Natural frequencies of heated annular and circular plates

Advances in Thermoelastic Damping in Micro- and Nano- Mechanical Resonators: a Review

“…The model was simplified by using the Helmholtz decomposition technique and the resulting equations were solved by using separation of variable method to obtain the secular equations in isolated mathematical conditions for the plates with stress-free or rigidly fixed, thermally insulated and isothermal boundary surfaces. Arafat et al [48] analytically analyzed the behavior of annular plates with clamped-clamped immovable boundary conditions subjected to axisymmetric in-plane thermal loads. The mode shapes and natural frequencies were calculated.…”

Modal interactions in the vibrations of a heated annular plate

“…The linear free vibrations problem for the heated annular plate was solved by Arafat et al [27] for both axisymmetric and asymmetric modes. In general, unless the temperature gradient is zero across the plate (i.e., = 1), analytical solutions are not available.…”

“…Among other internal resonances, Arafat et al [27] found that, as the thermal load p is increased and prior to buckling, a three-to-one internal resonance between the ÿrst and second axisymmetric modes is possible; that is, ! 02 ≈3!…”

“…In studying the linear free vibration problem of circular and annular plates under an axisymmetric thermal loading, as shown in Fig. 1, Arafat et al [27] found that, as the thermal load is increased, the natural frequencies can become near-commensurate with each other, possibly giving rise to internal resonances. For example, one-to-one, three-to-one, and internal combination resonances could be activated among di erent axisymmetric and asymmetric modes at di erent pre-buckling levels of the thermal load.…”

Analysis of Dynamic Characteristics of a Rotating, Themally Loaded Circular Saw Subjected to Tensioning over a Double Annular Domain

“…They are normally subjected to a thermal load resulting from the friction between the blade and the workpiece. This thermal load generally produces compressive in-plane forces, which in turn change dynamic characteristics such as natural frequencies (1) . Flexural vibrations that are excited by the interaction between the teeth of saw and the workpiece tend to become unstable and degrade the working accuracy (2) .…”