Galerkin Truncation Method Research Articles

Dynamic characteristics and vibration control of composite laminate wall panels in electric aircraft using NiTi shape memory alloys

This paper investigates the dynamic characteristics and vibration control of a new-energy electric aircraft cockpit wall panel. An innovative boundary-splitting Rayleigh-Ritz method is introduced for modal analyses of wall panel structures in thermal environments, verified with the hammering and finite element methods, showing a maximum error of 5.42 %. NiTi shape memory alloy (SMA) wires are embedded in composite structures for vibration control, with the first theoretical-experimental validation performed. A parametric fitting-computational model for NiTi-SMA is developed to extract mechanical parameters from hysteresis data. Theoretical models are decoupled into dynamical equations using the Galerkin truncation method. The damping capability of NiTi-SMA is demonstrated by solving the amplitude using the harmonic balance and Runge-Kutta methods. Room-temperature and variable-temperature vibration tests show over 30 % vibration control capability with optimized embedding methods. These experimental results confirm the theoretical findings, providing support for the analysis and vibration control of new-energy aircraft.

Forced vibration analysis of a beam-plate system coupled through multiple nonlinear single-freedom-degree systems

A beam-plate coupling system is widely used in the construction of different frameworks across industries according to engineering application requirements. In this paper, a nonlinear single-degree-of-freedom structure is introduced as a coupling element between the beam and the plate. Therefore, a coupling system connected through a plurality of the nonlinear single-degree-of-freedom structures is proposed with exposure of its operational mechanism and the dynamic behavior analysis as required. The nonlinear coupled governing equations of the beam-plate system with series nonlinear energy sinks are established according to the Hamilton principle and solvable by the Galerkin truncation method. Meanwhile, the nonlinear characteristics of transverse vibration of the coupling system are studied for discussing the effect of nonlinear stiffness on a low-frequency response of the beam-plate system. Further, the transverse vibration of the system may vary with changes in the nonlinear stiffness, the external viscous damping, and the motion mass of the nonlinear single-degree-of-freedom structures. Accordingly, the variations along with these factors are investigated. The results indicate that a reasonable nonlinear single-degree-of-freedom structure can have a significant influence on the transverse vibration characteristics of the beam-plate system.

The influence of coupling nonlinearities on the dynamic behavior of a beam-plate system

Considering the wide application of beam-plate systems in engineering, this work studies the dynamic behavior of the beam-plate system connected by coupling nonlinearities. The governing equations of the beam-plate system are derived and then solved by employing the Galerkin truncation method (GTM). The correctness and stability of dynamic responses of the beam-plate system with coupling nonlinearities are studied, where a four-truncation number of the beam combined with a six-truncation number of the plate can guarantee the stability of the system's dynamic responses. Then, the influence of coupling nonlinearities on the dynamic behavior of the beam-plate system is studied. By analyzing the numerical results, the dynamic behavior of the beam-plate system with coupling nonlinearities can be effectively controlled by adjusting the parameters of coupling nonlinearities in a reasonable range. Complicated responses of the beam-plate system are motivated by coupling nonlinearities, where the working state of complex responses is quasi-periodic. The employment of coupling nonlinearities provides a possible way to control the frequency components of the vibration signal belonging to the beam-plate system. A reasonable use of coupling nonlinearities plays a positive effect on the vibration control of the beam-plate system.

Longitudinal vibration control of a double-rod system by employing nonlinear energy sinks

This study aims to potential the potential utilization of nonlinear energy sinks (NESs) for controlling longitudinal vibrations in a double-rod system. The research introduces a longitudinal vibration prediction model for a double-rod system equipped with NESs. The generalized Hamilton principle is employed to derive governing equations of the double-rod system. The longitudinal vibration responses of the double-rod system are numerically solved through the application of Galerkin truncation method. The longitudinal vibration responses of the double-rod system are impacted by NESs, as they yield accurate numerical results. The installation of both NES 1 and NES 2 concurrently is recommended for mitigating the vibration of the double-rod system. Under reasonable single-frequency excitations, modifying the parameters of NESs can significantly alter both the vibration state and magnitudes of vibration in the double-rod system. Furthermore, the synchronous optimization of parameters in NES 1 and NES 2 is crucial for effectively controlling vibrations in the double-rod system. Sensitive parameter areas of NESs provide the possibility of controlling the vibration of the double-rod system by utilizing NESs.

Nonlinear vibration control of interconnected functionally graded fluid-conveying pipeline

In this study, an exploratory vibration experiment is established and an interconnected functionally graded material (FGM) pipeline system is installed. This experiment represents the first time that NiTiNOL-steel wire ropes (NiTi-ST) are applied to FGM pipeline system. And the experimental results demonstrate that the NiTi-ST has a practical vibration damping effect. Based on the experiment, a further theoretical pipeline system model is created. The effects of fluid velocity, temperature difference, and clamp stiffness on the modal and response of the pipeline system are added. The modal analysis part is solved through the implementation of the Rayleigh-Ritz approach using a set of modified orthogonal polynomials. In order to verify the accuracy of the Raylei-Ritz calculations, the finite element method (FEM) is also used. The response analysis part is solved through the Galerkin truncation method (GTM) and harmonic balance method (HBM). To match the harmonic balance solutions, the Runge-Kutta method (RK) are adopted. The results of the response analysis reveal that NiTi-ST demonstrates significant nonlinear vibration control capabilities to the interconnected FGM pipeline system under various conditions.

A study of controlling the transverse vibration of a beam-plate system by utilizing a nonlinear coupling oscillator

In the engineering field, prolonged exposure of complex structures to high-strength vibration environments can lead to various engineering challenges. This underscores the importance of effectively managing the vibration of complex structures. Given the extensive utilization of beams and plates in the construction of intricate structures, this study incorporates a nonlinear coupling oscillator into the beam-plate system and formulates a vibration analysis model for this system. The governing equations of the system are derived theoretically and solved numerically using the Galerkin truncation method. This study focuses on the operational modes of the nonlinear coupling oscillator based on accurate numerical results. Meanwhile, the influence of parameters belonging to the nonlinear coupling oscillator on the transverse vibration responses of the beam-plate system is systemically studied. Upon meticulous examination of the numerical results, the operational states of the nonlinear coupling oscillator encompass the normal vibration suppression mode and the quasi-periodic vibration suppression mode. The alteration in the linear elastic coupling stiffness of the beam-plate system impacts the effectiveness of vibration suppression in the nonlinear coupling oscillator. This influence stems from the fact that changes in the linear elastic coupling stiffness directly affect the nonlinear forces exerted on the beam-plate system within a nonlinear coupling oscillator. Within a feasible range, the nonlinear stiffness and viscous damping can be chosen as the adjustable parameters for the nonlinear coupling oscillator to regulate the vibration of the beam-plate system by adjusting the motion mass of the nonlinear coupling oscillator, which presents a challenge. The appropriate utilization of the nonlinear coupling oscillator demonstrates favorable control effectiveness in managing the transverse vibration of the beam-plate system. The study presented in this work offers a method to simultaneously control the vibration of each component within beam-plate coupling systems.

A study of analyzing longitudinal dynamic behavior of a double-rod system with longitudinal nonlinear supports

In engineering, shafting systems are typically subjected to longitudinal vibration excitations, which may result in unwanted vibration. To study the control of longitudinal vibration in shafting systems, they can be simplified to rod structures. Currently, engineers have attempted to apply the nonlinear principle to design nonlinear supports to control the vibration of flexible structures. However, the flexible structures referenced in the literature are usually composed of a single component, which limits the application of nonlinear supports to more complex structures. To explore the potential application of nonlinear supports in marine engineering, this work introduces a longitudinal vibration prediction model for a double-rod system equipped with longitudinal nonlinear supports. The generalized Hamilton principle is used to derive the governing equations for the double-rod system with longitudinal nonlinear supports. The longitudinal vibration responses of the double-rod system are numerically solved using the Galerkin truncation method. The numerical results confirm that a 1-term truncation number guarantees the stability of the longitudinal vibration prediction model. Under certain conditions, the longitudinal vibration responses are significantly affected by longitudinal nonlinear supports. It is recommended to install longitudinal nonlinear supports on both Rod 1 and Rod 2 simultaneously to suppress vibration in the first two main resonance orders. With reasonable excitations, the vibration state and magnitudes of the double-rod system can be effectively controlled by adjusting the longitudinal nonlinear supports. Complex longitudinal vibration responses are more readily induced by altering the parameters of the longitudinal nonlinear support installed on Rod 1. Choosing appropriate parameters for the nonlinear supports on Rod 1 and Rod 2 positively contributes to the reduction of vibration in the double-rod system.

Transient Response of Sandwich Plates with Corrugated Core Under Mechanical-Thermal Loads

The nonlinear dynamic responses of the corrugated sandwich plate under mechanical-thermal loads are studied and analyzed. Based on hyperbolic parabola shear deformation plate theory (HPSDT), the partial differential equation of the sandwich plate with the corrugated core is established. Using the Galerkin truncation method, the nonlinear motion equation is derived. By the solution, the response of the corrugation plate with simply supported four edges at different temperatures is obtained and validated. Then the transient responses of the corrugated sandwich plate under different impact loads are analyzed, and the effects of the base materials, the corrugation types and the structural parameters of the corrugated plate on the transient responses are discussed in detail. The research provides a reference for improving the impact resistance of the sandwich plate in practical application.

Dynamic analysis of a plate system coupled through several nonlinear spring-mass couplers

Nowadays, plates have been widely used in various engineering fields, where nonlinear additional equipment is designed to control the vibration of plates. However, the current studies mainly concentrate on the single-plate system with nonlinear additional equipment. Seldom studies use nonlinear couplers to connect the plate system, making the working mechanism of nonlinear couplers on the vibration responses of the plate system unclear, limiting the application of nonlinear couplers on vibration control of the plate system. To explore the potential application of nonlinear couplers, this work introduces nonlinear spring-mass couplers to connect the plate system, where the transverse vibration analysis model of the plate system coupled through nonlinear couplers is established. The Galerkin truncation method (GTM) is used to predict vibration responses of the plate system coupled through nonlinear couplers. After ensuring the validation of numerical results, forced vibration responses of the plate system coupled by nonlinear couplers are investigated. According to numerical results, responses in the vibration of the plate system are influenced by nonlinear couplers greatly. Choosing suitable parameters for nonlinear couplers plays an important role in the vibration control of the plate system. Under a certain parameter range of nonlinear couplers, responses in vibration present complex characters. Under complex responses, the target energy transfer is present in the vibration responses of the plate system.

Vibration energy characters study of a soft-core beam system coupled through nonlinear coupling layers

To ascertain the potential impact of nonlinear coupling layers on the vibration control of composite beam systems, this study develops a vibration analysis model of a nonlinear coupling-layered soft-core beam system. Ascertain the accuracy of the vibration responses computed by the soft-core beam system using the Galerkin truncation method (GTM) in comparison to the results obtained through alternative methods. The correct calculation results indicate that nonlinear coupling layers are the cause of nonlinear phenomena in the soft-core beam system. Nonlinear coupling layers are responsible for nonlinear phenomena of the soft-core beam system, such as peak-jumping and the region with multiple continuous amplitudes, according to numerical results. To reduce vibrations in the soft-core beam system's resonance regions, reasonable parameters for the nonlinear coupling layers are advantageous. When the soft-core beam system is excited at a single frequency, the kinetic energy responses include converted values of nonlinear coupling layers. Using nonlinear coupling layers, it is possible to establish an effective parameter range for the vibration energy control of the soft-core beam system. The utilization of reasonable parameters in nonlinear coupling layers can effectively mitigate the vibration energy of a soft-core beam system that is excited at a single frequency. The soft-core beam system induces the desired energy transfer phenomena at specific time intervals and under particular conditions.

Vibration control of composite laminate via NiTiNOL-steel wire ropes: Modeling, analysis, and experiment

To meet the requirements of vibration control for composite structures, the NiTiNOL steel wire ropes (NiTi-ST) is selected to control the vibration of the composite laminate. The dynamical modeling is built by integrating the composite laminate with the modified Bouc-Wen model. The dynamical model is discretized by Galerkin truncation method (GTM), and theoretical analysis is carried out using the harmonic balance method (HBM), Runge-Kutta method (RK) and finite element method (FEM). Based on the results of dynamical modeling and theoretical analysis, an experimental device is designed and manufactured. The results show that the NiTi-ST has excellent vibration control ability for composite laminate, and the impact on the natural frequency of the system is small.

A novel way for vibration control of FGM fluid-conveying pipes via NiTiNOL-steel wire rope

In this study, a coupling model of fluid-conveying pipes made of functionally graded materials (FGMs) with NiTiNOL-steel (NiTi-ST) for vibration absorption is investigated. The vibration responses of the FGM fluid-conveying pipe with NiTi-ST are studied by the Galerkin truncation method (GTM) and harmonic balance method (HBM). The harmonic balance solutions and the numerical results are consistent. Also, the linearized stability of the structure is determined. The effects of the structure parameters on the absorption performance are also studied. The results show that the NiTi-ST is an effective means of vibration absorption. Furthermore, in studying the effect of the NiTi-ST, a closed detached response (CDR) is first observed. It is noteworthy that the CDR may dramatically change the vibration amplitude and that the parameters of the NiTi-ST may determine the emergence or disappearance of the CDR. This vibration absorption device can be extended to offer more general vibration control in engineering applications.

Transverse Dynamic Responses and Optimization of a Flexibly Constrained Beam with Multiple Nonlinear Supports that Present Cubic Stiffness

This paper establishes a simplified model of a flexible beam with multiple nonlinear supports that present cubic stiffness to study the potential application of cubic nonlinearities in the vibration control of beams. The Lagrangian method (LM) is used to predict transverse dynamic responses of the flexible beam with multiple nonlinear supports that present cubic stiffness, whereas the harmonic balance method (HBM) and Galerkin truncation method (GTM) are utilized to study the accuracy of the LM. Against the background, the effect of nonlinear supports that present cubic stiffness on the nonlinear transverse dynamic responses of the beam is studied. For this study, the accuracy of the LM is guaranteed under the 4-term truncation number. Nonlinear transverse dynamic responses of the flexibly constrained beam with multiple nonlinear supports that present cubic stiffness are sensitively influenced by their initial calculation values. For different boundary conditions or working statuses, the maximum restoring forces at both ends of the flexible beam can be suppressed effectively by optimizing the parameters of nonlinear supports that present cubic stiffness. Appropriate parameters of nonlinear supports that present cubic stiffness are good at vibration control of the flexibly constrained beam. In addition, complicated dynamic responses of the flexible beam with multiple nonlinear supports that present cubic stiffness are motivated under some inappropriate parameters of nonlinear supports.

Vibration isolation performance of a beam clamped with torsional quasi-zero-stiffness isolator

Elastic beam structures connected by hinges are a common basic component of many mechanical systems. In this study, the nonlinear transverse vibration of a beam with one end free and the other end simply supported with a torsion spring is investigated. A torsional quasi-zero-stiffness vibration isolator is employed at the fixed end of the beam for isolating the vibration excitation from the base. Using Hamilton’s principle, the differential equation of motion for the transverse vibration of the beam is obtained, and the natural characteristics of the beam are determined. The dynamic equation of the system is derived by using Galerkin truncation method and the harmonic balance method is adopted to obtain the nonlinear vibration response. Compared with the vibration transmissibility of the beam supported by the equivalent linear torsion spring, the quasi-zero-stiffness vibration isolation system shows superior vibration isolation performance. It is found that the beam fixed with a torsional quasi-zero-stiffness isolator can reduce the first resonant frequency and provide a wide effective isolation range at low frequency, which is rarely affected by the strong nonlinearity. The results would provide an innovative method of introducing the torsional quasi-zero-stiffness isolator into the passive vibration control of continuum system.

An out-of-plane vibration model for in-plane curved pipes conveying fluid

Vibration characteristics of curved pipes conveying fluid have been studied extensively. However, most researches are limited to the in-plane transverse vibration. There is still a lack of effective modeling and corresponding research for the out-of-plane vibration of an in-plane curved pipe. For this purpose, a dynamic model of out-of-plane vibration of an in-plane curved pipe conveying fluid is established for the first time. Based on the force analysis of the pipe and the fluid, the governing equations of the out-of-plane vibration are derived analytically. By the transformation of arc length coordinates and global coordinates, a new description of the out-of-plane vibration of bending and torsion is introduced. The critical velocity, natural frequencies and modes of the curved pipe are calculated based on the Galerkin truncation method. The results show that the pipe with larger in-plane curvature has lower critical velocity of out-of-plane motion. Furthermore, there is a significant decrease in the natural frequencies with the increase of curvature and fluid velocity. In conclusion, this paper provides a new dynamic model for studying the out-of-plane vibration of in-plane curved pipes. Meanwhile, the dynamic characteristics of the out-of-plane vibration of the curved pipe are also revealed for the first time.

The performance of nonlinear vibration control via NiTiNOL–Steel wire ropes

Conventional nonlinear vibration absorbers must provide additional mass and motion space to achieve the expected vibration control effect, which will significantly alter the original design of the target structure. In this study, the shape memory NiTiNOL–steel (NiTi–ST) wire rope and composite plate are coupled to investigate the vibration control effect of such a configuration. The vibration response of the model is studied using the Galerkin truncation method (GTM) and the harmonic balance method (HBM), and the stability of the frequency response curve is determined using the linearized stability theory. NiTi–ST wire rope is found to have a good damping effect for both low- and high-order resonances. Under a strong external excitation, the appearance of the closed detached response (CDR) may cause system amplitude mutation; however, it can also achieve effective inhibition. Finally, the effect of the structural parameters on the vibration control is discussed. The nonlinear damping term in NiTi–ST determines the soft and hard characteristics of the system, while the equivalent linear damping term is mainly responsible for vibration suppression. Therefore, as a novel shock absorber, NiTi–ST can be applied to engineering structures in order to prevent vibration damage.

Transverse forced nonlinear vibration analysis of a double-beam system with a supporting nonlinearity

To exploit the potential application of supporting nonlinearity in marine engineering, an attempt is made to establish the transverse forced vibration analysis model of a double-beam system supported by a spring-mass system that is nonlinear. This kind of vibration system consists of two beam sections, boundary supports, a coupling component, and a nonlinear spring-mass arrangement. The variational approach and the generalized Hamiltonian concept are used to develop the governing equations of such a double-beam system. The Galerkin truncation method (GTM) is a technique for obtaining the governing equations’ residual equations. By solving the associated residual equations numerically, the nonlinear responses of the double-beam system can be figured out. The GTM has good solidity and correctness in the prediction of the vibration system’s forced transverse vibration. The dynamic responses of the double-beam structure supported by a spring-mass system that is nonlinear are subtle to their initial calculation values. Appropriate parameters of the nonlinear support will subdue the level of vibration at the boundary of the double-beam system. In contrast, unsuitable parameters of the nonlinear support motivate complex dynamic responses of the double-beam system and harmfully influence the vibration repression at the boundary of the vibration system.

Nonlinear transverse vibrations of a jointed structure with two slightly curved beams connected by complex elastic joints

The research of jointed structures has attracted extensive attention and most of the current research focuses on the dynamic modeling and analysis of straight-beam jointed structures, while sometimes the jointed structures are composed of curved beams in engineering practice. Therefore, this paper aims to propose a novel nonlinear dynamic modeling method regarding a jointed structure with two slightly curved beams (SCBs) connected by complex elastic joints. Firstly, the nonlinear dynamic analytic model of a single SCB is established as one part of the jointed structure. Secondly, the idea of the global mode method is used to establish the entire nonlinear dynamic analytic model of a two-SCB jointed structure and obtain its natural frequencies and mode shapes. The dynamic characteristics of the complex elastic joints are subsequently introduced into the entire nonlinear dynamic analytic model of the jointed structure by applying the equilibrium conditions and boundary conditions. Thirdly, a procedure to calculate the forced vibration responses of the two-SCB jointed structure is proposed by successively using the Galerkin truncation method, harmonic balance method and pseudo arc-length method. Finally, the proposed nonlinear dynamic modeling method is validated by carrying out case studies and the results calculated by the proposed method are in good agreement with the results of finite element analysis, which demonstrates the accuracy of the proposed nonlinear dynamic modeling method. Furthermore, the effects of stiffness at the complex elastic joints, external excitation amplitude, and initial curvature on the nonlinear dynamic characteristics of the jointed structure are also analyzed by the comparative case studies.

Measurement of axial motion yarn tension based on transverse vibration frequency

In the process of textile production, the detection of yarn tension is very important to ensure product quality. In order to detect yarn tension efficiently and conveniently, a non-contact detection method based on transverse frequency was established. Firstly, the governing equation of yarn motion was derived by using the Hamilton principle and solved by the Galerkin truncation method. Then, the natural frequencies of yarns with different linear density were calculated according to the yarn characteristic equation. Finally, the fitting formulas of tension, speed, and frequency of yarn axial motion were derived. By measuring the natural frequency of yarn, the yarn tension could be calculated conveniently and quickly. A high-speed camera was used to measure the transverse vibration displacement of yarn and detect the free vibration frequency of yarn. Through experiments, the results of tension sensor and yarn tension detection based on vibration frequency were compared, and the results show that the error was within 5%, which verifies the effectiveness and accuracy of this method.

Nonlinear Dynamic Behavior of a Generally Restrained Pre-Pressure Beam with a Partial Non-Uniform Foundation of Nonlinear Stiffness

For some transmission shafting, nonlinear supports are connected to beam structures through the pattern of the contact surface. Nonlinear foundations are typically installed to the beam structure for a limited range. The unsuitable installation mode of nonlinear foundations will make the parameters of nonlinear foundations no longer uniform. Most existing studies ignore the boundary rotational restraints of beam structures and concentrated mass introduced by the nonlinear supports or foundations. To improve the engineering acceptance of beam structures with nonlinearity, it is of great significance to study the dynamic behavior of the generally restrained pre-pressure beam structure with a partial non-uniform foundation of nonlinear stiffness. This study establishes the nonlinear vibration model of the beam structure with a local non-uniform nonlinear stiffness distribution. Nonlinear dynamic responses of the beam structure are predicted by the Galerkin truncation method (GTM). Mode functions of the generally restrained pre-pressure beam structure are set as the trail and weight function. The correctness of the GTM for dynamic prediction of the beam structure with a partial non-uniform nonlinear foundation is verified by using the harmonic balance method (HBM). The influence of sweeping ways of excitations and parameters of the partial non-uniform nonlinear foundation on nonlinear dynamic responses of the beam structure are investigated. Dynamic responses of the beam structure with a partial non-uniform nonlinear foundation are sensitive to their initial values. Vibration states of the beam structure are transformed effectively by changing parameters of the partial non-uniform nonlinear foundation. The vibration at both ends of the beam structure can be suppressed by applying suitable parameters of the partial non-uniform nonlinear foundation.