Cyclic shear behavior of en-echelon joints under constant normal stiffness conditions

To reveal the mechanism of shear failure of en-echelon joints under cyclic loading,such as during earthquakes,we conducted a series of cyclic shear tests of en-echelon joints under constant normal stiffness(CNS)conditions.We analyzed the evolution of shear stress,normal stress,stress path,dilatancy characteristics,and friction coefficient and revealed the failure mechanisms of en-echelon joints at different angles.The results show that the cyclic shear behavior of the en-echelon joints is closely related to the joint angle,with the shear strength at a positive angle exceeding that at a negative angle during shear cycles.As the number of cycles increases,the shear strength decreases rapidly,and the difference between the varying angles gradually decreases.Dilation occurs in the early shear cycles(1 and 2),while contraction is the main feature in later cycles(310).The friction coefficient decreases with the number of cycles and exhibits a more significant sensitivity to joint angles than shear cycles.The joint angle determines the asperities on the rupture surfaces and the block size,and thus determines the subsequent shear failure mode(block crushing and asperity degradation).At positive angles,block size is more greater and asperities on the rupture surface are smaller than at nonpositive angles.Therefore,the cyclic shear behavior is controlled by block crushing at positive angles and asperity degradation at negative angles.

A low-frequency pure metal metamaterial absorber with continuously tunable stiffness

To address the incompatibility between high environmental adaptability and deep subwavelength characteristics in conventional local resonance metamaterials,and overcome the deficiencies in the stability of existing active control techniques for band gaps,this paper proposes a design method of pure metal vibration damping metamaterial with continuously tunable stiffness for wideband elastic wave absorption.We design a dual-helix narrow-slit pure metal metamaterial unit,which possesses the triple advantage of high spatial compactness,low stiffness characteristics,and high structural stability,enabling the opening of elastic flexural band gaps in the low-frequency range.Similar to the principle of a sliding rheostat,the introduction of continuously sliding plug-ins into the helical slits enables the continuous variation of the stiffness of the metamaterial unit,achieving a continuously tunable band gap effect.This successfully extends the effective band gap by more than ten times.The experimental results indicate that this metamaterial unit can be used as an additional vibration absorber to absorb the low-frequency vibration energy effectively.Furthermore,it advances the metamaterial absorbers from a purely passive narrowband design to a wideband tunable one.The pure metal double-helix metamaterials retain the subwavelength properties of metamaterials and are suitable for deployment in harsh environments.Simultaneously,by adjusting its stiffness,it substantially broadens the effective band gap range,presenting promising potential applications in various mechanical equipment operating under adverse conditions.

Nonlinear Characteristic Analysis of Gas-Interconnected Quasi-Zero Stiffness Pneumatic Suspension System:A Theoretical and Experimental Study

Because of significantly changed load and complex and variable driving road conditions of commercial vehicles,pneumatic suspension with lower natural frequencies is widely used in commercial vehicle suspension system.How ever,traditional pneumatic suspension system is hardly to respond the greatly changed load of commercial vehicles To address this issue,a new Gas-Interconnected Quasi-Zero Stiffness Pneumatic Suspension(GIQZSPS)is presented in this paper to improve the vibration isolation performance of commercial vehicle suspension systems under frequent load changes.This new structure adds negative stiffness air chambers on traditional pneumatic suspension to reduce the natural frequency of the suspension.It can adapt to different loads and road conditions by adjusting the solenoid valves between the negative stiffness air chambers.Firstly,a nonlinear mechanical model including the dimensionless stiffness characteristic and interconnected pipeline model is derived for GIQZSPS system.By the nonlinear mechanical model of GIQZSPS system,the force transmissibility rate is chosen as the evaluation index to analyze characteristics.Furthermore,a testing bench simulating 1/4 GIQZSPS system is designed,and the testing analysis of the model validation and isolating performance is carried out.The results show that compared to traditional pneumatic suspension,the GIQZSPS designed in the article has a lower natural frequency.And the system can achieve better vibration isolation performance under different load states by switching the solenoid valves between air chambers.

Evaluation on Configuration Stiffness of Overconstrained 2R1T Parallel Mechanisms

Currently,two rotations and one translation(2R1T)three-degree-of-freedom(DOF)parallel mechanisms(PMs)are widely applied in five-DOF hybrid machining robots.However,there is a lack of an effective method to evaluate the configuration stiffness of mechanisms during the mechanism design stage.It is a challenge to select appropriate 2R1T PMs with excellent stiffness performance during the design stage.Considering the operational status of 2R1T PMs,the bending and torsional stiffness are considered as indices to evaluate PMs'configuration stiffness.Subsequently,a specific method is proposed to calculate these stiffness indices.Initially,the various types of structural and driving stiffness for each branch are assessed and their specific values defined.Subsequently,a rigid-flexible coupled force model for the over-constrained 2R1T PM is established,and the proposed evaluation method is used to analyze the configuration stiffness of the five 2R1T PMs in the entire workspace.Finally,the driving force and constraint force of each branch in the whole working space are calculated to further elucidate the stiffness evaluating results by using the proposed method above.The obtained results demonstrate that the bending and torsional stiffness of the 2RPU/UPR/RPR mechanism along the x and y-directions are larger than the other four mechanisms.

A vibration isolator with a controllable quasi-zero stiffness region based on nonlinear force design

To achieve stability optimization in low-frequency vibration control for precision instruments,this paper presents a quasi-zero stiffness(QZS)vibration isolator with adjustable nonlinear stiffness.Additionally,the stress-magnetism coupling model is established through meticulous theoretical derivation.The controllable QZS interval is constructed via parameter design and magnetic control,effectively segregating the high static stiffness bearing section from the QZS vibration isolation section.Furthermore,a displacement control scheme utilizing a magnetic force is proposed to regulate entry into the QZS working range for the vibration isolation platform.Experimental results demonstrate that the operation within this QZS region reduces the peak-to-peak acceleration signal by approximately 66.7%compared with the operation outside this region,thereby significantly improving the low frequency performance of the QZS vibration isolator.

Stiffness-tunable biomaterials provide a good extracellular matrix environment for axon growth and regeneration

Neuronal growth, extension, branching, and formation of neural networks are markedly influenced by the extracellular matrix—a complex network composed of proteins and carbohydrates secreted by cells. In addition to providing physical support for cells, the extracellular matrix also conveys critical mechanical stiffness cues. During the development of the nervous system, extracellular matrix stiffness plays a central role in guiding neuronal growth, particularly in the context of axonal extension, which is crucial for the formation of neural networks. In neural tissue engineering, manipulation of biomaterial stiffness is a promising strategy to provide a permissive environment for the repair and regeneration of injured nervous tissue. Recent research has fine-tuned synthetic biomaterials to fabricate scaffolds that closely replicate the stiffness profiles observed in the nervous system. In this review, we highlight the molecular mechanisms by which extracellular matrix stiffness regulates axonal growth and regeneration. We highlight the progress made in the development of stiffness-tunable biomaterials to emulate in vivo extracellular matrix environments, with an emphasis on their application in neural repair and regeneration, along with a discussion of the current limitations and future prospects. The exploration and optimization of the stiffness-tunable biomaterials has the potential to markedly advance the development of neural tissue engineering.

Iterative Subregion Correction Preconditioners with Adaptive Tolerance for Problems with Geometrically Localized Stiffness

We present a class of preconditioners for the linear systems resulting from a finite element or discontinuous Galerkin discretizations of advection-dominated problems.These preconditioners are designed to treat the case of geometrically localized stiffness,where the convergence rates of iterative methods are degraded in a localized subregion of the mesh.Slower convergence may be caused by a number of factors,including the mesh size,anisotropy,highly variable coefficients,and more challenging physics.The approach taken in this work is to correct well-known preconditioners such as the block Jacobi and the block incomplete LU(ILU)with an adaptive inner subregion iteration.The goal of these preconditioners is to reduce the number of costly global iterations by accelerating the convergence in the stiff region by iterating on the less expensive reduced problem.The tolerance for the inner iteration is adaptively chosen to minimize subregion-local work while guaranteeing global convergence rates.We present analysis showing that the convergence of these preconditioners,even when combined with an adaptively selected tolerance,is independent of discretization parameters(e.g.,the mesh size and diffusion coefficient)in the subregion.We demonstrate significant performance improvements over black-box preconditioners when applied to several model convection-diffusion problems.Finally,we present performance results of several variations of iterative subregion correction preconditioners applied to the Reynolds number 2.25×10^(6)fluid flow over the NACA 0012 airfoil,as well as massively separated flow at 30°angle of attack.

Multi-layer quasi-zero-stiffness meta-structure for high-efficiency vibration isolation at low frequency

An easily stackable multi-layer quasi-zero-stiffness(ML-QZS)meta-structure is proposed to achieve highly efficient vibration isolation performance at low frequency.First,the distributed shape optimization method is used to design the unit cel,i.e.,the single-layer QZS(SL-QZS)meta-structure.Second,the stiffness feature of the unit cell is investigated and verified through static experiments.Third,the unit cells are stacked one by one along the direction of vibration isolation,and thus the ML-QZS meta-structure is constructed.Fourth,the dynamic modeling of the ML-QZS vibration isolation metastructure is conducted,and the dynamic responses are obtained from the equations of motion,and verified by finite element(FE)simulations.Finally,a prototype of the ML-QZS vibration isolation meta-structure is fabricated by additive manufacturing,and the vibration isolation performance is evaluated experimentally.The results show that the vibration isolation performance substantially enhances when the number of unit cells increases.More importantly,the ML-QZS meta-structure can be easily extended in the direction of vibration isolation when the unit cells are properly stacked.Hence,the ML-FQZS vibration isolation meta-structure should be a fascinating solution for highly efficient vibration isolation performance at low frequency.

Theoretical and experimental investigations on an X-shaped vibration isolator with active controlled variable stiffness

A novel X-shaped variable stiffness vibration isolator(X-VSVI)is proposed.The Runge-Kutta method,harmonic balance method,and wavelet transform spectra are introduced to evaluate the performance of the X-VSVI under various excitations.The layer number,the installation angle of the X-shaped structure,the stiffness,and the active control parameters are systematically analyzed.In addition,a prototype of the X-VSVI is manufactured,and vibration tests are carried out.The results show that the proposed X-VSVI has a superior adaptability to that of a traditional X-shaped mechanism,and shows excellent vibration isolation performance in response to different amplitudes and forms of excitations.Moreover,the vibration isolation efficiency of the device can be improved by appropriate adjustment of parameters.

Performance of composite foundations with different load transfer platforms and substratum stiffness over silty clay

The semi-rigid pile-supported composite foundation is widely used in highway projects due to its effectiveness in increasing the bearing capacity and stability of foundations.It is crucial to understand the stress distribution across the embankment width and the behaviour of unreinforced foundations.Thus,five centrifuge tests were conducted to examine the bearing and deformation behaviours of NPRS(Non-Connected Piled Raft Systems)and GRPS(GeosyntheticReinforced Pile-Supported systems)with varying substratum stiffness,then a comparative analysis was conducted on embankment settlement,pressures underneath the embankments,and axial forces along the piles.The results indicated that greater substratum stiffness correlates with reduced settlement and deformation at various depths.Deformation occurring 5 meters from the embankment toe includes settlement in NPRS and upward movement in GRPS.The potential sliding surface is primarily located within the embankment in NPRS,whereas it may extend through both the embankment and foundation in GRPS.The pile-soil stress ratio and efficiency in NPRS are higher than in GRPS across the embankment.The axial force borne by end-bearing piles is significantly greater than that by floating piles.As the buried depth increases,the axial force in GRPS initially rises then declines,whereas in NPRS,it remains relatively constant within a certain range before decreasing.This study aids in assessing the applicability of composite foundations in complex railway environments and provides a reference for procedural measures under similar conditions.

Negative Stiffness Mechanism on An Asymmetric Wave Energy Converter by Using A Weakly Nonlinear Potential Model

Salter's duck,an asymmetrical wave energy converter(WEC)device,showed high efficiency in extracting energy from 2D regular waves in the past;yet,challenges remain for fluctuating wave conditions.These can potentially be addressed by adopting a negative stiffness mechanism(NSM)in WEC devices to enhance system efficiency,even in highly nonlinear and steep 3D waves.A weakly nonlinear model was developed which incorporated a nonlinear restoring moment and NSM into the linear formulations and was applied to an asymmetric WEC using a time domain potential flow model.The model was initially validated by comparing it with published experimental and numerical computational fluid dynamics results.The current results were in good agreement with the published results.It was found that the energy extraction increased in the range of 6%to 17%during the evaluation of the effectiveness of the NSM in regular waves.Under irregular wave conditions,specifically at the design wave conditions for the selected test site,the energy extraction increased by 2.4%,with annual energy production increments of approximately 0.8MWh.The findings highlight the potential of NSM in enhancing the performance of asymmetric WEC devices,indicating more efficient energy extraction under various wave conditions.

Ilizarov technique for treating elbow stiffness caused by myositis ossificans:A case report

BACKGROUND Myositis ossificans(MO)is a rare disease involving the formation of bone outside the musculoskeletal system.While surgical intervention is the main treatment approach,preventing recurrence and standardized rehabilitation are also crucial.Here,we present a surgical strategy to prevent the recurrence of MO.CASE SUMMARY A 28-year-old female patient was admitted for the first time for a comminuted fracture of the left olecranon.However,incorrect postoperative rehabilitation resulted in the development of elbow joint stiffness with ectopic ossification,causing a loss of normal range of motion.The patient was diagnosed with MO based on physical examination,X-ray findings,and clinical presentation.We devised a surgical strategy to remove MO,followed by fixation with an Ilizarov frame,and implemented a scientifically reasonable rehabilitation plan.The surgery lasted for 3 h with an estimated blood loss of 45 mL.A drainage tube was placed after surgery,and fluid was aspirated through ultrasound-guided puncture.The patient experienced a significant reduction in joint stiffness after surgery.In the final follow-up at 9 mouths,there was evident improvement in the range of motion of the elbow joint,and no other symptoms were reported.CONCLUSION The Ilizarov frame is an advantageous surgical technique for facilitating rehabilitation after MO removal.It offers benefits such as passive recovery,individualized treatment,and prompt recovery.

Systematic Elastostatic Stiffness Model of Over-Constrained Parallel Manipulators Without Additional Constraint Equations

The establishment of an elastostatic stiffness model for over constrained parallel manipulators(PMs),particularly those with over constrained subclosed loops,poses a challenge while ensuring numerical stability.This study addresses this issue by proposing a systematic elastostatic stiffness model based on matrix structural analysis(MSA)and independent displacement coordinates(IDCs)extraction techniques.To begin,the closed-loop PM is transformed into an open-loop PM by eliminating constraints.A subassembly element is then introduced,which considers the flexibility of both rods and joints.This approach helps circumvent the numerical instability typically encountered with traditional constraint equations.The IDCs and analytical constraint equations of nodes constrained by various joints are summarized in the appendix,utilizing multipoint constraint theory and singularity analysis,all unified within a single coordinate frame.Subsequently,the open-loop mechanism is efficiently closed by referencing the constraint equations presented in the appendix,alongside its elastostatic model.The proposed method proves to be both modeling and computationally efficient due to the comprehensive summary of the constraint equations in the Appendix,eliminating the need for additional equations.An example utilizing an over constrained subclosed loops demonstrate the application of the proposed method.In conclusion,the model proposed in this study enriches the theory of elastostatic stiffness modeling of PMs and provides an effective solution for stiffness modeling challenges they present.

Static-to-kinematic modeling and experimental validation of tendon-driven quasi continuum manipulators with nonconstant subsegment stiffness

Continuum robots with high flexibility and compliance have the capability to operate in confined and cluttered environments. To enhance the load capacity while maintaining robot dexterity, we propose a novel non-constant subsegment stiffness structure for tendon-driven quasi continuum robots(TDQCRs) comprising rigid-flexible coupling subsegments.Aiming at real-time control applications, we present a novel static-to-kinematic modeling approach to gain a comprehensive understanding of the TDQCR model. The analytical subsegment-based kinematics for the multisection manipulator is derived based on screw theory and product of exponentials formula, and the static model considering gravity loading,actuation loading, and robot constitutive laws is established. Additionally, the effect of tension attenuation caused by routing channel friction is considered in the robot statics, resulting in improved model accuracy. The root-mean-square error between the outputs of the static model and the experimental system is less than 1.63% of the arm length(0.5 m). By employing the proposed static model, a mapping of bending angles between the configuration space and the subsegment space is established. Furthermore, motion control experiments are conducted on our TDQCR system, and the results demonstrate the effectiveness of the static-to-kinematic model.

Liver stiffness in hepatocellular carcinoma and chronic hepatitis patients:Hepatitis B virus infection and transaminases should be considered

BACKGROUND Liver condition is a crucial prognostic factor for patients with hepatocellular carcinoma(HCC),but a convenient and comprehensive method to assess liver condition is lacking.Liver stiffness(LS)measured by two-dimensional shear wave elastography may help in assessing liver fibrosis and liver condition.Chronic hepatitis B(CHB)is an important risk factor for HCC progression,but LS was found to be less reliable in assessing liver fibrosis following hepatitis viral eradication.We hypothesize that the status of hepatitis virus infection would affect the accuracy of LS in assessing the liver condition.AIM To test the feasibility and impact factors of using LS to assess liver condition in patients with HCC and CHB.METHODS A total of 284 patients were retrospectively recruited and classified into two groups on the basis of serum CHB virus hepatitis B virus(HBV)-DNA levels[HBV-DNA≥100.00 IU/mL as Pos group(n=200)and<100.00 IU/mL as Neg group(n=84)].Correlation analyses and receiver operating characteristic analyses were conducted to evaluate the relationship between LS and liver condition.RESULTS A significant correlation was found between LS and most of the parameters considered to have the ability to evaluate liver condition(P<0.05).When alanine aminotransferase(ALT)concentrations were normal(≤40 U/L),LS was correlated with liver condition indices(P<0.05),but the optimal cutoff of LS to identify a Child-Pugh score of 5 was higher in the Neg group(9.30 kPa)than the Pos group(7.40 kPa).When ALT levels were elevated(>40 U/L),the correlations between LS and liver condition indices were not significant(P>0.05).CONCLUSION LS was significantly correlated with most liver condition indices in patients with CHB and HCC.However,these correlations varied according to differences in HBV-DNA and transaminase concentrations.

Shi-pi-xiao-ji formula suppresses hepatocellular carcinoma by reducing cellular stiffness through upregulation of acetyl-coA acetyltransferase 1

BACKGROUND Previous studies have shown that the Shi-pi-xiao-ji(SPXJ)herbal decoction formula is effective in suppressing hepatocellular carcinoma(HCC),but the underlying mechanisms are not known.Therefore,this study investigated whether the antitumor effects of the SPXJ formula in treating HCC were mediated by acetyl-coA acetyltransferase 1(ACAT1)-regulated cellular stiffness.Through a series of experiments,we concluded that SPXJ inhibits the progression of HCC by upregulating the expression level of ACAT1,lowering the level of cholesterol in the cell membrane,and altering the cellular stiffness,which provides a new idea for the research of traditional Chinese medicine against HCC.AIM To investigate the anti-tumor effects of the SPXJ formula on the malignant progression of HCC.METHODS HCC cells were cultured in vitro with SPXJ-containing serum prepared by injecting SPXJ formula into wild-type mice.The apoptotic rate and proliferative,invasive,and migratory abilities of control and SPXJ-treated HCC cells were compared.Atomic force microscopy was used to determine the cell surface morphology and the Young’s modulus values of the control and SPXJ-treated HCC cells.Plasma membrane cholesterol levels in HCC cells were detected using the Amplex Red cholesterol detection kit.ACAT1 protein levels were estimated using western blotting.RESULTS Compared with the vehicle group,SPXJ serum considerably reduced proliferation of HCC cells,increased stiffness and apoptosis of HCC cells,inhibited migration and invasion of HCC cells,decreased plasma membrane cholesterol levels,and upregulated ACAT1 protein levels.However,treatment of HCC cells with the water-soluble cholesterol promoted proliferation,migration,and invasion of HCC cells as well as decreased cell stiffness and plasma membrane cholesterol levels,but did not alter the apoptotic rate and ACAT1 protein expression levels compared with the vehicle control.CONCLUSION SPXJ formula inhibited proliferation,invasion,and migration of HCC cells by decreasing plasma membrane cholesterol levels and altering cellular stiffness through upregulation of ACAT1 protein expression.

Stiffness Characteristics of a Basic Nonlinear Air Spring Model

This study predicts the characteristics of a compressible polytropic air spring model. A second-order nonlinear autonomous air spring model is presented. The proposed model is based on the assumption that polytropic processes occur. Isothermal and isentropic compression and expansion of the air within the spring chambers are the two scenarios that are taken into consideration. In these situations, the air inside the spring chambers compresses and expands, resulting in nonlinear spring restoring forces. The MATLAB/Simulink software environment is used to build a numerical simulation model for the dynamic behavior of the air spring. To quantify the values of the stiffnesses of the proposed models, a numerical solution is run over time for various values of the design parameters. The isentropic process case has a higher dynamic air spring stiffness than the isothermal process case, according to the results. The size of the air spring chamber and the area of the air spring piston influence the air spring stiffness in both situations. It is demonstrated that the stiffness of the air spring increases linearly with increasing piston area and decreases nonlinearly with increasing air chamber length. As long as the ratio of the vibration’s amplitude to the air spring’s chamber length is small, there is good agreement in both scenarios between the linearized model and the full nonlinear model. This implies that linear modeling is a reasonable approximation of the complete nonlinear model in this particular scenario.

Pilot study on the effect of flavonoids on arterial stiffness and oxidative stress in chronic kidney disease

BACKGROUND Flavonoids,the main class of polyphenols,exhibit antioxidant and antihypertensive properties.AIM To prospectively investigate the impact of flavonoids on arterial stiffness in patients with chronic kidney disease(CKD)stagesⅠ-Ⅳ.METHODS In this prospective,single-arm study,CKD patients with arterial hypertension and diabetes mellitus were enrolled.Baseline demographic,clinical,and laboratory variables were recorded.Patients received daily treatment with a phenol-rich dietary supplement for 3 months.Blood pressure,arterial stiffness(carotidfemoral pulse wave velocity,central pulse pressure),and oxidative stress markers(protein carbonyls,total phenolic compound,total antioxidant capacity)were measured at baseline and at study end.RESULTS Sixteen patients(mean age:62.5 years,87.5%male)completed the study.Following intervention,peripheral systolic blood pressure decreased significantly by 14 mmHg(P<0.001).Carotid-femoral pulse wave velocity decreased from 8.9 m/s(baseline)to 8.2 m/s(study end)(P<0.001),and central pulse pressure improved from 59 mmHg to 48 mmHg(P=0.003).Flavonoids also reduced oxidative stress markers including protein carbonyls(P<0.001),total phenolic compound(P=0.001),and total antioxidant capacity(P=0.013).CONCLUSION Flavonoid supplementation in CKD patients shows promise in improving blood pressure,arterial stiffness,and oxidative stress markers.

Simulation Study on Dynamic Stiffness of Kick-reaction Link Actuator

Under the same aerodynamic load,the load transmitted by the kick-reaction link actuator to the aircraft structure is only 1/5 to 1/3 of that of the point-to-point actuator,which can significantly reduce the weight of the structure,and is widely used in the main flight control surface of large and medium-sized civil airliners.In order to restrain the flutter of aircraft rudder surface,it is necessary to design the servo stiffness of the kick-reaction link actuator in the development stage,so that it can meet the dynamic stiffness requirement within the frequency range.Taking the actuator of a civil airliner as the research object,the dynamic stiffness modeling of elevator actuator is carried out on MATLAB platform,and the dynamic stiffness test is carried out to verify the correctness of dynamic stiffness modeling.The simulation and test results show that the actuator can meet the dynamic stiffness design requirements.

Numerical Study of the Vibrations of Beams with Variable Stiffness under Impulsive or Harmonic Loading

The behavior of beams with variable stiffness subjected to the action of variable loadings (impulse or harmonic) is analyzed in this paper using the successive approximation method. This successive approximation method is a technique for numerical integration of partial differential equations involving both the space and time, with well-known initial conditions on time and boundary conditions on the space. This technique, although having been applied to beams with constant stiffness, is new for the case of beams with variable stiffness, and it aims to use a quadratic parabola (in time) to approximate the solutions of the differential equations of dynamics. The spatial part is studied using the successive approximation method of the partial differential equations obtained, in order to transform them into a system of time-dependent ordinary differential equations. Thus, the integration algorithm using this technique is established and applied to examples of beams with variable stiffness, under variable loading, and with the different cases of supports chosen in the literature. We have thus calculated the cases of beams with constant or variable rigidity with articulated or embedded supports, subjected to the action of an instantaneous impulse and harmonic loads distributed over its entire length. In order to justify the robustness of the successive approximation method considered in this work, an example of an articulated beam with constant stiffness subjected to a distributed harmonic load was calculated analytically, and the results obtained compared to those found numerically for various steps (spatial h and temporal τ ¯ ) of calculus, and the difference between the values obtained by the two methods was small. For example for ( h=1/8 , τ ¯ =1/ 64 ), the difference between these values is 17%.