On the three-dimensional vibrations of elastic prisms with skew cross-section

Three-dimensional (3-D) free vibration of an elastic prism with skew cross-section is investigated using an elasticity-based variational Ritz procedure. Specifically, the associated energy functional minimized in the Ritz procedure is formulated using a simple coordinate mapping to transform the solid skew elastic prism into a unit cube computational domain. The displacements of the prism in each direction are approximately expressed in the form of variable separation. As an enhancement to conventional use of algebraic polynomials trial series in related solid body vibration studies in the associated literature, the assumed skew prism displacement, u, v and w in the computational ξ–η–ζ skew coordinate directions, respectively, are approximated by a set of generalized coefficients multiplied by a finite triplicate Chebyshev polynomial series and boundary functions in ξ–η–ζ to ensure the satisfaction of the geometric boundary conditions of the prism. Upon invoking the stationary condition of the Lagrangian energy functional for the skew elastic prism with respect to the assumed generalized coefficients, the usual characteristic frequency equations of natural vibrations of the skew elastic prism are derived. Upper bound convergence of the first eight non-dimensional frequencies accurate to four significant figures is achieved by using up to 10–15 terms of the assumed skew prism displacement functions. First known 3-D vibration characteristics of skew elastic prisms are examined showing the effects of varying prism length ratios (ranging from skew solids to skew slender beams), as well as, varying cross-sectional side ratios and skewness, which collectively can serve as benchmark studies against which vibration modes predicted by classical Euler and shear deformable skew beam theories as well as alternative methodologies used in elastic prism vibrations of mechanical and structural components.     

Wake and cavitation characteristics of equilateral prisms at incidence

The characteristics of flow past a two-dimensional equilateral prism are experimentally studied at different angles of incidence to the approach flow. Tests were conducted in a water tunnel suitable for cavitation studies. The presence of cavitation facilitated excellent visual observations of the wake region. The mean separation pressure coefficients and the vortex-shedding frequency were determined at various degrees of cavitation for five different orientations of the prism. The latter determines, to a large extent, the vibration characteristics of the system in which the prismatic elements are located.     

Jump rule for edge impacts of rolling prisms

We study experimentally and theoretically the planar dynamics of purely rolling prisms on a rough ramp, where the rolling motion is interrupted intermittently by edge impacts. The experiments were carried out for prisms made of different materials and having different geometries. We found that the angular velocities of the rolling prisms are material-independent, but they change significantly with their geometry. We modelled the dynamics of edge impacts by considering a socalled detachment front propagating across the contact interface. The detachment front represents the moving boundary between a detached region and a stress region that coexist within the interface plane. The theoretical analysis indicates that the detachment front can be characterized by a scale number, whose value converges to 0.4050 for prisms having large number of edges. A new jump rule for edge impacts is then developed, by which we can accurately reproduce the experimental observations, and explain why the motion of the prism is material-independent.     

并列三柱体风致干扰DDES模拟分析

采用基于SSTk-ω湍流模型的延迟分离涡模拟方法DDES(Delayed detached eddy simulation),对三并列线性布置柱体的风荷载和流场特性进行了数值模拟.为验证方法的有效性,对单个三维方柱绕流进行了计算分析,比较计算值与试验结果的风荷载特征,两者符合较好.然后采用DDES研究了三并列方柱对称布置时,间距比对中间受扰建筑风致干扰的问题.数值结果表明,间距比较小时对受扰建筑风荷载分布有较大影响.总的来看,当间距比约为3.0时,受扰建筑的影响达到最大.同时,数值分析了三方柱周围瞬态流场特征,探讨了临近建筑之间的干扰机理.结果 表明,SST-DDES方法能够较好地预测绕方柱流动分离问题,可为群体高层建筑气动相互干扰研究提供参考.     

Three-dimensional vibration analysis of prisms with isosceles triangular cross-section

This paper studies the three-dimensional (3-D) free vibration of uniform prisms with isosceles triangular cross-section, based on the exact, linear and small strain elasticity theory. The actual triangular prismatic domain is first mapped onto a basic cubic domain. Then the Ritz method is applied to derive the eigenfrequency equation from the energy functional of the prism. A set of triplicate Chebyshev polynomial series, multiplied by a boundary function chosen to, a priori, satisfy the geometric boundary conditions of the prism is developed as the admissible functions of each displacement component. The convergence and comparison study demonstrates the high accuracy and numerical robustness of the present method. The effect of length-thickness ratio and apex angle on eigenfrequencies of the prisms is studied in detail and the results are compared with those obtained from the classical one-dimensional theory and the 3-D finite element method. Sets of valuable data known for the first time are reported, which can serve as benchmark values in applying various approximate beam and rod theories.     

Experimental and theoretical investigation of galloping of transversely inclined slender prisms

Galloping is characterized by large and periodical oscillations which may lead to collapse of slender structures. This study is the first attempt of a comprehensive experimental and theoretical investigation of galloping of transversely inclined prisms. A modified quasi-steady model is proposed with a constant term to estimate the galloping of a transversely inclined prism, which is later experimentally investigated by conducting a static Synchronous Multi-Pressure Sensing System (SMPSS) test and an aeroelastic test in a boundary layer wind tunnel. The galloping responses of the prisms were measured in the aeroelastic test, while the aerodynamic force coefficients were determined from the SMPSS test. These experimental results were subsequently utilized to validate the quasi-steady model. Based on the proposed model, the galloping responses of the prisms were predicted and compared with the experimental results. The experimentally measured and theoretically predicted galloping responses are discussed with respect to aerodynamic damping ratios, onset galloping wind speeds, distributed pressure coefficients, point pressure spectra and vortex shedding frequencies. Interesting findings are summarized.     

One-dimensional equations for coupled extensional,radial, and axial-shear motions of circular piezoelectric ceramic rods with axial poling

A system of approximate, one-dimensional partial differential equations with one spatial coordinate and time as independent variables is derived for axisymmetric motions of a piezoelectric ceramic rod of circular cross section. The equations take into account the couplings among extensional, radial and axial-shear modes. The dispersion curves for the three waves in an infinite rod are compared with analogous solutions of the three-dimensional equations. The equations obtained are useful in the modeling of ceramic rod piezoelectric transducers that are not very long and thin.     

Evolutive and nonlinear vibrations of rotor on aerodynamic bearings

The use of air as a lubricant in aerodynamic bearings is advantageous, particularly in the food industry. Aerodynamic bearings with tilting pads have complicated stiffness and damping properties and need a very detailed theoretical and experimental research. Response curves of rigid rotor supported on aerodynamic bearings are presented for a linear but evolutive mathematical model. Due to non-monotone properties of stiffness and damping matrices at variable revolutions, a new resonance appears. The mathematical model of rotor vibrations in the whole area of bearing clearance is also developed in the consideration of strongly nonlinear properties of aerodynamic bearing.     

Instantaneous flow field above the free end of finite-height cylinders and prisms

The flow above the free ends of surface-mounted finite-height circular cylinders and square prisms was studied experimentally using particle image velocimetry (PIV). Cylinders and prisms with aspect ratios of AR = 9, 7, 5, and 3 were tested at a Reynolds number of Re = 4.2 × 10⁴. The bodies were mounted normal to a ground plane and were partially immersed in a turbulent zero-pressure-gradient boundary layer, where the boundary layer thickness relative to the body width was δ/D = 1.6. PIV measurements were made above the free ends of the bodies in a vertical plane aligned with the flow centreline. The present PIV results provide insight into the effects of aspect ratio and body shape on the instantaneous flow field. The recirculation zone under the separated shear layer is larger for the square prism of AR = 3 compared to the more slender prism of AR = 9. Also, for a square prism with low aspect ratio (AR = 3), the influence of the reverse flow over the free end surface becomes more significant compared to that for a higher aspect ratio (AR = 9). For the circular cylinder, a cross-stream vortex forms within the recirculation zone. As the aspect ratio of the cylinder decreases, the reattachment point of the separated flow on the free end surface moves closer to the trailing edge. For both the square prism and circular cylinder cases, the instantaneous velocity vector field and associated in-plane vorticity field revealed small-scale structures mostly generated by the separated shear layer.     

Vehicle vibrating on a soft compacting soil half-space: Ground vibrations, terrain damage, and vehicle vibrations

The point of departure of the present work may be either an interest in vehicle vibrations themselves, or in ground vibrations and terrain damage due to vehicles traveling off-road. The vibrations of a vehicle traversing dry, soft terrain, which is either rough or undulating, may be significantly modified by the dynamic interaction of the vehicle with the soil, particularly due to losses of energy by soil compaction and as elastic waves. The present work provides a prediction methodology for both vehicle and soil vibrations, accounting for the effects mentioned above. An expedient linear method is compared to a rheologically-based non-linear method. In the linear method, the soil compaction is incorporated as a loss factor in the dynamic stiffness of the otherwise elastic half-space; the imaginary part of that dynamic stiffness already includes the effects of wave damping. The non-linear model treats the compaction using a general rheological model for soils exhibiting both viscous and thixotropic effects, and requires iterative solution. A key feature of the latter model is the hypothesis that the stress distribution may be approximately regarded as quasi-static when calculating compaction losses; that approximation is expected to hold at low frequencies, since the P-wavelength in the soil is then much greater than the dimensions of the zone in which most compaction occurs. The methods predict that the soil compaction and excited ground vibrations have maxima at the vehicle bounce and hop resonances, and at high frequencies at which the Rayleigh wavelength approaches the order of the contact patch diameter. Moreover, sufficiently soft, compactable soils, but fully realizable in nature, control the vehicle response at the hop resonance, and possibly also at the bounce resonance.     

The effect of incidence angle on the mean wake of surface-mounted finite-height square prisms

The mean wake of a surface-mounted finite-height square prism was studied experimentally in a low-speed wind tunnel to explore the combined effects of incidence angle (α) and aspect ratio (AR). Measurements of the mean wake velocity field were made with a seven-hole pressure probe for finite square prisms of AR = 9, 7, 5 and 3, at a Reynolds number of Re = 3.7 × 10⁴, for incidence angles from α = 0° to 45°. The relative thickness of the boundary layer on the ground plane, compared to the prism width, was δ/D = 1.5. As the incidence angle increases from α = 0° to 15°, the mean recirculation zone shortens and the mean wake shifts in the direction opposite to that of the mean lift force. The downwash is also deflected to this side of the wake and the mean streamwise vortex structures in the upper part of the wake become strongly asymmetric. The shortest mean recirculation zone, and the greatest asymmetry in the mean wake, is found at the critical incidence angle of α_critical ≈ 15°. As the incidence angle increases from α = 15° to 45°, the mean recirculation zone lengthens and the mean streamwise vortex structures regain their symmetry. These vortices also elongate in the wall-normal direction and become contiguous with the horseshoe vortex trailing arms. The mean wake of the prism of AR = 3 has some differences, such as an absence of induced streamwise vorticity near the ground plane, which support its classification as lying below the critical aspect ratio for the present flow conditions.     

Coupling effects on torsional vibrations of prismatic bars

Summary Due to the warping any torsional wave or disturbance in a noncircular cylinder is propagated with a higher velocity than that predicted by the elementary theory. This change of the wave velocity depending on wave length, i.e. the dispersion phenomenon, is accompanied by certain modifications of stresses and displacements. This paper presents a systematic investigation of these coupling effects within the framework of the linear theory of elasticity by means of the method of strained parameters and of the technique of Lindstedt-Poincaré leading to exact solutions in the asymptotic sense. The dispersion is found to be related to the shearing stresses of higher order of magnitude and to the distortion of cross section which are consequences not only of the axial stresses and accelerations hut also of the plane stresses. The coupling between torsional and flexural vibrations which occur in the case of cross sections not having two axes of symmetry is also determined explicitly.

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Kopplungseffekte bei Torsionsschwingungen zylindrischer Stäbe

Übersicht Infolge Verwölbung weisen Torsionswellen bei Stäben nichtkreisförmigen Querschnitts höhere Geschwindigkeiten auf als jene nach der klassischen St. Venantschen Theorie. Diese wellenlängenabhängige, als Dispersionsphänomen bekannte Änderung der Geschwindigkeit hängt unmittelbar mit Spannungen und Verschiebungen zusammen, welche über den Rahmen der elementaren Theorie hinausgehen. Die vorliegende Arbeit gibt eine systematische Untersuchung dieser Kopplungseffekte auf der Grundlage der linearen Elastizitätstheorie wieder, welche sich der Methode der asymptotischen Entwicklungen bedient und durch Anwendung der Technik von Lindstedt-Poincaré zu asymptotisch exakten Ergebnissen führt. Es wird gezeigt, daß die Dispersion im direkten Zusammenhang mit den Schubspannungen und Querverschiebungen sekundärer Größenordnung steht, welche als Folge von durch Verwölbung bedingten axialen Spannungen und Beschleunigungen, aber auch von ebenen Spannungen zustande kommen. Die je nach Symmetrieeigenschaft des Querschnitts zu erwartende Kopplung zwischen Torsions- und Biegeschwingungen wird ebenfalls explizit erfaßt.

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This paper has been presented at the XVIth International Congress of Theoretical and Applied Mechanics, held on August 19–25, 1984 in Lyngby/DK