Collision enhanced hyper-damping in nonlinear elastic metamaterial

Nonlinear elastic metamaterial, a topic which has attracted extensive attention in recent years, can enable broadband vibration reduction under relatively large amplitude. The combination of damping and strong nonlinearity in metamaterials may entail auxetic effects and offer the capability for low-frequency and broadband vibration reduction. However, there exists a clear lack of proper design methods as well as a deficiency in understanding properties arising from this concept. To tackle this problem, this paper numerically demonstrates that the nonlinear elastic metamaterials, consisting of sandwich damping layers and collision resonators, can generate very robust hyper-damping effect, conducive to efficient and broadband vibration suppression. The collision-enhanced hyper damping is persistently present in a large parameter space, ranging from small to large amplitudes, and for small and large damping coefficient. The achieved robust effects greatly enlarge the application scope of nonlinear metamaterials. We report the design concept, properties and mechanisms of the hyper-damping and its effect on vibration transmission. This paper reveals new properties offered by nonlinear elastic metamaterials, and offers a robust method for achieving efficient low-frequency and broadband vibration suppression.

Static analysis of multi-support spindle shafts with nonlinear elastic bearings

In this paper a mathematical model and computational tool are developed for the static analysis of multi-bearing spindle shafts with nonlinear elastic supports. Based on the Timoshenko beam theory a resolving system of equations is obtained that takes into account the nonlinear dependence of the bearing stiffness on the reaction forces acting upon them. A solution method is proposed and appropriate software is developed that implements the static analysis of multi-support spindle shafts with non-linearly elastic bearings in MATLAB environment. Key words: spindle, shaft, nonlinear elastic support, multi-bearing, nonlinear elastic stiffness, Timoshenko beam.

Optimization of two support spindle shaft on nonlinear elastic supports by rigidity characteristics

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.

Vibration suppression and dynamic behavior analysis of an axially loaded beam with NES and nonlinear elastic supports

Recently, dynamic analysis of a beam structure with nonlinear energy sink (NES) and various supports is attracting great attention. Most of the existing studies are about the beam structure with NES or nonlinear boundary supports with zero rotational restraint, respectively. However, there is little research accounting for such two types of complex factors simultaneously. In this work, the dynamic behavior of an axially loaded beam with both NES and general boundary supports is modeled and studied. The Galerkin truncated method (GTM) is employed to make the prediction of dynamic behavior of such a beam system, in which the mode functions of axially loaded Euler–Bernoulli beam with linear elastic boundary conditions are selected as the trail and weight functions. Then, the Galerkin condition is used to discretize the nonlinear governing equation of the beam system and establish the residual equations. The Runge–Kutta method is used to solve the residual matrix which consists of residual equations directly, and the harmonic balance method is also used to verify the results from the GTM. The influence of NES on vibration suppression and dynamic behavior of the beam structure is investigated and discussed. Results show that the vibration states of the beam structure can be transformed effectively through the change of NES parameters. On the other hand, the NES with suitable parameters has a beneficial effect on the vibration suppression at both ends of the beam structure.