985 V. Zhuravlev generalized the abovementioned results and shown that the Bryan’s effect has a three dimensional nature, i.e. that a vibrating pattern of an isotropic spherically symmetric body, arbitrary rotating in 3-D space, follows the inertial rotation of the solid body with a proportionality factor depending on the vibrating mode. This result had a qua...
In this paper we consider a model of an N-stepped bar. It is assumed that the longitudinal vibrat... more In this paper we consider a model of an N-stepped bar. It is assumed that the longitudinal vibrations in each section of the thick bar is described by a fourth order Rayleigh equation. The analytical tools of vibration analysis are based on finding eigenfunctions with piecewise continuous derivatives, which are orthogonal with respect, to a specially chosen weight function. The
Bryan's effect"-that is, the effect of a vibrating pattern's precession in the direction of inert... more Bryan's effect"-that is, the effect of a vibrating pattern's precession in the direction of inertial rotation of a vibrating ring-was discovered by G. Bryan in 1890. This effect has several applications in navigational instruments, such as cylindrical, hemispherical and planar circular disc rotational sensors. The model of a thin circular disc vibrating in its plane and subjected to inertial rotation is considered. The dynamics of the disc gyroscope are considered in terms of linear elasticity. Two models are considered: solid discs and a composite disc consisting of concentric annular discs with various boundary conditions on the inner and outer circumferences. It is assumed that the angular rate of inertial rotation of the composite disc is constant and has axial orientation. It is also assumed that this angular rate is much smaller than the lowest eigenvalue of the composite disk. Hence any centrifugal effects and quantities that are proportional to the square of the angular rate are negligible. Our model is formulated in general terms and then compared to a formulation in terms of Novozhilov-Arnold-Warburton's theory of thin shells. The system of equations of motion of the disc is separated and transformed into a pair of wave equations in polar coordinates. A solution is obtained in terms of Bessel and Neumann functions. Various non-axisymmetric modes of the composite disc are considered and the dependence of Bryan's effect on eigenvalues, mass densities of the composite disc, its modulii of elasticity, Poisson ratios, outer and inner radii of the disc, and for various types of boundary conditions, are investigated.
Students of mathematical physics, engineering, natural and biological sciences sometimes need to ... more Students of mathematical physics, engineering, natural and biological sciences sometimes need to use special functions that are not found in ordinary mathematical software. In this paper a simple universal numerical algorithm is developed to compute the Legendre function values of the first kind using the Legendre differential equation. The computed function values are compared to built-in values in Mathcad14 and Derive6. Error analysis is performed to test the accuracy of the algorithm. Graphical residuals are found to be of order 10 -12 . Finally, some physical application is presented.
In 1890 Bryan observed that when a vibrating structure is rotated the vibrating pattern rotates a... more In 1890 Bryan observed that when a vibrating structure is rotated the vibrating pattern rotates at a rate proportional to the rate of rotation. During investigations of the effect in various solid and fluid-filled objects of various shapes, an interesting commonality was found in connection with the gyroscopic effects of the rotating object. The effect has also been discussed in connection with a rotating fluid-filled wineglass. A linear theory is developed, assuming that the rotation rate is constant and much smaller than the lowest eigenfrequency of the vibrating system. The associated physics and mathematics are easy enough for undergraduate students to understand.
International Workshop on Advances in Sensors and Interface, 2011
In a recent article in the Journal of Sound and Vibration (JSV) we discussed the inDlence of mass... more In a recent article in the Journal of Sound and Vibration (JSV) we discussed the inDlence of mass imperfections and isotropic (viscous) damping on the vibrating pattern of a slowly rotating spherical body. Using the mathematical tools developed in the JSV article, in ad dition to mass imperfections, we demonstrate how to introduce prestress using a non-linear theory of elasticity
In this paper a model of an ñstepped bar with variable Cross-sections coupled with foundation by ... more In this paper a model of an ñstepped bar with variable Cross-sections coupled with foundation by means of lumped masses and springs is studied. It is assumed that the process of vibrations in each section of the bar is described by a wave equation. The analytical tools of vibration analysis are based on nding eigenfunctions with piecewise continuous derivatives, which
In 1890 Bryan observed that when a vibrating structure is rotated the vibrating pattern rotates a... more In 1890 Bryan observed that when a vibrating structure is rotated the vibrating pattern rotates at a rate proportional to the rate of rotation. During investigations of the effect in various solid and fluid-filled objects of various shapes, an interesting commonality was found in connection with the gyroscopic effects of the rotating object. The effect has also been discussed in connection with a rotating fluid-filled wineglass. A linear theory is developed, assuming that the rotation rate is constant and much smaller than the lowest eigenfrequency of the vibrating system. The associated physics and mathematics are easy enough for undergraduate students to understand.
Abstract. In this paper a model of an –stepped bar with variable Cross-sections coupled with foun... more Abstract. In this paper a model of an –stepped bar with variable Cross-sections coupled with foundation by means of lumped masses and springs is studied. It is assumed that the process of vibrations in each section of the bar is described by a wave equation. The analytical tools of vibration analysis are based on finding eigenfunctions with piecewise continuous derivatives, which are orthogonal with respect to a generalized weight function. These eigenfunctions automatically satisfy the boundary conditions at the end points as well as the non-classical boundary conditions at the junctions. The solution of the problems is formulated in terms of Green function. By means of the proposed algorithm a problem of arbitrary complexity could be considered in the same terms as a single homogeneous bar. This algorithm is efficient in design of low frequency transducers. An example is given to show the practical application of the algorithm to a two-stepped transducer.
Advances in Geotechnical Earthquake Engineering - Soil Liquefaction and Seismic Safety of Dams and Monuments, 2012
This constant of proportionality BF has come to be known as "Bryan's factor". The authors ) inves... more This constant of proportionality BF has come to be known as "Bryan's factor". The authors ) investigated Bryan's effect for an elastic, isotropic, spherically symmetric body, rotating in three-dimensional space. Among other results, they demonstrated that Bryan's factor depends on the vibration mode. Their concise formulation is given in general terms without computational detail or assumptions on the magnitude of rotation or illustrative examples. In "slow rotation" (explained below) was assumed for spherical bodies consisting of concentric layers of elastic and/or acoustic media. In that paper, some detail was supplied for computations and an illustrative example was presented. They did not assume a "thin shell theory", as in
ABSTRACT For an asymmetric rotor vibratory gyroscope that is oscillating in an elastic suspension... more ABSTRACT For an asymmetric rotor vibratory gyroscope that is oscillating in an elastic suspension means, the equation of motion is derived from the Eurler–Lagrange equation and the exact solution is obtained as Heun functions (Hfs). A fast and effective method for calculating Heun functions by direct calculation of solutions of the Heun differential equation (Hde) using standard numerical integration methods is developed. Three methods of accuracy check are employed in this case. The accuracy of the numerical solutions deteriorated in the vicinity of the singularity. To overcome this difficulty, an optimised method for calculating the Hfs is developed, which give a uniform accuracy of the calculated values on the interval. The optimised method for calculating the numerical Hfs by means of the solution of the governing initial value problem gave acceptable accuracy in modelling the behaviour of an asymmetric rotor gyroscope.
2011 4th IEEE International Workshop on Advances in Sensors and Interfaces (IWASI), 2011
In a recent article in the Journal of Sound and Vibration (JSV) we discussed the inDlence of mass... more In a recent article in the Journal of Sound and Vibration (JSV) we discussed the inDlence of mass imperfections and isotropic (viscous) damping on the vibrating pattern of a slowly rotating spherical body. Using the mathematical tools developed in the JSV article, in ad dition to mass imperfections, we demonstrate how to introduce prestress using a non-linear theory of elasticity
2007 International Conference - Days on Diffraction, 2007
Bryan's effect"-that is, the effect of a vibrating pattern's precession in the direction of inert... more Bryan's effect"-that is, the effect of a vibrating pattern's precession in the direction of inertial rotation of a vibrating ring-was discovered by G. Bryan in 1890. This effect has several applications in navigational instruments, such as cylindrical, hemispherical and planar circular disc rotational sensors. The model of a thin circular disc vibrating in its plane and subjected to inertial rotation is considered. The dynamics of the disc gyroscope are considered in terms of linear elasticity. Two models are considered: solid discs and a composite disc consisting of concentric annular discs with various boundary conditions on the inner and outer circumferences. It is assumed that the angular rate of inertial rotation of the composite disc is constant and has axial orientation. It is also assumed that this angular rate is much smaller than the lowest eigenvalue of the composite disk. Hence any centrifugal effects and quantities that are proportional to the square of the angular rate are negligible. Our model is formulated in general terms and then compared to a formulation in terms of Novozhilov-Arnold-Warburton's theory of thin shells. The system of equations of motion of the disc is separated and transformed into a pair of wave equations in polar coordinates. A solution is obtained in terms of Bessel and Neumann functions. Various non-axisymmetric modes of the composite disc are considered and the dependence of Bryan's effect on eigenvalues, mass densities of the composite disc, its modulii of elasticity, Poisson ratios, outer and inner radii of the disc, and for various types of boundary conditions, are investigated.
Students of mathematical physics, engineering, natural and biological sciences sometimes need to ... more Students of mathematical physics, engineering, natural and biological sciences sometimes need to use special functions that are not found in ordinary mathematical software. In this paper a simple universal numerical algorithm is developed to compute the Legendre function values of the first kind using the Legendre differential equation. The computed function values are compared to built-in values in Mathcad14 and Derive6. Error analysis is performed to test the accuracy of the algorithm. Graphical residuals are found to be of order 10-12. Finally, some physical application is presented.
In 1890 Bryan observed that when a vibrating structure is rotated the vibrating pattern rotates a... more In 1890 Bryan observed that when a vibrating structure is rotated the vibrating pattern rotates at a rate proportional to the rate of rotation. During investigations of the effect in various solid and fluid-filled objects of various shapes, an interesting commonality was found in connection with the gyroscopic effects of the rotating object. The effect has also been discussed in connection with a rotating fluid-filled wineglass. A linear theory is developed, assuming that the rotation rate is constant and much smaller than the lowest eigenfrequency of the vibrating system. The associated physics and mathematics are easy enough for undergraduate students to understand.
Buffelspoort TIME2008 Peer-reviewed Conference Proceedings, 22 – 26 September In this paper a mat... more Buffelspoort TIME2008 Peer-reviewed Conference Proceedings, 22 – 26 September In this paper a mathematical model describing the interaction of a lion population with that of the zebra and wildebeest populations is considered. The traditional method uses a model with known coefficients and a CAS numerical routine to determine a numerical solution that can be compared to historical data about the populations. The numerical values of the coefficients involved are usually "educated guesses" made by the team consisting of, for example, biologists, game rangers and experienced applied mathematicians. The coefficients are usually described in terms of quantities such as "carrying capacity", "birth rate" et cetera, and might mean little to the mathematician. In this paper an "inverse method" is considered, that is, a method easy enough for senior undergraduate and graduate mathematics majors to understand and apply as part of a "biomechanics"...
Proceedings of the International Conference Days on Diffraction-2005, 2005
In this paper we consider a model of an N-stepped bar. It is assumed that the longitudinal vibrat... more In this paper we consider a model of an N-stepped bar. It is assumed that the longitudinal vibrations in each section of the thick bar is described by a fourth order Rayleigh equation. The analytical tools of vibration analysis are based on finding eigenfunctions with piecewise continuous derivatives, which are orthogonal with respect to a specially chosen weight function. The solution of the problem is formulated in terms of Green function.
Days on Diffraction, St. Petersburg, Russia 2006 It was found by G. Bryan in 1890 that vibrating ... more Days on Diffraction, St. Petersburg, Russia 2006 It was found by G. Bryan in 1890 that vibrating pattern of a rotating ring follows to a direction of the inertial rotation of this ring with an angular rate of the vibrating pattern smaller than the inertial rate. In 1979 E. Loper and D. Lynch proposed a hemispherical vibrating bell gyroscope utilising the Bryan’s effect, which can measure an inertial angular rate and angle of rotation about the symmetry axis of the hemispherical shell. All these works exploited the precession properties of thin vibrating shells subjected to an inertial rotation around their axes of symmetry. In 1985 V. Zhuravlev generalized the abovementioned results and shown that the Bryan’s effect has a three dimensional nature, i.e. that a vibrating pattern of an isotropic spherically symmetric body, arbitrary rotating in 3-D space, follows the inertial rotation of the solid body with a proportionality factor depending on the vibrating mode. This result had a qua...
In this paper we consider a model of an N-stepped bar. It is assumed that the longitudinal vibrat... more In this paper we consider a model of an N-stepped bar. It is assumed that the longitudinal vibrations in each section of the thick bar is described by a fourth order Rayleigh equation. The analytical tools of vibration analysis are based on finding eigenfunctions with piecewise continuous derivatives, which are orthogonal with respect, to a specially chosen weight function. The
Bryan's effect"-that is, the effect of a vibrating pattern's precession in the direction of inert... more Bryan's effect"-that is, the effect of a vibrating pattern's precession in the direction of inertial rotation of a vibrating ring-was discovered by G. Bryan in 1890. This effect has several applications in navigational instruments, such as cylindrical, hemispherical and planar circular disc rotational sensors. The model of a thin circular disc vibrating in its plane and subjected to inertial rotation is considered. The dynamics of the disc gyroscope are considered in terms of linear elasticity. Two models are considered: solid discs and a composite disc consisting of concentric annular discs with various boundary conditions on the inner and outer circumferences. It is assumed that the angular rate of inertial rotation of the composite disc is constant and has axial orientation. It is also assumed that this angular rate is much smaller than the lowest eigenvalue of the composite disk. Hence any centrifugal effects and quantities that are proportional to the square of the angular rate are negligible. Our model is formulated in general terms and then compared to a formulation in terms of Novozhilov-Arnold-Warburton's theory of thin shells. The system of equations of motion of the disc is separated and transformed into a pair of wave equations in polar coordinates. A solution is obtained in terms of Bessel and Neumann functions. Various non-axisymmetric modes of the composite disc are considered and the dependence of Bryan's effect on eigenvalues, mass densities of the composite disc, its modulii of elasticity, Poisson ratios, outer and inner radii of the disc, and for various types of boundary conditions, are investigated.
Students of mathematical physics, engineering, natural and biological sciences sometimes need to ... more Students of mathematical physics, engineering, natural and biological sciences sometimes need to use special functions that are not found in ordinary mathematical software. In this paper a simple universal numerical algorithm is developed to compute the Legendre function values of the first kind using the Legendre differential equation. The computed function values are compared to built-in values in Mathcad14 and Derive6. Error analysis is performed to test the accuracy of the algorithm. Graphical residuals are found to be of order 10 -12 . Finally, some physical application is presented.
In 1890 Bryan observed that when a vibrating structure is rotated the vibrating pattern rotates a... more In 1890 Bryan observed that when a vibrating structure is rotated the vibrating pattern rotates at a rate proportional to the rate of rotation. During investigations of the effect in various solid and fluid-filled objects of various shapes, an interesting commonality was found in connection with the gyroscopic effects of the rotating object. The effect has also been discussed in connection with a rotating fluid-filled wineglass. A linear theory is developed, assuming that the rotation rate is constant and much smaller than the lowest eigenfrequency of the vibrating system. The associated physics and mathematics are easy enough for undergraduate students to understand.
International Workshop on Advances in Sensors and Interface, 2011
In a recent article in the Journal of Sound and Vibration (JSV) we discussed the inDlence of mass... more In a recent article in the Journal of Sound and Vibration (JSV) we discussed the inDlence of mass imperfections and isotropic (viscous) damping on the vibrating pattern of a slowly rotating spherical body. Using the mathematical tools developed in the JSV article, in ad dition to mass imperfections, we demonstrate how to introduce prestress using a non-linear theory of elasticity
In this paper a model of an ñstepped bar with variable Cross-sections coupled with foundation by ... more In this paper a model of an ñstepped bar with variable Cross-sections coupled with foundation by means of lumped masses and springs is studied. It is assumed that the process of vibrations in each section of the bar is described by a wave equation. The analytical tools of vibration analysis are based on nding eigenfunctions with piecewise continuous derivatives, which
In 1890 Bryan observed that when a vibrating structure is rotated the vibrating pattern rotates a... more In 1890 Bryan observed that when a vibrating structure is rotated the vibrating pattern rotates at a rate proportional to the rate of rotation. During investigations of the effect in various solid and fluid-filled objects of various shapes, an interesting commonality was found in connection with the gyroscopic effects of the rotating object. The effect has also been discussed in connection with a rotating fluid-filled wineglass. A linear theory is developed, assuming that the rotation rate is constant and much smaller than the lowest eigenfrequency of the vibrating system. The associated physics and mathematics are easy enough for undergraduate students to understand.
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Papers by Steve V Joubert