Vibration Differential Equation Research Articles

DETERMINATION OF FORCE FACTORS ACTING ON A SPRING WHEELED MACHINE BODY WITH HYDRO-PNEUMATIC SUSPENSION

A general mathematical model of the movement of a wheeled armored combat vehicle over unevenness is considered, which allows conducting research on improving their tactical and technical characteristics. The calculation scheme of the oscillating system "sprung body - suspension - unsprung masses - profile of road irregularities" was developed. A system of differential equations of vibrations of the sprung body has been compiled, which takes into account the fluctuations of unsprung masses, suspension breakdowns, separation of the wheels from bumps and the possibility of the sprung body hitting the intertrack space. With the use of previously developed methods for calculating the kinematics and dynamic load of the hydropneumatic suspension and its proposed kinematic scheme, dependencies were obtained for calculating the force factors that act on the sprung body from the side of the suspension nodes and when it hits the intertrack space. With the help of experimental dependencies obtained for military tracked vehicles when their front (guiding or leading) wheel hits the ground, a methodology for calculating the impact of a sprung body on the intertrack space has been developed. The obtained results make it possible to carry out research on improving the tactical and technical characteristics of these machines and to evaluate and reduce the loads on the armored hull when moving over bumps, by optimizing the characteristics of the suspension system, which is especially important for lightly armored vehicles.

Vibration characteristics analysis of multi-layer composite cylindrical shell casing with different layer thickness ratios under external pressure using energy method

PurposeAero-engine casings commonly use composite cylindrical shell structures with excellent properties such as corrosion resistance and fatigue resistance. Still, their vibration behavior is relatively complex and may cause fatigue vibration damage, so it is essential to analyze the vibration characteristics of composite cylindrical shells. The purpose of this paper is to analyze the vibration characteristics of multilayer composite cylindrical shells subjected to external pressures and having different interlayer thickness ratios and provide some theoretical basis for the fatigue damage prediction of cylindrical shell casing to ensure the safety and stability of the engine during flight.Design/methodology/approachFirstly, the vibration differential equation with external pressure is established based on Soedel theory considering nonlinear effects, while four symmetric boundary conditions are chosen to constrain the cylindrical shell. Then the Rayleigh–Ritz method, which is more efficient and accurate in calculating large structural systems, is applied to solve the problem, and the theoretical model of three-layer cylindrical shell under external pressure is established. The accuracy of the model is verified by comparing the data with the specialized literature. Subsequently, the effects of different external pressures and different thickness-to-diameter ratios, different length-to-diameter ratios and different interlayer thickness percentages on the natural frequency of multilayer composite cylindrical shells were investigated by control variable analysis.FindingsThe conclusions obtained show that the external pressure increases the natural frequency of the cylindrical shell and that the frequency characteristics of the cylindrical shell vary for different boundary conditions. The effect of length-to-diameter ratio, thickness-to-diameter ratio and the percentage of the thickness of the intermediate layer on the natural frequency of the cylindrical shell are significantly increased under external pressure. Because the presence of external pressure increases the frequency of the cylindrical shell by about 70%, it has almost no effect on the frequency at the minimum number of circumferential waves, and the effect on the frequency at the maximum number of circumferential waves is reduced to about 50%. The frequencies in the SL-SL boundary condition are all in perfect agreement with the S-S boundary condition under the influence of different influencing factors.Originality/valueIn this paper, the effect of external pressure and the natural properties of the cylindrical shell under external pressure on the cylindrical shell’s frequency is considered, emphasizing the effect of different layer thickness ratios on the frequency. This paper aims to summarize the changing law between the natural frequency of the cylindrical shell itself and different design parameters during the flight pressure process. Reliable theoretical predictions are provided for analyzing the vibrational behavior of shells subjected to external pressures in aerospace, as well as a database for the practical production of cylindrical shells.

Optimization of Dynamic Characteristics of Rubber-Based SMA Composite Dampers Using Multi-Body Dynamics and Response Surface Methodology

The suspension system of a commercial vehicle cab plays a crucial role in enhancing ride comfort by mitigating vibrations. However, conventional rubber suspension systems have relatively fixed stiffness and damping properties, rendering them inflexible to load variations and resulting in suboptimal ride comfort under extreme road conditions. Shape memory alloys (SMAs) represent an innovative class of intelligent materials characterized by superelasticity, shape memory effects, and high damping properties. Recent advancements in materials science and engineering technology have focused on rubber-based SMA composite dampers due to their adjustable stiffness and damping through temperature or strain rate. This paper investigates how various structural parameters affect the stiffness and damping characteristics of sleeve-type rubber-based SMA composite vibration dampers. We developed a six-degree-of-freedom vibration differential equation and an Adams multi-body dynamics model for the rubber-based SMA suspension system in commercial vehicle cabins. We validated the model’s reliability through theoretical analysis and simulation comparisons. To achieve a 45% increase in stiffness and a 64.5% increase in damping, we optimized the suspension system’s z-axis stiffness and damping parameters under different operating conditions. This optimization aimed to minimize the z-axis vibration acceleration at the driver’s seat. We employed response surface methodology to design the composite shock absorber structure and then conducted a comparative analysis of the vibration reduction performance of the optimized front and rear suspension systems. This study provides significant theoretical foundations and practical guidelines for enhancing the performance of commercial vehicle cab suspension systems.

Development of a novel circular saw blade substrate with high stiffness for mitigating vibration noise and improving sawing performance

The circular saw blade substrates featuring a large diameter-thickness ratio and multiple saw teeth are increasingly required to accomplish the first or second machining tasks with high efficiency in sawing operations. However, the weak bending stiffness of this substrate can easily cause intense vibrations and harsh noises, while its special structure characteristics pose further challenges to vibration mitigation. This research develops a novel circular saw blade substrate with high bending stiffness to improve dynamic stability. Taking the geometric features and boundary conditions of the substrate into account, a transverse vibration differential equation for high-speed sawing operation is established to investigate the relationship between the mode shapes of the substrate and the stability performance of the sawing system. Theoretical analysis shows that the laminated structure can enhance the bending stiffness. Using this finding, the novel substrate is designed equipped with an m-shaped damping cavity and a laminated core structure, and its dynamic performance is quantified by the complex modulus law and thin-plate vibration theory. Then, the optimized design of the relevant materials and geometrical dimensions of the novel substrate is executed by the pseudo-density method. Finally, the novel substrate is fabricated and then certified by sawtooth point dynamics tests and sawing experiments. The experimental results show that the sawing quality of the novel substrate is improved by more than 50.2 %, and the sawing vibration and weighted sound pressure level are reduced by about 35 % and 12 dB(A), respectively, compared with the conventional one.

Torsional vibration analysis and fuzzy adaptive PID control optimization of underwater vector propulsion shaft systems

To examine the torsional vibration of vector propulsion shaft system (VPSS), a four-degree-of-freedom vibration model of the VPSS is established by concentrated mass approach, and the torsional vibration differential equation of the VPSS is deduced based on the Lagrange approach. The vibration differential equations is addressed numerically applying the Runge-Kutta algorithm to investigate the effect of various working shaft angles β on the free and forced vibration of the VPSS under the excitation conditions. To further reduce the torsional vibration amplitude and avoid fatigue damage of the system, and considering the nonlinear relationship within the system, this study develops a fuzzy adaptive PID approach to control the VPSS. The results show that the maximum torsional vibration amplitude of the VPSS is reduced by more than 70% and the transient response time is shortened by more than 90% with this control approach, which effectively reduces the torsional vibration amplitude of the VPSS and thereby significantly improves the safety performance of the underwater vehicle. This may provide technical references for the vibration control theory of the VPSS as well as engineering applications.

Dynamics analysis of noncircular planetary gears

Noncircular planetary gear has simple structure and wide application, but its dynamics research is still blank. Therefore, this paper designs and establishes a 3–4 noncircular planetary gear model, and takes a deep dive into its dynamic characteristics. Firstly, the torsional vibration mechanical model of noncircular planetary gear is established, and the key factors in the planetary gear system are fully considered, such as time-varying meshing stiffness, backlash, meshing damping and excitation frequency. These factors have an important impact on the vibration characteristics of the system. Then, the vibration differential equation of the system is established, and the vibration characteristics of the system under different parameter conditions are analyzed. The results show that the noncircular planetary gear system with large damping and appropriate stiffness can maintain a more stable operating state at a larger excitation frequency. In addition, shortening the acceleration time of the system can effectively reduce the damage caused by the unstable state to the system. These findings provide a solid theoretical basis for the subsequent optimization and analysis of noncircular planetary gears.

Study on the coupling vibration characteristics of vertical unit and plant of large underground pumping station for water transfer project

Abstract To investigate the coupled vibration characteristics of vertical pumping unit and plant structure of large underground pumping station for water transfer project, the coupled vibration differential equations are established. The structure nodes of are simplified from horizontal direction and vertical direction respectively by force. The structural parameters and load parameters of the model are analyzed and calculated, and the vibration response of each node are calculated by solving the vibration differential equations. A spectral analysis of white noise is applied to the coupled model, and the natural frequencies and vibration structures of dominant vibration modals are obtained. Research indicates that the vertical vibration modes of the system exhibit a relatively low coupling degree with the plant structure. The shaft system and the plant structure mainly display combined vertical vibrations, while the plant structure predominantly exhibits independent vertical vibrations. This implies that vertical vibrations are jointly influenced by shaft system stiffness and plant structure properties. In horizontal vibration modes, a strong coupling characteristic is observed between the upper structure of the pump station unit and the plant structure. This is primarily manifested as coupled vibrations between the support and the plant structure, which are collectively influenced by support constraints, shaft system stiffness, and plant structural properties.

Early condition monitoring of circular saw blades with large diameter-to-thickness ratios under high-speed sawing of hard metals

Circular sawblades with large diameter-thickness ratios feature weak stiffness which results in self-excited and forced vibrations in high-speed sawing. The scenario brings uncertainty to the evolution of the multichannel signal features, which sometimes even exhibit contradictions, making it extremely challenging to predict the sawblade lifespan. To fill the gaps, a transverse vibration differential equation was firstly established to elucidate its complex behavior. Then, this study used an artificial intelligence algorithm based on multidomain features to predict the condition of the sawblade. Therefore, a hybrid prediction model was constructed using a BP neural network (BPNN) optimized by a genetic algorithm (GA). Cutting experiments were carried out to prove that the proposed model can effectively predict the sawblade condition. This study not only provides useful insights for the condition monitoring of sawblades, but also offers a solid foundation for monitoring wood, stone, and composite processing.

Design and Testing of the Double-Symmetric Eccentric Exciter for Fruit Tree Vibration Harvest

The amplitude of excitation force from exciters used in fruit tree vibration harvesting remains constant at a given frequency, leading to poor fruit detachment ratio and tree damage. A solution has been proposed through the development of a Double-Symmetric Eccentric Exciter (DSEE). This new exciter allows for the adjustment of excitation force amplitude while maintaining a constant frequency by varying the phase angle of the DSEE. To validate the effectiveness of the DSEE, vibration tests were conducted on fruit trees using different parameter exciting forces. Acceleration sensors were employed to measure the vibration accelerations of the tree branches. The experimental results revealed that when a fixed frequency excitation force with a constant phase angle was applied to the trunk, the vibration acceleration of branches exhibited inconsistent variations due to differences in the vibration differential equation parameters of each branch. Furthermore, it was observed that increasing the phase angle of the DSEE at a fixed frequency led to larger vibration accelerations in every branch. This signifies that adjusting the phase angle of the DSEE can effectively increase the amplitude of the exciting force. Consequently, the ability to control both the amplitude and frequency of the excitation force independently can mitigate issues such as low fruit harvest rates and minimize damage to fruit trees.

Vibration analysis of Timoshenko microbeams made of functionally graded materials on a Winkler-Pasternak elastic foundation

In this work, the free vibration analysis of Timoshenko microbeams made of the Functionally Graded Material (FGM) on the Winkler-Paternak elastic foundation based on the Modified Coupled Stress Theory (MCST) is investigated. Material characteristics of the beam vary throughout the thickness according to the power distribution and are estimated through Mori–Tanaka, Hashin-Shtrikman and Voigt homogenization techniques. The Timoshenko microbeam model considering the length scale parameter is applied. The free vibration differential equations of FGM microbeams are established based on the Finite Element Method (FEM) and Kosmatka’s shape functions. The influences of the size-effect, foundation, material, and geometry parameters on the vibration frequency are then analyzed. It is shown that the study can be applied to other FGMs as well as more complex beam structures.

Dynamic performance of multi‐layered composite sandwich open cylindrical shells with double‐layered soft cores

AbstractIn this paper, the dynamic performance of Multi‐layered Sandwich Composite Cylindrical Open Shells (MSCCOS) with double‐layered soft cores is analyzed for the first time by adopting the First‐order Shear Deformation Theory (FSDT). For accomplishing this aim, vibration differential equations of MSCCOS are derived based on Hamilton's variance principle. Then, the closed‐form Navier solution is introduced to obtain the numerical results of differential equations. Subsequently, the effectiveness and accuracy are discussed by several comparisons. Ultimately, the dramatic effect of different structural and material parameters on natural frequencies and associated loss factors is investigated and described vividly. Because shear deformation and rotary inertia of all layers are considered in vibration differential equations, thin and moderately thick MSCCOS could be researched, and some interesting discoveries for the dynamic performance of MSCCOS are derived.Highlights A vibration model of five‐layered composite open shell was obtained. The optimal structure parameters maximizing the damping properties are shown. Dynamics properties of five‐layered composite open shell are presented.

Study on Nonlinear Dynamic Characteristics of RV Reducer Transmission System

Rotate vector (RV) reducers have widely been used in high-performance precision drives for industrial robots. However, the current nonlinear dynamic studies on RV reducers are not extensive and require a deeper focus. To bridge this gap, a translational–torsional nonlinear dynamic model for an RV reducer transmission system is proposed. The gear backlash, time-varying mesh stiffness, and comprehensive meshing errors are taken into account in the model. The dimensionless vibration differential equations of the system were derived and solved numerically. By means of bifurcation diagrams, phase trajectories, Poincaré sections, and the power spectrum, the motion state of the system was studied with the bifurcation parameters’ variation, including excitation frequency and meshing damping. The results demonstrate that this system presents enriched nonlinear dynamic characteristics under different parameter combinations. The motion state of the system is more susceptible to change at lower frequencies. Increasing the meshing damping coefficient proves effective in suppressing the occurrence of chaos and reducing vibration amplitudes, significantly enhancing the stability of the transmission system.

Effect of moving loads on dynamic response of multi-span sandwich beams with porous functionally graded material

This paper analysis dynamic response of porous functionally graded material (p-FGM) multi-span sandwich beams subject to two moving loads. The core of sandwich beam is fully ceramic and skins are composed of a porous functionally graded material. The vibration differential equations of beams are established based on the finite element method (FEM). Using the Newmark direct integration method, the dynamic response of the beam is calculated. A detailed study is performed to investigate the influences of material, moving load speed, porous parameter and distance between two moving loads on the dynamic response of p-FGM multi-span sandwich beams.

Research on Nonlinear Vibration of Dual Mass Flywheel Considering Piecewise Linear Stiffness and Damping

Nonlinear torsional vibration differential equation of the nested arc-shaped short spring dual mass flywheel (DMF) is established, considering the piecewise linear stiffness and damping of the spring. The first-order approximate analytical solution under sinusoidal excitation and the amplitude–frequency characteristic function are obtained by means of the average method which verified by the Runge–Kutta (R–K) method. The effects of the parameters of input excitation, inertia, and piecewise linear stiffness and damping of DMF on the resonant amplitude, resonant frequency band, and equivalent linear natural frequency of the system are analyzed. The results show that the amplitude–frequency characteristic curve bending and jumping with the changes of excitation frequency and the peak of resonant amplitude can be obviously reduced by increasing the inertia of the primary flywheel and decreasing the inertia of the secondary flywheel. The complex nonlinear dynamic phenomena such as Period 1, quasi-periodic, and chaos are obtained by analyzing the forced vibration response under the different excitation frequencies.

Horizontal oscillations of the wood sawing support during the cutting process in a wood sawing line using a vertical bandsaw

The wooden trolley (in the sawing line using a vertical bandsaw) during work generally moves by iron wheels on two rails placed on the concrete floor. Due to the effect of noticeable cutting force applied by the saw blade and the vehicle's uneven weight on the wheels, it causes the vehicle to vibrate during the horizontal movement, affecting the stability of the wood sawing circuit. By modeling the problem of vibration of beams located on Winkler elastic foundations, subjected to mobile concentrated loads, the article has built a system of differential equations of vibration of two rails, giving an expression to calculate the vibration amplitude of the swan plane. The influence of the flexural rigidity EI of the rails and the elastic stiffness k of foundation on the vibration amplitude of the cutting plane is studied and analyzed. The use of the Dirac Delta function and the employed analytical solution allow the designers to evaluate the accuracy and reliability of the calculated results.

Modeling and braking analyses of longitudinal-vertical coupling towbarless aircraft taxiing system

The variation of tire vertical load has effects on the vehicle braking performance, which may lead to the misjudgment of braking distance and increasement of the accident risks. In order to evaluate the braking distance of the towbarless aircraft taxiing system (TLATS) accurately, a longitudinal-vertical coupling model of the TLATS is established, in which the effects of the vertical vibration on braking dynamics is considered. An 11-degrees of freedom (DOF) dynamic model of the TLATS, which consists of the system longitudinal motion, vertical vibration, and an advanced Lugre tire model, is proposed. The tire vertical dynamic loads are obtained by calculating system vibration differential equations as the input for the Lugre tire model, in which an arbitrary pressure distribution function over the contact patch and load related parameters are used. The Lugre tire model parameters are identified based on the virtual experiment results in ADAMS/Car software through a multi-objective optimization (MOO) method. The proposed tire model is more accurate by comparing with the models using other pressure distribution functions. The advanced Lugre model is integrated into the longitudinal-vertical coupling dynamic model of the TLATS, and the fuzzy PID control method is used to achieve the optimal slip rate during the braking procedure. The tire friction and TLATS’s braking performances in terms of the slip rate, braking speed and braking distance are compared between the coupling and uncoupling models under different traction velocities and the road excitations when a same fuzzy control method is used for the system braking. The results show that the proposed longitudinal-vertical coupling TLATS dynamic model could assess the braking behaviors of the TLATS more accurately than the simplified uncoupling model, which could decrease the potential safety risk of the aircraft towing operation on the airport apron.

An Investigation of Free Vibration Characteristics of a DFG-CNTRC Thin Laminated Shell in Thermal Environment

A dynamic model of dual-functional gradient carbon nanotube-reinforced composite (DFG-CNTRC) laminated shell in a thermal gradient environment is established. The carbon nanotubes (CNTs) are supposed to be FG distribution in the functionally graded material (FGM) consisting of metal and ceramic components, and its traveling wave vibration in thermal gradient environment environments is studied. The temperature between the internal and external sides of laminated shell varies linearly. Considering the impact of the thermal environment on material properties, the orthogonal polynomials are used as admissible function, and the vibration differential equation of the DFG-CNTRC laminated shell is obtained by using the Rayleigh–Ritz method. First-order shear deformation theory (FSDT) and artificial spring technique are applied to establish a general model of free vibration of rotating DFG-CNTRC laminated shells under arbitrary boundary in the thermal gradient environment. The effects of rotational speed, the environmental temperature, the thickness of the middle layer, and volume fraction of CNTs on the traveling waves are discussed.

Study on Nonlinear Vibration of Vertical Lifting Section of Bulk Grain Entrainment Ship Unloader

In view of the prominent problem that nonlinear vibration of a belt conveyor can easily occur during the vertical grain-conveyance process due to the coupling effect of airflow clamping and traction of the conveyor belt, which seriously affects the efficiency and stability of conveying materials by the belt conveyor, a method of solving the vibration analysis of the vertical lifting section of the bulk grain belt unloader by using nonlinear vibration is proposed. Firstly, based on the laminated plate theory, the vertical lifting belt and the grain material clamped by the belt are laminated. The nonlinear vibration differential equation of the vertical lifting section of the bulk grain-carrying ship unloader is established by elastic–plastic mechanics, and solved by perturbation theory and Galerkin discrete analysis. The vibration response curve and structural natural frequency of the vertical lifting section of the bulk grain-carrying machine are obtained by numerical solution, and the influence of the volume content of the clamped material on the vibration response and structural natural frequency of the lifting section is analyzed. This study provides theoretical support for the design of pressure-supply parameters, overall structure and operation parameters of the subsequent entrainment ship unloader, promotes the rapid development of the entrainment ship unloader, provides theoretical support for the design, manufacture, later operation, and maintenance of the entrainment ship unloader, and thus provides equipment and technical support for building an efficient and intelligent port.

Analysis on Thermo-Electrical Principal Parametric Resonance of an Axially Moving Piezoelectric Thin Plate

In this paper, principal parametric resonance of an axially moving piezoelectric rectangular thin plate under thermal and electric field is investigated. Based on Kirchhoff–Love plate theory and Von Karman theory, the transverse vibration differential equation of a piezoelectric rectangular thin plate under thermal and electric field is derived by using Hamilton’s principle. The dimensionless vibration equation of piezoelectric rectangular thin plate with parametric excitations is discretized by Galerkin’s method. Then, the multiple scales method is applied to derive amplitude-frequency response equation and the stability conditions of the steady-state solution are obtained by Lyapunov stability theory. Numerical method is used to find the influences of specific parameters on the vibration performance and stability of the system. Based on the global bifurcation diagram and corresponding response diagram, the influences of bifurcation control parameters on the nonlinear dynamic characteristics of the system are discussed. Numerical results illustrate that the system amplitude frequency characteristic curve presents soft spring characteristics. There are periodic and chaotic motions with the increase of velocity and the central temperature difference, and the decrease of plate thickness and velocity will result in the decrease of chaotic threshold. The results also show that increase the velocity perturbation amplitude can prolong the chaotic motion.

Dynamic Simulation Analysis of Truck Crane Based on ADAMS

Background: With a reduction in the weight of telescopic boom crane and an increase in its lifting capacity, the contradiction between its speed, stability and safety becomes more prominent. The telescopic arm also has flexible deformation during operation, which will cause vibration of the whole vehicle system. Therefore, in order to ensure safety, accuracy, and efficiency in the working process, it is necessary to accurately predict the dynamic characteristics of the telescopic boom load in the working process. Objective: The purpose of this study is to establish a rigid-flexible coupling dynamic model of the crane, which can accurately reflect the dynamic characteristics of the telescopic boom during its operation in the simulation test. Methods: In theory, this study analyzes the characteristics of a truck crane and simplifies it as a cantilever beam. Based on the Euler-Bernoulli beam theory, the vibration differential equation of the mobile masscantilever beam system is established. In the aspect of simulation, the three-dimensional modeling software Solidworks is used to establish the solid model of the truck telescopic boom, the finite element analysis software is used to establish the finite element model of the telescopic boom, and the modal neutral file is generated and imported into ADAMS to establish the dynamic model of the truck crane multi-level flexible telescopic boom. The dynamic simulation of the virtual prototype model of the truck telescopic boom crane is carried out in ADAMS software to obtain the dynamic characteristics of the key components of crane under specific working conditions. Results: The vibration differential equation derived in theory can be used to solve the dynamic response of the crane jib under specific conditions by MATLAB programming. In the aspect of simulation, the error fluctuation and its causes in the key components of the crane in the working state are analyzed, and the rationality of the virtual prototype model is verified. Finally, through the trajectory planning of the crane under typical working conditions, the purpose of stable control of the telescopic boom system of the truck crane is realized. Conclusion: The derived equation is universal for solving the vibration of crane jib in general cases. The rigid-flexible coupling model also provides a reference for the dynamic modeling, analysis, design, and manufacture of a multi-stage flexible telescopic boom. The findings of this study can provide a reference for related patent research and development.