Sliding Mode Control for Two-Degree-Of-Freedom Fractional Zener Oscillator

Fractional-order derivatives provide a powerful tool for the characterization of mechanical properties of viscoelastic materials. A fractional oscillator refers to mechanical model of viscoelastically damped structures, of which the viscoelastic damping is described by constitutive equations involving fractional-order derivatives. This paper proposes active control of vibration in a two-degree-of-freedom fractional Zener oscillator utilizing sliding mode technique. Firstly, with a state transformation, the fractional differential equations of motion are equivalently transformed into a relatively simple form. Meanwhile, a virtual fractional oscillator is generated, which is further used to analyze the original oscillator. Then, the stored energy in the two fractional derivative terms is derived based on the diffusive model of fractional integrator. Thus, the total mechanical energy in the virtual oscillator is determined as the sum of the kinetic energy, the potential energy and the fractional energy. Furthermore, sliding mode control of vibration in the fractional Zener oscillator is designed, of which the Lyapunov function is chosen as the total mechanical energy. Finally, numerical simulations are conducted to validate the effectiveness of the proposed controllers.

ԼՐԻՎ ՄԵԽԱՆԻԿԱԿԱՆ ԷՆԵՐԳԻԱՅԻ ՓՈՓՈԽՈՒԹՅԱՆ ԹԵՈՐԵՄԻ ԵՎ ՄՈՄԵՆՏՆԵՐԻ ԿԱՆՈՆԻ ՄԵԿ ՀԱՄԱՏԵՂ ԿԻՐԱՌՈՒԹՅԱՆ ՄԱՍԻՆ / ON THE JOINT APPLICATION OF A FULL MECHANICAL ENERGY CHANGE THEOREM AND THE RULES OF MOMENTS

Աշխատանքը նվիրված է ֆիզիկայի դպրոցական դասընթացում լրիվ մեխանիկական էներգիայի փոփոխության թեորեմի և մոմենտների կանոնի հնարավոր համատեղ կիրառությունների վերհանմանըֈ Դիտարկված է կոնկրետ խնդիր, որի շրջանակում վերոգրյալ երկու կանոների կիրառման արդյունքում հնարավոր է դառնում որոշել համասեռ շրջանագծային աղեղի զանգվածի կենտրոնըֈ Խնդրի շրջանակում ստացված արդյունքն ընդհանրացվել է կամայական կենտրոնային անկյունով համասեռ շրջանաձև աղեղի համարֈ: / The work is devoted to identifying possible joint applications of the rule of moments and the theorem for changing the total mechanical energy in a school physics course. A specific problem is considered, within the framework of which, as a result of the application of the above two rules, it becomes possible to determine the center of mass of a uniform circular arc. The result obtained in the framework of the problem is generalized to a homogeneous circular arc with an arbitrary central angle.

A biomechanical study of load carriage by two paired subjects in response to increased load mass

The biomechanics of load carriage has been studied extensively with regards to single individuals, yet not so much with regards to collective transport. We investigated the biomechanics of walking in 10 paired individuals carrying a load that represented 20%, 30%, or 40% of the aggregated body-masses. We computed the energy recovery rate at the center of mass of the system consisting of the two individuals plus the carried load in order to test to what extent the pendulum-like behavior and the economy of the gait were affected. Joint torque was also computed to investigate the intra- and inter-subject strategies occurring in response to this. The ability of the subjects to move the whole system like a pendulum appeared rendered obvious through shortened step length and lowered vertical displacements at the center of mass of the system, while energy recovery rate and total mechanical energy remained constant. In parallel, an asymmetry of joint moment vertical amplitude and coupling among individuals in all pairs suggested the emergence of a leader/follower schema. Beyond the 30% threshold of increased load mass, the constraints at the joint level were balanced among individuals leading to a degraded pendulum-like behavior.

Leg Joint Mechanics When Hopping at Different Frequencies

Although the dynamics of center of mass can be accounted for by a spring-mass model during hopping, less is known about how each leg joint (ie, hip, knee, and ankle) contributes to center of mass dynamics. This work investigated the function of individual leg joints when hopping unilaterally and vertically at 4 frequencies (ie, 1.6, 2.0, 2.4, and 2.8 Hz). The hypotheses are (1) all leg joints maintain the function as torsional springs and increase their stiffness when hopping faster and (2) leg joints are controlled to maintain the mechanical load in the joints or vertical peak accelerations at different body locations when hopping at different frequencies. Results showed that all leg joints behaved as torsional springs during low-frequency hopping (ie, 1.6 Hz). As hopping frequency increased, leg joints changed their functions differently; that is, the hip and knee shifted to strut, and the ankle remained as spring. When hopping fast, the body’s total mechanical energy decreased, and the ankle increased the amount of energy storage and return from 50% to 62%. Leg joints did not maintain a constant load at the joints or vertical peak accelerations at different body locations when hopping at different frequencies.

Evaluation of the Biomechanical Parameters of Human-Wheelchair Systems during Ramp Climbing with the Use of a Manual Wheelchair with Anti-Rollback Devices

Purpose: The main purpose of the research conducted was the analysis of kinematic and biomechanical parameters measured during manual wheelchair ramp-climbing with the use of the anti-rollback system and the comparison of the values tested with the manual wheelchair climbing the same ramp but without any modifications. The paper presents a quantitative assessment relating to the qualitative research of the anti-rollback system performed by another research team. Method and materials: The article presents the measurement results of the wheelchair motion kinematics and the activity of four upper limb muscles for eight subjects climbing a 4.58° ramp. Each subject propelled the wheelchair both with and without the anti-rollback system. The kinematic parameters were measured by means of two incremental encoders with the resolution of 500 impulses per single revolution of the measurement wheel. Whereas, the muscle activity was measured by means of surface electromyography with the use of Noraxon Mini DTS apparatus equipped with four measurement channels. Results: The surface electromyography measurement indicated an increase in the muscle activity for all four muscles, during the use of the anti-rollback system. The increase was: 18.56% for deltoid muscle anterior, 12.37% for deltoid muscle posteriori, 13.0% for triceps brachii, and 15.44% for extensor carpi radialis longus. As far as the kinematics analysis is concerned, a decrease in the measured kinematic parameters was observed in most participants. The medium velocity of the propelling cycle decreased by 26%. The ratio of the generated power and the loss power in a single propelling cycle λ had decreased by 18%. The least decrease was recorded for the measurement of mechanical energy E and the propelling cycle duration time. For the total mechanical energy, the decrease level was 3%, and for the propelling cycle duration it was 1%. The research carried out did not demonstrate any impact of the anti-rollback system use on the push phase share in the entire propelling cycle.

Depressed Cardiac Mechanical Energetic Efficiency: A Contributor to Cardiovascular Risk in Common Metabolic Diseases—From Mechanisms to Clinical Applications

Cardiac mechanical energetic efficiency is the ratio of external work (EW) to the total energy consumption. EW performed by the left ventricle (LV) during a single beat is represented by LV stroke work and may be calculated from the pressure–volume loop area (PVLA), while energy consumption corresponds to myocardial oxygen consumption (MVO2) expressed on a per-beat basis. Classical early human studies estimated total mechanical LV efficiency at 20–30%, whereas the remaining energy is dissipated as heat. Total mechanical efficiency is a joint effect of the efficiency of energy transfer at three sequential stages. The first step, from MVO2 to adenosine triphosphate (ATP), reflects the yield of oxidative phosphorylation (i.e., phosphate-to-oxygen ratio). The second step, from ATP split to pressure–volume area, represents the proportion of the energy liberated during ATP hydrolysis which is converted to total mechanical energy. Total mechanical energy generated per beat—represented by pressure–volume area—consists of EW (corresponding to PVLA) and potential energy, which is needed to develop tension during isovolumic contraction. The efficiency of the third step of energy transfer, i.e., from pressure–volume area to EW, decreases with depressed LV contractility, increased afterload, more concentric LV geometry with diastolic dysfunction and lower LV preload reserve. As practical assessment of LV efficiency poses methodological problems, De Simone et al. proposed a simple surrogate measure of myocardial efficiency, i.e., mechano-energetic efficiency index (MEEi) calculated from LV stroke volume, heart rate and LV mass. In two independent cohorts, including a large group of hypertensive subjects and a population-based cohort (both free of prevalent cardiovascular disease and with preserved ejection fraction), low MEEi independently predicted composite adverse cardiovascular events and incident heart failure. It was hypothesized that the prognostic ability of low MEEi can result from its association with both metabolic and hemodynamic alterations, i.e., metabolic syndrome components, the degree of insulin resistance, concentric LV geometry, LV diastolic and discrete systolic dysfunction. On the one part, an increased reliance of cardiomyocytes on the oxidation of free fatty acids, typical for insulin-resistant states, is associated with both a lower yield of ATP per oxygen molecule and lesser availability of ATP for contraction, which might decrease energetic efficiency of the first and second step of energy transfer from MVO2 to EW. On the other part, concentric LV remodeling and LV dysfunction despite preserved ejection fraction can impair the efficiency of the third energy transfer step. In conclusion, the association of low MEEi with adverse cardiovascular outcome might be related to a multi-step impairment of energy transfer from MVO2 to EW in various clinical settings, including metabolic syndrome, diabetes, hypertension and heart failure. Irrespective of theoretical considerations, MEEi appears an attractive simple tool which couldt improve risk stratification in hypertensive and diabetic patients for primary prevention purposes. Further clinical studies are warranted to estimate the predictive ability of MEEi and its post-treatment changes, especially in patients on novel antidiabetic drugs and subjects with common metabolic diseases and concomitant chronic coronary syndromes, in whom the potential relevance of MEE can be potentiated by myocardial ischemia.

Analysis of the blast furnace operations with a volume of 5000 m3 on tuyeres of different diameters from the positions of full mechanical energies of flows of combined blow and hearth gas

In the commissioning period of the development of pulverized coal injection technology (PCI) on a blast furnace No. 9 with a volume of 5000 m3 of PJSC “ArcelorMittal Kryvyi Rih”, frequent cases of burnout of refrigerators of the cooling system of the shoulders and air tyueres appeared due to the highly developed peripheral gas flow. An attempt to limit the gas flow at the periphery by controlling the distribution of charge materials on the top produced a short-term result. Based on the prevailing ideas, that to reduce the intensity of the peripheral gas flow, it is necessary to increase the speed of the blast and, accordingly, the kinetic energy of the blast flow, flowing out of the air tuyeres of a blast furnace, it was decided to reduce their diameter. As a result of analysis of the operation of the specified blast furnace using the technology of PCI on tuyeres with a diameter of 150 and 140 mm, increased peripheral gas flow with a smaller diameter was established. Based on the results of the analysis, conclusions were made by many researchers and it was shown that with constant kinetic energy of the blast, flowing from the tuyeres of different diameters, the dimensions of the combustion zone are always larger before the tuyeres of a larger diameter. This is explained by the fact that the kinetic energy of the gas flow is only a part of their total mechanical energy. It was shown that to analyze the change in the size of the combustion zones and the depth of penetration of the hearth gas, it is necessary to use the full mechanical energy of the flows of the combined blast on the cut of the tuyere and hearth gas. It was established that the transition to PCI in a blast furnace instead of natural gas, it always causes an increase in the peripheral gas flow. The main reason for this phenomenon is associated with a decrease in the total mechanical energy of blast and hearth gas. It was recommended on a blast furnace with a volume of 5000 m3 with a hearth diameter of 14.7 m and the PCI technology to maintain the total mechanical energy of the blast flow at least 2100–2600 kJ/s, and the full mechanical energy of the hearth gas flow at least 5100–5300 kJ/s.

Lyapunov Stability of a Fractionally Damped Oscillator with Linear (Anti-)Damping

In this paper, we develop a Lyapunov stability framework for fractionally damped mechanical systems. In particular, we study the asymptotic stability of a linear single degree-of-freedom oscillator with viscous and fractional damping. We prove that the total mechanical energy, including the stored energy in the fractional element, is a Lyapunov functional with which one can prove stability of the equilibrium. Furthermore, we develop a strict Lyapunov functional for asymptotic stability, thereby opening the way to a nonlinear stability analysis beyond an eigenvalue analysis. A key result of the paper is a Lyapunov stability condition for systems having negative viscous damping but a sufficient amount of positive fractional damping. This result forms the stepping stone to the study of Hopf bifurcations in fractionally damped mechanical systems. The theory is demonstrated on a stick-slip oscillator with Stribeck friction law leading to an effective negative viscous damping.

Mechanical Energy and Equivalent Viscous Damping for Fractional Zener Oscillator

This paper presents mechanical energy and equivalent viscous damping for a single-degree-of-freedom fractional Zener oscillator. Differential equation of motion is derived in terms of fractional Zener constitutive equation of viscoelastic materials. A virtual fractional oscillator is generated via a state transformation. Then, based on the diffusive model for fractional integrators, the stored energy in fractional derivatives with orders lying in (0, 1) and (2, 3) is determined. Thus, the total mechanical energy in the virtual oscillator is determined. Finally, fractional derivatives are split into three parts: the equivalent viscous damping, equivalent stiffness, and equivalent mass. In this way, the fractional differential equation is simplified into an integer-order differential equation, which is much more convenient to handle in engineering.