Differential Equation Of Heat Conduction Research Articles

Coupled responses of thermomechanical waves in functionally graded viscoelastic nanobeams via thermoelastic heat conduction model including Atangana–Baleanu fractional derivative

Accurately characterizing the thermomechanical parameters of nanoscale systems is essential for understanding their performance and building innovative nanoscale technologies due to their distinct behaviours. Fractional thermal transport models are commonly utilized to correctly depict the heat transfer that occurs in these nanoscale systems. The current study presents a novel mathematical thermoelastic model that incorporates a new fractional differential constitutive equation for heat conduction. This heat equation is useful for understanding the effects of thermal memory. An application of a fractional-time Atangana–Baleanu (AB) derivative with a local and non-singular kernel was utilized in the process of developing the mathematical model that was suggested. To deal with effects that depend on size, nonlocal constitutive relations are introduced. Furthermore, in order to take into consideration, the viscoelastic behaviour of the material at the nanoscale, the fractional Kelvin–Voigt model is utilized. The proposed model is highly effective in properly depicting the unusual thermal conductivity phenomena often found in nanoscale devices. The study also considered the mechanical deformation, temperature variations, and viscoelastic characteristics of the functionally graded (FG) nanostructured beams. The consideration was made that the material characteristics exhibit heterogeneity and continuous variation across the thickness of the beam as the nanobeam transitions from a ceramic composition in the lower region to a metallic composition in the upper region. The complicated thermomechanical features of simply supported viscoelastic nanobeams that were exposed to harmonic heat flow were determined by the application of the model that was constructed. Heterogeneity, nonlocality, and fractional operators are some of the important variables that contribute to its success, and this article provides a full study and illustration of the significance of these characteristics. The results that were obtained have the potential to play a significant role in pushing forward the design and development of tools, materials, and nanostructures that have viscoelastic mechanical characteristics and graded functions.

ANALYTICAL SOLUTION OF THE DIFFERENTIAL EQUATION OF HEAT CONDUCTIVITY FOR DAMAGED THERMAL INSULATION OF PIPELINES

For heat supply systems of Ukraine, the determination and forecasting of thermal energy losses during the transportation of heat carrier is an urgent problem. The thermal lines of the heating networks are long and its insulation is damaged. Installation of heat energy metering devices at all sources and all consumers (i.e. buildings) without exception allows determining and forecasting real heat losses in the heat network, but this is a difficult task, not all consumers and sources are actually covered by heat consumption accounting. The task of determining heat losses is also relevant for energy management systems of heat supply systems, energy companies and industry. The solution to the problem is proposed by a separate mathematical modeling of the temperature state of the section of the damaged insulating layer with the determination of the heat flow through it. It is proposed to solve the problem by the method of analytical solution of the differential equation of heat conduction with boundary conditions of the third kind. With the uniform distribution of characteristic damages along the total length of the pipeline, knowing the limits of damage influence, the share of insulation damage and the number of damages on the pipeline, it is possible to determine the real heat flow from the outer surface of the pipelines, including coefficient of increase of heat losses on the section of the heat pipe in relation to those determined by regulatory documents. Traditionally, one-dimensional models or determinations of the two-dimensional temperature field and actual heat fluxes in a cross-section to the axis of a characteristic damaged section of insulation are considered. But this does not take into account the two-dimensionality of the temperature field of the damaged layer of insulation along the length (that is, along the axis) of the pipeline. Therefore, the purpose of this work is to develop a methodology for determining heat losses through pipelines, taking into account the damage to their insulation and the distribution of characteristic damages along the length. The following simulation was carried out for one of the areas. Also, experimental and numerical studies using the finite difference method in combination with the method of running variable directions for similar models confirmed the coincidence of the analytical solution of the proposed model and the finite difference model within the permissible error.

IMPROVEMENT OF THE METHODOLOGY FOR OPTIMIZING THE PARAMETERS OF COMPLEX SYSTEMS UNDER THE INFLUENCE OF THERMAL LOAD SOURCES

The article investigates some aspects of solving the problem of constructing mathematical models for optimizing the parameters of local laser action on multilayer microbiological systems. The constraints on the desired laser parameters and on the temperature field in the material subject to laser fission are set. Verification of compliance with the restrictions on not exceeding the maximum value of the temperature field requires multiple calculations of the corresponding temperature field in the microbiological material. Control over the temperature field limits ensures the viability of its parts during the biotechnological process of laser fission. To construct adequate optimization mathematical models, the author substantiates the adequacy of the computational mathematical models describing the state of a microbiological system under the action of laser radiation sources. A multi-point boundary value problem with a system of inhomogeneous differential heat conduction equations for a multilayer microbiological medium is investigated and the correctness of this problem is substantiated for minor perturbations of the right-hand side of the differential equation. The results obtained in the article guarantee the adequacy of applied optimization mathematical models for finding rational values of technical parameters of laser emitters. In order to improve the accuracy of optimization of the technical parameters of laser emitters, the article presents a mathematical modeling of the preparatory stage of embryo defrosting. The presented mathematical models and methods of their implementation are necessary to improve the quality of embryo defrosting. Improvement of the methodology for solving applied biotechnological problems will certainly lead to the complication of mathematical models, but it will increase the accuracy of calculation and optimization of technical parameters of the biotechnological process of laser embryo division.

Heat transfer characteristics of subsea long-distance pipeline subject to direct electrical heating

For deep-water pipelines with long tie-back distances, high external pressure, and low ambient temperature, it is a key issue to ensure flow safety by controlling the generation of hydrates and waxes through electric heating technology. In-detail analysis of the heat transfer performance of electric heating pipelines is of great engineering significance for developing the safety of pipeline flow in offshore oil-gas fields. This paper proposed an electrical heating analytical method applicable to long-distance submarine pipelines. Based on the thermal equilibrium theory and the differential equation of heat conduction, a mathematical model of heat-fluid coupling for electric heating pipes is established, involving the theoretical model of the steady-state distribution of the electric heating in the axial and radial directions. Then, the definition of the average temperature of the pipeline is introduced and combined with the two-point Hermite approximation interpolation method, and an improved lumped parameter model of the transient heat transfer of the electric heating pipeline is established. Meanwhile, the finite difference method is applied to solve the ordinary differential equation of the pipeline and the energy equation of the internal fluid. The steady-state calculation result is utilized as the initial value of the transient heat transfer process, and the model is applied to analyze the influence variation on the transient temperature response characteristics of the pipeline under different working conditions, finally determining the prediction formula of the average temperature of the fluid under normal transportation and shutdown conditions.

Analysis of Varying Temperature Regimes in a Conductive Strip during Induction Heating under a Quasi-Steady Electromagnetic Field

Transition processes in a steel conductive strip are analyzed during its induction heating under a quasi-steady electromagnetic field. In particular, the temperature field in the strip is studied. A method of solving corresponding initial boundary problems in a two-dimensional mathematical model for differential equations of electrodynamics and heat conduction is developed. The Joule heat and the temperature are determined with a high level of accuracy. The defining functions are the temperature and component of the magnetic field intensity vector tangent to the bases and end planes of the strip. To find them, we use cubic approximation of the defining functions’ distribution along the thickness coordinate. The original two-dimensional initial boundary value problems for the defining functions are reduced to one-dimensional initial boundary value problems on their integral characteristics. General solutions for these problems are obtained using the finite integral transformation by the transverse variable and the Laplace transform of the integral by time. Integral characteristics’ expressions are represented as convolutions for functions that describe homogeneous solutions of one-dimensional initial boundary value problems and limiting values of defining functions on the bases and end planes of the strip. The change of temperature under a varying regime in the dimensionless Fourier time and temperature distribution over the strip cross-section in a steady state depending on the parameters of induction heating and the Biot number are numerically analyzed. Varying and constant temperature regimes of the strip under conditions of the near-surface and continuous induction heating are studied.

Physical and mathematical model of friction welding

The physical and mathematical model of the friction welding process in the form of a system of differential nonlinear heat conduction equation and a variational problem to determine the pressure distribution and shortening of the rods is proposed. The computer model of the friction welding process of solid cross-section rods on its basis is created. Computer simulation reproduces the main effects observed in the friction welding process: equalization of heat generation intensity in the joint plane; stabilization of the joint temperature when the plastic state of the metal is reached; characteristic sequence of increase, decrease and stabilization of torque during welding.

Subsurface temperature change attributed to climate change at the northern latitude site of Kapuskasing, Canada

Subsurface temperatures have been measured in different regions of the world, usually near the surface up to a depth of about a hundred meters. In this work a forward model calculation for a Northern Hemisphere soil temperature site at Kapuskasing, Canada, is presented, employing the solution of the differential equation of heat conduction through a semi-infinite homogeneous solid, subject to surface boundary conditions determined by surface air temperature. In this way, a detailed analysis is made of the subsurface temperature as a function of ground depth and for the time interval ranging from 1970 to the future (including the next century), for different scenarios of climate change. From these results, it was possible to determine the following characteristic quantities: (a) the depth where the surface perturbation (practically) finishes (in the range of about 180-200 m); (b) the depth where the subsurface temperature changes its slope from negative to positive; (c) the temperature change at the surface for the years where data exist; (d) the thermal gradient at steady state in the starting year (1880); (e) the temperature differences extrapolated at surface and at a 20 m depth, this last value corresponding to the depth at which seasonal and diurnal temperature variations are negligible; (f) the heat flow at surface to the inner part of the soil attributed to climate change, and (g) the temperature changes at surface for the 100 years interval (1980-2080) and mainly for the next century (2080-2180), for each site and for each IPCC Representative Concentration Pathway (RCP) scenario. As an example, the impact of the change in mean annual soil temperature due to global warming in near-surface geothermal energy is described.

Heat transfer process and reaction mechanism of AP/HTPB propellant under transient high temperature

For the safe use of solid rocket motors, it is crucial to investigate the thermal effect of AP/HTPB propellants. Transient high temperature may cause AP/HTPB propellants to burn or explode, resulting in serious damage. The fundamental equations that describe the thermal effect of AP/HTPB propellants are the heat conduction differential equation and Arrhenius law. In this paper, the thermal safety of AP/HTPB at transient high temperature is obtained by solving the heat conduction differential equation. The results show that the chemical source term S(T) is the main control factor of thermal runaway. When the action duration of the external high-temperature is short (ms-s magnitude), the first explosion occurs closest to the high-temperature source. When the action duration of the external high-temperature action is long (min-h magnitude), the first explosion occurs inside the propellant. The higher the high-temperature intensity of external action, the shorter the explosion delay time, but the explosion temperature does not change accordingly. The critical temperature of thermal runaway is 458 K under the condition of no shell and a high-temperature action duration of 2 h. At high-temperature intensities of 458 K, 459 K, and 500 K, the propellant explosion temperatures are 533 K, 539 K, and 540 K, and the explosion delay time is 1592s, 1102s, and 160 s, respectively. The critical duration of thermal runaway is 430 ms, 650 ms, 1350 ms, and 9820 ms, and the stable explosion delay time is 450 ms, 680 ms, 1400 ms, and 9920 ms, respectively, under the conditions of 2000 K, 1500 K, 1000 K, and 600 K without the shell. The propellant explosion temperature remains constant as the shell thickness grows while the explosion delay time gradually increases. When the propellant is thick enough, the temperature distribution of the system remains constant. The results of this study may be useful as a guide for assessing the thermal safety of AP/HTPB propellant.

Rapid and nondestructive testing for simultaneous measurement of thermal conductivity and thermal diffusivity of flat materials based on thermography

This study presents a novel experimental method that simultaneously supplies the thermal conductivity and thermal diffusivity of flat materials, which solves the limitations of traditional methods that have strict demands for thermal excitation, boundary, and size of samples. In this method, IR thermal imaging is used to collect data from the material surface temperature field caused by a laser excitation, and the differential terms in the basic heat-conduction differential equation are estimated by curve fitting. The differential equation is so transformed in an algebraic equation, which allows to simultaneously determine the thermal conductivity and thermal diffusivity. The measurement experiment does not require a strict control of the boundary or initial conditions, or thermal excitation modalities. The hypothesis this method is based on is that the spatial derivative of temperature in the Z-direction is linear with the temperature difference between the material surface and environment. Using this constraint the optimal operation conditions can be identified. The influence of operation conditions, including the heat transfer distance, excitation duration, material thickness, and number of sampling points on the measured surface, together with the results, were analysed by numerical simulation. The optimal operation conditions when testing two 304 stainless steel samples were found from this simulation analysis. Results of 304 stainless steel samples 1 and 2 mm thick show a deviation between the reference and measured values of the two thermal properties within 4.2 %, and repeatability within 5.5 %. Therefore, this method can realise rapid, nondestructive, and simultaneous measurements of thermal conductivity and thermal diffusivity.

Complex Three-Dimensional Mathematical Model of the Ignition of a Coniferous Tree via a Cloud-to-Ground Lightning Discharge: Electrophysical, Thermophysical and Physico-Chemical Processes

Thunderstorms are the main natural source of forest fires. The ignition mechanism of trees begins with the impact of cloud-to-ground lightning discharge. A common drawback of all predicting systems is that they ignore the physical mechanism of forest fire as a result of thunderstorm activity. The purpose of this article is to develop a physically based mathematical model for the ignition of a coniferous tree via cloud-to-ground lightning discharge, taking into account thermophysical, electrophysical, and physicochemical processes. The novelty of the article is explained by the development of an improved mathematical model for the ignition of coniferous trees via cloud-to-ground lightning discharge, taking into account the processes of soot formation caused by the thermal decomposition phase of dry organic matter. Mathematically, the process of tree ignition is described by a system of non-stationary nonlinear differential equations of heat conduction and diffusion. In this research, a locally one-dimensional method is used to solve three-dimensional partial differential equations. The finite difference method is used to solve one-dimensional heat conduction and diffusion equations. Difference analogues of the equations are solved using the marching method. To resolve nonlinearity, a simple iteration method is used. Temperature distributions in a structurally inhomogeneous trunk of a coniferous tree, as well as distributions of volume fractions of phases and concentrations of gas mixture components, are obtained. The conditions for tree trunk ignition under conditions of thunderstorm activity are determined. As a result, a complex three-dimensional mathematical model is developed, which makes it possible to identify the conditions for the ignition of a coniferous tree trunk via cloud-to-ground lightning discharge.

MODELS OF FIRE EXTINGUISHING WHEN FLAMMABLE LIQUID COMBUSTION

The process of extinguishing a class B fire with sprayed water is described by the differential equation of heat conduction with boundary conditions of the second kind. The solution of this differential equation in a dimensionless form allows to make a transition to the operative form of representation of the mathematical model of the quenching process - to the transfer function. The peculiarity of such a fire transfer function when extinguishing it with sprayed water is its irrationality, which causes difficulties in its use. One of the ways out of this situation is to switch to an equivalent representation of an irrational function of a complex variable in the form of a small-rational function of such a variable. Such a transition is carried out with the help of the minimax approximation using the Remez algorithm. Determination of the maximum order of the characteristic polynomial of the fire transfer function is carried out using the Hurwitz stability criterion. It is shown that for the transfer function of class B fire, the order of the Hurwitz polynomials does not exceed four, the approximation error does not exceed 3.6% using the method of undetermined coefficients, the transfer functions of the fire when they are extinguished with sprayed water are constructed in the form of a superposition of the transfer functions of aperiodic links. Numerical values of the parameters of such transfer functions were obtained. For the case when the intensity of the sprayed water supply is described by the Heaviside function, expressions describing the temperature of the surface of the burning liquid were constructed. It is noted that these expressions correspond to the expressions that describe the dynamic properties of class B fire in the time domain. It is shown that the maximum value of the relative error when using such transfer functions does not exceed 4.0%, if the general transfer function represents the superposition of the transfer functions of two aperiodic links. It is noted that when using the Padé approximation, the transfer function in the form of the transfer functions of eight aperiodic links corresponds to this value of the divergence error. Keywords: flammable liquid, fire extinguishing, sprayed water, transfer function.

Effect of Transverse Base Width Restraint on the Cracking Behavior of Massive Concrete

The effect of considering the third dimension in mass concrete members on its cracking behavior is investigated in this study. The investigation includes thermal and structural analyses of mass concrete structures. From thermal analysis, the actual temperature distribution throughout the mass concrete body was obtained due to the generation of heat as a result of cement hydration in addition to the ambient circumstances. This was performed via solving the differential equations of heat conduction and convection using the finite element method. The finite element method was also implemented in the structural analysis adopting the concept of initial strain problem. Drying shrinkage volume changes were calculated using the procedure suggested by ACI Committee 209 and inverted to equivalent temperature differences to be added algebraically to the temperature differences obtained from thermal analysis. Willam-Warnke model with five strength parameters is used in modeling of concrete material in which cracking and crushing behavior of concrete can be included. The ANSYS program was employed in a modified manner to perform the above analyses. A thick concrete slab of 1.5m in thickness and 10m in length was analyzed for different widths 2, 4, 8, and 10m to produce different aspect ratios (B/L) of 0.2, 0.4, 0.8, and 1.0 respectively. The results of the analyses show an increase in cracking tendency of mass concrete member as the aspect ratio of the same member is increased due to the effect of transverse base restraint. Accordingly, such effect cannot be ignored in the analysis of base restrained mass concrete structures subjected to temperature and drying shrinkage volume changes.

Analytical Solution of Transient Heat Conduction through a Hollow Cylindrical Thermal Insulation Material of a Temperature Dependant Thermal Conductivity

The one-dimensional, cylindrical coordinate, non-linear partial differential equation of transient heat conduction through a hollow cylindrical thermal insulation material of a thermal conductivity temperature dependent property proposed by an available empirical function is solved analytically using Kirchhoff’s transformation. It is assumed that this insulating material is initially at a uniform temperature. Then, it is suddenly subjected at its inner radius with a step change in temperature. Four thermal insulation materials were selected. An identical analytical solution was achieved when comparing the results of temperature distribution with available analytical solution for the same four case studies that assume a constant thermal conductivity. It is found that the characteristics of the thermal insulation material and the pressure value between its particles have a major effect on the rate of heat transfer and temperature profile.

Semi-analytical solution for interfacial debonding of high-speed railway ballastless track under thermal loading using a quasi-dynamic method

Interfacial debonding of multi-layered ballastless track systems has proved to be one of the most common diseases in the high-speed railway during long-term operations. This paper focuses on the theoretical modeling of track temperature field and consequent bond-slip behavior between track layers under thermal loading. By establishing heat conduction differential equations, whose boundary conditions are combined with complicated geographical and meteorological factors, the temperature field of a multi-layered track structure is obtained via the integral transform technique. The reliability of the heat conduction model is well-validated by an on-site experiment in a high-speed railway line. Then, we originally extend an explicit integration algorithm for dynamics analysis to solve nonlinear statics problems with small deformation by proposing a quasi-dynamic method (QDM). Equilibrium equations of a track slab subjected to quasi-dynamic equivalent thermal loads and mixed-mode interfacial cohesive forces are constructed based on the Mindlin plate in-plane and flexural vibration equations. The application of the QDM to solve interface responses is verified by comparing results obtained from finite element (FE) software. Finally, numerical discussions are conducted to reveal the nonlinear distribution characteristics of temperature field, evolution process of the interface damage and failure modes under time-varying thermal loads, which provide a better understanding of the interfacial debonding phenomenon encountered in ballastless track systems.

Fast response characteristics of fiber Bragg grating temperature sensors and explosion temperature measurement tests

Transient temperature measurement technology with fast response, harsh measurement environment, and high technical challenges has attracted research interest in the field of temperature measurement. Based on the differential equation of heat conduction at the interface of an optical fiber, the dynamic-response-time formula of the optical fiber was derived, the dynamic response capability of the optical fiber with different diameters was analyzed, and the transient thermal response simulation of the optical fiber was conducted. The theoretical dynamic test response times of optical fibers with diameters of 80, 125, and 145 µm were 30.21, 60.78, and 85.97 ms, respectively. Fast-response fiber Bragg grating (FR-FBG) temperature sensors were designed and manufactured by femtosecond-fabricated FBGs with diameters of 80 and 125 µm. The average response times of the 80 and 125 µm-FR-FBG temperature sensors were 33.2 and 65.3 ms, and the errors between the data and theoretical values were 9.89 % and 7.44 %, respectively. From explosion experiments, it is concluded that the femtosecond grating can be applied to the monitoring of rapid temperature response during explosion, and the response time is related to the diameter of the grating.

Coupled Thermoelasticity of Hemitropic Media. Pseudotensor Formulation

The present paper deals with the problem of deriving the constitutive equations for the micropolar thermoelastic continuum GN-I in terms of the standard pseudotensor formalism. In most cases, the pseudotensor approach is justified in modeling hemitropic micropolar solids, the thermomechanical properties of which are sensitive to mirror reflections of three-dimensional space. The requisite equations and notions from the theory of pseudotensors are revisited. General thermodynamic approaches are used, entropy and energy balance equations are discussed. The weights of the main thermomechanical pseudotensors are established. In a linear approximation, the constitutive equations of the hemitropic micropolar thermoelastic continuum (GN-I) of the first type are derived. A coupled system of differential equations of heat conduction and dynamic equations of the micropolar thermoelastic continuum GN-I is obtained.

Research of Tribological and Thermodynamic Parameters of the WC-Cu Braking System of the Experimental Modular Electric Vehicle

Abstract The authors of this manuscript present the development of a braking system with friction material base WC-Cu coating for the electric vehicle. This manuscript follows on from the original development of an AGV multi-disc braking system and an experimental investigation of the friction factor of WC-Cu coatings. In addition to developing the mechanical elements and construction of the electric vehicle, the tribological parameters of three samples of the steel substrate, the C45 with WC-Cu coating, were investigated in the tribological laboratory. A metallic coating of the WC-Cu base was applied on the C45 steel substrate using electro-spark deposition coating technology. The experiment used three samples with different percentage ratios of chemical elements in the coating structure. The tribometer working on a “Ball on Plate” principle was an investigation of the friction factor of all samples during the experiment. Subsequently, the surface of the samples was modified structure WC-Cu with laser technology. The microhardness of modified and unmodified coatings according to the Vickers methodology was investigated in the next stage. At the end of the experimental investigation, a braking simulation was created in the programming environment of the Matlab® software, considering all driving resistances. The researchers also focused on the simulation of heat conduction during braking for some considered driving modes with braking on a level and with a 20% slope roadway. The simulation of heat flow was carried out in the Matlab® programming environment using the Fourier partial differential equation for non-stationary heat conduction.

Analytical Solution of Transient Heat Conduction through a Hollow Spherical Thermal Insulation Material of a Temperature Dependant Thermal Conductivity

The one-dimensional, spherical coordinate, non-linear partial differential equation of transient heat conduction through a hollow spherical thermal insulation material of a thermal conductivity temperature dependent property proposed by an available empirical function is solved analytically using Kirchhoff’s transformation. It is assumed that this insulating material is initially at a uniform temperature. Then, it is suddenly subjected at its inner radius with a step change in temperature. Four thermal insulation materials were selected. An identical analytical solution was achieved when comparing the results of temperature distribution with available analytical solution for the same four case studies that assume a constant thermal conductivity. It is found that the characteristics of the thermal insulation material and the pressure value between its particles have a major effect on the rate of heat transfer and temperature profile.

Heat and mass transfer in the processes of drying thin natural leathers pasted on smooth surfaces

The results of the study of drying thin natural leather pasted on a smooth surface are presented. The results of solving the differential equation of non-stationary heat conduction with constant thermophysical coefficients are used to determine the average temperature during the period of decreasing drying rate. The calculated values of temperatures for moisture-proof and moisture-free skin surfaces are given. Methods for simplifying solutions of nonlinear equations with the aim of linearizing these equations for approximating solutions with variable transport coefficients are considered. The use of the method of piecewise stepwise approximation of the transfer coefficients with constant values of these coefficients over short time intervals showed that for low-intensity drying processes, the solutions of the linear heat equation are implemented quite accurately, confirming the regularities obtained empirically. The calculation of the duration of drying according to the method of B. S. Sazhin is given. The results of the processing of experimental curves for drying leathers pasted are presented. The reliability of the obtained equations is verified and the experiment is compared with the calculated values using the formulas. The obtained approximate analytical solutions with reliable values of the transfer coefficients, confirmed by experimentally established regularities, are of practical importance. Together with experimental methods, analytical methods make it possible to establish optimal drying regimes and more accurately generalize experimental data.

Combined Thermal and Wear Analysis of Linear Rolling Guide Subjected to Rigid–Flexible Coupling Motion Stage

The local heat and wear generated due to friction in a precision motion stage in electronic packaging equipment have a significant impact on its positioning precision. In this paper, we studied the temperature changes over time and the wear of a linear rolling guide (LRG) in a rigid–flexible coupling motion stage (RFCMS), both analytically and experimentally, and we proposed an evaluation method for LRG wear in an RFCMS. According to Fourier’s law and the law of conservation of energy, the differential equation of heat conduction for the LRG and the thermal boundary conditions were established. Steady-state and transient thermal simulations were carried out using ANSYS Workbench to predict the temperature increase in the LRG due to friction. Finally, a test apparatus was built to demonstrate that an RFCMS reduced the operating temperature of the LRG, which also reduced the wear on the contact surface. Through response surface methodology, the levels of the influence of different flexure hinge thicknesses, strokes, velocities, and accelerations on the temperature change rate (TCR) of the LRG were obtained, as well as the approximate regression equation of four variables of the TCR. This provided a new research method for precision maintenance, life design, and operational parameter selection for high-precision motion stages.