Comparison of 1D and 2D Theories of Thermoelastic Damping in Flexural Microresonators
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1052-DD06-22
Comparison of 1D and 2D Theories of Thermoelastic Damping in Flexural Microresonators Sairam Prabhakar, and Srikar Vengallatore Mechanical Engineering, McGill University, Montreal, H3A 2K6, Canada ABSTRACT Thermoelastic damping (TED) represents the lower limit of material damping in flexural mode micro- and nanoresonators. Current predictive models of TED calculate damping due to thermoelastic temperature gradients along the beam thickness only. In this work, we develop a two dimensional (2D) model by considering temperature gradients along the thickness and the length of the beam. The Green’s function approach is used to solve the coupled heat conduction equation in one and two dimensions. In the 1D model, curvature information is lost and, hence, the effects of structural boundary conditions and mode shapes on TED are not captured. In contrast, the 2D model retains curvature information in the expression for TED and can account for beam end conditions and higher order modes. The differences between the 1D and 2D models are systematically explored over a range of beam aspect ratios, frequencies, boundary conditions, and flexural mode shapes. INTRODUCTION Flexural mode microresonators with low damping are the building blocks of microelectromechanical systems (MEMS) used for sensing, communications and energy harvesting applications. Energy dissipation due to thermoelastic coupling represents the lower bound on damping in microscale flexural resonators [1]. In a beam undergoing flexural vibrations, the thermoelastic effect implies that an oscillating stress gradient will generate a corresponding temperature gradient such that the compressive regions are hotter, and the tensile regions colder, than the equilibrium temperature of the beam. This finite temperature gradient will inevitably lead to irreversible heat conduction, entropy generation, and energy dissipation. This mode of energy dissipation is called thermoelastic damping (TED). In 1937, Zener provided a closed form expression for TED by considering irreversible heat conduction in one-dimension (through the thickness of a vibrating Euler-Bernoulli beam of thickness h ). Zener’s formula for TED is given by [2]: Eα 2T0 ωn τ Eα 2T0 Ω h2 C −1 , (1) QZener τ = = = C 1 + ωn2 τ 2 C 1 + Ω2 π 2k Here E is the Young’s modulus, α is the linear coefficient of thermal expansion, T0 is the equilibrium temperature of the beam, C is the specific heat per unit volume, ωn is the undamped natural frequency of vibration of the beam, Ω = ωn τ is a normalized frequency, and τ is a time constant. Recently, Lifshitz and Roukes [3] improved upon Zener’s work by developing exact closed-form expressions for thermoelastic damping in a beam resonator within the context of a 1D theory. Their expression is given by:
−1 =6 QLR
ωn C E α 2 T0 1 ⎛ 1 ⎧ sinh ξ + sin ξ ⎫ ⎞ ⎬⎟, ξ = h ⎜1 − ⎨ 2 C ξ ⎝ ξ ⎩ cosh ξ + cos ξ ⎭ ⎠ 2k
(2)
These 1D models are now widely used to assess whether thermoelastic damping is dominant in microresonators by comparing theoretical predictions against ex
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