elastic supports
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2022 ◽  
Vol 12 (2) ◽  
pp. 746
Author(s):  
Qingyu Zhu ◽  
Qingkai Han ◽  
Xiaodong Yang ◽  
Junzhe Lin

This paper presents the dynamic characteristics analysis of a rigid body system with spatial multi-point elastic supports, as well as the sensitivity analysis of support parameters. A rigid object is characterized by six degrees-of-freedom (DOFs) motions and considering the spatial location vector decomposition of elastic supports, a rigid body system dynamic model with spatial multi-point elastic supports is derived via the Lagrangian energy method. The system modal frequencies are calculated, and to be verified by finite element modal analysis results. Next, based on the above-mentioned model, system modal frequencies are obtained under different support locations, where the support stiffness components are different. Interpolate the stiffness components corresponding to each support location, calculate system modal frequencies, and the response surface model (RSM) for system modal frequencies is established. Further, based on the RSM modal analysis results, the allowable support location for the system modal insensitive area can be obtained. At last, a lubricating oil-tank system with four supports is taken as an example, and the effects of support spatial locations and stiffness components on the system inherent characteristics are discussed. This present work can provide a basis for the dynamic design of the spatial location and stiffness for this type of installation structures.


Author(s):  
Sergij Yurijovich Pogorilov ◽  
Valerij Lvovich Khavin ◽  
Inna Petrovna Khavina

One of the main structural elements of metalworking machines is the spindle assembly (spindle), which is used to hold cutting tools or workpieces. The rigidity of the spindle assembly plays a decisive role in ensuring the accuracy and efficiency of the machine as a whole. The assessment of the spindle shaft stiffness is carried out on the basis of the analysis of the static bending of the spindle shaft, which made it possible to formulate and solve the problems of optimizing the spindle shaft according to the stiffness characteristics for two supporting structures on nonlinear elastic supports. To determine the stiffness of roller bearings, the work uses the dependence obtained on the basis of solving the problem of contact interaction of an elastic steel cylinder with curvilinear elastic steel half-spaces. For the considered design scheme, the optimization goals were chosen for the conditions of the smallest displacement of the end section of the spindle shaft console, the achievement of the minimum angle of rotation in this section or the minimum of their normalized superposition, which ensures maximum rigidity in the processing zone. Consideration has also been given to minimizing the swing angle at the front support to maximize bearing life. Mathematically, the problem is presented in the form of minimizing one of the 4 proposed objective functions by changing the variable parameters - the length of the cantilever and the value of the inter-support distance, represented as dimensionless quantities - the cantilever coefficient and the inter-support distance coefficient. Minimum and maximum values ​​of the cantilever length and shaft span were considered as constraints on the variable parameters. Varying the console coefficients and the inter-support distance was carried out by the method of sequential enumeration within the specified constraints, the solution of optimization problems is presented in a graphical form. The solution to the problem of shaft bending was carried out on the basis of the equation of the bent axis of the beam in the framework of the Euler - Bernoulli hypotheses and presented in an analytical form together with analytical dependencies for calculating the radial stiffness of a roller bearing as a function of the supporting force acting on it. The algorithm for solving optimization problems is implemented in the MatLAB package. Optimal solutions have shown that the minimum of the combined functions, consisting of the sum of the relative deflection values ​​at the end of the console and the angles of rotation at the end of the console and on the front support, is achieved at the same variable parameters as the minima of the angles of rotation at the end of the console and on the front support. The proposed approach to the design of the shafts of spindle units of metal-cutting machines, which are optimal in terms of rigidity characteristics, forms a tool for a reasonable choice of bearings and design parameters of spindle shafts.


2021 ◽  
Vol 3 (2) ◽  
pp. 88-102
Author(s):  
S. Bekshaev ◽  

The article analytically investigates the behavior of the frequencies and modes of natural vibrations of a rigid body, based on point elastic supports, when the position of the supports changes. It is assumed that the body is in plane motion and has two degrees of freedom. A linear description of body vibrations is accepted. The problems of determining such optimal positions of elastic supports at which the fundamental frequency of the structure reaches its maximum value are considered. Two groups of problems were studied. The first group concerns a body supported by only two supports. It was found that in the absence of restrictions on the position of the supports to maximize the fundamental natural frequency, these supports should be positioned so that the basic natural vibrations of the body are translational. Simple analytical conditions are formulated that must be satisfied by the corresponding positions of the supports. In real practical situations, these positions may be unreachable due to the presence of various kinds of restrictions due to design requirements. In this paper, optimization problems are considered taking into account a number of restrictions on the position of supports, typical for practice, expressed analytically by equations and inequalities. For each of the considered types of constraints, results are obtained that determine the optimal positions of the supports and the corresponding maximum values of the main natural frequencies. The approach applied allows us to consider other types of restrictions, which are not considered in the article. In the second group of problems for a body resting on an arbitrary number of supports, the optimal position of an additional elastic support introduced in order to maximize the fundamental frequency in fixed positions and the stiffness coefficients of the remaining supports was sought. It was found that this position depends on the value of the stiffness coefficient of the introduced support. Results are obtained that qualitatively and quantitatively characterize this position and the corresponding frequencies and modes of natural oscillations, including taking into account practically established limitations. The research method uses a qualitative approach, systematically based on the well-known Rayleigh theorem on the effect of imposing constraints on the free vibrations of an elastic structure.


Lubricants ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 121
Author(s):  
Vladimir Kodnyanko ◽  
Andrey Kurzakov ◽  
Olga Grigorieva ◽  
Maxim Brungardt ◽  
Svetlana Belyakova ◽  
...  

The design is considered and theoretical research of operability of the active radial gas-static bearing with restrictors of output flow rate in the form of mobile rings with an elastic supports and the dampers working by Helmholtz acoustic resonator principle is done. The mathematical model of the bearing dynamics and method of calculating its degree of stability are developed. The device is steady against vibrations; it has smaller power consumption compared to the known devices with input regulators, a zero and negative compliance of a gas-lubricated film.


2021 ◽  
Vol 19 (2) ◽  
pp. 6-12
Author(s):  
R. V. Guchinsky

Improving dynamic performance is a priority when designing new rolling stock. The frequency of natural bending vibrations of the body is one of the most important standardised indicators, a preliminary assessment of which allows obtaining optimal body designs.The objective of the work is to assess the prospects for the use of elastic fastenings of equipment to increase the natural vibration frequency of wagon bodies of suburban electric trains. Calculations were based on the finite element method and block Lanczos method. It is shown that it is advisable to use the rigid area tool and linearly elastic finite elements to calculate the frequencies in the simulation. The main ranges of fastening stiffness are highlighted, where the effect of using elastic supports is different. It is proposed to determine the stiffness of fastenings according to a given vibration frequency of the equipment. When the equipment is rigidly attached, the relative mass of the equipment does not affect the body bending vibration frequency. With elastic fastening, a greater effect can be achieved with a larger relative weight of the equipment. The effect of using resilient mounts increases with heavier equipment located closer to the centre of the body.It is shown that the effect of shear admittance of fastenings on the body vibration frequency is within 1 % and may not be considered in the simulation. In the considered example of a wagon body of a suburban electric train, the use of elastic supports allows an increase in the frequency of oscillations of the body by 3–10 %. 


2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Guoying Pang ◽  
Shuqian Cao ◽  
Yushu Chen ◽  
Huizheng Chen

To analyze the problem of vibration and bifurcation in the rotor system of the aeroengine with the elastic ring squeeze film damper (ERSFD) and elastic supports, the theoretical equation of the dynamic rotor system is developed in this paper, based on the rotor system, elastic ring squeeze film damper (ERSFD), and three elastic supports. The estimated analytical solution of the oil film force is solved using the short bearing approximation theory and the semi-oil film inference theory in the suspension and the inner and outer boss contact. Considering the oil film stiffness and damping of rolling bearings, the rolling bearing force model is established based on the elastohydrodynamic lubrication (EHL) theory. By the average method, the vibration and bifurcation modes are obtained concerning the bearing coefficient and parameters. The range optimization of parameters can be appropriately improved to enhance the dynamic characteristics of the device given different parameters of the hole of oil seepage, the stiffness, the position of elastic supports, and other structural parameters.


2021 ◽  
Vol 11 (19) ◽  
pp. 9273
Author(s):  
Publio Pintado ◽  
Carmen Ramiro ◽  
Eduardo Palomares ◽  
Angel L. Morales ◽  
Antonio J. Nieto ◽  
...  

This paper presents a new formulation for analyzing a beam on elastic supports traveling on irregular profiles. The model is a first approximation of a passenger railway vehicle car body. The main difference with previous works is the use of a complex modulus to represent structural damping rather than relying on equivalent viscous terms. The formulation groups rigid body modes with flexible modes and proposes a matrix form that is easy to interpret and solve in the frequency domain. Comfort indexes are readily obtained from weighted response spectral densities. The model is used to assess the influence of structural damping and stiffness on comfort. It will be shown that the evolution of comfort with stiffness is non-monotonic and, therefore, comfort does not always improve as stiffness increases.


2021 ◽  
Vol 11 (19) ◽  
pp. 9141
Author(s):  
Yeqing Jin ◽  
Ruiping Yang ◽  
Hengxu Liu ◽  
Haiwei Xu ◽  
Hailong Chen

Free vibration analyses of lattice sandwich beams with general elastic supports have rarely been discussed in this field’s literature. In this paper, a unified method is proposed to study the free vibration characteristics of lattice sandwich beams under various boundary conditions. The proposed method is to convert the three truss cores of lattice sandwich beams into an equivalent homogeneous layer and introduce two different types of constraint springs to simulate the general elastic support boundary at both ends of lattice sandwich beams. By changing the rigidity of the boundary restraint spring, various boundary conditions can be easily obtained without modifying the solving algorithm and solving process. In order to overcome all the discontinuities or jumps associated with the elastic boundary support conditions, the displacement function of lattice sandwich beams is usually obtained as an improved Fourier cosine series along with four sine terms. On this basis, the unknown series coefficients of the displacement function are treated as the generalized coordinates and solved using the Rayleigh–Ritz method. The correctness of the present method is verified through comparison with existing literature. The calculation results of the present method are highly accurate, indicating that the present method is suitable for analyzing the vibration characteristics of lattice sandwich beams with general elastic supports. In addition, the effects of beam length, panel thickness, core height, radius and truss inclination on the natural frequencies of lattice sandwich beams with arbitrary boundary conditions have been discussed in this paper.


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