Due to the unique magnetic, mechanical and thermal properties, magnetic nanoparticles(MNPs) have comprehensive applications as the contrast and therapeutic agents in biomedical imaging and magnetic hyperthermia. The l...Due to the unique magnetic, mechanical and thermal properties, magnetic nanoparticles(MNPs) have comprehensive applications as the contrast and therapeutic agents in biomedical imaging and magnetic hyperthermia. The linear and nonlinear magnetoacoustic responses determined by the magnetic properties of MNPs have attracted more and more attention in biomedical engineering. By considering the relaxation time of MNPs, we derive the formulae of second harmonic magnetoacoustic responses(2H-MARs) for a cylindrical MNP solution model based on the mechanical oscillations of MNPs in magnetoacoustic tomography with magnetic induction(MAT-MI). It is proved that only the second harmonic magnetoacoustic oscillations can be generated by MNPs under an alternating magnetic excitation. The acoustic pressure of the 2H-MAR is proportional to the square of the magnetic field intensity and exhibits a linear increase with the concentration of MNPs. Numerical simulations of the 2H-MAR are confirmed by the experimental measurements for various magnetic field intensities and solution concentrations using a laser vibrometer. The favorable results demonstrate the feasibility of the harmonic measurements without the fundamental interference of the electromagnetic excitation, and suggest a new harmonic imaging strategy of MAT-MI for MNPs with enhanced spatial resolution and improved signal-to-noise ratio in biomedical applications.展开更多
This paper analyzes the structure and transmission principles of a modulation permanent magnet gear transmission. Its 3D data model is built based on the known optimized parameters from research team. Its structure of...This paper analyzes the structure and transmission principles of a modulation permanent magnet gear transmission. Its 3D data model is built based on the known optimized parameters from research team. Its structure of the harmonic response is analyzed and discussed under the software ANSYS. The displacement response and the initial 6 order response frequency and phase angle are obtained. The change rule of these responses is known under the forced vibration.展开更多
Sub-harmonic component generated from microbubbles is proven to be potentially used in noninvasive blood pressure measurement. Both theoretical and experimental studies are performed in the present work to investigate...Sub-harmonic component generated from microbubbles is proven to be potentially used in noninvasive blood pressure measurement. Both theoretical and experimental studies are performed in the present work to investigate the dependence of the sub-harmonic generation on the overpressure with different excitation pressure amplitudes and pulse lengths. With 4-MHz ultrasound excitation at an applied acoustic pressure amplitude of 0.24 MPa, the measured sub-harmonic amplitude exhibits a decreasing change as overpressure increases; while non-monotonic change is observed for the applied acoustic pressures of 0.36 MPa and 0.48 MPa, and the peak position in the curve of the sub-harmonic response versus the overpres- sure shifts toward higher overpressure as the excitation pressure amplitude increases. Furthermore, the exciting pulse with long duration could lead to a better sensitivity of the sub-harmonic response to overpressure. The measured results are ex- plained by the numerical simulations based on the Marmottant model. The numerical simulations qualitatively accord with the measured results. This work might provide a preliminary proof for the optimization of the noninvasive blood pressure measurement through using sub-harmonic generation from microbubbles.展开更多
In order to extend the rail life and improve the firing accuracy,the electromagnetic launcher's rail can be modeled as a beam on elastic foundation with simply supported beam with moving load.Euler beam theory is ...In order to extend the rail life and improve the firing accuracy,the electromagnetic launcher's rail can be modeled as a beam on elastic foundation with simply supported beam with moving load.Euler beam theory is applied to build the mechanical model and the analytical solution of the equation subjected to harmonic magnetic pressure is derived in details,which has successfully avoided the errors caused by using the uniform pressure to approximately replace the variable force.Numerical analysis of the dynamic response on rail by using the MATLAB software shows that the peak values of maximal deflection and vibration velocity increase gradually as the exciting frequency increases.Taking the same speed of load into account,the dynamic response of rail is obviously smaller than that under constant force.Therefore the reliable theory basis is provided for the design and control of rail to promote the practical application of electromagnetic launcher.展开更多
Aiming at the shortcomings of the existing electric energy metering method,combining with the harmonic responsibility analysis model based on the reference impedance method and the idea of apparent power decomposition...Aiming at the shortcomings of the existing electric energy metering method,combining with the harmonic responsibility analysis model based on the reference impedance method and the idea of apparent power decomposition in IEEE Std 1459-2010 standard,two new metering indicators—billing active power and billing power factor are defined.A new electric energy metering method is proposed and its specific implementation steps are given.The simulation model is built in Matlab/Simulink,and three different examples are set up.Using the simulation data,the various metering indicators need to be examined by the existing electric energy metering method and the new electric energy metering method are calculated.The calculation results show that the new electric energy metering method not only overcomes the shortcomings of the existing electric energy metering method,but also is very easy to be popularized and applied.展开更多
Comparisons of the common methods for obtaining the periodic responses show that the harmonic balance method with alternating frequency/time (HB-AFT) do- main technique has some advantages in dealing with nonlinear ...Comparisons of the common methods for obtaining the periodic responses show that the harmonic balance method with alternating frequency/time (HB-AFT) do- main technique has some advantages in dealing with nonlinear problems of fractional exponential models. By the HB-AFT method, a rigid rotor supported by ball bearings with nonlinearity of Hertz contact and ball passage vibrations is considered. With the aid of the Floquet theory, the movement characteristics of interval stability are deeply studied. Besides, a simple strategy to determine the monodromy matrix is proposed for the stability analysis.展开更多
An efficient multi-harmonic method is proposed for studying the effects of mistuning on resonant features of bladed disks with blade-to-blade dry friction damping. This method is able to predict accurately the forced ...An efficient multi-harmonic method is proposed for studying the effects of mistuning on resonant features of bladed disks with blade-to-blade dry friction damping. This method is able to predict accurately the forced response of bladed disks in frequency domain, which is validated by numerical integration method in time domain. The resonant features of both tuned and mistuned systems are investigated by using this method under various system coupling strengths, viscous dampings, and dry friction darnpings, etc. The results demonstrate that the proposed multi-harmonic method is very efficient for studying the mistuning effects on the resonant response of bladed disks with blade-to-blade dry friction damping, especially considering the combined effects of various system parameters.展开更多
A rotating packed bed is a typical chemical process enhancement equipment that can strengthen micromixing and mass transfer.During the operation of the rotating packed bed,the nonreactants and products irregularly adh...A rotating packed bed is a typical chemical process enhancement equipment that can strengthen micromixing and mass transfer.During the operation of the rotating packed bed,the nonreactants and products irregularly adhere to the wire mesh packing in the rotor,thus resulting in an imbalance in the vibration of the rotor,which may cause serious damage to the bearing and material leakage.This study proposes a model prediction for estimating the bearing residual life of a rotating packed bed based on rotor imbalance response analysis.This method is used to determine the influence of the mass on the imbalance in the vibration of the rotor on bearing damage.The major influence on rotor vibration was found to be exerted by the imbalanced mass and its distribution radius,as revealed by the results of orthogonal experiments.Through implementing finite element analysis,the imbalance response curve for the rotating packed bed rotor was obtained,and a correlation among rotor imbalance mass,distribution radius of imbalance mass,and bearing residue life was established via data fitting.The predicted value of the bearing life can be used as the reference basis for an early safety warning of a rotating packed bed to effectively avoid accidents.展开更多
A study on dynamic response of transversely isotropic saturated poroelastic media under a circular non-axisymmetrical harmonic source has been presented by Huang Yi et al. using the technique of Fourier expansion and ...A study on dynamic response of transversely isotropic saturated poroelastic media under a circular non-axisymmetrical harmonic source has been presented by Huang Yi et al. using the technique of Fourier expansion and Hankel transform. However, the method may not always be valid. The work is extended to the general case being in the rectangular coordinate. The purpose is to study the 3-d dynamic response of transversely isotropic saturated soils under a general source distributing in arbitrary rectangular zoon on the medium surface. Based on Biot's theory for fluid- saturated porous media, the 3-d wave motion equations in rectangular coordinate for transversely isotropic saturated poroelastic media were transformed into the two uncoupling governing differential equations of 6-order and 2-order respectively by means of the displacement functions. Then, using the technique of double Fourier transform, the governing differential equations were easily solved. Integral solutions of soil skeleton displacements and pore pressure as well as the total stresses for poroelastic media were obtained. Furthermore, a systematic study on half-space problem in saturated soils was performed. Integral solutions for surface displacements under the general harmonic source distributing on arbitrary surface zone, considering both case of drained surface and undrained surface, were presented.展开更多
As to motorized spindle system, this paper builds a simplified 3D model of spindle and bearing, performs structure modal analysis, reveals its dynamic characteristics under the free model; furthermore, modifies bearin...As to motorized spindle system, this paper builds a simplified 3D model of spindle and bearing, performs structure modal analysis, reveals its dynamic characteristics under the free model; furthermore, modifies bearing radial stiffness and number of model and studies the change of modal parameters. On this basis, through the harmonic response analysis of the finite element model, dy- namic response characteristic caused by imbalance of monitored spindle system and law of vibration response to different amount of unbalance is analyzed.展开更多
Many studies have been done on the heave-pitch unstable coupling response for a spar platform by a 2-DOF model.In fact,in addition to the heave and pitch which are in one plane,the nonlinear unstable motion will also ...Many studies have been done on the heave-pitch unstable coupling response for a spar platform by a 2-DOF model.In fact,in addition to the heave and pitch which are in one plane,the nonlinear unstable motion will also occur in roll.From the results of the experiments,the unstable roll motion plays a dominant role in the motion of a spar platform which is much stronger than that of pitch.The objective of this paper is to study 3-DOF coupling response performance of spar platform under wave and vortex-induced force.The nonlinear coupled equations in heave,roll and pitch are established by considering time-varying wet surface and coupling.The first order steady-state response is solved by multi-scales method when the incident wave frequency approaches the heave natural frequency.Numerical integration of the motion equations has been performed to verify the first-order perturbation solution.The results are confirmed by model test.There is a saturation phenomenon associated with heave mode in 3-DOF systems and all extra energy is transferred to roll and pitch.It is observed that sub-harmonic response occurs in roll and pitch when the wave force exceeds a certain value.The energy distribution in roll and pitch is determined by the initial value and damping characteristics of roll and pitch.The energy transfers from heave to pitch and then transfers from pitch to roll.Due to the influence of the low-frequency vortex-excited force,the response of roll is more complicated than that of pitch.展开更多
Einstein claimed Bohr’s theory is incomplete: “the wave function does not provide a complete description of the physical reality” [1]. Their views represent two physics in schism [2] [3]. Quanta are fundamental. Th...Einstein claimed Bohr’s theory is incomplete: “the wave function does not provide a complete description of the physical reality” [1]. Their views represent two physics in schism [2] [3]. Quanta are fundamental. The theory of diffraction in quasicrystals, that is summarized here, is falsifiable and verified. The quanta are not only harmonic;but harmonic in dual series: geometric and linear. Many have believed the quantum is real;rather than conceptual and axiomatic. The quasicrystal proves its reality.展开更多
Based on the requirements of the two-phase rapier loom’s beat-up system characteristics, the dynamic responses of its beat-up system to three different types of cam input motion are studied in this paper. Also, their...Based on the requirements of the two-phase rapier loom’s beat-up system characteristics, the dynamic responses of its beat-up system to three different types of cam input motion are studied in this paper. Also, their corresponding analytical comparisons are made. At the end of the paper, the authors put forward a proposal of new type cam beat-up motion for future practice.展开更多
Long-period ground motion has become an important consideration because of the increasing number of large and long-period structures.Therefore,a thorough investigation on the formation and characteristics of longperio...Long-period ground motion has become an important consideration because of the increasing number of large and long-period structures.Therefore,a thorough investigation on the formation and characteristics of longperiod ground motion is desirable for engineering applications.In this work,an analytical study is performed to examine the effect of several parameters and the combining mode for equivalent harmonic components on the dynamic response of systems.The results of the work show that the harmonic components in equivalent ground motion are evidently influenced by the intensity rise time,duration,phase and combining mode.Moreover,the long-period ground motions are simplified and simulated by separate harmonic components through proper combination.The findings of the work are believed to be useful in the selection of input ground motion in structural seismic analysis.展开更多
Diffraction in quasicrystals is in logarithmic order and icosahedral point group symmetry. Neither of these features are allowed in Bragg diffraction, so a special theory is required. The present work displays exact a...Diffraction in quasicrystals is in logarithmic order and icosahedral point group symmetry. Neither of these features are allowed in Bragg diffraction, so a special theory is required. The present work displays exact agreement between the analytic metric with a numeric description of diffraction in quasicrystals that is based on quasi-structure factors. So far, we treated the hierarchic structure as ideal;now, we detail the theory by including two significant features: firstly, the steady state wave function of the incident radiation demonstrates how harmonics, in metrical space and time, enable coherent interaction between the periodic wave packet and hierarchic quasicrystal;secondly, mapping of the hierarchic structure for any influence of defects will allow estimation of possible error margins in the analysis. The hierarchic structure has the required logarithmic periodicity: superclusters, containing about 10<sup>3</sup> atoms, convincingly map phase contrast images;while higher orders leave space for subsidiary speculation. The diffraction is completely explained for the first time.展开更多
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11934009,11974187,and 11604156)。
文摘Due to the unique magnetic, mechanical and thermal properties, magnetic nanoparticles(MNPs) have comprehensive applications as the contrast and therapeutic agents in biomedical imaging and magnetic hyperthermia. The linear and nonlinear magnetoacoustic responses determined by the magnetic properties of MNPs have attracted more and more attention in biomedical engineering. By considering the relaxation time of MNPs, we derive the formulae of second harmonic magnetoacoustic responses(2H-MARs) for a cylindrical MNP solution model based on the mechanical oscillations of MNPs in magnetoacoustic tomography with magnetic induction(MAT-MI). It is proved that only the second harmonic magnetoacoustic oscillations can be generated by MNPs under an alternating magnetic excitation. The acoustic pressure of the 2H-MAR is proportional to the square of the magnetic field intensity and exhibits a linear increase with the concentration of MNPs. Numerical simulations of the 2H-MAR are confirmed by the experimental measurements for various magnetic field intensities and solution concentrations using a laser vibrometer. The favorable results demonstrate the feasibility of the harmonic measurements without the fundamental interference of the electromagnetic excitation, and suggest a new harmonic imaging strategy of MAT-MI for MNPs with enhanced spatial resolution and improved signal-to-noise ratio in biomedical applications.
文摘This paper analyzes the structure and transmission principles of a modulation permanent magnet gear transmission. Its 3D data model is built based on the known optimized parameters from research team. Its structure of the harmonic response is analyzed and discussed under the software ANSYS. The displacement response and the initial 6 order response frequency and phase angle are obtained. The change rule of these responses is known under the forced vibration.
基金Project supported by the National Basic Research Program from Ministry of Science and Technology,China(Grant No.2011CB707900)the National Natural Science Foundation of China(Grant Nos.81271589,81227004,11174141,11374155,11612032,and 81301616)the Natural Science Foundation of Jiangsu Province,China(Grant Nos.BE2011110 and BK20131017)
文摘Sub-harmonic component generated from microbubbles is proven to be potentially used in noninvasive blood pressure measurement. Both theoretical and experimental studies are performed in the present work to investigate the dependence of the sub-harmonic generation on the overpressure with different excitation pressure amplitudes and pulse lengths. With 4-MHz ultrasound excitation at an applied acoustic pressure amplitude of 0.24 MPa, the measured sub-harmonic amplitude exhibits a decreasing change as overpressure increases; while non-monotonic change is observed for the applied acoustic pressures of 0.36 MPa and 0.48 MPa, and the peak position in the curve of the sub-harmonic response versus the overpres- sure shifts toward higher overpressure as the excitation pressure amplitude increases. Furthermore, the exciting pulse with long duration could lead to a better sensitivity of the sub-harmonic response to overpressure. The measured results are ex- plained by the numerical simulations based on the Marmottant model. The numerical simulations qualitatively accord with the measured results. This work might provide a preliminary proof for the optimization of the noninvasive blood pressure measurement through using sub-harmonic generation from microbubbles.
基金Sponsored by the National Natural Science Foundation of China(Grant No.50875230)
文摘In order to extend the rail life and improve the firing accuracy,the electromagnetic launcher's rail can be modeled as a beam on elastic foundation with simply supported beam with moving load.Euler beam theory is applied to build the mechanical model and the analytical solution of the equation subjected to harmonic magnetic pressure is derived in details,which has successfully avoided the errors caused by using the uniform pressure to approximately replace the variable force.Numerical analysis of the dynamic response on rail by using the MATLAB software shows that the peak values of maximal deflection and vibration velocity increase gradually as the exciting frequency increases.Taking the same speed of load into account,the dynamic response of rail is obviously smaller than that under constant force.Therefore the reliable theory basis is provided for the design and control of rail to promote the practical application of electromagnetic launcher.
基金National Natural Science Foundation of China(No.51367010)Science and Technology Program of Gansu Province(No.17JR5RA083)+1 种基金Program for Excellent Team of Scientific Research of Lanzhou Jiaotong University(No.201701)Scientific Research Program of Colleges and Universities of Gansu Province(No.2016B-032)。
文摘Aiming at the shortcomings of the existing electric energy metering method,combining with the harmonic responsibility analysis model based on the reference impedance method and the idea of apparent power decomposition in IEEE Std 1459-2010 standard,two new metering indicators—billing active power and billing power factor are defined.A new electric energy metering method is proposed and its specific implementation steps are given.The simulation model is built in Matlab/Simulink,and three different examples are set up.Using the simulation data,the various metering indicators need to be examined by the existing electric energy metering method and the new electric energy metering method are calculated.The calculation results show that the new electric energy metering method not only overcomes the shortcomings of the existing electric energy metering method,but also is very easy to be popularized and applied.
基金supported by the National Natural Science Foundation of China(No.10632040)
文摘Comparisons of the common methods for obtaining the periodic responses show that the harmonic balance method with alternating frequency/time (HB-AFT) do- main technique has some advantages in dealing with nonlinear problems of fractional exponential models. By the HB-AFT method, a rigid rotor supported by ball bearings with nonlinearity of Hertz contact and ball passage vibrations is considered. With the aid of the Floquet theory, the movement characteristics of interval stability are deeply studied. Besides, a simple strategy to determine the monodromy matrix is proposed for the stability analysis.
基金National Nature Science Foundation of China (NO.50275121)
文摘An efficient multi-harmonic method is proposed for studying the effects of mistuning on resonant features of bladed disks with blade-to-blade dry friction damping. This method is able to predict accurately the forced response of bladed disks in frequency domain, which is validated by numerical integration method in time domain. The resonant features of both tuned and mistuned systems are investigated by using this method under various system coupling strengths, viscous dampings, and dry friction darnpings, etc. The results demonstrate that the proposed multi-harmonic method is very efficient for studying the mistuning effects on the resonant response of bladed disks with blade-to-blade dry friction damping, especially considering the combined effects of various system parameters.
基金the High-Performance Computing Platform of Beijing University of Chemical Technology(BUCT)for supporting this papersupported by the Fundamental Research Funds for the Central Universities(JD2319)+2 种基金the CNOOC Technical Cooperation Project(ZX2022ZCTYF7612)the National Natural Science Foundation of China(51775029,52004014)the Chinese Universities Scientific Fund(XK2020-04)。
文摘A rotating packed bed is a typical chemical process enhancement equipment that can strengthen micromixing and mass transfer.During the operation of the rotating packed bed,the nonreactants and products irregularly adhere to the wire mesh packing in the rotor,thus resulting in an imbalance in the vibration of the rotor,which may cause serious damage to the bearing and material leakage.This study proposes a model prediction for estimating the bearing residual life of a rotating packed bed based on rotor imbalance response analysis.This method is used to determine the influence of the mass on the imbalance in the vibration of the rotor on bearing damage.The major influence on rotor vibration was found to be exerted by the imbalanced mass and its distribution radius,as revealed by the results of orthogonal experiments.Through implementing finite element analysis,the imbalance response curve for the rotating packed bed rotor was obtained,and a correlation among rotor imbalance mass,distribution radius of imbalance mass,and bearing residue life was established via data fitting.The predicted value of the bearing life can be used as the reference basis for an early safety warning of a rotating packed bed to effectively avoid accidents.
文摘A study on dynamic response of transversely isotropic saturated poroelastic media under a circular non-axisymmetrical harmonic source has been presented by Huang Yi et al. using the technique of Fourier expansion and Hankel transform. However, the method may not always be valid. The work is extended to the general case being in the rectangular coordinate. The purpose is to study the 3-d dynamic response of transversely isotropic saturated soils under a general source distributing in arbitrary rectangular zoon on the medium surface. Based on Biot's theory for fluid- saturated porous media, the 3-d wave motion equations in rectangular coordinate for transversely isotropic saturated poroelastic media were transformed into the two uncoupling governing differential equations of 6-order and 2-order respectively by means of the displacement functions. Then, using the technique of double Fourier transform, the governing differential equations were easily solved. Integral solutions of soil skeleton displacements and pore pressure as well as the total stresses for poroelastic media were obtained. Furthermore, a systematic study on half-space problem in saturated soils was performed. Integral solutions for surface displacements under the general harmonic source distributing on arbitrary surface zone, considering both case of drained surface and undrained surface, were presented.
基金Special Topic of the Ministry of Education about Humanities and Social Sciences(12JDGC007)National Science and Technology Support Project(2011BAF09B01)Key State Science and Technology Projects(2009ZX04010-021)
文摘As to motorized spindle system, this paper builds a simplified 3D model of spindle and bearing, performs structure modal analysis, reveals its dynamic characteristics under the free model; furthermore, modifies bearing radial stiffness and number of model and studies the change of modal parameters. On this basis, through the harmonic response analysis of the finite element model, dy- namic response characteristic caused by imbalance of monitored spindle system and law of vibration response to different amount of unbalance is analyzed.
基金supported by the National Natural Science Foundation of China(Grant No.51279130)
文摘Many studies have been done on the heave-pitch unstable coupling response for a spar platform by a 2-DOF model.In fact,in addition to the heave and pitch which are in one plane,the nonlinear unstable motion will also occur in roll.From the results of the experiments,the unstable roll motion plays a dominant role in the motion of a spar platform which is much stronger than that of pitch.The objective of this paper is to study 3-DOF coupling response performance of spar platform under wave and vortex-induced force.The nonlinear coupled equations in heave,roll and pitch are established by considering time-varying wet surface and coupling.The first order steady-state response is solved by multi-scales method when the incident wave frequency approaches the heave natural frequency.Numerical integration of the motion equations has been performed to verify the first-order perturbation solution.The results are confirmed by model test.There is a saturation phenomenon associated with heave mode in 3-DOF systems and all extra energy is transferred to roll and pitch.It is observed that sub-harmonic response occurs in roll and pitch when the wave force exceeds a certain value.The energy distribution in roll and pitch is determined by the initial value and damping characteristics of roll and pitch.The energy transfers from heave to pitch and then transfers from pitch to roll.Due to the influence of the low-frequency vortex-excited force,the response of roll is more complicated than that of pitch.
文摘Einstein claimed Bohr’s theory is incomplete: “the wave function does not provide a complete description of the physical reality” [1]. Their views represent two physics in schism [2] [3]. Quanta are fundamental. The theory of diffraction in quasicrystals, that is summarized here, is falsifiable and verified. The quanta are not only harmonic;but harmonic in dual series: geometric and linear. Many have believed the quantum is real;rather than conceptual and axiomatic. The quasicrystal proves its reality.
文摘Based on the requirements of the two-phase rapier loom’s beat-up system characteristics, the dynamic responses of its beat-up system to three different types of cam input motion are studied in this paper. Also, their corresponding analytical comparisons are made. At the end of the paper, the authors put forward a proposal of new type cam beat-up motion for future practice.
基金Supported by Major Research Plan of National Natural Science Foundation of China(No.91215301)National Natural Science Foundation of China(No.51238012,No.51178152,No.51008208)the Special Fund for Earthquake Scientific Research in the Public Interest(No.201208013)
文摘Long-period ground motion has become an important consideration because of the increasing number of large and long-period structures.Therefore,a thorough investigation on the formation and characteristics of longperiod ground motion is desirable for engineering applications.In this work,an analytical study is performed to examine the effect of several parameters and the combining mode for equivalent harmonic components on the dynamic response of systems.The results of the work show that the harmonic components in equivalent ground motion are evidently influenced by the intensity rise time,duration,phase and combining mode.Moreover,the long-period ground motions are simplified and simulated by separate harmonic components through proper combination.The findings of the work are believed to be useful in the selection of input ground motion in structural seismic analysis.
文摘Diffraction in quasicrystals is in logarithmic order and icosahedral point group symmetry. Neither of these features are allowed in Bragg diffraction, so a special theory is required. The present work displays exact agreement between the analytic metric with a numeric description of diffraction in quasicrystals that is based on quasi-structure factors. So far, we treated the hierarchic structure as ideal;now, we detail the theory by including two significant features: firstly, the steady state wave function of the incident radiation demonstrates how harmonics, in metrical space and time, enable coherent interaction between the periodic wave packet and hierarchic quasicrystal;secondly, mapping of the hierarchic structure for any influence of defects will allow estimation of possible error margins in the analysis. The hierarchic structure has the required logarithmic periodicity: superclusters, containing about 10<sup>3</sup> atoms, convincingly map phase contrast images;while higher orders leave space for subsidiary speculation. The diffraction is completely explained for the first time.