Element Free Method In Vibration Computing Of Elasticity Structure And Precision Analysis

Meshless method is numerical approach method based on the nodes, it constructed interpolation function directly but not elements. It only requires to separate the result demain into a series of regular or irregular nodes, approaches the number to the nodal date, disposes the geometry characteristic change of calculation area more easily. It has much more merits that most finite element method can't compare when to handle the great transform problem, high degree problem and dynamic boundary problem.As to practical engineering vibration system, it usually involves some special power function, such as pulse function, jump function and period function. They can often be met in the vibration of the transportation and carrying tool, a great deal of practice indicated that it is important to research the vibration system under the pulse, jump and period power. Certainly, all these researches involve the dynamic state characteristic of mechanical system.In the beam structure, in addition to support the moving loading function by car or train, and also support various shake loading function by earthquake, wind loading, crowd loading, ship and channel bump loading etc, causing the beam structure response (structure transform, inner stress, and acceleration over loading) seriously, sometimes make the bridge structure heavy damage, even take place vital accident. So analyzing all kinds of dynamic response being aimed at beam structure is an important link in the process of beam design and calculation. Currently, the finite element theory in the process of solving structure dynamic problem has made remarkable achievements, But with the increasing development of computational mechanics, many other researchers still try to carry out research in other field to make up for the deficiencies of finite element theory. Meshless method has being researched extensively in the world, its scope of application broads increasingly. In the field of dynamic research, putting meshless method to use in the expansion propagation calculation of dynamic crack has appeared on the literature, the same to the dynamic analysis of the beams and the plates. Meshless method is not perfect in the field of mathematical theory argumentation, the boundary conditions disposal, efficiency and effectiveness of calculation, a large number of engineering examples and mature common software can't be seen. But it is being the hot engineering research field relying on the more merit than finite element in recent years.In this paper a large number of numerical calculations and research work has been done in the field of the basis theory about meshless method, influence factors of calculated precision and the inherent characteristics analysis about the beam structure vibration, along with solving the static and dynamic problem of plate structure. Conclusions can be summarized as follows:1. In meshless method, MLS method is numerical approximation method based on the nodes, which require information of nodes but not elements. The shape function in the mobile field function is structure based on the interpolation function and weight function, they have great influence on the calculation precision by meshless method. So the main task to analyze the influence of the shape function on the calculation precision is the influence of constructing shape function and analyzing weight function on calculation precision. In this paper the test function that take the displacement function as solving partial differential equations was constructed by meshless method, including the selection of base function and weight function, the calculation of shape function along with derivative terms, as well as analyzing size of the influence field of the weight function and node layout on the calculation precision of shape function along with its displacement. Cited examples showed the reasonable of the shape function and its precision analysis.2. In order to compare the influence that different types of weight function on calculation precision by meshless method, in this paper sample type weight function and exponential weight functions along with its different radius of the influence domain were selected separately, interpolation function was constructed respectively by using moving least—square method. To build meshless equation of beam structure dynamic by using this interpolation function as displacement field function. To calculate the natural frequency and modal vector of the beam and get two types of weight function about the beam free vibration as well as its EFM result of different the influence domain radius along with comparative analysis by using penalty function method to meet the essential boundary conditions. To test the reliability of the method by integrating concrete calculated sample of the beam structure.3. In order to compare the influence of the base function on the calculation precision by meshless method, in this paper orthogonal polynomials sequence was constructed by Schmidt orthogonal method. By using orthogonal polynomial base and power polynomial base respectively, to select sample type weight function to build displacement field function, by using structure dynamic base equation to separate by meshless method similar to finite element method, to get meshless dynamic equation. By using penalty function method to meet the essential boundary conditions, to deserved two types of meshless solutions, compare with analytical solutions.4. In order to compare the influence of the base function order and the size of weight function affected region radius on the calculation precision and efficiency by meshless method, in this paper the first, the second, the third base function and simply weight function under different affected radius construct function respectely. To derive the meshless dispersing equation without discrete of plate bending problem based on Mindlin plate element format. To calculate the meshles solution of elastic plate bending problem combining the calculatal programming, as well as compare its calculatal precision and efficiency, the results showed that meshless method to calculate elastic plate bending problem is feasible and effective.5. In this paper the second basal function and sample weight function with good precision and efficiency were selected, and contract field function under certain influence radius. To derive the meshless discrete equation of plate dynamic problem based on Mindlin plate element format. By using Newmark in direct integration method, to calculate vertical dynamic response problem of the elastic plate combining the calculation programming, to obtain meshless solution of the displacement, velocity and acceleration response from the plate under pulse-induced, the result indicate that it is feasible and effective to calculate the dynamic response of elastic plate by meshless method. This paper has the following innovation points about researching meshless method and the applied significance in this field:1. Further expanded the applied scope of meshless method, and make it in common use. This paper has been done some research in the field of beam and plate vibration, along with solved some more simple practical engineering problem with meshless method calculation, proved the theoretical basis for some more complex practical engineering issues.2. Develop the related meshless theory of in particular in mathematical theory. This paper has introduced the mathematical theory based on meshless method, including the structure of approximate function and the discrete programs of elastic structure equation.3. Continue to research and search for the more applicable shape function method, do a large number of studies in the field of types of base function and weight function as well as the impact to calculated precision, this paper deduced Schmidt orthogonal formula of orthogonal base function and tectonic theory of radial base meshless method, as well as to compare three types base function and the effects that fitting the order of base function into the calculated precision and efficiency of meshless method.4. Look for the method to raise the calculated efficiency of meshless method, approach the most optimization usage of the memory space in the field of compiling procedures, and meet the needs of large scale computing. Meshless method of radial basis has good computational precision and efficiency.5. All the calculated result date were derived from common meshless method software which be developed independently, and applied it in the practical engineering analysis. The opening up and usage of the software expanded new develop space for meshless method.