Impulse/momentum equation

Integrating Equation 2-30 with respect to time yields the impulse/momentum equation... 

While the modified energy equation provides for calculation of the flowrates and pressure drops in piping systems, the impulse-momenlum equation is required in order to calculate the reaction forces on curved pipe sections. I he impulse-momentum equation relates the force acting on the solid boundary to the change in fluid momentum. Because force and momentum are both vector quantities, it is most convenient to write the equations in terms of the scalar components in the three orthogonal directions. 

The collisional impulse can be obtained by using the impulse-momentum equations [Goldsmith, I960]. However, this stereomechanical approach does not yield the transient stresses, collisional forces, impact duration, or collisional deformation of the colliding objects. These quantities have to be obtained from the analysis of stresses and strains of the solids due to the impact, which is introduced later. 

Lankarani, H. M. and P. E. Nikravesh, Canonical Impulse-Momentum Equations for Impact Analysis of Multibody Systems, Journal of Mechanical Design, Trans. ASME, Vol. 114, Mar. 1992, pp. 180-186. 

ABSTRACT. Analytical evaluation of the performance of multibody mechanical systems becomes rapidly unmanageable as the complexity of the systems increase. For problems that involve intermittent motion due to an impact, prediction of the responses is even more difficult. In an impact, nonlinear contact forces of unknown nature are created, which act and disappear over a short period of time. In this paper, different contact force models are formulated, with which a continuous analysis method is developed for a simple two-particle impact. The procedure is then generalized to impact in multibody systems using the concept of effective mass. A piecewise analysis method is discussed, which is based on a canonical form of the system impulse/momentum equations. The suitability of these methods are discussed by application of these procedures to some examples. An optimization methodology is then discussed for the selection of proper parameters in a given contact force model. The use of this technique in the selection of the most suitable materials, which are impact-resistant, is also discussed. 

The presented continuous analysis methods, both the multibody system and the two-particle model, have been compared to the piecewise analysis method. For the piecewise analysis, the impulse-momentum equations used at the time of impact are of canonical... 

In the first simulation, a piecewise analysis was performed for any contact between the vehicle rollbar cage and the ground. The canonical impulse-momentum equations, nation (22), were used at the time of impact. The frictional impulse was also introduced in the analysis. 

The foregoing discussion of impulse and momentum applies only when no change in rotational motion is involved. There is an analogous set of equations for angular impulse and impulse momentum. The angular momentum about an axis through the center of mass is defined as... 

To understand how an appropriate momentum equation can be derived, consider first a stationary tank into which solid masses are thrown, Figure 1.7a. Momentum is a vector and each component can be considered separately here only the x-component will be considered. Each mass has a velocity component vx and mass m so its x-component of momentum as it enters the tank is equal to mvx. As a result of colliding with various parts of the tank and its contents, the added mass is brought to rest and loses the x-component of momentum equal to mvx. As a result there is an impulse on the tank, acting in the x-direction. Consider now a stream of masses, each of mass m and with a velocity component vx. If a steady state is achieved, the rate of destruction of momentum of the added masses must be equal to the rate at which momentum is added to the tank by their entering it. If n masses are added in time t, the rate of addition of mass is nmJt and the rate of addition of x-component momentum is (nm/t)vx. It is convenient to denote the rate of addition of mass by Af, so the rate of addition of x-momentum is Mvx. 

There is one important question why are tectonic stresses changing relatively slowly as we can see in the field Definitely, there is a special mechanism of their redistribution typical for a wave energy transfer in a cataclastic medium. Its elements (blocks) can rotate. This adds the balance of the moment of momentum to the conventional impulse balance equations as well as spin (surplus) velocity of an individual block. The constitutive laws were suggested (Nikolaevskiy, 1996), that led to the Sin-Goidon equation with its soliton solution. 

In mechanics, a body is an object made up of several elements and its total mass is the sum of the mass of each part. This is in contradistinction with the notion of particle, which is an elementary entity endowed with an indissociable mass (without losing the nature of the particle). For distinguishing the two systems, the scheme in the case study abstract shows a body which is a cluster of particles the variability of the entity number [the momentum (impulse) which is the sum of all momentum associated with every particle] and, in the equations, the variables featuring the impulse (momentum) and the mass are in uppercase for a body and in lowercase for a particle. 

The impulse (momentum) of the dipole corresponds therefore, by integration, to the common difference of each pole. From this, it may be deduced that the conservation of momenta is expressed by the following balance equation... 

Then the general equations written above are transformed into the so-called Kirch-hoff equations which describe the motion of a rigid body in an ideal boundless liquid. In this case, the vectors e and u = are usually called an impulsive force and an impulsive momentum, respectively. 

The third term on the rhs of Eq. 202 is zero due to axi-symmetry. We further restrict ourselves to a quasi steady-state solution, i.e. we assume that at any given time /, the flow can be approximated as being steady. This would mean, for instance, that the impulsive loading involved in the start-up of the squeeze would not be covered by the solution. The quasi steady-state assumption allows us to discard the term on the Ihs of Eq. 202. The simplified z-momentum equation is thus... 

After introduction of an impulsive source U <5(R — R ) <5(f) of momentum into the suspension at position R = R (with R = Rn- + r ), at time t = 0 and vector strength U, subsequent transport of this momentum tracer through the suspension (the latter assumed to be initially at rest) is governed by the system of equations (Mauri and Brenner, 1991a,b)... 

These are physically the correct equations of motion, only the momentum degree of freedom is spreading diffusively. This is expected classically from the medium giving the system minute impulse kicks changing its momentum only. If we have a harmonic oscillator,... 

Here, as in Ehod et al. [5], we have identified the incident energy pulse E as the product of the acoustic intensity (see Eq. 15) with the pulse duration and cross-sectional area. This represents the total energy delivered by the beam while it is on. Equating the impulse to the initial momentum of the ejected cylinder, one finds the scaling for the initial velocity... 

The M. (, t) terms are the nine time-dependent components of the moment tensor. This equation can be simplified dramatically by assuming that the source time function is an impulse and that all of the moment tensor components have the same time-dependency (synchronous source approximation). In addition, since the equivalent body forces conserve angular momentum. My = only six of the components ate required. Applying these assumptions, Eq. 5.8 reduces to the following linear relationship ... 

Inversion Both inductive relations have the same shape between electromagnetism and mechanics however, the role of variables is inverted In Equation 4.4, the cmrent / (which is a flow) is replaced by the momentum P (which is an impulse) and the quantity (flux) of induction 0g (which is an impulse) by the velocity v (which is a flow). The same distribution of roles is not always found in an inductive relation from one energy variety to the other. One finds again this same inversion of roles in the capacitive relationship between the thermal energy and the other varieties. 

The third region, the high momentum transfer, is the one where the single particle properties of the electron are prominent. This region is investigated mainly by Y-ray Compton scattering experiments and equation (1) is reduced within the limits of Impulse Approximation to ... 

Forces R and K are the global pressure and friction forces, rcspectively, exerted by the flow on solid walls (a minus sign appears in equation [2.17] in front of K). The left-hand side of equation [2.18] represents the impulse flux. While applying the momentum theorem, it is important to specify whether the forces under consideration are the ones exerted by the flow on solid walls or the other way round. Sign errors frequently occur when there is lack of specificity in this respect. To apply the momentum theorem, the first step is to specify domain D, throughflow surfaces Sjand solid surfaces S. Care must then be taken to orientate the normals in the proper direction. 

A collision between the two bodies is known as an impact during which forces are created that act and disappear over a short period of time. The duration of the contact period governs the choice of the method used to analyze the impact. The methods for predicting the impact responses can primarily be classified into two groups. In one, the impact is treated as a discontinuous event. Momentum-balance/impulse equations are usually formulated by integrating the acceleration-based form or the canonical form of the governing equations of motion. The solution to these equations gives the jump in the... 

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