Erratum: “The Band Gap Formation in Rotors With Longitudinal Periodicity and Quasi-Periodicity” [ASME J. Eng. Gas Turbines Power, 2022, 144(5), p. 051003; DOI: 10.1115/1.4053193]

Equation (10) gives

\sum_{j = 1}^{n} f (x_{j}) \approx \frac{1}{Δ x} \int_{0}^{L} f (x) d x

where

f (x_{j}) = \frac{m_{j}}{ρ A L} ϕ_{i} (x_{j}) ϕ_{k} (x_{j})

Considering that all disks have the same mass m_D, then

f (x_{j}) = \frac{m_{D}}{ρ A L} ϕ_{i} (x_{j}) ϕ_{k} (x_{j})

Hence, from Eq. (10)

\begin{matrix} \sum_{j = 1}^{N_{D}} \frac{m_{D}}{ρ A L} ϕ_{i} (x_{j}) ϕ_{k} (x_{j}) \approx \frac{1}{Δ x} \int_{0}^{L} \frac{m_{D}}{ρ A L} ϕ_{i} (x) ϕ_{k} (x) d x \\ = \frac{\hat{m}}{Δ x} L δ_{i k} = \hat{m} (N_{D} + 1) δ_{i k} = μ δ_{i k} \end{matrix}

(11)

where $\hat{m} = m_{D} / ρ A L$ and $μ = \hat{m} (N_{D} + 1)$ ⁠, which we can call the specific disk mass.

Similarly, if we consider that all disks have the same inertia J_D, then

\sum_{j = 1}^{N_{D}} \frac{J_{D}}{ρ A L} ϕ_{i} (x_{j}) \frac{d^{2} ϕ_{k} (x_{j})}{d x^{2}} \approx \frac{1}{Δ x} \int_{0}^{L} \frac{J_{D}}{ρ A L} ϕ_{i} (x) \frac{d^{2} ϕ_{k} (x)}{d x^{2}} d x

(12)

Therefore, considering Eqs. (13) and (14)

\begin{matrix} \sum_{j = 1}^{N_{D}} \frac{J_{D}}{ρ A L} ϕ_{i} (x_{j}) ϕ_{k} (x_{j}) \approx \frac{1}{Δ x} \int_{0}^{L} - \frac{J_{D}}{ρ A L} {(\frac{k π}{L})}^{2} ϕ_{i} (x) ϕ_{k} (x) d x \\ = - \hat{J} (N_{D} + 1) {(\frac{k π}{L})}^{2} δ_{i k} = λ {(\frac{k π}{L})}^{2} δ_{i k} \end{matrix}

(15)

where $\hat{J} = J_{D} / ρ A L$ and $λ = \hat{J} (N_{D} + 1)$ ⁠, which we can call the specific disk inertia.

The results shown in Figs. 15–17 were obtained using the present formulation of μ and λ.