This paper introduces the results of selecting and breeding a micro-organism, Strain I, and its core model experiment investigation for microbial enhanced oil recovery (MEOR). Strain I was separated from the formation...This paper introduces the results of selecting and breeding a micro-organism, Strain I, and its core model experiment investigation for microbial enhanced oil recovery (MEOR). Strain I was separated from the formation water of the Dagang oil field, with analytical results showing that Strain I is a gram-positive bacillus. A further study revealed that this strain has an excellent tolerance of environmental stresses: It can survive in conditions of 70℃, 30 wt% salinity and pH3.5-9.4. Strain I can metabolize biosurfactants that could increase the oil recovery ratio, use crude oil as the single carbon source, and decompose long-chain paraffin with a large molecular weight into short-chain paraffin with a small molecular weight. The core model experiment shows that Strain I enhances oil recovery well. Using 2 vol% of the fermentation solution of Strain I to displace the crude oil in the synthetic plastic bonding core could increase the recovery ratio by 21.6%.展开更多
A three-dimensional mathematical model was established to predict the multiphase flow,motion and dispersion of desulfurizer particles,and desulfurization of hot metal during the Kanbara reactor(KR)process.The turbulen...A three-dimensional mathematical model was established to predict the multiphase flow,motion and dispersion of desulfurizer particles,and desulfurization of hot metal during the Kanbara reactor(KR)process.The turbulent kinetic energy-turbulent dissipation rate(k-ε)turbulence model,volume-of-fluid multiphase model,discrete-phase model,and unreacted core model for the reaction between the hot metal and particles were coupled.The measured sulfur content of the hot metal with time during the actual KR process was employed to validate the current mathematical model.The distance from the lowest point of the liquid level to the bottom of the ladle decreased from 3170 to2191 mm when the rotation speed increased from 30 to 110 r/min,which had a great effect on the dispersion of desulfurizer particles.The critical rotation speed for the vortex to reach the upper edge of the stirring impeller was 70 r/min when the immersion depth was 1500 mm.The desulfurization rate increased with the increase in the impeller rotation speed,whereas the influence of the immersion depth was relatively small.Formulas for different rotation parameters on the desulfurization rate constant and turbulent energy dissipation rate were proposed to evaluate the variation in sulfur content over time.展开更多
The kinetics of H<sub>2</sub>S removal by zinc oxide desulfurizer was studied through thermogravimetricanalysis.The experimental results show that desulfurization rate was controlled,at high temperatureand...The kinetics of H<sub>2</sub>S removal by zinc oxide desulfurizer was studied through thermogravimetricanalysis.The experimental results show that desulfurization rate was controlled,at high temperatureand low conversion,by the chemical reaction rate,and at low temperature and high conversion by thegrain diffusion rate.The reaction is first order with respect to H<sub>2</sub>S concentration in the differentcontrolled stages.The kinetic behavior can be modeled through the employment of the shrinking coremodel.The values of the model parameters were determined.The variation tendencies with temperatureand concentration of H<sub>2</sub>S at the controlled stages were discussed.展开更多
The effect of rotation on the shape (figure) and gravitational quadrupole of astronomical bodies is calculated by using an approximate point core model: A point mass at the center of an ellipsoidal homogeneous fluid. ...The effect of rotation on the shape (figure) and gravitational quadrupole of astronomical bodies is calculated by using an approximate point core model: A point mass at the center of an ellipsoidal homogeneous fluid. Maclaurin’s analytical result for homogenous bodies generalizes to this model and leads to very accurate analytical results connecting the three observables: oblateness (ò), gravitational quadrupole (J2), and angular velocity parameter (q). The analytical results are compared to observational data for the planets and a good agreement is found. Oscillations near equilibrium are studied within the model.展开更多
Controlling inclusion composition, from the point of view of thermodynamics, only explains the probability and limit of reaction. However, kinetics makes the nucleation and the velocity of growth of inclusions clear, ...Controlling inclusion composition, from the point of view of thermodynamics, only explains the probability and limit of reaction. However, kinetics makes the nucleation and the velocity of growth of inclusions clear, and these kinetic factors are very important to the quality of slab. The basic kinetic theory of unreacted core model was used to build the mathematical model for the growth of inclusions and the concerned software was developed through Visual Basic 6.0. The time that different radius inclusions attain saturation was calculated to determine the controlling step of reaction between steel and inclusions. The time for the growth of inclusion obtained from the model was in good agreement with the data measured by Japanese Okuyama G, which indicated that the model is reasonable.展开更多
The kinetics oflignocellulose hydrolysis under the conditions of high temperature and dilute acid (mass fraction 0.05%) was investigated in this paper. By studying the reducing sugar concentration versus reaction te...The kinetics oflignocellulose hydrolysis under the conditions of high temperature and dilute acid (mass fraction 0.05%) was investigated in this paper. By studying the reducing sugar concentration versus reaction tempera ture (170℃-220℃) and reaction time (150-1800 s) during the hydrolysis process of five kinds of crop straw (rice, wheat, cotton, rape and corn), the shrinking core model was established, and the differential equation of the model and its analytical solution were obtained. With a numerical calculation method, the kinetic equation was estimated, and the degradation of reducing sugar obeyed firstorder kinetics was obtained. The calculated results from the equations agreed well with the original experimental data. The calculation by the model showed that the reducing sugar concentration increases as the size of the particles decrease, and the uniform particles increase.展开更多
Based on the MASNUM wave-tide-circulation coupled numerical model, the temperature structure along 35°N in the Yellow Sea was simulated and compared with the observations. One of the notable features of the tempe...Based on the MASNUM wave-tide-circulation coupled numerical model, the temperature structure along 35°N in the Yellow Sea was simulated and compared with the observations. One of the notable features of the temperature structure along 35°N section is the double cold cores phenomena during spring and summer. The double cold cores refer to the two cold water centers located near 122°E and 125°E from the depth of 30m to bottom. The formation, maintenance and disappearance of the double cold cores are discussed. At least two reasons make the temperature in the center (near 123°E) of the section higher than that near the west and east shores in winter. One reason is that the water there is deeper than the west and east sides so its heat content is higher. The other is invasion of the warm water brought by the Yellow Sea Warm Current (YSWC) during winter.This temperature pattern of the lower layer (from 30m to bottom) is maintained through spring and summer when the upper layer (0 to 30m) is heated and strong thermocline is formed. Large zonal span of the 35°N section (about 600 km) makes the cold cores have more opportunity to survive. The double cold cores phenomena disappears in early autumn when the west cold core vanishes first with the dropping of the thermocline position.展开更多
A simple Core-SOL-Divertor (C-S-D) model has been developed to investigate qualitatively the overall features of the operational space for the integrated core and edge plasma. To construct the simple C-S-D model, a ...A simple Core-SOL-Divertor (C-S-D) model has been developed to investigate qualitatively the overall features of the operational space for the integrated core and edge plasma. To construct the simple C-S-D model, a simple core plasma model of ITER physics guidelines and a two-point SOL-divertor model are used. The simple C-S-D model is applied to the study of the EAST operational space with lower hybrid current drive experiments under various kinds of trade-off for the basic plasma parameters. Effective methods for extending the operational space are also presented, From this study for the EAST operational space, it is evident that the C-S-D model is a useful tool for understanding qualitatively the overall features of the plasma operational space.展开更多
The kinetics equation of the Mg-based hydrogen storage alloys (Mg-Ni-MO) was established by the shell and shrinking core model. The total coefficients of the kinetics equation of the hydrogen absorption and desorption...The kinetics equation of the Mg-based hydrogen storage alloys (Mg-Ni-MO) was established by the shell and shrinking core model. The total coefficients of the kinetics equation of the hydrogen absorption and desorption process with shell diffusion as the controlling step were determined by semi-empirical and semi-theoretical methods, and the apparent activation energy of the hydrogen absorption process was obtained. The calculation results can well accord with the experimental data, and can well forecast the hydrogen storage capacity and absorption rate at different times. By using the kinetics equation, the effects of temperature and pressure on the hydrogen storage process can also be well understood. The kinetics equation is helpful for the design of the hydrogen storage container.展开更多
Scattering attenuation in short wavelengths has long been interesting to geophysicists. Ultrasonic coda waves, observed as the tail portion of ultrasonic wavetrains in laboratory ultrasonic measurements, are important...Scattering attenuation in short wavelengths has long been interesting to geophysicists. Ultrasonic coda waves, observed as the tail portion of ultrasonic wavetrains in laboratory ultrasonic measurements, are important for such studies where ultrasonic waves interact with smallscale random heterogeneities on a scale of micrometers, but often ignored as noises because of the contamination of boundary reflections from the side ends of a sample core. Numerical simulations with accurate absorbing boundary can provide insight into the effect of boundary reflections on coda waves in laboratory experiments. The simulation of wave propagation in digital and heterogeneous porous cores really challenges numerical techniques by digital image of poroelastic properties, numerical dispersion at high frequency and strong heterogeneity, and accurate absorbing boundary schemes at grazing incidence. To overcome these difficulties, we present a staggered-grid high-order finite-difference (FD) method of Biot's poroelastic equations, with an arbitrary even-order (2L) accuracy to simulate ultrasonic wave propagation in digital porous cores with strong heterogeneity. An unsplit convolutional perfectly matched layer (CPML) absorbing boundary, which improves conventional PML methods at grazing incidence with less memory and better computational efficiency, is employed in the simulation to investigate the influence of boundary reflections on ultra- sonic coda waves. Numerical experiments with saturated poroelastic media demonstrate that the 2L FD scheme with the CPML for ultrasonic wave propagation significantly improves stability conditions at strong heterogeneity and absorbing performance at grazing incidence. The boundary reflections from the artificial boundary surrounding the digital core decay fast with the increase of CPML thick- nesses, almost disappearing at the CPML thickness of 15 grids. Comparisons of the resulting ultrasonic coda Qsc values between the numerical and experimental ultrasonic S waveforms for a cylindrical rock sample demonstrate that the boundary reflection may contribute around one-third of the ultrasonic coda attenuation observed in laboratory experiments.展开更多
On the basis of the dyeing and weaving corporations’ characters,this paper put forward the dimensionality and index system to analyze the core competence. This paper divided the core competence into three layers and ...On the basis of the dyeing and weaving corporations’ characters,this paper put forward the dimensionality and index system to analyze the core competence. This paper divided the core competence into three layers and gave out the Dimensional-Hierarchical structure of core competence through combining the analysis dimensionalities with the competence layers. The model was described to evaluate,analyze and judge the dyeing and weaving corporation’s competence.展开更多
The liquid-liquid extraction method using reverse micelles can simultaneously extract lipid and protein of oilseeds,which have become increasingly popular in recent years.However,there are few studies on mass transfer...The liquid-liquid extraction method using reverse micelles can simultaneously extract lipid and protein of oilseeds,which have become increasingly popular in recent years.However,there are few studies on mass transfer processes and models,which are helpful to better control the extraction process of oils and proteins.In this paper,mass transfer process of peanut protein extracted by bis(2-ethylhexyl)sodium sulfosuccinate(AOT)/isooctane reverse micelles was investigated.The effects of stirring speed(0,70,140,and 210 r/min),temperature of extraction(30,35,40,45,and 50℃),peanut flour particle size(0.355,0.450,0.600,and 0.900 mm)and solidliquid ratio(0.010,0.0125,0.015,0.0175,and 0.020 g/mL)on extraction rate were examined.The results showed that extraction rate increased with temperature rising,particle size reduction as well as solid-liquid ratio increase respectively,while little effect of stirring speed(P>0.05)was observed.The apparent activation energy of extraction process was calculated as 10.02 kJ/mol and Arrhenius constant(A)was 1.91 by Arrhenius equation.There was a linear relationship between reaction rate constant and the square of the inverse of initial particle radius(1/r_(0)^(2))(P<0.05).This phenomenon and this shrinking core model were anastomosed.In brief,the extraction process was controlled by the diffusion of protein from the virgin zone interface of particle through the reacted zone and it was in line with the first order reaction.Mass transfer kinetics of peanut protein extracted by reverse micelles was established and it was verified by experimental results.The results provide an important theoretical guidance for industrial production of peanut protein separation and purification.展开更多
The paper is devoted to mathematical modelling of static and dynamic stability of a simply supported three-layered beam with a metal foam core. Mechanical properties of the core vary along the vertical direction. The ...The paper is devoted to mathematical modelling of static and dynamic stability of a simply supported three-layered beam with a metal foam core. Mechanical properties of the core vary along the vertical direction. The field of displacements is for- mulated using the classical broken line hypothesis and the proposed nonlinear hypothesis that generalizes the classical one. Using both hypotheses, the strains are determined as well as the stresses of each layer. The kinetic energy, the elastic strain energy, and the work of load are also determined. The system of equations of motion is derived using Hamilton's principle. Finally, the system of three equations is reduced to one equation of motion, in particular, the Mathieu equation. The Bubnov-Galerkin method is used to solve the system of equations of motion, and the Runge-Kutta method is used to solve the second-order differential equation. Numerical calculations are done for the chosen family of beams. The critical loads, unstable regions, angular frequencies of the beam, and the static and dynamic equilibrium paths are calculated analytically and verified numerically. The results of this study are presented in the forms of figures and tables.展开更多
文摘This paper introduces the results of selecting and breeding a micro-organism, Strain I, and its core model experiment investigation for microbial enhanced oil recovery (MEOR). Strain I was separated from the formation water of the Dagang oil field, with analytical results showing that Strain I is a gram-positive bacillus. A further study revealed that this strain has an excellent tolerance of environmental stresses: It can survive in conditions of 70℃, 30 wt% salinity and pH3.5-9.4. Strain I can metabolize biosurfactants that could increase the oil recovery ratio, use crude oil as the single carbon source, and decompose long-chain paraffin with a large molecular weight into short-chain paraffin with a small molecular weight. The core model experiment shows that Strain I enhances oil recovery well. Using 2 vol% of the fermentation solution of Strain I to displace the crude oil in the synthetic plastic bonding core could increase the recovery ratio by 21.6%.
基金financially supported by the National Science Foundation China(No.52104343)the Natural Science Foundation of Hebei Province,China(No.E2021203222)+1 种基金support from the High Steel Center(HSC)at Yanshan UniversityNorth China University of Technology,China。
文摘A three-dimensional mathematical model was established to predict the multiphase flow,motion and dispersion of desulfurizer particles,and desulfurization of hot metal during the Kanbara reactor(KR)process.The turbulent kinetic energy-turbulent dissipation rate(k-ε)turbulence model,volume-of-fluid multiphase model,discrete-phase model,and unreacted core model for the reaction between the hot metal and particles were coupled.The measured sulfur content of the hot metal with time during the actual KR process was employed to validate the current mathematical model.The distance from the lowest point of the liquid level to the bottom of the ladle decreased from 3170 to2191 mm when the rotation speed increased from 30 to 110 r/min,which had a great effect on the dispersion of desulfurizer particles.The critical rotation speed for the vortex to reach the upper edge of the stirring impeller was 70 r/min when the immersion depth was 1500 mm.The desulfurization rate increased with the increase in the impeller rotation speed,whereas the influence of the immersion depth was relatively small.Formulas for different rotation parameters on the desulfurization rate constant and turbulent energy dissipation rate were proposed to evaluate the variation in sulfur content over time.
基金Supported by the National Natural Science Foundation of China.
文摘The kinetics of H<sub>2</sub>S removal by zinc oxide desulfurizer was studied through thermogravimetricanalysis.The experimental results show that desulfurization rate was controlled,at high temperatureand low conversion,by the chemical reaction rate,and at low temperature and high conversion by thegrain diffusion rate.The reaction is first order with respect to H<sub>2</sub>S concentration in the differentcontrolled stages.The kinetic behavior can be modeled through the employment of the shrinking coremodel.The values of the model parameters were determined.The variation tendencies with temperatureand concentration of H<sub>2</sub>S at the controlled stages were discussed.
文摘The effect of rotation on the shape (figure) and gravitational quadrupole of astronomical bodies is calculated by using an approximate point core model: A point mass at the center of an ellipsoidal homogeneous fluid. Maclaurin’s analytical result for homogenous bodies generalizes to this model and leads to very accurate analytical results connecting the three observables: oblateness (ò), gravitational quadrupole (J2), and angular velocity parameter (q). The analytical results are compared to observational data for the planets and a good agreement is found. Oscillations near equilibrium are studied within the model.
基金Item Sponsored by National Natural Science Foundation of China and Shanghai Baoshan Steel Group(50674013)
文摘Controlling inclusion composition, from the point of view of thermodynamics, only explains the probability and limit of reaction. However, kinetics makes the nucleation and the velocity of growth of inclusions clear, and these kinetic factors are very important to the quality of slab. The basic kinetic theory of unreacted core model was used to build the mathematical model for the growth of inclusions and the concerned software was developed through Visual Basic 6.0. The time that different radius inclusions attain saturation was calculated to determine the controlling step of reaction between steel and inclusions. The time for the growth of inclusion obtained from the model was in good agreement with the data measured by Japanese Okuyama G, which indicated that the model is reasonable.
文摘The kinetics oflignocellulose hydrolysis under the conditions of high temperature and dilute acid (mass fraction 0.05%) was investigated in this paper. By studying the reducing sugar concentration versus reaction tempera ture (170℃-220℃) and reaction time (150-1800 s) during the hydrolysis process of five kinds of crop straw (rice, wheat, cotton, rape and corn), the shrinking core model was established, and the differential equation of the model and its analytical solution were obtained. With a numerical calculation method, the kinetic equation was estimated, and the degradation of reducing sugar obeyed firstorder kinetics was obtained. The calculated results from the equations agreed well with the original experimental data. The calculation by the model showed that the reducing sugar concentration increases as the size of the particles decrease, and the uniform particles increase.
文摘Based on the MASNUM wave-tide-circulation coupled numerical model, the temperature structure along 35°N in the Yellow Sea was simulated and compared with the observations. One of the notable features of the temperature structure along 35°N section is the double cold cores phenomena during spring and summer. The double cold cores refer to the two cold water centers located near 122°E and 125°E from the depth of 30m to bottom. The formation, maintenance and disappearance of the double cold cores are discussed. At least two reasons make the temperature in the center (near 123°E) of the section higher than that near the west and east shores in winter. One reason is that the water there is deeper than the west and east sides so its heat content is higher. The other is invasion of the warm water brought by the Yellow Sea Warm Current (YSWC) during winter.This temperature pattern of the lower layer (from 30m to bottom) is maintained through spring and summer when the upper layer (0 to 30m) is heated and strong thermocline is formed. Large zonal span of the 35°N section (about 600 km) makes the cold cores have more opportunity to survive. The double cold cores phenomena disappears in early autumn when the west cold core vanishes first with the dropping of the thermocline position.
基金supported in part by the JSPS-CAS Core-University Program in the field of Plasma and Nuclear Fusionalso carried out as a joint project under the Facility Utilization Program of JAERI.
文摘A simple Core-SOL-Divertor (C-S-D) model has been developed to investigate qualitatively the overall features of the operational space for the integrated core and edge plasma. To construct the simple C-S-D model, a simple core plasma model of ITER physics guidelines and a two-point SOL-divertor model are used. The simple C-S-D model is applied to the study of the EAST operational space with lower hybrid current drive experiments under various kinds of trade-off for the basic plasma parameters. Effective methods for extending the operational space are also presented, From this study for the EAST operational space, it is evident that the C-S-D model is a useful tool for understanding qualitatively the overall features of the plasma operational space.
文摘The kinetics equation of the Mg-based hydrogen storage alloys (Mg-Ni-MO) was established by the shell and shrinking core model. The total coefficients of the kinetics equation of the hydrogen absorption and desorption process with shell diffusion as the controlling step were determined by semi-empirical and semi-theoretical methods, and the apparent activation energy of the hydrogen absorption process was obtained. The calculation results can well accord with the experimental data, and can well forecast the hydrogen storage capacity and absorption rate at different times. By using the kinetics equation, the effects of temperature and pressure on the hydrogen storage process can also be well understood. The kinetics equation is helpful for the design of the hydrogen storage container.
基金supported by the National Natural Science Foundation of China (40925013)the Strategic Leading Science and Technology Programme (Class B) of the Chinese Academy of Sciences (Grant No.XDB10010400)the China National Major Science and Technology Project (2011ZX05023-005004)
文摘Scattering attenuation in short wavelengths has long been interesting to geophysicists. Ultrasonic coda waves, observed as the tail portion of ultrasonic wavetrains in laboratory ultrasonic measurements, are important for such studies where ultrasonic waves interact with smallscale random heterogeneities on a scale of micrometers, but often ignored as noises because of the contamination of boundary reflections from the side ends of a sample core. Numerical simulations with accurate absorbing boundary can provide insight into the effect of boundary reflections on coda waves in laboratory experiments. The simulation of wave propagation in digital and heterogeneous porous cores really challenges numerical techniques by digital image of poroelastic properties, numerical dispersion at high frequency and strong heterogeneity, and accurate absorbing boundary schemes at grazing incidence. To overcome these difficulties, we present a staggered-grid high-order finite-difference (FD) method of Biot's poroelastic equations, with an arbitrary even-order (2L) accuracy to simulate ultrasonic wave propagation in digital porous cores with strong heterogeneity. An unsplit convolutional perfectly matched layer (CPML) absorbing boundary, which improves conventional PML methods at grazing incidence with less memory and better computational efficiency, is employed in the simulation to investigate the influence of boundary reflections on ultra- sonic coda waves. Numerical experiments with saturated poroelastic media demonstrate that the 2L FD scheme with the CPML for ultrasonic wave propagation significantly improves stability conditions at strong heterogeneity and absorbing performance at grazing incidence. The boundary reflections from the artificial boundary surrounding the digital core decay fast with the increase of CPML thick- nesses, almost disappearing at the CPML thickness of 15 grids. Comparisons of the resulting ultrasonic coda Qsc values between the numerical and experimental ultrasonic S waveforms for a cylindrical rock sample demonstrate that the boundary reflection may contribute around one-third of the ultrasonic coda attenuation observed in laboratory experiments.
文摘On the basis of the dyeing and weaving corporations’ characters,this paper put forward the dimensionality and index system to analyze the core competence. This paper divided the core competence into three layers and gave out the Dimensional-Hierarchical structure of core competence through combining the analysis dimensionalities with the competence layers. The model was described to evaluate,analyze and judge the dyeing and weaving corporation’s competence.
基金This study was supported by the National Natural Science Foundation of China(No.U21A20270 and 32202079)Postdoctoral Science and Technology Project of Henan,Grant No.HN2022046+2 种基金Science and Technology Project of Henan Province(232103810064)the Innovative Funds Plan of Henan University of Technology(2021ZKCJ03)the Key Scientific Research Projects of Colleges and Universities of Henan(23A550012).
文摘The liquid-liquid extraction method using reverse micelles can simultaneously extract lipid and protein of oilseeds,which have become increasingly popular in recent years.However,there are few studies on mass transfer processes and models,which are helpful to better control the extraction process of oils and proteins.In this paper,mass transfer process of peanut protein extracted by bis(2-ethylhexyl)sodium sulfosuccinate(AOT)/isooctane reverse micelles was investigated.The effects of stirring speed(0,70,140,and 210 r/min),temperature of extraction(30,35,40,45,and 50℃),peanut flour particle size(0.355,0.450,0.600,and 0.900 mm)and solidliquid ratio(0.010,0.0125,0.015,0.0175,and 0.020 g/mL)on extraction rate were examined.The results showed that extraction rate increased with temperature rising,particle size reduction as well as solid-liquid ratio increase respectively,while little effect of stirring speed(P>0.05)was observed.The apparent activation energy of extraction process was calculated as 10.02 kJ/mol and Arrhenius constant(A)was 1.91 by Arrhenius equation.There was a linear relationship between reaction rate constant and the square of the inverse of initial particle radius(1/r_(0)^(2))(P<0.05).This phenomenon and this shrinking core model were anastomosed.In brief,the extraction process was controlled by the diffusion of protein from the virgin zone interface of particle through the reacted zone and it was in line with the first order reaction.Mass transfer kinetics of peanut protein extracted by reverse micelles was established and it was verified by experimental results.The results provide an important theoretical guidance for industrial production of peanut protein separation and purification.
基金Project supported by the Ministry of Science and Higher Education of Poland(Nos.04/43/DSPB/0085and 02/21/DSPB/3464)
文摘The paper is devoted to mathematical modelling of static and dynamic stability of a simply supported three-layered beam with a metal foam core. Mechanical properties of the core vary along the vertical direction. The field of displacements is for- mulated using the classical broken line hypothesis and the proposed nonlinear hypothesis that generalizes the classical one. Using both hypotheses, the strains are determined as well as the stresses of each layer. The kinetic energy, the elastic strain energy, and the work of load are also determined. The system of equations of motion is derived using Hamilton's principle. Finally, the system of three equations is reduced to one equation of motion, in particular, the Mathieu equation. The Bubnov-Galerkin method is used to solve the system of equations of motion, and the Runge-Kutta method is used to solve the second-order differential equation. Numerical calculations are done for the chosen family of beams. The critical loads, unstable regions, angular frequencies of the beam, and the static and dynamic equilibrium paths are calculated analytically and verified numerically. The results of this study are presented in the forms of figures and tables.