An energy-based Rayleigh-Ritz formulation is presented for the free axial vibration of tapered nanorods subjected to uniform thermal loading, represented by a prescribed effective thermal axial force, within Eringen’s nonlocal elasticity. Crucially, the weak-form approach rigorously preserves the exact gradient-dependent nonlocal mass operator, avoiding the artificial truncation of cross-sectional spatial derivatives common in the literature. A gap is addressed where prior studies have largely examined tapering, thermal, or nonlocal effects in isolation. To bridge this, the present study concurrently evaluates symmetric ( β _b=β _h=β ) and asymmetric ( β _bβ _h ) tapers under clamped-free and clamped-clamped boundary conditions. Within a consistent nondimensional setting, the combined influence of the taper parameter ( β ), temperature increment ( Δ T ), and the nonlocal parameter ( α =e_0a/L ) on the first three natural frequencies is rigorously quantified. The results indicate that uniform thermal loading and nonlocality soften the response and primarily rescale the frequency levels without changing the relative ranking among geometries. The usefulness of geometric asymmetry is governed by the boundary condition. For clamped–free support, profiles narrowing toward the free end lead to clear, temperature–insensitive increases in the nondimensional natural frequencies across Modes 1–3 (yielding gains of up to 44 β raises the nondimensional frequencies, and increasing L also raises them toward the classical local limit, since L inversely scales the nonlocal parameter ( α =e_0a/L ), making short rods more sensitive to nonlocal softening. These trends provide practical guidance for nanoscale resonators and AFM–like cantilevers operating under uniform temperature fields and moderate nonlocality by indicating which taper strategies increase resonance robustly, and they supply benchmark data for future thermo–nonlocal modeling.
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