Equivalent Mechanical Model Research Articles

Dynamic Magnification Factor of Pile-Supported Wharf under Horizontally Bi-Directional Ground Motion

ABSTRACT This study proposes a generally applicable equation for dynamic magnification factor (DMF) based on the conditions for static equilibrium of an equivalent mechanical model for actual wharf under horizontally bi-directional ground motion. The new equation can be expressed as a function of eccentricity ratio and aspect ratio for wharf segment. The applicability and accuracy of proposed equation are validated by conducting two case studies that compare DMF determined by equation with that determined by nonlinear time history analyses using bi-directional earthquake ground motion records. The validation shows that the new DMF equation has favorable applicability and accuracy.

Dynamic Analysis of Slider-Crank Mechanism with a Cracked Rod

The dynamical equation of the slider-crank mechanism is established by using Lagrange equation and Newton’s second law. The slider-crank mechanism with an open crack rod is investigated and then establishes the equivalent mechanics model by a massless torsional spring to simulate the influence of the crack in the rod, and the mechanism of a cracked rod is divided into two subsystems. The dynamical equation of the slider-crank mechanism with a crack rod is established. Comparing the dynamic analysis results between with and without crack in the rod, the results show that the existence of the crack leads to a great change in the motion characteristics of the slider. The calculated maximum Lyapunov exponent is positive, which shows that the movement of the slider in the crank slider mechanism with a cracked rod is chaotic.

Computational simulation of nonlinear liquid sloshing in a tank truck

Liquid sloshing in the tank of a tank truck is one of the important factors affecting the driving stability of the tank truck. When the external motivation is larger, the liquid sloshing phenomenon shows the nonlinear characteristics of the movement, causing rollover accidents. Therefore, it is of practical significance to study the movement of a tank car to improve its stability. Based on the equivalent pendulum mechanical model under nonlinear condition, the liquid sloshing nonlinear motion of a tank was investigated. A simulated liquid sloshing test bench was used to verify the validity of the nonlinear model. This study provides a theoretical basis to improve the driving stability of a tank truck.

Permeability Anisotropy of Columnar Jointed Rock Masses

This paper investigated the anisotropic permeability of columnar jointed rock masses at Baihetan hydropower station in Southwest China. Discrete element analysis is conducted to evaluate the hydraulic Representative Elemental Volume (REV) size of jointed rock masses. It is found to be 5m×5m and it is seven times of the column length. Research shows that permeability tensor of the jointed rocks exists. The maximum principle permeability is 3.47 × 10−14 m2 and the minimum principle permeability is 1.39 × 10−14 m2. The direction of maximum principle permeability is 72 degrees anticlockwise from the positive x-axis direction, which is along the same direction of the column length. A continuum hydro-mechanical coupling analysis is carried out on water infill of a tailrace tunnel in columnar basalt. An equivalent anisotropic mechanical model is applied in the numerical simulation. Anisotropic deformation and anisotropic permeability characteristics are investigated under the condition that the angle between maximum principle stress axis and maximum principle permeability axis is 72 degrees. Results show that anisotropic seepage field has an effort on anisotropic deformation field.

Dynamic response of tank trains to random track irregularities

This paper presents a dynamic analytical model for tank train vibrations. The train is considered as a system of 27 degrees of freedom consisting of lateral, roll, yaw, vertical, and pitch motions for the vehicle body and its two bogies and lateral, roll and vertical motions for the four wheel-sets. Liquid sloshing in the tank is modeled using an equivalent mechanical mass-spring model. Coupling between the vehicle system and the railway track is realized through the interaction forces between the train and the rail, where the vertical and lateral irregularity profiles of the track are regarded as stationary ergodic Gaussian random processes and simulated by polynomial functions. Random vibration theory is used to obtain the response power spectral densities. Finally, numerical results for a typical test case including natural frequencies of a coupled system, frequency response functions, and output power spectral densities are presented.

Response regimes in equivalent mechanical model of moderately nonlinear liquid sloshing

The paper considers non-stationary responses in reduced-order model of partially liquid-filled tank under external forcing. The model involves one common degree of freedom for the tank and the non-sloshing portion of the liquid, and the other one -- for the sloshing portion of the liquid. The coupling between these degrees of freedom is nonlinear, with the lowest-order potential dictated by symmetry considerations. Since the mass of the sloshing liquid in realistic conditions does not exceed 10% of the total mass of the system, the reduced-order model turns to be formally equivalent to well-studied oscillatory systems with nonlinear energy sinks (NES). Exploiting this analogy, and applying the methodology known from the studies of the systems with the NES, we predict a multitude of possible non-stationary responses in the considered model. These responses conform, at least on the qualitative level, to the responses observed in experimental sloshing settings, multi-modal theoretical models and full-scale numeric simulations.

Nonlinear modelling of tuned sloshing dampers with large internal obstructions: Damping and frequency effects

Internal obstructions are commonly employed in tuned sloshing damper (TSD) tanks to increase the inherent damping of the sloshing liquid closer to its optimal value for the targeted excitation amplitude. However, these obstructions also produce added mass, which acts to decrease the natural sloshing frequency of the tank. Using potential flow theory, this study develops both a linearized equivalent mechanical model and a fourth-order nonlinear multimodal model that are capable of accommodating the fluid drag and added mass that are generated by the paddles. The linearized equivalent mechanical model represents the sloshing liquid as an equivalent spring–mass–dashpot system with amplitude-dependent damping. The fourth-order nonlinear multimodal model captures the nonlinear coupling that occurs among the first four sloshing modes. Shake table tests are then conducted on a scale model TSD tank equipped with vertical paddles of cruciform plan section. The sloshing forces predicted by the linearized and nonlinear models are in reasonable agreement with the experimental results. The nonlinear model is generally in good agreement with the experimental wave heights, whereas the linearized model often underestimates the peak wave heights considerably. A plot is created which shows the frequency shift produced by the paddles as a function of the width of the paddles. If the paddles’ widths are greater than approximately 5% of the tank width, the effect on the natural sloshing frequency is noticeable.

MATHEMATICAL MODEL OF DPKR-2 DYZEL TRAIN CAR

Purpose. In order to study the dynamic phenomena arising when rolling stock moves along a rail track both in the straight and curved track sections, the article is aimed to construct a mathematical model of DPKr-2 diesel train car. It will be constructed on the basis of mechanical model of this car of Kryukiv Railway Car Building Works. Methodology. To construct a mathematical model a system of 38 differential equations of the diesel train movement is formed. When it is used a pneumatic spring in the core stage of spring suspension, its equivalent mechanical mo-del is presented as Kelvin-Voigt knot. It includes a parallel elastic element and an element of viscous friction. Rail track flexibility is taken into account by elastic and dissipative elements. During simulation it was assumed that the wheel pair and the track weight interacting with it were moving intact. Geometric inequalities of the left and right rails were accepted as disturbances when studying the forced vertical and horizontal oscillations. Findings. On the basis of the adopted mechanical model of the diesel train car we constructed the mathematical model consisting of 38 differential equations of motion . Originality. For the first time, for the DPKr-2 diesel train car we developed its spatial mathematical model taking into account the features of the interaction of individual elements of its construction and the possibilities of the rail track depression. When constructing the mathematical model, it was proposed to take into account the flexibility of the rail track by elastic and dissipative elements. Originality. The mathematical model of the diesel train car will be used for studying the dynamic phenomena and determining the dynamic loads of structural elements during operation. The study of these phenomena is necessary for optimal choice of the scheme and parameters of rolling stock equipment, in particular antivibration devices (spring suspension, horizontal, longitudinal and transverse joints of wheel pairs with the bogie frame, bogie with the body), as well as for reduction of dynamic forces acting on the elements of rolling stock construction and rail track.

Assessment of ground vehicle tankers interacting with liquid sloshing dynamics

This paper presents an assessment of the problems encountered with the design, dynamics and stability of road tankers. Generally, the research activities pertaining to these problems may be classified into three groups, liquid sloshing dynamics in moving containers, trucks dynamics carrying solids, and dynamic coupling of liquid-vehicle systems. The modal analysis of liquid free surface for circular, elliptic and generic cross-section geometries is presented together with the corresponding equivalent mechanical models. Considering popularity of horizontal elliptic cross-section tanks, the trammel equivalent pendulum received extensive research activities and the main results are discussed. Design optimisation of the liquid container cross-section and the passive and active control of liquid sloshing approaches are used by engineers to minimise rollover problem. The most difficult problem of road tankers is the coupling dynamics of liquid and vehicle dynamics under different conditions such as braking and lateral acceleration; in this view, computer numerical simulations have been developed.

Modeling Problem of Equivalent Mechanical Models of a Sloshing Fluid

Because the defects in the existing modeling methods for the equivalent mechanical model of a sloshing fluid have led to incorrect or inaccurate results in the existing equivalent models, this paper discusses three different modeling methods for the equivalent models: the traditional method, Housner’s method, and the modified method. The equivalent models obtained by the three methods are, respectively, presented and compared with each other for a liquid in rectangular and upright cylindrical tanks. The results show that the traditional method cannot provide the correct location expressions of the equivalent masses because the two types of different excitations are simultaneously used in one equivalent model. An equivalent model is exclusively applicable to a certain excitation (a translational excitation in a certain direction or a rotational excitation about a certain axis). Housner’s method is based on physical intuition, instead of fluid dynamics theory, therefore the calculation precision of Housner’s solution is not satisfactory. Housner’s method is only suitable for vertical tanks with a flat bottom subjected to a horizontal excitation. Based on a conceptual mistake in the traditional method, the concept of the equivalent model is reclarified, and the modified equivalence method is therefore suggested. A supplementary solution for the equivalent model in a cylindrical tank is presented. The correct models can be acquired using the modified equivalence method, which is applicable to tanks of arbitrary shape.

Comparative research on two equivalent mechanical models of liquid sloshing in tanks

Using a quasi-static model, it was found that the mass centre of the liquid trajectory is the same as the length-width ratio of the ellipse of the tank. According to the principle of mechanical equivalence, two types of equivalent mechanical models of liquid sloshing in a tank were established, and the relevant parameters of the two models were derived. The simulation results show that the two models can more accurately describe the linear motion of liquid sloshing. The fluid dynamics simulation results of tank liquid vibration frequency division were compared with those obtained using ANSYS Fluent software under similar conditions; it was concluded that the model is more suitable for a liquid vibration frequency range. A simulated test bench was used to verify the two models; the results show the pendulum model is more suitable.

Comparative research on two equivalent mechanical models of liquid sloshing in tanks

Using a quasi-static model, it was found that the mass centre of the liquid trajectory is the same as the length-width ratio of the ellipse of the tank. According to the principle of mechanical equivalence, two types of equivalent mechanical models of liquid sloshing in a tank were established, and the relevant parameters of the two models were derived. The simulation results show that the two models can more accurately describe the linear motion of liquid sloshing. The fluid dynamics simulation results of tank liquid vibration frequency division were compared with those obtained using ANSYS Fluent software under similar conditions; it was concluded that the model is more suitable for a liquid vibration frequency range. A simulated test bench was used to verify the two models; the results show the pendulum model is more suitable.

Method for constructing an equivalent mechanical model of an elastic system of a machine with various technological equipment

The article deals with the construction of an equivalent mechanical model of the elastic system of the machine for turning non-rigid shafts with various technological equipment, including machining in centers and machining in a chuck with the rear rotating center being clamped without using additional support and using a vibration-damper lunette. The relevance of this study due to the fact that in view of low stiffness processed non-rigid shaft technology system machine-tool-tool-workpiece is extremely sensitive to the deformations caused by the action of cutting forces. As a result, significant processing errors are generated. High sensitivity to deformations serves as a source of intensive vibrations of the workpiece, which leads to deterioration of the quality of the machined surface of the part, a decrease in the durability and durability of the cutting tool, accelerated wear and loss of guaranteed machine accuracy parameters. This makes it essential to significantly reduce the cutting conditions, to resort to multi-pass processing, which greatly increases the laboriousness and cost of manufacturing products. In this context, the aim of this study – the creation of an equivalent mechanical model of the machine, allowing the process of studies to determine the boundaries of sustainable non-rigid turning shafts considering the different ways of mounting the workpiece on the machine. The materials of the article can be useful for engineers and scientists in the field of engineering.

Modeling of cohesive fracture interacting with a stationary capillary fluid

Understanding cohesive fracture interacting with a stationary capillary fluid is essential for fracture mechanics based analysis and design of many quasibrittle structures in contact with fluids. In this study, an equivalent linear elastic fracture mechanics (LEFM) model is used to determine the equilibrium state for a cohesive crack partially filled with a capillary fluid. The model is shown to deliver predictions similar to the cohesive zone model (CZM) at a much lower computational cost. The equivalent LEFM model is applied to determine the instantaneous failure pressure of high-density polyethylene (HDPE) pipes as well as the wetting zone size for modeling the stress corrosion cracking of HDPE pipes.

A vibration amplitude model for the giant magnetostrictive ultrasonic processing system

For the purposes of estimation and control of the vibration amplitude in rotary ultrasonic machining, the relation between vibration amplitude and excitation signal was examined. The mechanical components outside the Terfenol-D were regarded as a single-degree-of-freedom mass spring damping system, and an equivalent mechanical model of the giant magnetostrictive ultrasonic processing system was addressed. After that, the vibration amplitude was modeled as a function of the frequency and amplitude of the excitation current by considering both magnetization and magnetostriction of Terfenol-D and ultimately achieved through parameter identification using the methods of impedance analysis and nonlinear least-squares fitting. In addition, the relation between resonant frequency of the giant magnetostrictive ultrasonic processing system and the magnitude of the excitation current was studied so that the vibration amplitude model was improved. Thus, the vibration amplitude of the giant magnetostrictive ultrasonic processing system excited by various electric signals with different frequencies and amplitudes can be calculated. As demonstrated by comparison with experimental results, the vibration amplitude model can accurately estimate the vibration amplitude of the giant magnetostrictive ultrasonic processing system. It indicates that the model is meaningful for the design and control of the giant magnetostrictive ultrasonic processing system.

Seismic responses of base-isolated flexible rectangular fluid containers under horizontal ground motion

This paper focuses on analyzing, the results and arguments about the analysis of seismic base-isolated flexible rectangular containers. An equivalent mechanical model of rectangular containers with three lumped masses and six degrees of freedom is used. Lead rubber bearing (LRB) and friction pendulum system (FPS) are two types of seismic isolators, used to isolate the storage tank base. Results show that seismic base isolation can be an efficient way to reduce seismic responses such as base shear, wall deflection and hydrodynamic pressure, but it can adversely affect the free surface sloshing height. The seismic responses of isolated tanks under unidirectional excitation (No interaction) and bidirectional excitation (Interaction) are not considerably different and the results are not very sensitive to the interaction effect of the bearing forces. In any case, the careful choice of the number and the mechanical characteristics of seismic base isolation systems is suggested to achieve good responses.

Response regimes in equivalent mechanical model of strongly nonlinear liquid sloshing

Equivalent mechanical model of liquid sloshing in partially-filled cylindrical vessel is treated in the cases of free oscillations and of horizontal base excitation. The model is designed to cover both regimes of linear and essentially nonlinear sloshing. The latter regime involves hydraulic impacts applied to the walls of the vessel. These hydraulic impacts are commonly simulated with the help of high-power potential and dissipation functions. For analytical treatment, we substitute this traditional approach by consideration of the impacts with velocity-dependent restitution coefficient. The resulting model is similar to recently explored vibro-impact nonlinear energy sink (VI NES) attached to externally forced linear oscillator. This similarity allowed exploration of possible response regimes. Steady-state and chaotic strongly modulated responses are encountered. Besides, we simulated the responses to horizontal excitation with addition of Gaussian white noise, and related them to reduced dynamics of the system on a slow invariant manifold (SIM). All analytical predictions are in good agreement with direct numerical simulations of the initial reduced-order model.

Assembly deformation prediction of riveted panels by using equivalent mechanical model of riveting process

The assembly deformation of skin panels can be introduced by the riveting process. However, it is computationally costly and time consuming to conduct the numerical simulation of the riveting process for each rivet in skin panels. Here, an equivalent mechanical model of the riveting process is established to improve the computational efficiency of the deformation prediction for the riveted structure. The axial pressure beneath the rivet driven head is solved in the equivalent mechanical model. The assembly deformation of riveted panels is found to be influenced by the axial pressure significantly, challenging the traditional analogy of riveting to the radial expansion process. In addition, the dependence of the radial pressure upon the radial displacement of the rivet hole is established. The characteristic of non-uniform distribution of the radial pressure along the thickness direction is taken into consideration. The deformations of riveted structures are predicted considering various heights of the rivet driven head and different materials of the panel. The effectiveness of the proposed model is evaluated by comparing the deformation results of the numerical simulation and those of the experimental measurement. This investigation can help to predict the deformation of large riveted structures and improve the awareness of the effect of axial pressure on the assembly deformation.

Nonlinear model and attitude dynamics of flexible spacecraft with large amplitude slosh

This paper is focused on the nonlinearly modelling and attitude dynamics of spacecraft coupled with large amplitude liquid sloshing dynamics and flexible appendage vibration. The large amplitude fuel slosh dynamics is included by using an improved moving pulsating ball model. The moving pulsating ball model is an equivalent mechanical model that is capable of imitating the whole liquid reorientation process. A modification is introduced in the capillary force computation in order to more precisely estimate the settling location of liquid in microgravity or zero-g environment. The flexible appendage is modelled as a three dimensional Bernoulli-Euler beam and the assumed modal method is employed to derive the nonlinear mechanical model for the overall coupled system of liquid filled spacecraft with appendage. The attitude maneuver is implemented by the momentum transfer technique, and a feedback controller is designed. The simulation results show that the liquid sloshing can always result in nutation behavior, but the effect of flexible deformation of appendage depends on the amplitude and direction of attitude maneuver performed by spacecraft. Moreover, it is found that the liquid sloshing and the vibration of flexible appendage are coupled with each other, and the coupling becomes more significant with more rapid motion of spacecraft. This study reveals that the appendage's flexibility has influence on the liquid's location and settling time in microgravity. The presented nonlinear system model can provide an important reference for the overall design of the modern spacecraft composed of rigid platform, liquid filled tank and flexible appendage.

Internal Resonances and Dynamic Responses in Equivalent Mechanical Model of Partially Liquid-Filled Vessel

This study treats oscillations of a liquid in partially filled vessel under horizontal harmonic ground excitation. When exposed to external disturbances, the liquid is divided into two portions: oscillating sloshing portion, and a-’static’ non-sloshing portion, located at the bottom of the tank. Such excitation may lead to hydraulic impacts applied by the sloshing mass on the tank walls.Different equivalent mechanical models are suggested to mimic the liquid sloshing mass motion and essential dynamical regimes of the overall tank-liquid system, such as a series of pendula or mass-spring-dashpot systems which can impact the vessel walls. We use parameters of the equivalent mass-spring system for the well-explored case of cylindrical vessels. The hydraulic impacts are modeled by high-power potential function. Finite-Element (FE) method is used to determine and verify the model parameters and to identify dominant dynamical regimes, natural modes and frequencies. The tank failure modes and critical locations are identified. Mathematical relation is found between degrees-of-freedom (DOFs) motion and the mechanical stress applied in the tank critical section. This is the prior attempt to take under consideration large-amplitude nonlinear sloshing and tank structure elasticity effects for design, regulation definition and resistance analysis purposes. Both linear (tuned mass damper, TMD) and nonlinear (nonlinear energy sink, NES) passive energy absorbers contribution to the overall system mitigation is firstly examined.