Rotary Inertia Research Articles

Influence of a Pair of Unequal Rotational Fluxes on Entrained Gaseous Filament

Abstract Efforts are made to elucidate a comprehensive analysis of entrainment dynamics triggered by a couple of unequal rotational fluxes within a viscous pool. Cylindrical rollers are employed to establish the rotational field. The top drum is equally submerged in both phases and also it provides a constant rotational inertia. Concomitantly, the bottom roller is completely submerged in the viscous bath, and it provides an unequal rotational strength in reference to top roller. The average rotational strength of both rollers is measured using average Capillary number (Caavg). The entrainment phenomenon is strongly influenced by both Caavg and gap between the rollers (W/D). Characterization of entrained filament is elucidated by predicting the horizontal distance (L*), radial distance (r*), temporal vertical displacement (Y*), maximum vertical displacement (Ymax*), width (H*), and location of V-shaped diversion (Øs*). Characterization of liquid tip is performed by measuring the travel rate (γ*) along periphery of drum from receding to advancing junction. Air mass ejection from filament tip is analyzed by estimating the first bubble ejection time (t1st*), volume of accumulated of ejected gaseous masses (v*), and ejection frequency (f). Furthermore, the effect of gravitational pull (specified by Archimedes number, Ar) and viscous drag (measured by Morton number, Mo) on the pattern of entrained air filament is described. Lastly, an analytical framework is established to determine the width of the V-junction by balancing the important influencing forces, which are strongly affecting the filament. Analytical observations show a satisfactory agreement with numerical findings.

Asymptotic behavior of solutions of a viscoelastic Shear beam model with no rotary inertia: General and optimal decay results

Abstract In this study, we consider a viscoelastic Shear beam model with no rotary inertia. Specifically, we study ρ 1 φ t t − κ ( φ x + ψ ) x + ( g ∗ φ x x ) ( t ) = 0 , − b ψ x x + κ ( φ x + ψ ) = 0 , \begin{array}{rcl}{\rho }_{1}{\varphi }_{tt}-\kappa {\left({\varphi }_{x}+\psi )}_{x}+\left(g\ast {\varphi }_{xx})\left(t)& =& 0,\\ -b{\psi }_{xx}+\kappa \left({\varphi }_{x}+\psi )& =& 0,\end{array} where the convolution memory function g g belongs to a class of L 1 ( 0 , ∞ ) {L}^{1}\left(0,\infty ) functions that satisfies g ′ ( t ) ≤ − ξ ( t ) ϒ ( g ( t ) ) , ∀ t ≥ 0 , g^{\prime} \left(t)\le -\xi \left(t)\Upsilon \left(g\left(t)),\hspace{1.0em}\forall t\ge 0, where ξ \xi is a positive nonincreasing differentiable function and ϒ \Upsilon is an increasing and convex function near the origin. Using just this general assumptions on the behavior of g g at infinity, we provide optimal and explicit general energy decay rates from which we recover the exponential and polynomial rates when ϒ ( s ) = s p \Upsilon \left(s)={s}^{p} and p p covers the full admissible range [ 1 , 2 ) \left[1,2) . Given this degree of generality, our results improve some of earlier related results in the literature.

Natural Characteristics of a Marine Two-Stage Tandem Hybrid Planetary System

This study focuses on a marine two-stage tandem hybrid planetary system. Natural frequencies and vibration modes are determined using a translational–torsional coupled dynamic model. Based on the motion characteristics of the transmission system, free vibration is categorized into three typical modes. The parameter sensitivity of natural frequencies is computed, and the effects of structural parameters such as unequally spaced planet, mesh stiffness, planet mass and rotational inertia on the natural frequencies are analyzed. Utilizing the coupling factor, the mode transition criterion for the natural frequencies response to parameters is formulated. The results demonstrate that the vibration modes of the two-stage tandem hybrid planetary system can be classified as the fixed-axis train vibration mode, the differential train vibration mode, and the coupled vibration mode. Unequally spaced planet primarily disrupts vibration modes without significantly affecting natural frequencies. In contrast, the effects of mesh stiffness, planet mass and rotational inertia on the modes are opposite to those of unequally spaced planets. The validity of the parameter sensitivity and mode transition criterion is substantiated through illustrative examples.

Vibration behavior prediction of submerged nanobeams with axially traveling supports: numerical, analytical, and machine learning approaches

This work surveyed the vibration behavior and stability analysis of an underwater moving flexible nanosize beam, implementing numerical and analytical methods as well as tree-based machine learning (ML) algorithms. Dynamic modeling is performed based on the nonlocal stress-strain gradient theory (NSSGT), incorporating variable environmental conditions, surface energy, and rotational inertia effects. The natural frequencies and instability threshold are computed numerically. The exact closed-form mathematical expression for the critical towing velocity of the nanosize beam is determined analytically. Also, decision tree regression and least-squares boosting tree (LSBT) regression algorithms are exploited to predict stability boundaries and the fundamental vibration frequency, and their efficiency is surveyed accordingly. Comparative studies and parametric investigations are conducted. The impressions of geometry, added mass coefficient, scale ratio parameter, surface layer characteristics, fluid mass ratio, humidity, temperature rise, and external magnetic field intensity on the nanosystem dynamics are examined and illuminated. It is detected that the results of the analytical method and ML-based approaches are consistent with those reported by the numerical technique and literature. The results asserted that the proposed ML algorithms have acceptable performance and excellent effectiveness in computational time and cost. It is comprehended that the dynamic features of submerged traveling nanobeams drastically depend on the rotational inertia effects and thickness-dependent scale effects. The stability of the immersed movable nanoscale beams is perceived to improve by increasing the scale ratio parameter and the added mass coefficient. From the perspective of the optimal design of engineering instrumentation tools, the outcomes of the current article can be applied as an inclusive benchmark.

Optimal Rotor Design and Analysis of Energy-Efficient Brushless DC Motor-Driven Centrifugal Monoset Pump for Agriculture Applications

The agricultural sector emphasizes sustainable development and energy efficiency, particularly in optimizing water pumping systems for irrigation. Brushless DC (BLDC) motors are the preferred prime mover over induction motors due to their high efficiency in such applications. This article details the rotor design and analysis of an energy-efficient BLDC motor with specifications of 1 hp, 3000 rpm, and 48 V, specifically tailored for a centrifugal monoset pump for irrigation. The focus lies in achieving optimal energy efficiency through grey wolf optimization (GWO) algorithm in the rotor design to determine optimal dimensions of the Neodymium Iron Boron (NdFeB) magnet as well as its grade. The finite element method analysis software, MagNet, is used to model and analyze the BLDC motor. The motor parameters, such as speed, torque, flux functions, temperature, and efficiency, are analyzed. For performance comparison, the same model with different magnet models is also analyzed. Validation via 3D finite element analysis highlights improvements in magnet flux linkage, stator tooth flux density, and rotor inertia with increased magnet thickness. Simulation results affirm the consistent performance of the designed BLDC motor, preferably when efficiency is increased. This efficiency and the constant speed lead to an improvement in the overall conversion efficiency of 7% within its operating range, affirming that the motor pump system is energy-efficient.

Asymptotic Analysis of an Elastic Layer under Light Fluid Loading

Asymptotic analysis for an elastic layer under light fluid loading was developed. The ratio of fluid and solid densities was chosen as the main small parameter determining a novel scaling. The leading- and next-order approximations were derived from the full dispersion relation corresponding to long-wave, low-frequency, antisymmetric motions. The asymptotic plate models, including the equations of motion and the impenetrability condition, motivated by the aforementioned shortened dispersion equations, were derived for a plane-strain setup. The key findings included, in particular, the necessity of taking into account transverse plate inertia at the leading order, which is not the case for heavy fluid loading. In addition, the transverse shear deformation, rotation inertia, and a number of other corrections appeared at the next order, contrary to the previous asymptotic developments for fluid-loaded plates not assuming a light fluid loading scenario.

Origami-inspire quasi-zero stiffness structure for flexible low-frequency vibration isolation

As an emerging technology, origami exhibits rich nonlinear static and dynamic properties. However, simulating creases remains a key challenge hindering practice application, especially for volumetric (deformable) origami. Inspired by the bistable behavior and axial-rotational coupling mechanism of triangulated cylindrical origami (TCO), this paper proposes a novel combined truss-spring structure with additional inertial mechanism (TCTS-RIM) for low-frequency vibration isolation. Unlike pure TCO, the TCTS-RIM uses elastic linkages to simulate all creases and modifies crease positions, enhancing physical realization in engineering applications and offering tunable stiffness and inertia on demand. Static analysis shows the modified truss-spring structure in TCTS-RIM has superior quasi-linear negative stiffness compared to pure TCO, which enables a very wide quasi-zero stiffness (QZS) range via linear spring compensation. The wide QZS range for different loads can be realized by setting appropriate structural parameters. From system dynamic equations, nonlinear inertia, nonlinear quadratic force, and nonlinear damping due to RIM are identified and discussed. The Alternating frequency–time harmonic balance method is used to obtain the displacement transmissibility, and its parametric influencing investigations verify that the proposed TCTS-RIM has excellent low-frequency vibration isolation performance and rich tunability. Comparative discussions show the wide QZS range significantly reduces the resonance frequency, broadens the isolation frequency band, and yields superior dynamic performance versus pure TCO isolators. Moreover, equivalent mass, anti-resonance and softening behavior induced by the rotational inertia mechanism further enhance low-frequency vibration isolation performance. An experimental prototype is built, and experimental results validate the correctness of the theoretical analyses. This work presents a simple mechanical implementation of volumetric origami, which may progress the engineering applications of origami and provide a viable approach to low-frequency vibration control.

Using your head - cranial steering in pterosaurs.

The vast majority of pterosaurs are characterized by relatively large, elongate heads that are often adorned with large, elaborate crests. Projecting out in front of the body, these large heads and any crests must have had an aerodynamic effect. The working hypothesis of the present study is that these oversized heads were used to control the left-right motions of the body during flight. Using digital models of eight non-pterodactyloids ("rhamphorhyncoids") and ten pterodactyloids, the turning moments associated with the head + neck show a close and consistent correspondence with the rotational inertia of the whole body about a vertical axis in both groups, supporting the idea of a functional relationship. Turning moments come from calculating the lateral area of the head (plus any crests) and determining the associated lift (aerodynamic force) as a function of flight speed, with flight speeds being based on body mass. Rotational inertias were calculated from the three-dimensional mass distribution of the axial body, the limbs, and the flight membranes. The close correlation between turning moment and rotational inertia was used to revise the life restorations of two pterosaurs and to infer relatively lower flight speeds in another two.

Tunnel rockbursts induced by dynamic disturbances: mechanism and mitigation

Rockbursts are characterized by violent rock fractures and pose a significant threat to hard-rock tunnels, potentially resulting in casualties and damage to excavation spaces. Globally recognized as one of the least understood and most feared challenges in underground excavations, rockbursts are often triggered by dynamic disturbances such as engineering activities or nearby vibrations. This study conceptualizes rockbursts as dynamic buckling or instability issues inherent in rock structures. It specifically investigates the mechanism of tunnel rockbursts induced by ambient blasting. The derivation of the governing equation of motion, which incorporates shear deformation and rotary inertia of the rock column, results in coupled Mathieu equations. By employing the proposed numerical method, the conditions triggering rockburst were established using instability diagrams. The study examines the effects of static components, dynamic loading, and frequency on a tunnel example, revealing that the amplitude and frequency of dynamic disturbances are critical in influencing the occurrence of tunnel rockbursts through perturbation effects and parametric resonance mechanisms. These insights offer valuable understanding into the mechanisms, mitigation, and control of tunnel rockbursts.

Free vibration analysis of tall buildings using a generalized continuous model: simplified approach to soil-structure interaction

Currently, due to the simplifications imposed on continuous models, there is an absence of a generalized approximate solution to investigate all possible dynamic behaviors of a tall building. In an attempt to address this deficiency, this paper proposes a new generalized continuous model and its solution methods to generalize all types of behavior that may be encountered in structural systems and tall buildings. The continuous model results from a composite beam that serially couples the classical sandwich beam and a pure shear beam. Unlike existing continuous models, the serial coupling, inclusion of new kinematic fields, and inclusion of rotational inertia allow for the resolution of complex interaction among different types of bending and shear behaviors. Specifically, it is shown how local shear behavior simultaneously interacts with global bending behavior, global shear, and local bending. Two solutions to the dynamic problem are presented: a closed-form analytical solution method for structures with uniformity along their height, and a numerical method that combines the advantages of the continuous method and the transfer matrix method to resolve structural and geometric variability in height. The transfer matrix is analytically solved by maintaining a fixed 6 × 6 matrix range that is independent of the number of levels of the structure, drastically reducing computational cost by eliminating the classical need to calculate the inverse of the zero matrix. Numerical applications investigate the effect of rotational inertia and local shear on the accuracy of the proposed results compared to results from the finite element method. To this end, a parametric analysis of 1560 different cases covering a wide range of typical behaviors in individual structural systems and tall buildings is conducted. The results exhibit promising accuracy, showing that the qualities of the proposed continuous model significantly improve precision compared to classical continuous models and therefore justify its practical application by academics and practicing engineers. Furthermore, the proposed mathematical approach is easily applicable in various domains of composite materials engineering.

Frequency Regulation in Nanocomposite Sandwich Joined Conical-Conical Shells with Open Cell Foam Partially Supported by Pasternak Elastic Foundation Using GDQ

This study investigated the natural vibration of a sandwich conical-conical shell that is partially supported by Winkler–Pasternak foundations. The sandwich system comprises an open-cell foam (OCF) core and laminated composite face sheets reinforced with graphene platelets using functionally graded models (FG-GPLRC). To account for through-the-thickness shear deformations and rotary inertias, the first-order theory of shells is combined with Donnell-type kinematic assumptions. Hamilton’s principle is utilized to establish the general motion equations and related boundary and continuity conditions. The resulting system of equations is discretized using the semi-analytical generalized differential quadrature (GDQ) approach. An eigenvalue problem is formulated to analyze the corresponding mode shapes and vibration frequencies for the shell ends and continuity criteria under various boundary conditions. Parametric experiments are conducted for sandwich conical-conical shells partially supported by Winkler–Pasternak foundations. This study examined the impact of multiple factors on the frequency of a joined shell structure, including the total thickness, FG-GPL face sheet thickness, various boundary conditions, the length of each cone segment, and the Winkler and shear elastic foundation. Notably, this research also analyzed the effects of a partial elastic foundation on the structure’s frequency for the first time.

Packing optimization of practical systems using a dynamic acceleration methodology

System design is a challenging and time-consuming task which often requires close collaboration between several multidisciplinary design teams to account for complex interactions between components and sub-systems. As such, there is a growing demand in industry to create better performing, efficient, and cost-effective development tools to assist in the system design process. Additionally, the ever-increasing complexity of systems today often necessitates a shift away from manual expertise and a movement towards computer-aided design tools. This work narrows the scope of the system design process by focusing on one critical design aspect: the packaging of system components. The algorithm presented in this paper was developed to optimize the packaging of system components with consideration of practical, system-level functionalities and constraints. Using a dynamic acceleration methodology, the algorithm packages components from an initial position to a final packed position inside of a constrained volume. The motion of components from initial to final positions is driven by several acceleration forces imposed on each component. These accelerations are based on physical interactions between components and their surrounding environment. Various system-level performance metrics such as center of mass alignment and rotational inertia reduction are also considered throughout optimization. Results of several numerical case studies are also presented to demonstrate the functionality and capability of the proposed packaging algorithm. These studies include packaging problems with known optimal solutions to verify the efficacy of the algorithm. Finally, the proposed algorithm was used in a more practical study for the packaging of an urban air mobility nacelle to demonstrate the algorithm’s prospective capabilities in solving real-world packaging problems.

Vibrational Analysis of Axially Loaded Rayleigh Beams on Variable Winkler Foundations with Implications for Structural Design

In this study, we explore the complex vibration problem of a Rayleigh beam under axial loads, positioned over a variable Winkler foundation with elastic properties changing along its length. The investigation delves into how factors like rotational inertia, axial forces, and foundation properties affect the beam’s dimensionless eigenvalues and eigenmodes. Employing the Differential Transform Method (DTM), we tackle the governing equations across various foundation conditions, including constant, linear, and parabolic variations. The results align well with the previous research, affirming their precision between scenarios considering or neglecting the elastic foundation’s effects. Notably, we emphasize the influence of linear spring stiffness on vibration characteristics, overshadowing rotary spring stiffness. Thus, the important novelty of the study lies in its comprehensive investigation into the vibration dynamics of a Rayleigh beam under axial loads on a variable Winkler foundation with elastic properties varying along its length. The findings illuminate a previously unexplored dynamic: the intricate interplay between linear and rotary spring stiffness, showcasing the dominant impact of linear stiffness on vibration characteristics. Of significant note is the profound effect of axial forces on fundamental eigenvalues, underscoring their critical role in structural behavior. These developments possess substantial practical implications, providing invaluable guidance for the accurate modeling of beams supported by elastic foundation, thereby enhancing structural design and performance.

A numerical assembly technique for free vibration of plane frames using a shifted and deflated Newton method

A semi-analytic method, namely the Numerical Assembly Technique (NAT) is extended to compute natural frequencies of arbitrary planar frame structures. The frame structure is systematically defined by a set of nodes, beams, bearings, springs, and external loads and the corresponding boundary and interface conditions are formulated. We consider the Timoshenko-Ehrenfest beam theory and take axial, bending and shear deformations as well as linear and rotational inertia into account. Since we use the exact solutions of the governing differential equations in the NAT, the accuracy of the results is independent from the spatial discretization. However, the resulting eigenvalue problem is non-linear and requires a numerical algorithm to solve. We consider three different representations of exact homogeneous solutions and study the numerical stability of the resulting system matrix. To solve for the eigenvalues we propose a Newton method enhanced with a shifted deflation algorithm. The stability and the efficiency of this approach is illustrated in numerical examples.

Water Droplet Collison and Erosion on High-Speed Spinning Wheels

<div>The water droplet erosion (WDE) on high-speed rotating wheels appears in several engineering fields such as wind turbines, stationary steam turbines, fuel cell turbines, and turbochargers. The main reasons for this phenomenon are the high relative velocity difference between the colliding particles and the rotor, as well as the presence of inadequate material structure and surface parameters. One of the latest challenges in this area is the compressor wheels used in turbochargers, which has a speed up to 300,000 rpm and have typically been made of aluminum alloy for decades, to achieve the lowest possible rotor inertia. However, while in the past this component was only encountered with filtered air, nowadays, due to developments in compliance with tightening emission standards, various fluids also collide with the spinning blades, which can cause mechanical damage. One such fluid is the condensed water in the low-pressure exhaust gas recirculation channel (LP-EGR) formulated at cold starts and low-speed high load conditions. This kind of design has been developed to reduce nitrogen oxide emissions and is used in both gasoline and diesel engines. This article presents a state-of-the-art review of this WDE process, focusing on the formation of the condensed water before the compressor wheel, summarizing the influencing factors of WDE and the effects of the damage including using component testbench experiences and simulation methodologies. Inspection possibilities such as high-speed camera measurement and vibration analysis are also an important part of the document.</div>

Longtime dynamics of solutions for higher-order (m1,m2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(m_{1},m_{2})$\\end{document}-coupled Kirchhoff models with higher-order rotational inertia and nonlocal damping

The Kirchhoff model is derived from the vibration problem of stretchable strings. This paper focuses on the longtime dynamics of a higher-order (m1,m2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(m_{1},m_{2})$\\end{document}-coupled Kirchhoff system with higher-order rotational inertia and nonlocal damping. We first obtain the state of the model’s solutions in different spaces through prior estimation. After that, we immediately prove the existence and uniqueness of their solutions in different spaces through the Faedo-Galerkin method. Subsequently, we prove their family of global attractors using the compactness theorem. Finally, we reflect on the subsequent research of the model and point out relevant directions for further research on the model. In this way, we systematically study the longtime dynamics of the higher-order (m1,m2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(m_{1},m_{2})$\\end{document}-coupled Kirchhoff model with higher-order rotational inertia, thus enriching the relevant findings of higher-order coupled Kirchhoff models and laying a theoretical foundation for future practical applications.

Sliding-mode Active Disturbance Rejection Permanent Magnet Synchronous Motor Control System

Aiming at the permanent magnet synchronous motor with external perturbations and internal uncertainties such as external load, rotational inertia and armature resistance changes, a servo control system based on sliding mode self-immunity controller is designed. By constructing a mathematical model of the servo system of the permanent magnet synchronous motor, considering the internal uncertainty caused by the change of the system parameters and the external random disturbance as the "total disturbance", and designing the transition process and the linear expansion state observer to observe and compensate for it, the system response can track the input signal quickly and without overshooting, and the sliding-mode state feedback makes the closed-loop servo system achieve fast and stable control, and the sliding-mode Active Disturbance Rejection control controller can be used to control the system. The sliding mode state feedback enables the closed-loop servo system to realize fast and stable control, and its consistent stability is proved by the Lyapunov method. Simulations show that this sliding- mode Active Disturbance Rejection control control strategy improves the system response speed and load carrying capability compared with the traditional sliding-mode control.

Active Vibration Damping Using Rotor Torque Control in Electric Aircraft

Vibration damping in aerospace structures can decrease the likelihood of failures and instabilities and improve passenger comfort. This paper introduces the novel idea of damping vibration using the electric proprotors on aircraft without compromising flight control. The equations of motion of a cantilevered beam with a propeller at the tip driven by an electric motor are obtained using Hamilton’s principle, solved analytically in the frequency domain, and approximately in the time domain. Feeding back the beam tip angular rate to the motor torque is shown to asymptotically stabilize all transverse beam vibration modes. The overall vibration control consists of the inner rate feedback damping loop with an outer rotor speed control loop. Experimental frequency response and step response validate the models and show that the closed-loop damping in the first mode is three times higher than open loop with less than 1% rotor speed change for a 3% initial tip displacement. Theoretical results give good agreement with experiments. Parametric studies based on a 12 kg quadcopter indicate that if the rotor inertia is sufficiently large the proposed control can provide 8% damping with minimal impact on rotor speed.

Modeling and analysis of a macro-fiber piezoelectric bimorph energy harvester operating in d33-Mode using Timoshenko theory

Piezoelectric bimorph cantilevered beams are frequently employed in energy harvesting applications. The use of macro-fiber composites, which combine piezoelectric fibers with rectangular cross-sections and interdigitated electrodes, has significantly enhanced their performance by enabling operation in the 33-mode of piezoelectricity, thereby increasing conversion efficiency. In this article, the Timoshenko beam theory is applied to overcome the limitations of the Euler-Bernoulli theory in modeling transverse vibrations of harvesters with low slenderness ratios (length to thickness ratios). The mixture rule formulation is applied to determine the equivalent properties of the macro-fiber composite (MFC) structure. The electromechanical properties of a representative volume element (RVE) bounded by two successive interdigitated electrodes are coupled to the overall electro-elastic dynamic model of the structure using the Timoshenko theory. Frequency response functions for the power and tip vibration displacement of the MFC bimorph beam are determined from the model. Experimental data from the literature are used to validate the improved results predicted by the model developed in this article. A comparative analysis between the electromechanical responses predicted using a model based on Timoshenko and a model based on Euler-Bernoulli theory is conducted showing agreement, for harvesters with large slenderness ratios and discrepancies of up to 30% for low slenderness ratios. It has also been shown that the model based on Euler-Bernoulli theory overestimates both the mechanical and the electrical responses for low slenderness ratios as the shear and rotary inertia effects are not negligible in this case.

Response of a Stretched String Subjected to a Moving Mass

In this paper, the dynamics of a taut horizontal string with a constant velocity moving mass including its rotary inertia, is modelled. The equation of motion is solved using Galerkin’s approach, employing appropriate comparison functions. A discontinuity or jump in the trajectory of the mass has been established when the mass is about to leave the string. The consideration of rotary inertia in the model is found to affect the spatial location of the jump in the trajectory of the moving mass.