Lumped Mass Method Research Articles

Quasi-static testing based structural modal parameter estimation

ABSTRACT Structural modal identification is generally implemented within a dynamic framework, requiring multidisciplinary knowledge and adequate operational experience. This preliminary study attempts to explore an easy-to-handle quasi-static alternative method for dynamics-based modal identification of beam-type structures. Acceleration time-history data are replaced by quasi-static deflections induced by slow moving loads. A concept of quasi-static deflection influence surface is proposed based on the theory of influence lines and the principle of virtual work. Its analytical expression is simplified into a deflection matrix by choosing specific points on the surface according to deflection measurement coordinates. The matrix form is divided by external loads to obtain a generalised quasi-static flexibility matrix, which is used to replace the inversion of the global stiffness matrix in the modal eigenvalue equation. The lumped-mass method is also employed to establish the mass matrix. Subsequently, the eigenvalue equation is analytically solved seeking for modal frequencies and mode shapeswithout performing modal tests. The feasibility of the proposed method has been successfully verified against three experimental examples including a continuous box girder with variable cross sections. It was observed that the modal frequencies and mode shapes estimated by the proposed method were very close to those given by the dynamically modal tests.

Nonlinear dynamics of full-ceramic bearing-rotor system considering structural defects in cage pockets

Full ceramic bearings provide exceptional characteristics, including elevated accuracy and substantial load-bearing capability, rendering them indispensable in high-end rotating machinery. Still, the long-term use of the bearing will lead to the gradual wear and defects of the cage pocket, which may affect the unstable operation of the rotor system. This study examines the vibration characteristics of cage failure through simulation and experiments. To study the dynamic characteristics of the bearing-rotor system, a dimensionless model is established by using the lumped mass method, and a cage failure model is added to the model, which can study the influence of cage failure on the system stability. Phase trajectories, Poincaré plots, and bifurcation diagrams are employed to analyze the impact of individual parameters on the nonlinear vibration of the system. Ultimately, to gather cage vibration data from two different types of faulty, the experimental approaches are used. The findings indicate that in the event of cage failure, the system’s amplitude amplifies and the time-domain vibration signal exhibits distinct periodicity. Then, the vibration energy gradually focuses on low-frequency vibrations. Additionally, the spectrograms allow for easier identification of frequency components associated with the cage frequency, such as nfc and mfs ± nfc, the largest error is 3.32% when comparing the two models’ simulated results with the experimental data. The model accurately depicts the frequency characteristics of failures resulting from wear and defects in full ceramic bearing cages. It offers crucial information for detecting failures and monitoring the health of full ceramic bearings.

Surrogate Modeling of a Lumped-Mass Multibody Structure Using Hamiltonian Neural Networks

The complexity of highly flexible structures restricts their use in real-time simulations. To address this challenge, we investigate the use of Hamiltonian neural networks (HNNs) as an alternative method for modeling a highly flexible cantilever beam. We derived the reference structural model using a lumped-mass rigid multibody method considering the Hamiltonian formalism and used it to generate a dataset consisting of generalized coordinates and momenta as inputs and their respective time derivatives as outputs. The trained neural networks are used as surrogate models to simulate the cantilever beam under free and forced conditions. Preliminary findings indicate that HNNs create accurate and efficient surrogate models whilst learning conservation laws. For forced-response simulations, our approach requires analytical calculation of external forces, offsetting the computational Efficiency gains of our surrogate models. The outcomes of this study give initial perspectives and limitations of the use of surrogate models based on HNNs as a means to efficient simulations of highly flexible structures.

A numerical analysis on the drainage capacity of a ropax vessel in the garage deck, in still water and in waves

The ropax vessels are designed to provide the transfer of vehicles, cargo, and passengers and to ensure maximum comfort and safety. The absence of a transversal subdivision for all vessel length poses serious threats to ship safety in case of flooding of the garage deck. When the floodwater is not properly and efficiently drained overboard it can have detrimental consequences both on ship stability, due to large free surface effects and, on loads on deck structures due to the pressure of the accumulated water.In this paper, we propose and apply a fast numerical method for the study of the drainage capacity in the garage deck of a ropax vessel assuming a passive drainage system i.e. comprising only scuppers, whereas the investigated flooding scenario depends on fire fighting operations.The effectiveness of the drainage system is simulated accounting for ship dynamics in calm water and in waves, by adopting a hybrid non-linear numerical model. Flooded water is modelled by lumped mass method; the couplings between ship and floodwater dynamics is achieved by the apparent gravity technique. Two scupper configurations are considered, by modifying scupper dimensions and positions on board. Several numerical simulations are carried out assuming also partially inefficient scuppers. The analysis of the obtained results can provide a general support for the design of passive drainage systems. Based on the outcomes of the case study, this research discloses an improvement in the drainage system when more scuppers are allocated at the stern zone.

An effective modelling approach for multi-body dynamic analysis of epicyclic gear train of GTF considering both structural flexibility and mechanical interactions

In this paper, a new modelling approach is proposed for the dynamic investigation of epicyclic gear train, and the novelty of this work lies in consideration of both structural flexibility and mechanical interactions during the analysis procedure. The method is of capacity to directly present the dynamic results of the supporting structure for convenient practical engineering evaluation and reduce the dimension of the system reasonably with appropriate assumptions for better computational efficiency. Firstly, the mechanical interactions among components are discussed in detail, and the principle that the structural flexibility works is also explained at length. Secondly, taking the epicyclic gear train of the geared turbofan (GTF) engine; for example, the dynamic model of the system is then established based on the developed hybrid user-defined elements. For model validation, the governing equations of the system are also derived by the lumped mass method. Thirdly, with the same values of the parameters, the results of normal dynamic meshing force obtained by the proposed model are compared with the ones by the lumped mass model. It can be stated from the data that (1) the maximum relative error between the theoretical value and the average value calculated by the two models is 8.28%, (2) the gear mesh frequency obtained by the two models are sufficiently close to the theoretical value, and (3) the fluctuation trends of the dynamic force keep basically consistent with each other. In summary, the comparison presented clearly indicates that the proposed model is indeed reasonable, which provides a new way for dynamic investigation and structural redesign of a large epicyclic gear train. Finally, as a practical engineering application, the vibration result of the deformable supporting structure of the GTF gear train is also presented, which directly provides valuable reference for vibration monitoring, fault diagnosis and other engineering problems in practice.

Reduced resonance frequency and enhanced coupling coefficient in fishtailing magnetoelectric resonator

Reducing the resonance frequency of the first-order longitudinal vibration mode for a bulk magnetoelectric (ME) resonator has been a challenge for very low frequency (VLF) magnetic sensing and transmitting application. In this work, we design and fabricate a fishtailing ME resonator that reduces the anti-resonance frequency from 18.8 to 12.2 kHz and the resonance frequency from 18.2 to 11.5 kHz compared with conventional 1–1 typed resonator. Meanwhile, a significant enhancement of the resonant ME coupling coefficient [from ∼8271 to ∼11 400 V/(cm Oe)] is also realized. Finite element simulation and lumped mass method are used to calculate and analyze the principles of the reduced resonance frequency and the reasons for enhanced coupling performance. The geometry tailoring method in this work is believed to inspire applications of the resonant ME effect at very low frequency.

Hydrodynamic behaviors of a spring-mounted fishing net in wave-current combined flows

The flexible fishing nets used in the deep-sea aquaculture cages are usually very large and they are tied tightly on the steel frames of the cages. Under the action of currents and waves, the fishing nets are easily broken at the connections. In this paper, a large fishing net mounted on a steel frame with linear and nonlinear springs is modelled theoretically and its hydrodynamic behaviors in the wave-current combined flows are analysed. The wave-current combined flows are described with the potential flow theory. The flexible fishing net is simulated with the lumped-mass method. The viscous drag forces caused by the fluid flows are calculated with the screen model. The simulation results demonstrate that when the fishing net is mounted with linear or nonlinear springs which have larger stiffnesses, the vibration displacements of the fishing net will become smaller while the dynamic tensions of the net twines will become larger.

Study on the vibration characteristics of worm helical gear drive

The application of worm helical gear in smart car, smart home and other fields can significantly reduce the weight of products. Accordingly, the requirements for vibration and noise are also increased. In this paper, the lumped mass method is used to establish the dynamic model of point contact bending-twisting shaft meshing coupling worm helical gear drive. In this model, the contact ellipse is transformed into an equivalent rectangle, and the nonlinear factors such as time-varying meshing stiffness, tooth meshing error and meshing damping are considered. The vibration characteristics of worm helical gear drive in three directions and at meshing point are obtained for the first time. Machining accuracy, tooth surface modification and assembly errors are transformed into the error in the dynamic model respectively, and their effects on the vibration characteristics are analyzed. The closed vibration and noise performance test bench of worm helical gear drive is built, and the influence of three kind of errors on the vibration characteristics of worm helical gear drive are tested to verify the correctness of the dynamic model and its numerical analysis results. Both the theoretical analysis and the experimental results show that machining accuracy, tooth surface modification and the assembly error have a great influence on the vibration characteristics.

Application of polynomial chaos expansion in sensitivity analysis of towed cable parameters of the underwater towing system

The key to achieving the optimal design of towed cables, maintaining numerical simulation accuracy, and achieving precise control of the towed body lies in sensitivity analysis. However, the traditional global sensitivity analysis method presents challenges such as high calculation costs and low accuracy. To address these issues, this paper introduces polynomial chaos expansion (PCE) to quantitatively analyze the impact of uncertainties in physical and environmental parameters on the position and attitude of the towed cable. Latin hypercube sampling is employed to obtain sample sets of input parameters, and these samples are applied to the lumped mass method to calculate the end position coordinates of the towed cable, which serves as the output response. PCE is utilized to quantitatively compute the Sobol global sensitivity index of the towed cable parameters. The accuracy of the PCE model is verified, and the optimal degree of basis functions is selected using the bias-variance trade-off. The advantages of PCE are demonstrated by comparing it with the Monte Carlo and Morris methods. The results indicate that PCE accurately calculates the global sensitivity index of towed cable parameters even with a limited sample size. Under the condition of a fixed cable length, the position and attitude of the towed cable are sensitive to the current rate, liquid density, cable diameter, normal drag coefficient, and specific gravity. The feasibility and efficiency of PCE applied to the sensitivity analysis of towed cable parameters is verified, and recommendations for the engineering application of towed cables are summarized.

Study on hydrodynamic characteristics and depth control of the towed sensors array system

The hydrodynamic characteristics and depth control of the towed sensors array system (TSAS) are investigated based on lumped mass method and computational fluid dynamics (CFD). Firstly, the flexible towed cable is discretized into multiple rigid links with the lumped mass point, and the terminal point is the towed underwater vehicle (TUV). Coupling analysis of each sensors integrated module to a finite length of towed cable is performed, taking the module as the lumped mass point. The variation of normal and tangential forces with the change of angle are discussed for different towed speeds. Then, the dynamics model of the TUV is established and solved. The equilibrium equation for each lumped mass point is established using the recursive method from the bottom up. The angle between each link and the horizontal plane is calculated. Finally, the coordinate of each node is derived, and the position of the towed cable is fitted. The study shows that the hydrodynamic characteristics of the module and towed cable significantly impact the depth control and profile measurement. Sea trials validate the reliability of the method proposed in this paper, which is a guide to the design and manipulation of the TSAS.

Study on the seismic model of a special shelf structure

In this paper, the shelf structure is discretized, reduced in order, examined, and studied for seismic response by using the concentrated mass method, Lagrange method, transfer function method, and fourth-order Runge-Kutta method based on MATLAB numerical calculation. A simplified model of special equipment including supporting frame structure is proposed, and the calculation methods of equivalent mass, stiffness, and damping of supporting frame structure are given. In addition, the seismic response time-history curve of each position of the shelf structure is quickly obtained by using the lumped mass method in this paper. The conclusion is that the seismic response of the upper floor is greater than that of the lower floor, and the seismic response of the middle position of the same floor is greater than that of both sides. This paper provides a research foundation for the follow-up work. Research in this paper verifies the rationality of simplifying the complex dynamic model of the shelf structure continuum into a simple basic model, which provides a simplified model with a high fitting for the follow-up rub-impact analysis and research, and greatly improves the calculation efficiency.

Simulation of anchor chain based on lumped mass method

In order to develop a anchoring operation simulation system and improve safety during anchoring operations, a relatively accurate mathematical model of anchoring operations needs to be established. In this paper, the stress condition of anchor chain under environmental and subsea geological conditions is further studied and the stress condition of anchor chain is analyzed based on the previous research. In this paper, a quasi-static model based on catenary method is used as the basis of dynamic analysis, and the dynamic model of anchor chain is established based on the concentrated mass method, which fully considers the influence of anchor chain weight, hydrodynamic force, ocean current and interaction with the seabed. The fourth-order Runge Kutta method was used to solve the model numerically, and a calculation procedure was developed. The accuracy of the model was verified by comparing the calculated results with the experimental results, indicating that the constructed anchor chain dynamics model has a high accuracy.

A novel multi-module coupled algorithm for the wreck off-bottom simulation

This paper proposes a novel multi-module coupled algorithm for the wreck off-bottom simulation (MCAWOS), involving several structures, multiple mediums, and three modules: the wreck-soil interaction module, the lifting-buffering module, and the barge-cable coupled module. In the present algorithm, the finite element method, the potential flow method and the lumped mass method are integrated and used to simulate the off-bottom process. The dynamic response characteristics of a typical salvage case with rated lifting tensions are investigated. It is found that the salvage system in waves exhibits a special breakout phenomenon, where the wreck displacements increase abruptly, and uneven distributions of the lifting tensions are observed. Case studies are conducted to further ascertain the mechanism of the breakout. As the rated lifting tension increases, the salvage efficiency rises, while the maximum roll and pitch amplitudes of the wreck decrease. Meanwhile, larger rated tension may result in more waste in equipment lifting capacity, thus 229 ton is finally selected as the appropriate preset lifting tension. In addition, when the wave direction angle increases from 90∘ to 180∘, the maximum roll value of the wreck shows a roughly decreasing trend and the planar deflection of the salvage barge relative to the wreck decreases. However, the maximum values of pitch, yaw, lifting tension and Mises seabed soil stress show irregular fluctuations. Taking all dynamic indicators into account, the wave heading interval of 150∘ to 180∘ is expected to be an ideal operating window. Conversely the 105∘ case is a dangerous state.

Dynamic analysis of the umbilical cable pull-in operation through J-tube under different wave directions

Pipelines and cables are widely used for transporting oil resources in offshore oil exploitation. J-tubes are a well-known means of placing small-diameter pipeline risers and umbilical connections on offshore platforms. In this paper, OrcaFlex develops a model of an umbilical cable J-tube pull-in installation based on an umbilical cable cross section design. The hydrodynamic response to various wave directions is studied using the lumped mass method. The results show that the longitudinal waves have little effect on the maximum effective tension of the section under water. The transverse waves will cause the cable to vibrate violently in the transverse direction. The mean shear force under the wave direction of 120° is the largest, followed by 60° and 75°, and finally 90° and 105°. In addition to the hydrodynamic response data, the mechanical characteristics of the umbilical cable and the factors that should be considered in the pull-in installation of the J-tube are provided with significance for practical application of engineering.

Impact Dynamic Analysis of Suspended Marine Cables

Objective: It is often impossible to avoid collisions between marine cables, the impact dynamic response of the collisions between marine cable needs to be obtained. Methods: In this paper, we derive the lumped mass method based on the bending moment and torque; using the lumped parameter method, the suspended cable is discretized into lumped mass models, and the impact force of the suspended cable is considered in the dynamic analysis; the impact dynamics model of the suspension cable is developed by referring to the specific parameters of the suspension cable and combining them with offshore construction processes. Results: There is a relationship between the impact force and the tension of the cable. The tension and impact force are related in cable collisions. While the maximum effective tension occurs at the upper end of the vertical cable and at the collision point, the horizontal cable's tension at the collision position is minimal. The distribution of curvature mutates at the collision point, causing bending deformation. The horizontal cable experiences greater bending change at the collision position than the vertical cable. Conclusion: In summary, collision of different parts of vertical cable is not obvious, while horizontal cable is more prone to self-collision and collision with vertical cable. Collision impact on horizontal cable is greater, with minimum tension occurring at collision point. Maximum velocity for horizontal cable appears at collision position, and for vertical cable at bottom. Maximum acceleration for both cables occurs at collision zone.

Research on the Design of Phononic Crystal Rod Band Gaps Based on Lumped Mass Method

Research on the Design of Phononic Crystal Rod Band Gaps Based on Lumped Mass Method

Influence of Different Static Equilibrium Calculation Methods on the Dynamic Response of Marine Cables during the Releasing Process: Review and a Case Study

When analyzing the dynamic characteristics of marine cables, the static equilibrium state must first be calculated; a dynamic analysis can then be carried out based on the static equilibrium. Since the calculation of the static equilibrium is the basis of dynamic calculation, different methods for the calculation of the static equilibrium will have important impacts on the dynamic response characteristics of cables, which result in cables having different dynamic characteristics. This paper summarizes the research progress in the effects of different methods for the calculation of the static equilibrium on the dynamic response of marine cables during the releasing process, and current methods for their static calculation are more comprehensively developed as well as being more accurate. To study the influence of different static equilibrium calculation methods, with the reference of some specific parameters of a mooring cable, combined with specific sea conditions, through the necessary simplification of the cable releasing process, based on the lumped mass method, mooring cables are discretized into the lumped mass model, and dynamic analysis models of the releasing process of mooring cables under three methods for the calculation of the static equilibriums (the fast static equilibrium calculation method, the analytic catenary calculation method, and the catenary calculation method) are established. The dynamic characteristics of the spatial configurations of mooring cables based on different static equilibrium calculation methods are obtained through time domain coupling analyses. It was found that if the static equilibrium of a cable is calculated, taking into account gravity, buoyancy, wave current resistance, inertia forces, and the axial stiffness of a cable, then the characteristics of a cable during dynamic descent are closest to that of a real situation. The calculation results have a certain guiding significance for specific engineering practices.

Vibration energy accumulation and absorption characteristics of pseudo acoustic black hole wedge

The Acoustic Black Hole (ABH) is an effective structure for harvesting and dissipating energy from flexural waves. It works by reducing the thickness of the ABH wedge in a power law relationship with m ≥ 2, which rapidly decreases the local phase velocity of flexural waves and leads to an energy convergence effect. However, the vibration performance may differ if the tapered wedge does not meet the requirements of a typical ABH. This article introduces and discusses two types of “pseudo-ABH” wedges: linear-degenerate ABH and step-degenerate ABH. These wedges do not follow the power-law and continuity conditions of typical ABH, respectively. To establish the dynamic models of pseudo-ABH wedges, the lumped-mass transfer matrix method (LTMM) and traveling wave analysis are introduced and validated using analytic solutions of the non-uniform one-dimensional Euler-Bernoulli beam and finite element method. Meanwhile, a good match between the results of the Timoshenko and Euler-Bernoulli beam theory also indicates the correctness of our model. The energy ratio and reflection coefficient are defined to characterize the energy accumulation effect and vibration absorption performance of the pseudo-ABH wedges. The vibration performance of the linear-degenerate ABH is closer to the typical ABH than the step-degenerate ABH. The step-degenerate ABH has more effective energy accumulation and reduction performance for individual frequencies. However, as the number of segments in the step-degenerate ABH increases, its dynamic performance becomes closer to that of the typical ABH. The novelty of this article is to study vibration energy accumulation and absorption characteristics of pseudo acoustic black hole wedges using LTMM model, disclose the differences of linear and step degenerate ABH for their performance, and recommend the step degenerate ABH in place of the typical ABH of a power law thickness relationship.

Research on the design of phononic crystal shaft bandgaps based on lumped mass method

Phononic crystals tend to mature in the research of forward problems due to their unique bandgap characteristics, but it is difficult to realize reasonable parameter design at specific bandgap frequencies in practical engineering applications. Thus, based on the lumped mass method, a research method for the inverse problem of phononic crystal shaft bandgaps is proposed in this paper. On the basis of the discretization model, the relationship between polynomials and the characteristic of bandgaps is discussed through derivation of the dispersive polynomial, and the feasible regions of the corresponding parameters are solved according to the bandgap existence criterion established by the characteristic rules for analytical verification. The results show that the bandgap existence criterion for frequency points and frequency bands can accurately and effectively invert the relevant parameters that meet the requirements, and the method is of great practical significance for promoting the development of phononic crystal shaft parameter design under the target bandgap in engineering applications.

Dynamic analysis of towed cable with variable length during turning maneuvers

The configuration of marine towing cable changes significantly during the turning process, with the rotating procedure with fixed cable length being the most frequent. To overcome these challenges, the configuration and dynamic properties of the marine towing cable must be addressed. However, under some particular operating situations, the tugboat must release the marine towed cable during rotation, resulting in a constant change in the length of the marine cable. In view of this, the towed cable is discretized into a lumped mass model based on the lumped mass method, and the dynamic analysis model of the rotation process of towed cable with variable length under different release speeds and different depths is established. This is done with reference to the specific parameters of a towed system, combined with the specific sea conditions of a particular sea area. Time-domain coupling analysis is used to determine the dynamic changes in configuration and stress of marine towing cable at various release speeds and depths. The results of the calculations have some guiding relevance for a certain engineering practice.