Kinematical differential equation

Kinematics is based on one-dimensional differential equations of motion. Suppose a particle is moving along a straight line, and its distance from some reference point is S (see Figure 2-6a). Then its linear velocity and linear acceleration are defined by the differential equations given in the top half of Column 1, Table 2-5. The solutions... 

Comparing with the solution in Section 6.4, we observe that instead of a single differential equation for the velocity profile, two coupled (through y) differential equations are obtained. However, the kinematics can be well approximated by assuming dP/dy = 0, which then will... 

The kinematics of the deformation is described by the following set of differential equations [Poitou, 1988] ... 

Equation 1 is the continuum equilibrium condition, Eq. 2 are the constitutive equations, which specify the material behavior, and Eq. 3 are the kinematics, the displacement-deformation conditions. The forming machine defines the boundary conditions for this set of partial differential equations. It can be seen without going into details that those equations are essentially grade two in the displacements, which means that boundary conditions are in the displacements or in the first derivative of displacements. Eirst derivatives of displacements are in all constitutive equations connected to stress at least due to elasticity, which is common for all materials. Erom these two types of press machines can be derived, namely, path-driven machines, where boundary conditions for the displacements are prescribed, and force-driven machines, where stress boundary conditions are prescribed, which are integrated to the press force. Erom this it follows also that despite the possibilities of servo presses to operate under different modes of the drive, path, force, or energy, nothing really new is added, because the drive can only introduce either boundary condition at a time. 

Numerical solutions to the coupled heat and mass balance equations have been obtained for both isothermal and adiabatic two- and three-transition systems but for more complex systems only equilibrium theory solutions have so far been obtained. In the application of equilibrium theory a considerable simplification becomes possible if axial dispersion is neglected and the plug flow assumption has therefore been widely adopted. Under plug flow conditions the differential mass and heat balance equations assume the hyperbolic form of the kinematic wave equations and solutions may be obtained in a straightforward manner by the method of characteristics. In a numerical simulation the inclusion of axial dispersion causes no real problem. Indeed, since axial dispersion tends to smooth the concentration profiles the numerical solution may become somewhat easier when the axial dispersion terra is included. Nevertheless, the great majority of numerical solutions obtained so far have assumed plug flow. 

In the foregoing demonstration, we had limited ourselves to include only the kinematic aspects of bubble motion. A dynamic model including force balances on bubble motion would have called for adding the bubble velocity also as a particle state variable. Such a model could also have been considered allowing for bubble velocity to be a random process satisfying a stochastic differential equation of the type (2.11.14). The basic objective of this example has been to demonstrate applications in which particle state can be a random process. The next and the last example in this chapter considers a similar application, but with a distinction that can help address an entirely different class of problems. 

Values of the kinematics factor were reported by Lee and Liu [6]. The coat-hanger die with the above geometric parameters can theoretically deliver a liquid film with perfect flow uniformity. The first-order differential equations that describe the variations of l iy) can be solved by the fourth-order Runge-Kutta method [24]. 

The examples we have studied thus far have had rather simple kinematics flow parallel or nearly parallel to a wall and ideal or nearly ideal extension. Thus, we have been able to obtain exact solutions for the flow or to obtain approximate solutions based on the small difference between the actual flow and an ideal case for which an exact solution is available. Even for the case of fiber spinning, where an analytical solution to the thin filament equations cannot be obtained under conditions relevant to industrial practice, we simply need to obtain a numerical solution to a pair of ordinary differential equations, which is a task that can be accomplished using elementary and readily available commercial software. 

One may recall the approach employed to establish these three solutions, by making assumptions about the kinematics of the flow to simplify the system of differential equations to be solved. In fact, we have merely checked that such solutions verify Navier-Stokes equations and the boundary conditions for each of the problems considered. 

As with the simple models from Chapter 3, each different mechanical model can be described by a differential equation. The differential equation governing the response for any mechanical model may be obtained by considering the constitutive equations for each element as well as the overall equilibrium and kinematic constraints of the network. Once the differential equation is obtained, the response of the model to any desired loading can be examined by solving the differential equation for that particular loading. The solution for simple creep or relaxation loading will provide the creep compliance or the relaxation modulus for the given model. In this... 

The objective is to find the constitutive equation (governing differential equation) for the three-parameter model. The kinematic equation for the three-parameter solid is,... 

Note that this Four-Parameter Fluid model is composed of a Kelvin element (subscripts 1) and a Maxwell element (subscripts 0). Thus, the constitutive laws (differential equations) for the Kelvin and Maxwell elements need to be used in conjunction with the kinematic and equilibrium constraints of the system to provide the governing differential equation. Again, treating the time derivatives as differential operators will allow the simplest derivation of Eq. 5.12. The derivation is left as an exercise for the reader as well as the determination of the relations between the pi and q, coefficients and the spring moduli and damper viscosities (see problem 5.1). 

A differential equation for either of the series of Kelvin elements can be found using the same procedure described in developing the differential equation for a series of Maxwell elements. The equilibrium constraint, kinematic constraint and constitutive equations are given by... 

The standard procedure (Rose 1961), for heavy atoms, is to solve the wave equation in a spherical potential with relativistic kinematics. In open-shell systems the charge density is spherically symmetrized by averaging over all azimuthal quantum numbers. The wave equations to be solved in practice are, therefore, the coupled first-order differential equations for the radial components of the Dirac equation (Rose 1961)... 

Different approaches are possible for the description of the motion of a multibody system. A review of methods commonly in use may be found in [13] or [17]. A minimal description, e.g. with Lagrange s method, provides the smallest system of differential equations but suffers a few drawbacks. Some of these are the impossibility of describing the whole configuration space in a single coordinate chart without singularities in the presence of kinematical loops the difficulty inherent in the calculation of the eliminated constraint reactions and the consideration of friction forces the fact that the equations may be extremely complicated, i.e., the functions describing the differential equations are difficult to derive and expensive to compute the lack of modularity, in the sense that the knowledge of the equations of motion of subsystems is difficult to use in the derivation of the equations of the full system. 

The kinematic model is derived from the geometrical model by differentiating equation... 

The student is often left with the impression that the difference between open systems and closed systems is simply the replacement of the internal energy, U, by the enthalpy, H. In addition, the student may leave this brief encounter with bodies and control volumes with the idea that the kinematics required to accomplish the transformation consists of nothing more than a sketch on a piece of paper. If we want our students to believe this type of development, then we want them to believe in the tooth fairy. And if we want them to believe In the tooth fairy, what happens when they come face-to-face with a tough problem Instead of dealing with one entrance-one exit systems in which "a pound goes in and a pound goes out," let us use kinematics and the concepts of stress to derive the differential equation associated with Eq. 4-4 (Whitaker, Sec. 10.1, 1968)... 

The kinematical unknowns of the problem depend only on the variable x (a, 6x, Ox) ) and will be established from the solution of global differential equations of motion, which will be formulated employing the calculus of variations. In order to facilitate the analysis, the following stress resultants are defined as... 

This model may be extended to any porous body with constant cross section and porosity p, if r is considered as an equivalent capillary radius or, simply, as an unspecified geometrical parameter. The differential equation for the liquid uptake per unit area /i of a porous body is now eq. 3 with the parameters b and being defined by eq. 4 and 5. The dynamic viscosity r) was replaced by the kinematic viscosity t using... 

How do we derive the differential equations for the higher order polymers, that is, how do we find the values for p and ql The principles of statics and kinematics are applied to the composite elements, and a creep test and a relaxation test are applied. 

A kinematic wave may be called linear if the relationship between the flow and the concentration can be expressed by one or more linear equations, algebraic or differential. The term linear may also be applied when a diffusion term is included in the continuity equation as is done in 3 of Lighthill ... 

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