One-dimensional Model Research Articles

Electron kinetics near the minimum of Paschen's curves of pulsed breakdown

The main objective of the present paper is the detailed analysis of the electron kinetics near the Paschen's curve minimum of pulsed breakdown of nitrogen gas. Three main questions are to be answered. First, whether the breakdown curve minimum corresponds to the threshold electric field necessary for the runaway electron generation. Second, what is the role of these electrons in the vicinity of the Paschen's curve minimum. Third, what is the ionization cost near the minimum. To answer these questions, the one-dimensional model is used in which electrons are modeled using the Particle-in-Cell approach, while ions are modeled in the drift-diffusion approximation.

Investigating the formation of soot in CH4 pyrolysis reactor: A numerical, experimental, and characterization study

Methane pyrolysis is a promising method for eco-friendly hydrogen production, but soot formation and carbon interaction pose challenges for scaling up. Therefore, understanding the dynamics of soot formation and carbon deposition is crucial. This study delves into the intricacies of soot formation in methane pyrolysis under industrially relevant conditions, namely operations at atmospheric pressure, employing a H2:CH4 ratio of 2 and exploring a range of hot zone temperatures (1473 K, 1573 K, 1673 K, and 1773 K) with a 5 s residence time. Utilizing a detailed gas-phase kinetic model with direct carbon deposition reactions, the research adopts the method of moments coupled with a one-dimensional plug flow reactor model to simulate soot formation. The model is validated by characterizing soot particles that were produced in a pyrolysis reactor by means of transmission electron microscopy, Dynamic light scattering (DLS), and Raman spectroscopy. Results show that lower temperatures lead to nucleation-dominated growth, whereas higher temperatures significantly restrain particle growth due to carbon deposition. DLS data indicate a complex balance between particle growth and deposition processes. These findings provide insights into operational parameters that can enhance reactor performance and sustainability in hydrogen production processes by mitigating soot and carbon deposition.

Estimating pulmonary arterial remodeling via an animal-specific computational model of pulmonary artery stenosis.

Pulmonary artery stenosis (PAS) often presents in children with congenital heart disease, altering blood flow and pressure during critical periods of growth and development. Variability in stenosis onset, duration, and severity result in variable growth and remodeling of the pulmonary vasculature. Computational fluid dynamics (CFD) models enable investigation into the hemodynamic impact and altered mechanics associated with PAS. In this study, a one-dimensional (1D) fluid dynamics model was used to simulate hemodynamics throughout the pulmonary arteries of individual animals. The geometry of the large pulmonary arteries was prescribed by animal-specific imaging, whereas the distal vasculature was simulated by a three-element Windkessel model at each terminal vessel outlet. Remodeling of the pulmonary vasculature, which cannot be measured in vivo, was estimated via model-fitted parameters. The large artery stiffness was significantly higher on the left side of the vasculature in the left pulmonary artery (LPA) stenosis group, but neither side differed from the sham group. The sham group exhibited a balanced distribution of total distal vascular resistance, whereas the left side was generally larger in the LPA stenosis group, with no significant differences between groups. In contrast, the peripheral compliance on the right side of the LPA stenosis group was significantly greater than the corresponding side of the sham group. Further analysis indicated the underperfused distal vasculature likely moderately decreased in radius with little change in stiffness given the increase in thickness observed with histology. Ultimately, our model enables greater understanding of pulmonary arterial adaptation due to LPA stenosis and has potential for use as a tool to noninvasively estimate remodeling of the pulmonary vasculature.

A generalized mathematical model for the damped free motion of a liquid column in a vertical U-tube

A one-dimensional mathematical model is developed to analyze the unsteady characteristics of arbitrary damped free motion, denoted as Z(τ), in a rigid, constant-diameter vertical column of a U-tube filled with an incompressible Newtonian liquid, where τ represents time. This model utilizes a simplified unsteady momentum equation derived from the Navier–Stokes equations in a circular coordinate system. Moreover, it incorporates assumptions about the periodicity of arbitrary damped free oscillations and employs the Fourier series representation to characterize the damped free motion. The combination of assumptions made for periodicity, the simplified momentum equation, and the Fourier series representation makes the current mathematical model unique and novel compared to prevailing models in the literature. In this model, the governing partial differential equation contains two dependent variables: Z(τ) is the known variable, as one can measure from experiments, and the instantaneous velocity uz is the unknown variable. Fitting the experimental data into the Fourier series provides the Fourier coefficients associated with the specific experiment. The Laplace transform method is used to determine the analytical solution for uz corresponding to the known Z(τ). The analytical expressions for instantaneous flow characteristics of practical importance, including area-averaged velocity, wall shear stress, and acceleration/deceleration, are deduced from uz. The analytical solutions presented are valid for generalized unsteady motions, including underdamped oscillations with varying amplitudes and periods, underdamped oscillations with varying amplitudes and constant period, and overdamped motion that does not exhibit a single oscillation. The findings from the present model offer insights for formulating a new friction model.

Development of optimal energy management strategy for proton exchange membrane fuel cell-battery hybrid system for drone propulsion

Drones powered by fuel cells have been developed to ensure long flight distances. Due to the low dynamic responsiveness of proton exchange membrane fuel cell (PEMFC) systems, they must be integrated with batteries to fully meet the power demand profiles of drones. In this study, a dynamic model of a PEMFC-battery hybrid system for drone propulsion is developed in MATLAB-Simulink® to establish the optimal energy management strategy (EMS) for ensuring efficient and safe operation. The proposed PEMFC system comprises two modules of an open-cathode PEMFC stack, which are connected in parallel. To estimate the dynamic behavior and performance of PEMFC, one-dimensional dynamic model of PEMFC considering two-phase transport through the gas diffusion layer is developed and simulated. A zero-dimensional dynamic model of the battery was also developed based on an equivalent circuit model. The resulting PEMFC-battery hybrid system model was simulated to determine the power required for the propulsion of the drone according to a flight scenario comprising five phases: takeoff hover, climb, cruise, descent, and landing hover. Three different EMSs are developed and simulated to define the optimal one for the PEMFC-battery hybrid system by comparing the total hydrogen consumption and the final SOC of the battery As a result, the EMS that equally splits the required power between two PEMFCs exhibits the lowest hydrogen consumption during the flight simulation. This study helps provide the optimal system design and control scheme of drones powered by fuel cells.

The solar beryllium abundance revisited with 3D non-LTE models

The present-day abundance of beryllium in the solar atmosphere provides clues about mixing mechanisms within stellar interiors. However, abundance determinations based on the Be II313.107 nm line are prone to systematic errors due to imperfect model spectra. These errors arise from missing continuous opacity in the UV, a significant unidentified blend at 313.102 nm, departures from local thermodynamic equilibrium (LTE), and microturbulence and macroturbulence fudge parameters associated with one-dimensional (1D) hydrostatic model atmospheres. Although these factors have been discussed in the literature, no study has yet accounted for all of them simultaneously. To address this, we present 3D non-LTE calculations for neutral and ionised beryllium in the Sun. We used these models to derive the present-day solar beryllium abundance, calibrating the missing opacity on high resolution solar irradiance data and the unidentified blend on the centre-to-limb variation. We find a surface abundance of 1.21 ± 0.05 dex, which is significantly lower than the value of 1.38 dex that has been commonly adopted since 2004. Taking the initial abundance via CI chondrites, our result implies that beryllium has been depleted from the surface by an extra 0.11 ± 0.06 dex, or 22 ± 11%, on top of any effects of atomic diffusion. This is in tension with standard solar models, which predict negligible depletion, as well as with contemporary solar models that have extra mixing calibrated on the abundances of helium and lithium, which predict excessive depletion. These discrepancies highlight the need for further improvements to the physics in solar and stellar models.

Modeling the hundreds-of-nanoseconds-long irradiation of tin droplets with a 2 µm-wavelength laser for future EUV lithography

Abstract The properties of an extreme ultraviolet (EUV) source driven by hundreds-of-nanoseconds-long laser pulses of λ laser = 2 µm wavelength are investigated through radiation-hydrodynamic simulations. We show that single-pulse irradiation of 30 µm-diameter tin droplets can generate in-band energies ∼ 30 mJ directed in the 2π sr solid angle subtended by the collector mirror, yielding energies ∼100 mJ produced from 45 µm-diameter droplets. Time-integrated in-band EUV emission profiles, generated by taking Abel transforms of the local net in-band emissivity, reveal EUV source sizes in the axial (laser) direction × radial direction of ∼600 (1000) × 100 (150) µm2 for 30 (45) µm-diameter droplets. We find that approximately 74% of the total in-band emission is produced during the first half of the total propulsion distance irrespective of droplet size or laser intensity. Furthermore, we propose a one-dimensional analytical propulsion model to qualitatively explain the simulated droplet trajectory and to predict the time taken for the droplet to vaporize. These findings offer motivation for the development of high-power EUV sources based on single-pulse, hundreds-of-nanoseconds-long irradiation of tin droplets.

One-dimensional N-layer thermal modelling as a basis for effective machine learning training data generation for nondestructive testing of composite parts.

The analytical solution of the one-dimensional N-layer thermal model (NLM) was successfully employed to generate training data for a machine learning (ML) based procedure for the nondestructive inspection of carbon fiber reinforced composite parts with infrared thermography (IRT). The main objective was to identify a reliable correlation between the experimental data of a pulsed IRT experiment and the NLM prediction, thereby enabling the use of simulated data for ML training. This paper focuses on the initial stages of this process, in more detail on the analytical modelling and experimental data preprocessing, such as normalization, correction of experimental shortcomings and feature selection for machine learning. For pulse phase thermography (PPT) simulated and experimentally derived phase data was compared directly in the frequency domain. Therefore, the features for training and validation of ML were defined from those phase spectra in frequency domain. The suitability of these features for automated and reliable defect depth and/or defect material detection was investigated in both simulated and measured IRT test data. As a basis for feature selection, we used referenced and normalized phase-frequency curves as a function of defect depth. A correlation was identified between the results of the experimental and the simulated feature sets, both qualitatively and quantitatively. To demonstrate the practical applicability of this method, two different, generic ML techniques, multilayer perceptron and random forest regression, were tested as examples. The investigation was performed on plates made of multidirectional carbon fiber reinforced polymer (CFRP) with artificial defects made from three different materials.

Design optimization and off-design performance analysis of one-dimensional supercritical CO2 Brayton cycle

Supercritical carbon dioxide (SCO2) recompression Brayton cycle (RBC) is an encouraging technology for thermal power conversion and heat harvest for a wide range of heat sources due to its high-efficiency, low auxiliary power consumption, and compact size. However, most studies on its design and off-design performance analysis have been limited to the simple zero-dimensional component models without considering aerothermodynamics and ambient conditions. In this study, we developed a set of fully one-dimensional components models for SCO2 compressors, turbines, and heat exchangers, and, then, applied them to the multi-objective design optimization of a SCO2 cycle targeting a 50 MW net power output. The cycle off-design performance analysis methods and the control strategies were developed to countermeasure the performance degradation against the varying cooling water temperatures and part loads. The multi-objective cycle design optimization demonstrates that the cycle thermal efficiency and total product unit cost can be 0.476 and 11.652$/GJ, respectively, for the 50 MW SCO2 RBC cycle. The choke margin of the main compressor is 0.136 less than that of the re-compressor, and condensation can lead to a sharp decline in efficiency under high-flow conditions, causing the earlier onset of choking. The off-design performance analysis shows that the turbine inlet pressure control improves cycle efficiency below the design point of 317 K but does otherwise above the point. In contrast, inventory control strategies can maintain stable cycle efficiency and achieve adjustable net power output between 10 % and 110 %. The findings have the potential to advance further the commercial application of SCO2 cycles in high-temperature thermal energy systems.

Design of a tubular segmented-in-series solid oxide fuel cell (SOFC): One-dimensional steady state modeling

In this study, a one-dimensional steady-state model has been established for rapidly predicting the output performance of tubular segmented-in-series solid oxide fuel cells (SOFCs), which will be used for the structural and material design of T-SIS-SOFC to improve output performance. This model simplifies ohmic polarization, activation polarization, and concentration polarization into equivalent resistances, while also considering the impact of contact resistance, interconnects, supports, and current collectors on the output performance. After validation, the model has proven reliable in predicting the output performance of cells with different structures. For a single cell, ohmic polarization is the primary influencing factor, and increasing conductivity through methods such as raising temperature, modifying materials, or reducing thickness can significantly reduce ohmic polarization. For the entire tubular SOFC system, when the total length is fixed, there exists an optimal number of cells and an optimal length for each cell. Based on the material and structure of the single cell unit employed in this paper, the maximum output power, approximately 48 W, is achieved when the number of cell units is 30.

Deep structural insights into the origin of the Heyuan ms 4.5 revealed by deep seismic sounding profiles in southeast China

This study presents an interpretation of a deep seismic sounding (DSS) profile that carried out along the Cathaysia Block in southeast china, aiming to explore the crustal velocity structure. Data used in the survey were obtained from three controlled-source explosions conducted along the 320 km long Lianping-Heyuan-Shanwei profile. The modeling was based on ray tracing, using the extrapolation of seismic wave arrival times with the help of travel times predicted from a one-dimensional velocity model. The average velocity structure of the middle crust is 6.0–6.4 km/s, while a low velocity anomaly of approximately 0.1–0.2 km/s in the vicinity of the Heyuan-Shaowu fault zone. The resulting 2D velocity model indicates that steeply dipping low-velocity zones that correlate with the projection of two major fault zones. These zones, together with a flat LVZ at a depth of 12 km, define a triangular region that correlates with numerous hypocenters. This tectonic setting is favorable for the accumulation and release of strain in high-velocity media within the triangular region. The unique triangular structure in the upper crust provides necessary shallow medium conditions for seismic activity. This indicates that increased seismicity within this area is partially attributed to heightened stress within higher-velocity material. The triangular annular low-velocity body, situated in the upper crust, is influenced by dynamic environmental factors caused by deep thermal disturbances. The deep-seated fault serves as a conduit for the historical migration of thermal material, likely contributing to the seismogenic conditions for earthquakes in Heyuan’s region through deep-seated thermal disturbances. These findings provide a novel geophysical reference model for the regional seismicity near the Xinfengjang reservoir and significantly contribute to understanding the causal relationship between tectonic setting and seismicity. In comparison with previous studies, our research is dedicated to investigating the causes of shallow earthquakes in the region and exploring the relationship between deep and shallow structures.

Integration of the exploding foil initiator with capacitor discharge unit and its performance characterization

This study presents a novel integrated exploding foil initiator system (EFIs), consisting of a planar switch, an EFI chip, and a capacitor. Additionally, we propose an evaluation method for integrated EFIs that combines numerical simulations with experimental testing, enabling a thorough assessment of their performance. Under test conditions of 900 V/0.22 μF, the EFI demonstrated an inductance of 10.2 nH and a resistance of 107.5 mΩ, measured via the sampling resistance method. The operational behavior of the EFIs was analyzed through a two-dimensional metal electrical explosion model and a one-dimensional flyer propulsion model, with the results compared to experimental data. Results indicated that under 900–1200 V/0.22 μF, the deviations between measured and calculated values for electrical explosion performance and flyer velocity were both within 5 %, validating the accuracy of the evaluation approach. Firing tests further confirmed that the EFIs successfully detonated HNS-IV at 1000 V/0.22 μF.

Electric transport and topological properties of binary heterostructures in topological insulators

The design of devices with coupled hybrid structures offers an approach to creating synthetic topological materials. This work discusses the topological and transport properties of low-dimensional binary heterostructures of topological and trivial materials. By adjusting the parameters of each component, we control the global topological properties to enhance tunneling and optimize the transmission of the topological edge states (TES). Considering a one-dimensional tight-binding model, we build heterostructures of coupled chains employing Green’s functions (GF) formalism. We determine the topological characteristics of chains and couple them together, applying Dyson’s equation to generate the heterostructure. The intensity and decay length of the TES vary depending on the coupling parameters and the size of each chain. We investigate the topological diagrams phase using the energy bands of the periodic system and calculating the invariant from the Zak phase. Using cross-band condition, we derive analytical functions of the parameter space to get the phase topological diagram, which can be compared with the LDOS maps at zero energy. Finally, we calculate the differential conductance with the Keldysh GF technique to demonstrate the tunneling of the TES at the zero bias voltage and discuss potential design and experimental applications.

Helium Abundance of the Sun: A Spectroscopic Analysis

Determining the He/H ratio in cool stars presents a fundamental astrophysical challenge. While this ratio is established for hot O and B stars, its extrapolation to cool stars remains uncertain due to the absence of helium lines in their observed spectra. We address this knowledge gap by focusing on the Sun as a representative cool star. We conduct spectroscopic analyses of the observed solar photospheric lines by utilizing a combination of MgH molecular lines and neutral Mg atomic lines including yet another combination of CH and C2 molecular lines with neutral C atomic lines. Our spectroscopic analyses were further exploited by adopting solar model atmospheres constructed for distinct He/H ratios to determine the solar photospheric helium abundance. The helium abundance is determined by enforcing the fact that for an adopted model atmosphere with an appropriate He/H ratio, the derived Mg abundance from the neutral Mg atomic lines and that from the MgH molecular lines must be the same. The same holds for the C abundance derived from neutral C atomic lines and that from CH lines of the CH molecular band and C2 lines from the C2 Swan band. The estimated He/H ratio for the Sun is discussed based on the one-dimensional local thermodynamic equilibrium model atmosphere. The helium abundance (He/H = 0.091−0.014+0.019 ) obtained for the Sun serves as a critical reference point to characterize the He/H ratio of cool stars across the range in their effective temperature.

Optimization of thermal shock response spectrum as infrared thermography post-processing methodology using Latin hypercube sampling and analytical thermal N-layer model

In this work, we continue to develop and investigate the Thermal Shock Response Spectrum (TSRS) method as an alternative data processing method for infrared thermography (IRT). We focus on improving the current TSRS algorithm and present an optimization methodology for finding the optimal thermal Q-factor and characteristic frequency pair, which is based on the widely applied random sampling method. We show the qualitative relationship between the determined optimal characteristic frequency and the corresponding maximum difference in diffusion length between reference and defective models, as calculated by selecting a specific one-dimensional thermal N-layer model The investigations were performed on an inhomogeneous plate made of carbon fiber reinforced polymer (CFRP) with artificial square defects at different depths. Furthermore, two different heat sources were used: a xenon flash lamp and a laser. These sources are not only distinct by their underlying physics but also generate inherently different pulse shapes. To quantitatively estimate the contrast between defect and non-defect areas, and to compare these results with commonly used infrared thermography (IRT) data post-processing methods such as Pulse Phase Thermography (PPT) and Thermographic Signal Reconstruction (TSR), the Tanimoto criterion (TC) and signal-to-noise ratio (SNR) were used.

Impact of ice growth on the physical and chemical properties of dense cloud cores

We investigated the effect of time-dependent ice growth on dust grains on the opacity and hence on the dust temperature in a collapsing molecular cloud core, with the aim of quantifying the effect of the dust temperature variations on ice abundances as well as the evolution of the collapse. To perform the simulations, we employed a one-dimensional collapse model that self-consistently and time-dependently combines hydrodynamics with chemical and radiative transfer simulations. The dust opacity was updated on the fly based on the ice growth as a function of the location in the core. The results of the fully dynamical model were compared against simulations run with different values of fixed ice thickness. We found that the ice thickness increases quickly and reaches a saturation value (as a result of a balance between adsorption and desorption) of approximately 90 monolayers in the central core (volume density ~104 cm−3), and several tens of monolayers at a volume density of ~103 cm−3, after only a few 105 yr of evolution. The results thus exclude the adoption of thin (approximately ten monolayer) ices in molecular cloud simulations except at very short timescales. However, the differences in abundances and the dust temperature between the fully dynamic simulation and those with a fixed dust opacity are small; abundances change between the solutions generally within a factor of two only. The assumptions on the dust opacity do have an effect on the collapse dynamics through the influence of the photoelectric effect on the gas temperature, and the simulations take a different time to reach a common central density. This effect is, however, small as well. In conclusion, carrying out chemical simulations using a dust temperature corresponding to a fixed opacity seems to be a good approximation. Still, although at least in the present case its effect on the overall results is limited – as long as the grains are monodisperse – ice growth should be considered to obtain the most accurate representation of the collapse dynamics. We have found in a previous work that considering a grain size distribution leads to a complicated ice composition that depends on the grain size nonlinearly. With this in mind, we will carry out a follow-up study where the influence of the grain size on the present simulation setup is investigated.

Analysis of cooperative cooling performance between the battery and cabin of pure electric vehicles

Abstract The cooperative cooling of the cabin and battery in pure electric vehicles (PEVs) is crucial for the comfort, thermal safety of the occupants, and the lifespan of the batteries. Therefore, a well-designed thermal management system (TMS) is particularly important. Thus, this paper constructs an cooperative cooling for the battery pack circuit and the cabin circuit in parallel, aiming to enhance the vehicle system’s integration and reduce energy consumption. Firstly, a one-dimensional simulation model of the cooperative TMS is established, and the model is calibrated through experiments. Subsequently, the system’s performance is analyzed under different ambient temperatures. The study finds that as the ambient temperature increases, the system’s energy consumption rises, and the Coefficient of Performance (COP) decreases. Moreover, when the system switches from the simultaneous cooling mode of the cabin and battery to the sole cooling mode of the cabin, the temperature of the cabin will fluctuate to a certain extent, within a range of 2°C. At same time, the COP value also increases with the switch of the mode. Finally, the analysis of the simultaneous cooling mode of the cabin and battery side under different ambient temperatures reveals that as the ambient temperature increases from 34°C to 42°C, the exhaust temperature, pressure ratio, and compressor speed increase accordingly, while the compressor efficiency decreases. The condenser and chiller experience a reduction in their heat transfer efficiency by 59.4 watts per degree Celsius and 218.5 watts per degree Celsius, respectively. Conversely, the evaporator’s heat transfer efficiency undergoes an enhancement of 6.77 watts per degree Celsius.

Flood simulation and impact analysis of Xun River under backwater condition

Abstract The intersection of river channels is easy to cause the backside of the main flow, which greatly influences the change of hydrological conditions of river channels. To study the changing characteristics of river flood under the condition of uplift, a one-dimensional hydrodynamic model was established to simulate the evolution process of river flood under the condition of seeking river uplift. Factors such as distance, height, flood speed, inundation area, inundation depth, and flood speed were selected to analyze the changing rules of the influence of river bullying. The results show that the influence degree of the river following is closely related to the size of the flood. Under the same earthquake magnitude, the backwater distance, backwater height, inundation area, inundation depth, and inundation speed of sub-river flood increase with the increase of flood magnitude, while the flood speed decreases. When the magnitude of the flood is once in 200 years, the flood elements reach the maximum, the reflux distance is 8, 368 m, the reflux height is 3.2 m, and the velocity is reduced by 43.2%. This study can provide relevant data for the flood control work in the Sun River basin and provide a reference for the scheme design of tributary interchanges.

Predictive modeling of flow characteristics in supersonic separators using machine learning

In the pursuit of cleaner energy sources, pre-processing fuel streams is crucial before distribution. Challenges arise during this process, leading to operational issues and diminished transmission efficiency. Among different technologies, supersonic separators are the ones that separate undesired components from the main gas stream more effectively. Precise forecasting of nozzle flow rate and shockwave location at a low cost can markedly enhance the efficiency of manufacturing and rating. In this research, the one-dimensional method has been developed to solve flow equations simultaneously by implementing equations of state. CFD has been used for the verification of the developed model, and the effect of different equations of state is investigated. Results show that the one-dimensional model significantly reduces the calculation time. Different EOSs result in up to 2.0 % and 3.5 % variation in the prediction of shock location and the mass flow rate, respectively.A machine learning methodology is employed to predict the location of shockwaves and the choked flow rate at an even lower cost. A dataset of 160 entries, generated using Peng Robinson's EOS, serves as the basis for this analysis. Prediction tasks are executed using Random Forest, k-nearest Neighbors, and Multilayer Perceptron models. While all three models demonstrate proficiency in data interpolation with R2 scores up to 0.9962 and 0.9951 for the shockwave location and mass flow rate, respectively, discerning their performance in extrapolation appears contingent on the specific case. In the case of extrapolation, Multilayer Perceptron and k-Nearest Neighbors show better match to one-dimensional results.

An analog of the pendulum/rotor system with elementary solutions

Abstract We explore a novel one-dimensional model potential, V(x), which mimics many aspects of the classic pendulum/rotor system, including low- and high-energy limits which are exactly solvable, as well as a divergent period separating the two. Using energy methods, the classical period, T(E), is evaluated in closed form for all values of the energy (0 < E < ∞) in terms of simple logarithmic functions, in contrast to the pendulum/rotor system where the results require elliptic integrals. We find an unexpected duality between the exact expressions for the period on either side of the divergence. Using both energy methods, and a Newton’s law differential equation approach, we present closed form expressions for trajectory solutions, x(t) and p(t) = mv(t) (or the linear momentum), in terms of hyperbolic functions, which explicitly give the same values for T(E).