A doubly-clamped nanobeam behaves partly like a stiff bending beam and partly like a tensioned string. Its fundamental resonant frequency combines both contributions in quadrature:
f_bend = (λ1² / 2πL²) · √(EI / ρA) [pure bending, λ1 = 4.730]
f_tens = (1 / 2L) · √(T / ρA) [pure string, T = tension]
f1 = √(f_bend² + f_tens²) [combined resonator]
T = E · ε · A (axial force from applied strain ε)
Here E is Young's modulus, I = wh³/12 the second moment of area, A = wh the cross-section, ρ the density, and L the suspended length. At zero strain the beam vibrates in the pure bending regime (f_bend only); as tensile strain ε is dialed up, the built-in axial tension T stiffens the beam and pulls its resonance upward toward the tension-dominated ("string") regime — the same effect used to tune NEMS resonators and to null out fabrication-induced frequency spread.
- Axial strain ε — the applied tensile pre-strain; raising it increases T and pushes f1 up (tension stiffening).
- Beam length L — longer beams are softer in bending (f_bend ∝ 1/L²) but also softer as a string (f_tens ∝ 1/L), so both terms drop as L grows.
- Thickness h — sets the bending stiffness EI ∝ h³; thin beams are bending-soft and become tension-dominated at much lower strain.
- Material — sets E and ρ, which set both the intrinsic bending stiffness and the speed of sound in the beam.
The 3D view shows the beam's fundamental mode shape oscillating (visualization speed is compressed for watchability; the numeric f1 readout is the true physical value in MHz). Beam width is fixed at 200 nm; strain is restricted to the tensile (ε ≥ 0) range to avoid the buckling instability that occurs under compression.