Constrained variational approach for dynamic analysis of elastic contact problems

Flat punch of arbitrary shape on an elastic half-space

A new method is proposed for the analysis of elastic contact problems for a curved punch of non-elliptic planform under the action of a normal force. The punch base is assumed to be a quadratic surface. The method is based on an integral representation for the reciprocal distance between two points obtained by the author earlier. Some general relationships are established between the applied force and the punch settlement. Specific formulae are derived for a punch whose planform has a shape of a polygon, a rectangle, a rhombus and a cross. An example of a finite rigid cylinder lying on its generator and pressed against an elastic half-space is considered in detail. The method allows to have singular stresses at the cylinder edges and zero stresses at the rest of the boundary of the contact domain. The last condition serves for defining the width of the domain of contact. All the formulae are checked against the solutions known in the literature, and a good accuracy is confirmed in a sufficiently wide range of the aspect ratio.

2 - Governing Equations for Motion and Deformation of Block Systems and Heat Transfer

Numerous studies about the modelization of multibody systems with flexible parts undergoing large rigid body motions and small elastic deformations have recently been performed [[1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31]]. Some of them [[10][11][12][13][14][15][16][17]], [[20], [[25][26]], [[28]] focused on the influence of geometric nonlinearities, and others [[18][19]], [[21][22][23][24]], [[30][31]] focused on the terms describing dynamic nonlinearities. In this work [[30][31]], we look at the dynamic nonlinearities. The method usually used to model the motion of such systems consists of separating the motion of each component of the system of rigid body motion and small elastic deformations and keeps only the linear terms connected to them. In some cases of fast motion, the problem of the accuracy of this linearization occurs. Precisely, it has been noted that small deformations do not imply small rates of deformation or small accelerations of deformation [[18]]. The question then raised is to know if this linearization is too premature. After establishing a method to allow the evaluation of the relative importance of nonlinearity terms, it appears that these terms must be kept in some cases. The obtained method is general and does not concern only the dynamic terms. A slider-crank mechanism was used to test the theory. Kane's method [[32]] was used to establish the equations of motion, and the software AUTOLEV (Online Dynamics, Sunnyvale, CA) was used for numerical study. *Communicated by E. Haug.

For efficient dynamic simulation of mechanical and aerospace systems, the use of different sets of coordinate types may be necessary. The components of multibody systems can be rigid, flexible or very flexible and can be subject to contact forces. Examples of challenging problems encountered when multibody systems are considered are crashworthiness, problems of cables used in rescue operations and heavy load handling, belt drives, leaf spring system design, tire deformations, large deformation of high-speed rotors, stability problems, and contact problems. Most large displacement problems in structural mechanics are being solved using incremental solution procedures. On the other hand, general purpose flexible multibody computer tools and methodologies in existence today are not, in general, capable of systematically and efficiently solving applications that include, in addition to rigid bodies and bodies that undergo small deformations, bodies that experience very large deformations. The objective of this paper is to discuss the development of new computational algorithms, based on non-incremental solution procedures, for the computer simulation of multibody systems with flexible components. These new algorithms that do not require special measures to satisfy the principles of work and energy and lead to optimum sparse matrix structure can be used as the basis for developing a new generation of flexible multibody computer programs. The proposed non-incremental algorithms integrate three different formulations and three different sets of generalized coordinates for modeling rigid bodies, flexible bodies with small deformations, and very flexible bodies that undergo large deformations. The implementation of a general contact model in multibody algorithms is also presented as an example of mechanical systems with non-generalized coordinates. The kinematic equations that describe the contact between two surfaces of two bodies in the multi body system are formulated in terms of the system-generalized coordinates and the non-generalized surface parameters. Each contact surface is defined using two independent parameters that completely determine the tangent and normal vectors at an arbitrary point on the body surface. In the contact model presented in this study, the points of contact are determined on line during the dynamic simulation by solving the non-linear differential and algebraic equations of the constrained multibody system. The augmented form of the equations of motion expressed in terms of the generalized coordinates and non-generalized surface parameters is presented in this paper.

The statical behaviour of a spatial bar of an elastic and isotropic material under arbitrary distributed loads having a non‐circular helicoidal axis and cross‐section supported elastically by single and/or continuous supports is studied by the stiffness matrix method based on the complementary functions approach. By considering the geometrical compatibility conditions together with the constitutive equations and equations of equilibrium, a set of 12 first‐order differential equations having variable coefficients is obtained for spatial elements of helicoidal axes. The stiffness matrix and the element load vector of a helicoidal bar with a non‐circular axis and arbitrary cross‐section are obtained taking into consideration both the presence of an elastic support and the effects of the axial and shear deformations. For helicoidal staircases, the significance of both axial and shear deformations and eccentricities existing in wide and shallow sections are also investigated. The developed model has been coded in Fortran‐77, which has been applied to various example problems available in the relevant literature, and the results have been compared.

A new method for the element stiffness matrix of arbitrary planar bars

Constrained motion of a planar rigid-flexible manipulator is considered in this study. a dynamic model with zero tip deformation constraint of constrained rigid-flexible 608

A theoretical analysis of free dry and wet vibration of a trapezoidal, 2-way tapered, marine spade rudder, is presented. The rudder is considered as a hollow Kirchhoff’s plate, with the chord section as a NACA profile. The chord length and the thickness taper from the top to the bottom, over the vertical span. The rudder is pivoted at the top, with the pivot behind the leading edge. The pivot is modeled as a combination of a translational and a rotational spring, in order to include the rigid body modes of the rudder vibration. The span-wise and chord-wise non-uniform beam vibration is first analyzed by the Rayleigh-Ritz method, in order to establish the non-uniform beam mode shapes. The span-wise beam is a linearly tapered vertical cantilever, with non-classical edge at the top and free at the bottom. The chord-wise section is a 2-span beam with the ends free, and four continuity conditions at the pivot. The non-uniform mode shapes, in either direction, are a weighted summation of the uniform beam mode shapes, which also satisfy the boundary/continuity conditions. They now act as admissible spatial functions to the plate vibration, which is analyzed by the Galerkin’s method. Eigenvalue analysis generates the plate natural frequencies. A weighted superposition, of the product of the beam mode shapes, in either direction, generates the plate mode shapes. Alternately, uniform beam mode shapes are used as admissible functions into the Galerkin’s method for the plate natural frequencies and mode shapes. The natural frequencies are generated for various positions of the rudder stock along the chord length. The pivot conditions (in both translational and rotational rigid body degree of freedom) influence the prominence of the rigid body mode shapes. The natural frequencies are analyzed for various pivot fixities, taper ratios, and aspect ratios of the plate. This is followed by the wet vibration analysis of the rudder. First, 2D strip theory is used to generate the added mass of each chord section. Constant strength source distribution technique is used to generate the added mass in sway and yaw of a 2D aerofoil. Each flexural and torsional mode is associated with its own added mass. Various empirical corrections are done to account for the 3D flow. Finally, 3D panel method is used to generate the modal added masses, and hence the wet natural frequencies. The added mass coefficient is generated for various aerofoil fineness ratios, pivot fixities, taper ratios, and aspect ratios of the plate.

Greenwood and Williamson (GW) and Greenwood and Tripp (GT) elastic contact models have extensively been used previously for the mixed-lubrication analysis of water-lubricated journal bearings (WLJBs). The approximate expressions of a parabolic cylinder function involved in these contact models are available in the literature which makes these models easy to implement in the mixed-lubrication analysis. However, approximate expressions available so far are valid only for the Gaussian distribution of asperity heights. Moreover, elastic-plastic contact models with few exceptions have rarely been used in the mixed-lubrication analysis of WLJBs. The present work demonstrates a simple numerical procedure to predict the mixed-lubrication parameters for WLJBs. The average Reynolds equation considering non-Gaussian flow factors is solved along with film thickness and load balance equations in a coupled manner. A routine is developed to predict the pressure curve (a plot between average asperity pressure and rigid body displacement) for different contact models and non-Gaussian surface topography, and the asperity pressure for a particular surface topography is predicted utilizing the concept of a macro-micro approach. The effect of non-Gaussian surface topography on the tribological performance of WLJBs is discussed in detail. The influence of different contact models on the performance of WLJBs is also presented. It is shown that the GT elastic contact model fails to simulate the effect of kurtosis and skewness properly. However, the Kogut and Etsion (KE) elastic-plastic contact model which is based on finite element method (FEM)-based analysis excellently simulates the non-Gaussian effect on the performance of the WLJB.

Formulation and implementation of beam contact problems under large displacement by a mathematical programming

This paper is concerned with the large‐deflection (bending and free vibration)analysis of thin elastic curved beams by conventional and mixed finite element methods. The conventional finite element method is based on the total potential energy expression, whereas the mixed method is based on a Reissner‐type variational statement and involves the bending moments and deflections as primary dependent variables. Incremental and direct formulations are presented for both methods. The nonlinearity is included in the numerical method via an iterative procedure (i.e. the in‐plane force due to the large deflection is not treated as constant but included under the integral in the energy expression). In the case of vibrations, the transverse shear and rotary inertia effects are also included. A number of numerical examples of beams with various edge conditions are analysed for deflections and natural frequencies, and the results are compared with those reported in the literature.

Natural vibration analysis of cables is required in various fields, such as transmission lines and elevator ropes. Actual cables are often non-uniform. Because an exact solution is difficult to calculate in this case, an approximate solution must be calculated through numerical analysis. It is important to define the admissible function in the assumed method, which is widely used among numerical analysis methods. However, selecting an admissible function for a problem in which only geometric boundary conditions exist, such as spring-end support cable, is difficult. In this study, we propose the admissible and quasi-comparison functions for the addressed problem and compare the results to the finite element method results. This study can be used for the analysis of various structures in which cables are used.

This paper presents a study of wave propagation through an infinite periodic structure that consists of elastic Timoshenko beams interconnected with rigid bodies. This is a generalized approach in which the beams are not coaxial and the centre of mass of each rigid body is placed away from the intersection of their neutral axes. An analytical approach is used by applying the transfer matrix method (TMM), along with the Floquet-Bloch theorem for elastic wave propagation. Subsequent parametric analysis is performed with visualization of resulting band diagrams of a representative structure. These results are verified through comparison with solutions obtained using the finite-element method (FEM). In this manner, a comprehensive dynamical analysis of tailored metastructures is provided.This article is part of the theme issue 'Current developments in elastic and acoustic metamaterials science (Part 2)'.

The handling and control of towed cable and body systems onboard surface ships and submarines presents a significant technical challenge to design engineers in the defense and ocean industries. The current approaches rely heavily on the empirical methods and the time-consuming and costly prototype testing. Computer simulation provides a cost effective way to reduce the high risks associated with the towed cable/body system. However, the current dynamic analysis of towed cables is mostly done by the finite difference (FD) method in stead of the finite element (FE) method that is widely used in almost all engineering fields. This paper presents an alternative FE method to simulate the dynamics of towed cable and body system, in which the large rigid body motion is coupled with small elastic deformation. The newly derived FE method is formulated in terms of element nodal positions, which is different from the existing FE methods that use displacements. The alternative FE method solves for the cable position directly in order to eliminate accumulated numerical errors arising from existing FE methods that solve for displacements first in order to obtain the cable position over very long period of time. The alternative FE formulation is implemented and applied to real applications to demonstrate its robustness by comparing simulation results with the experimental and sea trial data.