A quaternion-based mathematical model for geometrically exact dynamic analysis of cantilevered pipe conveying fluid

For the first time, the nonlinear geometrically exact governing equations and corresponding boundary conditions of hanging cantilevered flexible pipe conveying fluid in the framework of the quaternion system are developed. The linear model is also derived from the nonlinear one for the stability analysis. The linear and nonlinear mathematical formulations of the system according to the geometrically exact rotation-based model are extracted from the current model. The integro-partial differential algebraic equations of the system are converted to a set of ordinary differential algebraic equations via the Galerkin discretization technique and the resulting equations are numerically solved to determine the self-excited oscillation behavior of the system in the post-flutter region. Geometrically exact time traces, bifurcation diagrams, phase planes, and deformed configurations, along with the stability characteristics of the system are determined and compared with those reported based on the geometrically exact rotation-based model. The comparative studies divulge that the present model is capable of successfully capturing the stability and dynamic characteristics of the system. An interesting feature of the current ordinary differential algebraic equations is that their coefficients are time-independent, unlike the coefficients of geometrically exact rotation-based equations in the Galerkin form. The results show this feature leads to a remarkable decrease in the computational cost, although the number of equations increases and the nonlinearity becomes stronger. The challenging issues with the numerical solution utilized in this study are also discussed.Comment: 34 pages; 15 figure

Size-dependent behaviour of functionally graded microbeams using various shear deformation theories based on the modified couple stress theory

This study investigates the mechanical behaviours of functionally graded (FG) microbeams based on the modified couple stress theory. The material properties of these beams are varied through beam’s depth and calculated by using classical rule of mixture and Mori–Tanaka scheme. The displacement fields are presented by using a unified framework which covers various theories including classical beam theory, first-order beam theory, third-order beam theory, sinusoidal beam theory, and quasi-3D beam theories. The governing equations of bending, vibration and buckling problems are derived using the Hamilton’s principle and then solved by using Navier solutions with simply-supported boundary conditions. A number of numerical examples are conducted to show the validity and accuracy of the proposed approaches. Effects of Poisson’s ratio, material length scale parameter, power-law index, estimation methods of material properties and slenderness ratio on deflections, stresses, natural frequencies and critical buckling loads of FG microbeams are examined

Nonlinear thermo-resonant behavior of fluid-conveying FG pipes

In the current paper, an attempt is made to analyze the moderately large oscillations of a geometrically nonlinear functionally graded pipe conveying hot fluid subjected to a harmonic lateral excitation. The material properties of functionally graded pipe are presumed to vary continuously and smoothly through its radial direction according to a power law function. In addition, the temperature-dependency of material properties for both the pipe and fluid are taken into account. The equations of motion of the system in the form of partial differential equations (PDEs) are derived by implementing the Euler-Bernoulli beam hypothesis and the von-Karman geometric nonlinearity. The achieved PDEs are discretized to a set of nonlinearly coupled ordinary differential equations via the Galerkin technique. In order to assess the nonlinear thermo-resonant characteristics of the system, the method of harmonic balance is employed. Furthermore, the temperature distribution in the radial direction of pipe is calculated by use of the one-dimensional steady stead heat conduction model in conjunction with the Galerkin technique. The nonlinear thermo-resonant behavior of the system accompanied by bifurcations is examined via constructing the frequency-amplitude, force-amplitude, and backbone curves. In addition, the role of gyroscopic damping in the nonlinear resonant responses of system is explored. Eventually, the comparative studies for a homogeneous isotropic pipe conveying fluid in the reference temperature are conducted by employing numerical results available in the scientific literature