Browsing by Author "Von Hertzen, Raimo"
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Item Interior formulation of axisymmetric Levinson plate theory(2016-06-01) Karttunen, Anssi T.; Von Hertzen, Raimo; Department of Mechanical Engineering; Marine Technology; Solid MechanicsIn this study, we show that the axisymmetric Levinson plate theory is exclusively an interior theory and we provide a consistent variational formulation for it. First, we discuss an annular Levinson plate according to a vectorial formulation. The boundary layer of the plate is not modeled and, thus, the interior stresses acting as surface tractions do work on the lateral edges of the plate. This feature is confirmed energetically by the Clapeyron's theorem. The variational formulation is carried out for the annular Levinson plate by employing the principle of virtual displacements. As a novel contribution, the formulation includes the external virtual work done by the tractions based on the interior stresses on the inner and outer lateral edges of the Levinson plate. The obtained plate equations are consistent with the vectorially derived Levinson equations. Finally, we develop an exact plate finite element both by a force-based method and from the total potential energy of the Levinson plate.Item On the foundations of anisotropic interior beam theories(2016-02-15) Karttunen, Anssi T.; Von Hertzen, Raimo; Department of Mechanical Engineering; Marine Technology; Solid MechanicsThis study has two main objectives. First, we use the Airy stress function to derive an exact general interior solution for an anisotropic two-dimensional (2D) plane beam. Second, we cast the solution into the conventional form of 1D beam theories to clarify some basic concepts related to anisotropic interior beams. The derived general solution provides the exact third-order interior kinematic description for the plane beam and includes the Levinson/Reddy-kinematics as a special case. By applying the Clapeyron's theorem, we show that the stresses acting as surface tractions on the lateral end surfaces of the interior beam need to be taken into account in all energy-based considerations related to the interior beam in order to avoid artificial end effects. Exact 1D interior beam equations are formed from the general 2D solution. Finally, we develop an exact interior beam finite element based on the general solution. With full anisotropic coupling, the stiffness matrix of the element becomes initially asymmetric due to the interior nature of the plane beam. By redefining the generalized nodal axial forces of the element, the stiffness matrix takes a symmetric form.Item Steady-state vibration of a viscoelastic cylinder cover subjected to moving loads(2016-07-01) Karttunen, Anssi T.; Von Hertzen, Raimo; Department of Mechanical Engineering; Marine TechnologyThe dynamic steady-state response of a viscoelastic cylinder cover subjected to circumferentially moving constant point and distributed loads is studied using a 1D Pasternak-type foundation model. The cover material is modeled according to the generalized Maxwell model as an incompressible frequency-dependent viscoelastic material spanning a wide relaxation spectrum. The vibration response of the cover for a moving twin point load is obtained using a modal expansion approach. On the basis of the solution, additional moving load cases are derived. In the case of a single moving point load, representing a load resultant due to rolling contact, numerical calculations show that regardless of the viscoelastic damping in the model, the critical load speed for the system can be well estimated by a resonance condition. In the vicinity of the critical speed, an incipient traveling wave arises behind the moving load. The viscoelastic cover stiffens for increasing excitation frequencies, thus, the cover response divides into two separate mode branches, of which the low-mode branch is dominant. A method to suppress the traveling wave vibrations in the cover at supercritical speeds using a moving twin point load, adjusted according to a dominant resonating mode, is presented. Using a distributed moving load, it is shown that depending on the wavelength, a traveling wave generated at the leading edge of the load may be reinforced at the trailing edge, the lift-off point, of the load. The developed model offers a fast and reliable way for practitioners to estimate the critical speeds of rolling contact machines with viscoelastic covers.