Lumped mass method

Finally, it is possible to reduce the dimensionality of a problem by determining which rate processes in series is the controlling step. As shown for l>i -C 1, the convection controls the cooling process and conduction is so fast that the solid is considered isothermal, reducing the dimensionality from (x, y, z) to a zero dimensional problem or lumped mass method. 

In a complete coupled analysis (Fig. 24.17) the 6 DOF motions of the floating vessel is solved at a given time step. The loads and motions of the top of each of the riser/mooring fine are determined. A finite element (or lumped mass) method for... 

The system of differential equations is too complex to be solved analytically. Assumptions of a linear adsorption isotherm can be used to obtain analytical solutions, but this approach is generally not applicable to describe affinity chromatography experiments. Several numerical techniques arc used to solve the system of partial differential equations. The other method is to use an analytical solution with simplifying approaches [32] that describe the adsorption process with a single step and a lumped mass transfer coefficient [27],... 

The analysis results in Figure 1 are from a numerical transient thermal and structural collapse analysis, where heat conduction is included and where the structural stiffness (and hence the load path) is computed for every time step in the solution process. This analysis is more accurate than the former because it includes the composite action of the three dimensional imevenly heated structure, where the load path is shed from the hot to the cold members. The method therefore represents the structural redimdancy and gives (more accurately) longer times to collapse than the lumped thermal mass method. 

The finite element method using consistent mass provides an excellent result for the fundamental frequency, but the accuracy deteriorates for higher modes. The lumped mass approximation provides only the fundamental frequency and the result is less accurate. Note that the finite element method overestimates the fundamental frequency whereas the lumped mass approximation provides an under estimate. 

When all the knots in the mesh and at the bar are used as mass points, there are more than 10,000 mass points for a common plane net, which requires too long a calculation time for practical simulations. To reduce the computational effort, a mesh grouping method can be employed in calculations. The method consists of modeling a given number of actual meshes as a fictitious equivalent mesh that has the same physical qualities as the actual meshes, such as the projected area of the net, specific mass, weight, etc. Figure 23.12 shows an example of a 4 x 4 mesh with 65 lumped mass points that is approximated to a 2 x 2 mesh with 21 lumped mass points. 

Static Analysis for Natural Frequency of an Overhung Shaft. The elements that determine the lateral natural frequency are the magnitudes and locations of concenbated and distributed masses, the tensile modulus of elasticity of the material, and the moment of inertia of the shaft. Ramsey and Zoller (1976) presented the basic elements of natural frequency for a shaft and impeller system like the one shown in Figure 21-32. That method uses a lumped mass, static technique for computing the critical speed of a shaft and impeller system. The mass of the individual impellers and the distributed mass of the shaft is lumped into a single mass at the end of... 

Determination of the vertical distribution of seismic forces along the height of the structure, where the force at a given level is proportional to the height h times the lumped mass m at that height (simplified method)... 

The structural analysis involves the definition of the model and selection of the analysis type. The model should represent the stiffness, the mass, and the loads of the structure. The structures can be represented using simplified models, such as the lumped mass models, and advanced models resorting the finite element method (FEM) and discrete element method (DEM). Depending on the characteristics of the structure, different types of analysis can be used such as limit analysis, linear and nonlinear static analysis, and linear and nonlinear dynamic analysis. 

The complex soil-structure interaction of underground structures during seismic loading can be simulated using numerical analysis tools which include lumped mass/stif iess methods and finite-element/difference methods. 

Lumped mass/stiffriess methods are useful to analyze the 3D behavior of a tuiuiel lining in a simplified manner. Many parameters for the springs that represent the structure stiffness and the soil stiffness must be defined to have a realistic model. 

This matrix is usually diagonalized using a simple mass lumping technique (Pittman and Nakazawa, 1984) to minimize the computational cost of pressure calculations in this method. 

STRESS. Applies the variational recovery method to calculate nodal values of pressure and, components of the stress. A mass lumping routine is called by STRESS to diagonalize the coefficient matrix in the equations to eliminate the... 

In most cases of interest, however, the system represented by equation (5.3.3) is overdetermined and we must enforce the closure condition with a different method. Let us return to a standard mass-balance least-square problem, such as, for instance, calculating the mineral abundances from the whole-rock and mineral chemical compositions. If xu x2,.. -,x are the mineral fractions, which may be lumped together in a vector x, the closure condition... 

Typically, a non-linear system dynamic model is made up of individual lumped models of the components which at a minimum conserve mass and energy across the given component, but may also have a momentum equation if pressure drops must also be analyzed. For most dynamic problems of interest in hybrid studies, however, the momentum equation may be taken as quasi-steady (unless the solver requires the dynamic form to perform the numerical solution). Higher fidelity individual models or reduced order models (ROMs) can also be used, where the connection to the system model would be made at each subcomponent boundary. Since dynamic systems modeling is not as common as steady-state modeling, some discussion of modeling approaches will be given. There are two primary methods used to provide solutions for the pressure-flow dynamics of a system model. 

The decision process of putting together models based the above methods is helped by some observations about timescale and solution stiffness. The modeler should decide on a timescale of interest. The characteristic times, or first order time constants, for a typical flow and a pressure element based on lumped momentum and mass conservation respectively are shown in the following equations [7, 8] ... 

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