Free flexural vibrations of homogeneous beams with symmetrically variable depths

The subject of the paper are homogeneous beams of symmetrically variable depth and bisymmetrical cross sections. Free flexural vibrations of these beams are analytically and numerically studied. Based on Hamilton’s principle, the differential equations of motion of these beams are obtained. The equations of motion are analytically solved with consideration of the bending lines of these beams subjected to their own weight. The fundamental natural frequency for exemplary beams is derived and presented in Tables and Figures.

An extended Hamilton principle as unifying theory for coupled problems and dissipative microstructure evolution

An established strategy for material modeling is provided by energy-based principles such that evolution equations in terms of ordinary differential equations can be derived. However, there exist a variety of material models that also need to take into account non-local effects to capture microstructure evolution. In this case, the evolution of microstructure is described by a partial differential equation. In this contribution, we present how Hamilton’s principle provides a physically sound strategy for the derivation of transient field equations for all state variables. Therefore, we begin with a demonstration how Hamilton’s principle generalizes the principle of stationary action for rigid bodies. Furthermore, we show that the basic idea behind Hamilton’s principle is not restricted to isothermal mechanical processes. In contrast, we propose an extended Hamilton principle which is applicable to coupled problems and dissipative microstructure evolution. As example, we demonstrate how the field equations for all state variables for thermo-mechanically coupled problems, i.e., displacements, temperature, and internal variables, result from the stationarity of the extended Hamilton functional. The relation to other principles, as the principle of virtual work and Onsager’s principle, is given. Finally, exemplary material models demonstrate how to use the extended Hamilton principle for thermo-mechanically coupled elastic, gradient-enhanced, rate-dependent, and rate-independent materials.

Motion equations and non-Noether symmetries of Lagrangian systems with conformable fractional derivative

In this paper, we present the fractional motion equations and fractional non-Noether symmetries of Lagrangian systems with the conformable fractional derivatives. The exchanging relationship between isochronous variation and fractional derivative, and the fractional Hamilton's principle of the holonomic conservative and non-conservative systems under the conformable fractional derivative are proposed. Then the fractional motion equations of these systems based on the Hamilton's principle are established. The fractional Euler operator, the definition of fractional non-Noether symmetries, non-Noether theorem and Hojman's conserved quantities for the Lagrangian systems are obtained with conformable fractional derivative. An example is given to illustrate the results.

Stochastic Variational Formulations of Fluid Wave–Current Interaction

We are modelling multiscale, multi-physics uncertainty in wave–current interaction (WCI). To model uncertainty in WCI, we introduce stochasticity into the wave dynamics of two classic models of WCI, namely the generalised Lagrangian mean (GLM) model and the Craik–Leibovich (CL) model. The key idea for the GLM approach is the separation of the Lagrangian (fluid) and Eulerian (wave) degrees of freedom in Hamilton’s principle. This is done by coupling an Euler–Poincaré reduced Lagrangian for the current flow and a phase-space Lagrangian for the wave field. WCI in the GLM model involves the nonlinear Doppler shift in frequency of the Hamiltonian wave subsystem, which arises because the waves propagate in the frame of motion of the Lagrangian-mean velocity of the current. In contrast, WCI in the CL model arises because the fluid velocity is defined relative to the frame of motion of the Stokes mean drift velocity, which is usually taken to be prescribed, time independent and driven externally. We compare the GLM and CL theories by placing them both into the general framework of a stochastic Hamilton’s principle for a 3D Euler–Boussinesq (EB) fluid in a rotating frame. In other examples, we also apply the GLM and CL methods to add wave physics and stochasticity to the familiar 1D and 2D shallow water flow models. The differences in the types of stochasticity which arise for GLM and CL models can be seen by comparing the Kelvin circulation theorems for the two models. The GLM model acquires stochasticity in its Lagrangian transport velocity for the currents and also in its group velocity for the waves. However, the CL model is based on defining the Eulerian velocity in the integrand of the Kelvin circulation relative to the Stokes drift velocity induced by waves driven externally. Thus, the Kelvin theorem for the stochastic CL model can accept stochasticity in its both its integrand and in the Lagrangian transport velocity of its circulation loop. In an “Appendix”, we also discuss dynamical systems analogues of WCI.

Dynamic Analysis of Encastre Beams by Modification of the System’s Stiffness Distribution

Numerical and energy methods are used to dynamically analyze beams and complex structures. Hamilton’s principle gives exact results but cannot be easily applied in frames and complex structures. Lagrange’s equations can easily be applied in complex structures by lumping the continuous masses at selected nodes. However, this would alter the mass distribution of the system, thus introducing errors in the results of the dynamic analysis. This error can be corrected by making a corresponding modification in the systems’ stiffness matrix. This was achieved by simulating a beam with uniformly distributed mass with the force equilibrium equations. The lumped mass structures were simulated with the equations of motion. The continuous systems were analyzed using the Hamilton’s principle and the vector of nodal forces {P} causing vibration obtained. The nodal forces and displacements were then substituted into the equations of motion to obtain the modified stiffness values as functions of a set of stiffness modification factors. When the stiffness distribution of the system was modified by means of these stiffness modification factors, it was possible to predict accurately the natural fundamental frequencies of the lumped mass encastre beam irrespective of the position or number of lumped masses.