Nonlinear dynamic analysis of asymmetric bistable energy harvesters

The paper defines the asymmetric model and its “restoring force” Fr(x) with a tilt term as Fr(x) = −(1/2) x (1 + 2δx − x^2) − p sin φ (from the equation of motion in Eq. (4) and the explicit Fr statement) and then states the optimal angle as φopt = arcsin((8δ^3 + 9δ)/(27p)) without a derivation. Under the given Fr sign, enforcing Fr(x1) + Fr(x2) = 0 yields sin φ = −(8δ^3 + 9δ)/(27p), i.e., the opposite sign. The model’s solution derives this formula step-by-step and explains that the paper’s expression is recovered under the opposite sign convention for the tilt term or angle orientation. Notably, the paper later uses φopt ≈ −4.95° for δ = 0.15 and p = 0.59, consistent with the model’s negative sign, reinforcing that the paper’s presentation is sign-ambiguous and lacks the derivation. Hence: paper incomplete (sign convention not clarified; derivation omitted), model correct. See the model equation and Fr definition (Eq. (4)) and the paper’s stated φopt (Eq. (9)) and its numerical value used later in the text .