br The expressions for the

The expressions for the moment of inertia of this cross-section and for its area have the form
Further transformations lead to the expressions and we can then adduce the axial force in the following form:
As a result, substituting order Cabozantinib (21) into formula (20), we obtain the formula for the ultimate internal pressure in the bellows: where μ = 0.5 with a pinned connection, μ = 1 with a mixed connection and μ = 2 with a fixed connection.
Comparison of FE modeling results and analytical calculation We have considered several configurations of corrugation geometry and constructed bellows consisting of 3–12 corrugations for each of them, considering settings with different types of connections, namely:
Let us consider the FE solution scheme in details. First, the problem of static loading by internal pressure is solved, whence the stress matrix [σ] is derived:
Next, adjacent forms of equilibrium are searched for. Refs. [24–27] described registering the changes in a surface element\'s orientation in searching for adjacent forms of equilibrium. Let us write the equation of equality of external loads for adjacent forms of equilibrium: where [S] is the stiffness matrix, [] is the matrix of derivatives of shape functions. As a result, we obtain a set of multipliers of the initial load for each possible adjacent form of equilibrium:
We considered two sets of corrugation geometry parameters (see sets nos. 1 and 2 in Table 1). A curve of ultimate internal pressure distribution versus the number N of corrugations in the bellows was constructed for each of the parameter sets. The graphs also show the values obtained analytically by formula (22). Analysis of the obtained data revealed that the FE simulation results are close to theoretical ones (see Fig. 5a). However, this does not apply to bellows shells for which the local form of buckling occurs earlier than the global one. For example, it can be seen from Fig. 5b that there is a pronounced disagreement between the results of the analytical and finite-element simulations at the ultimate pressure of 20Mpa. This clearly demonstrates the effect of buckling in the corrugation plane. It should be noted that the ultimate pressure, at which adjacent steady local states of equilibrium occur, is virtually independent of the number of corrugations in the bellows. Let us consider some of the obtained adjacent equilibrium states in the bellows. Fig. 6 shows the bellows consisting of 4 and 7 corrugations. The main geometrical parameters of the corrugation are listed in Table 1 (set no. 3). By analyzing Fig. 6, it can be concluded that the global buckling form is characterized by the deviation of the bellows shell from the symmetry axis, whereas the local form is the bending of the shell in the corrugation plane.
Conclusion We have carried out a theoretical and numerical study of the stability of U-shaped expansion bellows under internal pressure, simulated the conditions in which adjacent states of equilibrium occur. Comparison of the theoretical results with the data of the numerical experiment conducted in the Ansys software revealed an agreement in the global buckling form for a considerable length of the shell.
Introduction Determining the aerodynamic characteristics of airfoil profiles is an important practical problem of computational aerodynamics. These characteristics are typically calculated using two-dimensional Reynolds-Averaged Navier–Stokes equations (RANS) in combination with various turbulence models, since scale-resolving simulation of such flows requires high computational costs [1], which is inadmissible for engineering computations. Thus, the two-dimensional RANS approach presents an acceptable compromise between the accuracy of calculations and computational resources required. Existing experience in simulating low Mach number flows (Mach number M ≤ 0.2) around airfoils indicates that for attached flow regimes the simulation results are in good agreement with experimental data, while for separated flows the accuracy is reduced significantly. Usually the value of the lift coefficient exceeds the experimental one, with the maximum error (which can achieve 40%) occurring at an angle of attack α that provides the maximum lift coefficient in the simulation. Note that these specific regimes are of significant practical interest for wind power and other industries.