Aiming at the problem of multibody system difficult expression using bond graph, Chapter four systemically promotes the bond graph expression methods of complex mechanical system based on Euler equation of Newtonian mechanics, Lagrange equations of the first kind and Lagrange equations of the second kind.

In this paper, the active vibration control system of a 3 Dimensional supporter is realized based on smart structures. The finite element control equations are derived from Hamilton principle and Euler equation. The experimental results are provided.

The technique proposed builds the sound field separation formula in wave number domain according to the superposition of particle velocity theorem and Euler equation in wave number domain, then the sound pressure caused by sources on one side of holographic plane can be obtained as expected by talcing inverse two-dimensional Fourier transform of this formula.

The formula of the sound field separation technique based on measurement of the three-dimensional acoustic intensity with single holographic plane was established according to the superposition theory of particle velocity and the Euler equation in wave number domain, and the acoustic pressure caused by the sound sources on one side of holographic plane was also obtained.

To the lowest-order approximation in h, an important relation between the vacuum expectation value of color gauge fields and scalar fields is also derived by solving the Euler equation for the gauge fields.

2. The C∞stability of Euler equation determines C∞stability of some models when viscosity (molecule viscosity、turbulent viscosity) is neglected during atmospheric motion.

At large Reynolds numbers, the problem of determining the pressure distribution comes down to solving the Euler equation with corresponding boundary conditions.

The fluid is assumed to be ideal and incompressible and its flow symmetric; the lateral bending of the beam is described by the Euler equation.

The singularities of the corresponding analog of the Jacobi equation (and of the Euler equation) are generated by the procedure of integration by parts, which leads to differentiation with respect to measures glued (joined) together.

These relationships are based on the Euler equation, the correspondence principle, and the rule of contiguity of regions.

The Euler equation for the energy functional leads to a second-order differential equation for the canonical profile of the function μ = 1/q.

To be able to locate the point of inception of air entrainment is of considerable significance in the design of ski-jump spillways for high dams, in that this is not only prerequisite to the theoretical analysis of jet diffusion in the air and the subsequent alleviation of erosion as the aerated jet dives to the bed, but also essential if the possible benefit of aeration in the reduction of concrete pitting is to be evaluated. Although past contributions to this problem are numerous, no method has yet been available...

To be able to locate the point of inception of air entrainment is of considerable significance in the design of ski-jump spillways for high dams, in that this is not only prerequisite to the theoretical analysis of jet diffusion in the air and the subsequent alleviation of erosion as the aerated jet dives to the bed, but also essential if the possible benefit of aeration in the reduction of concrete pitting is to be evaluated. Although past contributions to this problem are numerous, no method has yet been available to yield the correct prediction of the onset of aeration in or downstream of the curvilinear portion of the spillway which is known to take place much earlier than usual. Even for the straight portion of the spillway, calculated positions of aeration inception do not always match with the observed values (see table 1, and compare columns 4 to 7). In this paper is presented a rational and yet rather simple procedure with which one may treat the general problem of locating the position of aeration inception no matter if the spillway contains a curve or not. In the first place, the irrotational or "ambient" flow outside of the boundary layer is studied. In view of the fact that the flow over the spillway of a high dam is much smaller in extent laterally than longitudinally, an approximation similar to that used in the derivation of boundary layer equations from the Navier-Stokes equations is applied to the Euler equations. The resulting expressions indicate that the usual assumption of concentric streamlines is justifiable. The depth of flow is taken as that so calculated plus the displacement thickness of the boundary layer. Since on the plane of the complex potential, the flow over a spillway may be formulated as a simply-mixed boundaryvalue problem, Wood's exact method is applied to a numerical example with gravitational effect taken into consideration. The result of calculation indicates that both methods yield practically the same depth of flow. The inception of aeration is, as usual, assumed to occur as the boundary layer meets the free surface. Under the combined influence of gravity and boundary geometry, the flow over a dam is continuously accelerated or decelerated. In such case, boundary-layer computation by usual method is both involved and of doubtful accuracy. It is found, however, that in the case of flow over a spillway, the flow outside of the turbulent boundary layer conforms to a condition of self-preservation as proposed by Townsend. Since the Reynolds number for high dams may surely reach very high values, the turbulent boundary layer itself may be assumed to be approximately self-preserving. In short, this means that Townsend's theory of approximate self-preservation for boundary layers under the influence of longitudinal acceleration may be applied. This also means that the computation of boundary layer development may be much simplified. Based on an analysis of prototype data, it is found that in the present case involving air-entrainment inception, thickness of the boundary layer should be defined as that at which the mean velocity is within 0.1% of the velocity of flow outside the boundary layer. In the carrying out of the computations, boundary layer thickness at various sections are first estimated, then the irrotational flow outside the boundary layer is analysed by the simplified method to obtain the surface profile and the parameter "a" denoting the variation of velocity along the surface, and finally Townsend's theory of approximate self-preservation is applied to recompute the thickness of the boundary layer along the spillway. As any error made in the estimation of boundary-layer thickness has little effect on the computation of surface profile and hence on any subsequent computations, reasonably experienced computers should find it unnecessary to repeat the computations. Results of computations are found to be within 10% of the observed data obtained at two dams.

In the free-boundary case, the extremum of the potential functional is found from the variational principle. Thereby the equation and boundary conditions required for plasma equilibrium are derived. The Euler equation of the relevant functional is the magnetic surface function equation with the condition of free boundary. A variational functional suitable for numerical computation is given. This functional corresponds to a boundary value problem with an equal-value surface boundary condition....

In the free-boundary case, the extremum of the potential functional is found from the variational principle. Thereby the equation and boundary conditions required for plasma equilibrium are derived. The Euler equation of the relevant functional is the magnetic surface function equation with the condition of free boundary. A variational functional suitable for numerical computation is given. This functional corresponds to a boundary value problem with an equal-value surface boundary condition. For the case of a conducting wall of simple geometry (i.e., a rectangular wall), numerical computation has been carried out by using the Ritz method.

This paper uses the variational method to calculate the configurations of projectile nose with the minimum drag under the conditions that the sum of nose wave resistance and frictional resistance satisfies the extremes (Euler equation). This paper also analyses and computes numerically the optimum nose L/D (length-to-diameter ratio), secant parabolic nose and tail contraction ratio and presents the calculated curve.