Design optimization and stiffness-equivalent method for an integrated starter generator in aerospace applications

Abstract

This study proposes a computational framework for the optimization and vibration analysis of an integrated starter generator using the stiffness-equivalent method. The stiffness-equivalent method was formulated based on the principle of minimum total potential energy and allows for the compensation of the stiffness of dummy structures such as main generator, exciter, and permanent magnet generator. We validated the proposed method by comparing the fine model and the equivalent model using 3D finite element analysis. An optimization was performed to achieve lightweight designs by considering operational constraints using the stiffness-equivalent method and PyAnsys. This approach produces design optimization with minimal weight while fulfilling all specified requirements. Furthermore, the framework can be used as a foundation for building a database of structural and vibrational properties of integrated starter generators, relevant to mechanical and aerospace fields. The validation results confirm that the proposed method is appropriate for these applications, owing to its relative simplicity and computational efficiency.

Introduction

Recent advancements in air transportation have led to increased environmental pollution, which in turn has driven a paradigm shift in aircraft design towards more electric aircraft (MEA) systems. These systems offer significantly higher efficiencies compared to traditional pneumatic and hydraulic power systems^1,2,3. As depicted in Fig. 1, MEA architectures incorporate several modifications relative to conventional pneumatic and hydraulic power supply systems. Due to the heightened demand for electrical power within MEA systems, high-power energy sources are essential to meet operational requirements^4,5,6,7,8,9. Within MEA configurations, the integrated starter generator (ISG) plays a critical role as a key component of the gas turbine engine. Its primary functions include acting as a starter motor to accelerate the engine to the necessary idle speed for fuel combustion, and subsequently functioning as a generator to supply electrical power to other onboard systems once the engine reaches the idle condition. Given that the ISG operates as a high-speed rotational system, it is crucial during the initial design phase to conduct rotor dynamic characteristics analysis. This analysis is necessary to identify and mitigate the risk of resonance excitation at critical speeds, ensuring the reliable and safe operation of the system. These structural and rotor dynamic characteristics of high-speed rotational system have been investigated using numerical and experimental methods.

Flowers and Ryan¹⁰ developed rotordynamic equations that incorporate disk flexibility. Hsieh et al.¹¹ introduced a modified transfer matrix method to analyze the coupled lateral and torsional vibrations in symmetric rotor-bearing systems under external torque. Taplak and Parlak¹² conducted a dynamic analysis of a gas turbine rotor with specific geometrical and mechanical properties using the Dynrot program. Xu et al.¹³ described the mechanical and thermal design of a high-speed, high-power-density synchronous permanent magnet machine for an aero engine starter generator, with a power rating of 150 kW and a maximum speed of 32,000 rpm. Bartolo and Gerada¹⁴ presented their work on designing a starter-generator for a regional jet engine that meets specific torque and speed requirements. Khamari et al.¹⁵ designed a complex turboexpander rotor supported on gas foil bearings and investigated its modal behavior under high rotational speeds. Huang et al.^16,17 proposed a model for unbalanced magnetic pull based on the time-varying rotor center position, considering the coupling between rotor bending and dynamic misalignment of the bearing inner ring. This model combines a mixed eccentricity unbalanced magnetic pull approach with a force component accounting for the dynamic misalignment angle in the bearing. Arslan et al.¹⁸ introduced a novel, highly reliable design for synchronous radial magnetic couplings capable of delivering torque up to 12 N·m, aimed at isolating the Starter/Generator and piston engine shafts in model aircraft, validated through numerical and experimental analyses. Shoujun et al.¹⁹ performed a detailed numerical design of a high-speed switched reluctance starter/generator tailored for more or all-electric aircraft. Jin et al.²⁰ investigated the impact of thermal-expansion-induced dynamic air gap displacements on vibration characteristics in high-speed motorized spindle rotors, employing both numerical and experimental methods. Guan et al.²¹ proposed an active vibration control strategy based on rotor speed modulation, leveraging the advantages of direct-drive motors. Choi and Yang²² focused on the optimal shape design of rotor shafts to adjust critical speeds under the constraint of constant mass.

While 3D finite element analysis (FEA) and experimental approaches can precisely determine the detailed structural and vibrational characteristics of rotating systems, they are less practical for preliminary design stages that involve numerous iterations and modifications. To overcome this limitation, research has focused on developing analysis techniques that incorporate coupled structural and vibrational behavior. Cui et al.²³ proposed a theoretical model based on the dynamic equivalent method (DEM) to investigate the vibration performance of corrugated sandwich structures across a broad frequency range. Hua et al.²⁴ introduced a novel stiffness-equivalent approach utilizing a laminated wall beam model, which estimates the contribution of infill wall stiffness to vertical vibration modes by adjusting the beam’s elastic modulus, thereby simplifying the modeling process of wall elements. Lin et al.²⁵ developed a reduced-order, equivalent linearization method based on explicit time-domain analysis for nonstationary random vibration with local nonlinearities. Minfang et al.²⁶ constructed a two-dimensional equivalent plate model using the stiffness derived from variational asymptotic homogenization of the unit cell structure. Wang et al.²⁷ proposed an equivalent linearization method that incorporates higher-order statistics within a nonlinear reduced-order framework to analyze the geometrically nonlinear random vibrations of complex structures. Bai et al.²⁸ introduced an equivalent shape-preserving clipping method to control the spectrum, preventing over-testing of triaxial random vibrations. Grosso et al.²⁹ developed a novel approach employing an equivalent modal damping ratio within a circle fit technique, addressing difficulties in modal parameter identification for non-conventional viscoelastic models. Kuang et al.³⁰ proposed a hybrid-dimensional modeling strategy for analyzing the vibration of embedded one-dimensional (1D) structures within three-dimensional (3D) solids under rotating conditions. Yun et al.³¹ presented an equivalent model for predicting the mechanical behavior and failure strength of honeycomb cores with thick cell walls. In the critical speed calculation of the integrated starter generator, employing a lumped mass approach for computational efficiency poses challenges because the unbalanced mass extends over a lengthy segment, significantly affecting the overall stiffness of the rotor system. This leads to substantial discrepancies compared to detailed 3D finite element models. Therefore, an improved stiffness-equivalent method that accounts for the vibrational coupling and incorporates the stiffness of the unbalanced mass is necessary to perform efficient yet accurate finite element analyses for critical speed estimation. However, existing studies have typically considered only part of the disk’s stiffness and have focused solely on rotor dynamics analysis, without fully addressing the coupled vibrational behavior influenced by the unbalanced mass^{32,33,34,35,36}.

This study proposes a computational framework for the optimization and vibration analysis of an integrated starter generator utilizing a stiffness-equivalent method. The stiffness-equivalent method is formulated based on the principle of minimum total potential energy, enabling the compensation of the stiffness of auxiliary structures such as the main generator, exciter, and permanent magnet generator. We validated the proposed approach by comparing the finite element models specifically, the detailed 3D fine model and the equivalent model using 3D finite element analysis. An optimization process was conducted to achieve a lightweight design that satisfies operational constraints, employing the stiffness-equivalent method and PyAnsys. This methodology facilitates the development of a design with minimal weight while fulfilling all specified requirements. Additionally, we verified the performance of the optimized ISG through 3D thermo-structural and vibration analyses. The results confirmed that both the optimized ISG and its components maintain adequate thermo-mechanical stability and vibrational performance under typical operating conditions, demonstrating their suitability for practical applications. Furthermore, this framework can serve as a foundation for establishing a database of the structural and vibrational properties of integrated starter generators, pertinent to the mechanical and aerospace fields. The validation outcomes affirm that the proposed method is appropriate for such applications, owing to its relative simplicity and computational efficiency.

Fig. 1

Comparison of traditional aircrafts and more-electric aircrafts^1,2,3.

Methodology

Problem statement

The stiffness-equivalent method was formulated based on the principle of minimum total potential energy to achieve high computational efficiency in optimizing an integrated starter generator. The integrated starter generator consists of a main generator, exciter, permanent magnet generator (PMG), shaft, and bearings, as shown in Fig. 2. It is supported by bearings at both ends, with loading conditions that include centrifugal forces and thermal stresses generated during operation. L represents the bearing span, while L₁, L₂, L₃, and L₄ denote the distances between the left end bearing and the main generator, the main generator and the exciter, the exciter and the PMG, and the PMG and the right end bearing, respectively. L_M, L_E, and L_P indicate the lengths of the main generator, exciter, and PMG, respectively. k_B and k_H are the stiffness coefficients of the bearings and housings, respectively. The symbols m and J represent the dummy mass and moment of inertia, where the subscripts M, E, and P refer to the main generator, exciter, and PMG, respectively.

Fig. 2

A schematic of an integrated starter generator.

Stiffness-equivalent method

Since the unbalanced mass of the integrated starter generator is elongated, modeling it as a lumped mass results in reduced stiffness compared to a continuous model. To describe the vibrational orthogonality, the stiffness correction factor (l), which is applied to the regions requiring stiffness compensation as shown in Fig. 3, is incorporated. To determine this correction factor, the regions were modeled using both a continuous model and a lumped mass model consisting of four elements. To match the vibrational orthogonality between the two models, the mass and stiffness matrices of each must be defined. Accordingly, as shown in Fig. 4, each element is modeled as a Timoshenko beam element. In this context, v and w represent lateral displacements in the Y and Z directions, respectively, while y and φ denote the rotational displacements about the Z and Y axes, respectively. The length of each element is denoted by L_e.

Based on the principle of minimum total potential energy, the mass and stiffness matrices of both the continuous and lumped mass models can be formulated using the shape functions of the Timoshenko beam element, as shown below³⁷.

$$\left[ M \right]_{c}^{{SYS}}=\sum\limits_{{i=1}}^{4} {{{\left( {{{\left[ B \right]}^{\left( i \right)}}} \right)}^T}\left[ M \right]_{c}^{{\left( i \right)}}{{\left[ B \right]}^{\left( i \right)}}}$$

(1)

$$\left[ K \right]_{c}^{{SYS}}=\sum\limits_{{i=1}}^{4} {{{\left( {{{\left[ B \right]}^{\left( i \right)}}} \right)}^T}\left[ K \right]_{c}^{{\left( i \right)}}{{\left[ B \right]}^{\left( i \right)}}}$$

(2)

$$\left[ M \right]_{l}^{{SYS}}={\left( {{{\left[ B \right]}_d}} \right)^T}{\left[ M \right]_d}{\left[ B \right]_d}+\sum\limits_{{i=1}}^{4} {{{\left( {{{\left[ B \right]}^{\left( i \right)}}} \right)}^T}\left[ M \right]_{l}^{{\left( i \right)}}{{\left[ B \right]}^{\left( i \right)}}}$$

(3)

$$\left[ K \right]_{l}^{{SYS}}=\left( {\left[ I \right]+\left[ \lambda \right]} \right)\sum\limits_{{i=1}}^{4} {{{\left( {{{\left[ B \right]}^{\left( i \right)}}} \right)}^T}\left[ K \right]_{l}^{{\left( i \right)}}{{\left[ B \right]}^{\left( i \right)}}}, \left[ \lambda \right]={\left( {{{\left[ B \right]}_\lambda }} \right)^T}\left[ {{\lambda ^*}} \right]{\left[ B \right]_\lambda }$$

(4)

$${\left[ B \right]^{\left( 1 \right)}}=\left[ {\begin{array}{*{20}{c}} 1&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0 \\ 0&1&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0 \\ 0&0&1&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0 \\ 0&0&0&1&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0 \\ 0&0&0&0&1&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0 \\ 0&0&0&0&0&1&0&0&0&0&0&0&0&0&0&0&0&0&0&0 \\ 0&0&0&0&0&0&1&0&0&0&0&0&0&0&0&0&0&0&0&0 \\ 0&0&0&0&0&0&0&1&0&0&0&0&0&0&0&0&0&0&0&0 \end{array}} \right],$$

(5)

$${\left[ B \right]^{\left( 2 \right)}}=\left[ {\begin{array}{*{20}{c}} 0&0&0&0&1&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0 \\ 0&0&0&0&0&1&0&0&0&0&0&0&0&0&0&0&0&0&0&0 \\ 0&0&0&0&0&0&1&0&0&0&0&0&0&0&0&0&0&0&0&0 \\ 0&0&0&0&0&0&0&1&0&0&0&0&0&0&0&0&0&0&0&0 \\ 0&0&0&0&0&0&0&0&1&0&0&0&0&0&0&0&0&0&0&0 \\ 0&0&0&0&0&0&0&0&0&1&0&0&0&0&0&0&0&0&0&0 \\ 0&0&0&0&0&0&0&0&0&0&1&0&0&0&0&0&0&0&0&0 \\ 0&0&0&0&0&0&0&0&0&0&0&1&0&0&0&0&0&0&0&0 \end{array}} \right],$$

$${\left[ B \right]^{\left( 3 \right)}}=\left[ {\begin{array}{*{20}{c}} 0&0&0&0&0&0&0&0&1&0&0&0&0&0&0&0&0&0&0&0 \\ 0&0&0&0&0&0&0&0&0&1&0&0&0&0&0&0&0&0&0&0 \\ 0&0&0&0&0&0&0&0&0&0&1&0&0&0&0&0&0&0&0&0 \\ 0&0&0&0&0&0&0&0&0&0&0&1&0&0&0&0&0&0&0&0 \\ 0&0&0&0&0&0&0&0&0&0&0&0&1&0&0&0&0&0&0&0 \\ 0&0&0&0&0&0&0&0&0&0&0&0&0&1&0&0&0&0&0&0 \\ 0&0&0&0&0&0&0&0&0&0&0&0&0&0&1&0&0&0&0&0 \\ 0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&1&0&0&0&0 \end{array}} \right],$$

$${\left[ B \right]^{\left( 4 \right)}}=\left[ {\begin{array}{*{20}{c}} 0&0&0&0&0&0&0&0&0&0&0&0&1&0&0&0&0&0&0&0 \\ 0&0&0&0&0&0&0&0&0&0&0&0&0&1&0&0&0&0&0&0 \\ 0&0&0&0&0&0&0&0&0&0&0&0&0&0&1&0&0&0&0&0 \\ 0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&1&0&0&0&0 \\ 0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&1&0&0&0 \\ 0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&1&0&0 \\ 0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&1&0 \\ 0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&1 \end{array}} \right],$$

$${\left[ B \right]_d}=\left[ {\begin{array}{*{20}{c}} 0&0&0&0&0&0&0&0&1&0&0&0&0&0&0&0&0&0&0&0 \\ 0&0&0&0&0&0&0&0&0&1&0&0&0&0&0&0&0&0&0&0 \\ 0&0&0&0&0&0&0&0&0&0&1&0&0&0&0&0&0&0&0&0 \\ 0&0&0&0&0&0&0&0&0&0&0&1&0&0&0&0&0&0&0&0 \end{array}} \right],$$

$${\left[ B \right]_\lambda }=\left[ \begin{gathered} \begin{array}{*{20}{c}} 0&0&0&0&1&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0 \\ 0&0&0&0&0&1&0&0&0&0&0&0&0&0&0&0&0&0&0&0 \\ 0&0&0&0&0&0&1&0&0&0&0&0&0&0&0&0&0&0&0&0 \\ 0&0&0&0&0&0&0&1&0&0&0&0&0&0&0&0&0&0&0&0 \end{array} \hfill \\ \begin{array}{*{20}{c}} 0&0&0&0&0&0&0&0&1&0&0&0&0&0&0&0&0&0&0&0 \\ 0&0&0&0&0&0&0&0&0&1&0&0&0&0&0&0&0&0&0&0 \\ 0&0&0&0&0&0&0&0&0&0&1&0&0&0&0&0&0&0&0&0 \\ 0&0&0&0&0&0&0&0&0&0&0&1&0&0&0&0&0&0&0&0 \end{array} \hfill \\ \begin{array}{*{20}{c}} 0&0&0&0&0&0&0&0&0&0&0&0&1&0&0&0&0&0&0&0 \\ 0&0&0&0&0&0&0&0&0&0&0&0&0&1&0&0&0&0&0&0 \\ 0&0&0&0&0&0&0&0&0&0&0&0&0&0&1&0&0&0&0&0 \\ 0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&1&0&0&0&0 \end{array} \hfill \\ \end{gathered} \right]$$

$$\left[ M \right]_{j}^{{\left( i \right)}}=\int_{0}^{{L_{e}^{{\left( i \right)}}}} {\rho A\left( {{N_v}^{T}{N_v}+{N_w}^{T}{N_w}} \right)dX} +\int_{0}^{{L_{e}^{{\left( i \right)}}}} {\rho I\left( {{N_\phi }^{T}{N_\phi }+{N_\psi }^{T}{N_\psi }} \right)} dX,\,\,\,i=1,\,\,...\,\,,\,4,\,\,\,j=c,\,l$$

(6)

$${\left[ M \right]_d}=\left[ {\begin{array}{*{20}{c}} {{m_k}}&{2{m_k}{L_k}}&0&0 \\ {{m_k}{L_k}}&{{J_k}+4{m_k}{L_k}^{2}}&0&0 \\ 0&0&{{m_k}}&{ - 2{m_k}{L_k}} \\ 0&0&{ - 2{m_k}{L_k}}&{{J_k}+4{m_k}{L_k}^{2}} \end{array}} \right],\,\,k=M,\,E,\,P$$

(7)

$$\begin{gathered} \left[ K \right]_{j}^{{\left( i \right)}}=\int_{0}^{{L_{e}^{{\left( i \right)}}}} {EI\left( {\frac{{\partial {N_v}^{T}}}{{\partial X}}\frac{{\partial {N_v}}}{{\partial X}}+\frac{{\partial {N_w}^{T}}}{{\partial X}}\frac{{\partial {N_w}}}{{\partial X}}} \right)dX} +\int_{0}^{{L_{e}^{{\left( i \right)}}}} {\kappa GA} \left( {\frac{{\partial {N_v}^{T}}}{{\partial X}}\frac{{\partial {N_v}}}{{\partial X}}+} \right.\frac{{\partial {N_w}^{T}}}{{\partial X}}\frac{{\partial {N_w}}}{{\partial X}}+{N_\phi }^{T}{N_\phi }+{N_\psi }^{T}{N_\psi } \hfill \\ \left. {+\frac{{\partial {N_w}^{T}}}{{\partial X}}{N_\phi }+{N_\phi }^{T}\frac{{\partial {N_w}}}{{\partial X}} - \frac{{\partial {N_v}^{T}}}{{\partial X}}{N_\psi } - {N_\psi }^{T}\frac{{\partial {N_v}}}{{\partial X}}} \right)dX,\,\,\,i=1,\,\,...\,\,,\,4,\,\,\,j=c,\,l \hfill \\ \end{gathered}$$

(8)

where [M]_c^SYS and [K]_c^SYS represent the mass and stiffness matrices of the continuous model, respectively, while [M]_l^SYS and [K]_l^SYS denote the mass and stiffness matrices of the lumped mass model. [B] is a Boolean matrix whose elements are logical values. The matrices [M]_j⁽ⁱ⁾ and [K]_j⁽ⁱ⁾ correspond to the mass and stiffness matrices of individual elements. The superscript (i) indicates the element number, the subscript j distinguishes between the continuous model and the lumped mass model, and the subscript k differentiates among the components of the unbalanced mass in the integrated starter generator, namely the main generator, exciter, and PMG. [M]_d is the mass matrix associated with the unbalanced mass. The parameters ρ, E, and G are the material density, Young’s modulus, and shear modulus, respectively. A and I represent the cross-sectional area and the moment of inertia. k is the shear correction factor of the Timoshenko beam, and can be formulated as follows:

$$\kappa =\frac{{6\left( {1+\nu } \right){{\left( {1+{\mu ^2}} \right)}^2}}}{{\left( {7+6\nu } \right){{\left( {1+{\mu ^2}} \right)}^2}+\left( {20+12\nu } \right){\mu ^2}}}$$

(9)

where µ denotes the inner to outer radius ratio, and ν represents the poisson’s ratio of the material. N_v, N_w, N_φ, and N_y are the shape functions for lateral and rotational displacements, respectively, and are formulated as follows³⁷:

$$\begin{gathered} {N_v}\left( X \right)=\left[ {\begin{array}{*{20}{c}} {\left( {1 - \xi } \right)\left( {2 - \xi - {\xi ^2}+6r} \right)\frac{R}{4}}&{L_{e}^{{\left( i \right)}}\left( {1 - {\xi ^2}} \right)\left( {1 - \xi +3r} \right)\frac{R}{8}}&0&0&{\left( {1+\xi } \right)\left( {2+\xi - {\xi ^2}+6r} \right)\frac{R}{4}} \end{array}} \right. \hfill \\ {\left. {\begin{array}{*{20}{c}} { - L_{e}^{{\left( i \right)}}\left( {1 - {\xi ^2}} \right)\left( {1+\xi +3r} \right)\frac{R}{8}}&0&0 \end{array}} \right]^T} \hfill \\ \end{gathered}$$

(10)

$$\begin{gathered} {N_w}\left( X \right)=\left[ {\begin{array}{*{20}{c}} 0&0&{\left( {1 - \xi } \right)\left( {2 - \xi - {\xi ^2}+6r} \right)\frac{R}{4}}&{ - L_{e}^{{\left( i \right)}}\left( {1 - {\xi ^2}} \right)\left( {1 - \xi +3r} \right)\frac{R}{8}}&0 \end{array}} \right. \hfill \\ {\left. {\begin{array}{*{20}{c}} 0&{\left( {1+\xi } \right)\left( {2+\xi - {\xi ^2}+6r} \right)\frac{R}{4}}&{L_{e}^{{\left( i \right)}}\left( {1 - {\xi ^2}} \right)\left( {1+\xi +3r} \right)\frac{R}{8}} \end{array}} \right]^T} \hfill \\ \end{gathered}$$

(11)

$$\begin{gathered} {N_\phi }\left( X \right)=\left[ {\begin{array}{*{20}{c}} 0&0&{3\left( {1 - {\xi ^2}} \right)\frac{R}{{2L_{e}^{{\left( i \right)}}}}}&{ - \left( {1 - \xi } \right)\left( {1+3\xi - 6r} \right)\frac{R}{4}}&0 \end{array}} \right. \hfill \\ {\left. {\begin{array}{*{20}{c}} 0&{ - 3\left( {1 - {\xi ^2}} \right)\frac{R}{{2L_{e}^{{\left( i \right)}}}}}&{ - \left( {1+\xi } \right)\left( {1 - 3\xi - 6r} \right)\frac{R}{4}} \end{array}} \right]^T} \hfill \\ \end{gathered}$$

(12)

$$\begin{gathered} {N_\psi }\left( X \right)=\left[ {\begin{array}{*{20}{c}} { - 3\left( {1 - {\xi ^2}} \right)\frac{R}{{2L_{e}^{{\left( i \right)}}}}}&{ - \left( {1 - \xi } \right)\left( {1+3\xi - 6r} \right)\frac{R}{4}}&0&0&{3\left( {1 - {\xi ^2}} \right)\frac{R}{{2L_{e}^{{\left( i \right)}}}}} \end{array}} \right. \hfill \\ {\left. {\begin{array}{*{20}{c}} { - \left( {1+\xi } \right)\left( {1 - 3\xi - 6r} \right)\frac{R}{4}}&0&0 \end{array}} \right]^T} \hfill \\ \end{gathered}$$

(13)

$$\xi =2\frac{X}{{L_{e}^{{\left( i \right)}}}} - 1, R=\frac{1}{{1+3r}}, r=\frac{{4EI}}{{\kappa GA{{\left( {L_{e}^{{\left( i \right)}}} \right)}^2}}}$$

(14)

Since the vibrational orthogonality between the continuous model and the lumped mass model must be matched based on their natural frequencies, the mass and stiffness matrices of both models can be formulated according to the following relation:

$${\left( {\left[ M \right]_{c}^{{SYS}}} \right)^{ - 1}}\left[ K \right]_{c}^{{SYS}}={\left( {\left[ M \right]_{l}^{{SYS}}} \right)^{ - 1}}\left[ K \right]_{l}^{{SYS}}$$

(15)

Accordingly, this relation can be reformulated as an eigenvalue problem for the stiffness correction factor, as shown below:

$$\left[ M \right]_{l}^{{SYS}}{\left( {\left[ M \right]_{c}^{{SYS}}} \right)^{ - 1}}\left[ K \right]_{c}^{{SYS}}=\left( {\left[ I \right]+\left[ \lambda \right]} \right)\sum\limits_{{i=1}}^{4} {{{\left( {{{\left[ B \right]}^{\left( i \right)}}} \right)}^T}\left[ K \right]_{l}^{{\left( i \right)}}{{\left[ B \right]}^{\left( i \right)}}}$$

(16)

The stiffness correction factor can be determined by solving this eigenvalue problem, and its scalar value is defined based on the lowest-order structural mode, excluding rigid-body modes. Additionally, this correction factor is used to compensate for the stiffness in the finite element model of the integrated starter generator, which includes the lumped mass.

Fig. 3

Lumped mass modeling of the integrated starter generator for stiffness compensation.

Fig. 4

Timoshenko beam element.

Verification

Finite element model

To verify the stiffness-equivalent method, we performed vibration analysis of an integrated starter generator using the finite element method with fine, equivalent and lumped mass modeling. The models were created using ANSYS, as shown in Fig. 5. ANSYS (modal analysis) was used as the solver for 3D finite element analysis, with models built using twenty-node solid elements (SOLID186) and ten-node solid elements (SOLID187) to simulate the mechanical behavior. A convergence analysis determined that the fine model should consist of at least 52,000 elements to ensure accurate results. The geometric parameters and material properties of the integrated starter generator are listed in Tables 1 and 2. The geometric parameters are normalized: length values are based on the bearing span, mass values are normalized by the mass of the main generator, and moments of inertia are normalized according to those of the main generator. Additionally, the material properties of the main generator, exciter, and PMG were assigned based on equivalent properties calculated using the Voigt model based on the rule of mixtures. The equivalent models incorporated the proposed stiffness-equivalent method by applying a stiffness correction factor to the elastic moduli of the main generator, exciter, and PMG. The load condition of the integrated starter generator includes centrifugal force acting along the x-axis, with boundary conditions provided by bearing supports at both ends.

Fig. 5

Finite element models of the integrated starter generator.

Table 1 Geometric parameters of the integrated starter generator.

Table 2 Material properties.

Calculation of bearing stiffness

The integrated starter generator is supported at both ends by bearings, which are deep groove ball bearings. For the vibration analysis of the integrated starter generator, the bearing stiffness is required. In this section, the bearing stiffness was calculated using 3D structural analysis. The deep groove ball bearing consists of an outer cage, inner cage, and rolling elements. The finite element model was developed using ABAQUS, as shown in Fig. 6, and comprised linear solid elements (C3D8). To improve computational efficiency, a symmetric model for a single rolling element was employed, and a convergence analysis determined that the finite element model should include at least 72,000 elements to ensure accurate results. ABAQUS (general, static) was used as the solver for the 3D finite element analysis. The boundary condition involved a radial displacement constraint at the top of the outer ring. The initial conditions included the initial preload and centrifugal force, while the load condition for stiffness calculation was a radial displacement load. The displacement load was applied using multiple point constraints. A symmetry boundary condition was also applied to the symmetry plane. Contact conditions were imposed at the contact surfaces between the outer cage, inner cage, and rolling elements. The friction coefficient was set to 0.01, considering lubrication conditions.

Figure 7 presents the structural analysis results of the symmetric model representing a single rolling element of the bearing. As the radial displacement increases, the corresponding radial load also increases, as indicated by the load–displacement curve. This load–displacement relationship is subsequently used to evaluate the bearing stiffness, as formulated below.

$${F_r}={k_r}{\delta _r}^{{{n_b}}}$$

(17)

where F_r is the load applied to the symmetry model of a single rolling element of the bearing, δ_r is the radial displacement of the symmetry model in response to the applied load, k_r is the stiffness of the symmetry model of a single rolling element, and n_r is the degree of the polynomial used to describe the displacement. The values of k_r and n_r are determined through regression analysis.

As shown in Fig. 8, the deep groove ball bearing supports the load through contact between multiple rolling elements, the outer cage, and the inner cage. Using the calculated displacement-load relationship of the symmetry model for a single rolling element, the bearing stiffness can be expressed mathematically as follows³⁸.

$${k_B}={k_r}\left[ {1+2\sum\limits_{{i=1}}^{{{m_r}}} {{{\cos }^{{n_r}}}\left( {{\theta _r}i} \right)\cos \left( {{\theta _r}i} \right)} } \right]$$

(18)

where m_r is the number of contact rolling elements, and q_r is the angle between the rolling elements. This formulation is used to represent the load-sharing behavior among multiple rolling elements as a function of their angular positions. Ultimately, the polynomial degree is determined to be 1.1, and the bearing stiffness is calculated as 211,072 N/mm using the above relationship.

Fig. 6

Finite element model of the bearing.

Fig. 7

Result of the structural analysis of the symmetry model for a single rolling element of the bearing.

Fig. 8

Schematic of the deep groove ball bearing.

Vibration analysis results and discussion

Rotor vibration analyses were performed for the fine model, the stiffness-equivalent model, and the lumped-mass model of the integrated starter generator, with gyroscopic effects taken into account. Figure 9 presents the corresponding Campbell diagrams and natural mode shapes, with engine orders considered up to 1×. The natural modes obtained from the fine, stiffness-equivalent, and lumped-mass models exhibit identical mode shapes. Specifically, the first mode corresponds to the first bending mode of the shaft, while the second mode corresponds to the second bending mode. A forward mode is defined as a vibration mode in which the shaft whirls in the same direction as the rotational motion, which is the most commonly observed behavior in rotating machinery under normal operating conditions. In contrast, a backward mode refers to a vibration mode in which the shaft whirls in the direction opposite to the rotation. Such backward modes may arise due to asymmetric stiffness or damping characteristics, or under transient operating conditions such as deceleration from high rotational speeds. The differences in the first and second critical speeds predicted by the three models are within 8.8%, with the maximum discrepancy observed in the first backward mode of Case 1 (Table 3). Although additional discrepancies appear in Case 3 for the second backward mode, the integrated starter generator employs a single-shaft configuration, under which backward modes do not induce significant shaft stress. Therefore, backward modes are excluded from the design consideration³⁹. Furthermore, it was confirmed that the proposed stiffness-equivalent model maintains a comparable level of accuracy across different material cases, indicating limited sensitivity to material property variations. In contrast, the lumped-mass model exhibits errors exceeding 40% when compared with the fine model, thereby highlighting the practical effectiveness of the proposed method. This level of accuracy is considered sufficient for preliminary design applications. Since the preliminary design stage requires a large number of iterative calculations, it is essential to rapidly generate initial design candidates using computationally efficient methods. In engineering practice, a preliminary design approach is generally regarded as effective when its accuracy remains within 10% of that obtained from detailed three-dimensional analyses. The resulting designs are not intended to represent the final configuration, but rather to serve as intermediate solutions prior to detailed design. In terms of computational efficiency, the stiffness-equivalent model required only 66 s of computation time using the proposed method, representing a substantial reduction compared to the 1128 s required for finite element analysis of the fine model under identical CPU conditions.

Fig. 9

Result of the vibration analysis. (a) Campbell diagram. (b) Natural mode shape.

Table 3 Comparison of the fine model, the stiffness-equivalent model, and the lumped-mass model.

Design optimization

An optimization framework was established based on the specified design requirements and parameters. The optimization algorithm was developed in MATLAB using a genetic algorithm (GA), following the flowchart shown in Fig. 10. To integrate finite element analysis incorporating the stiffness-equivalent method into the optimization process, an automated PyMAPDL-based code was developed for the generation and analysis of finite element models corresponding to the design variables of the integrated starter generator. Data exchange between MATLAB and PyMAPDL is implemented by directly invoking Python from the MATLAB-based optimization algoithm. The outputs generated by PyMAPDL are stored in a text-file format and subsequently read and processed in MATLAB. This automated workflow was fully integrated into the optimization algorithm. Genetic algorithms are particularly effective in identifying global optima, as they iteratively propagate high-performing design candidates across successive generations^{40,41,42,43,44}. In this study, the algorithm employed a population size of 20 and was executed for 200 generations. The mutation rate was set to 0.2, and 50% of the population was retained at each generation. Constraint handling was performed using a penalty-function approach.

Fig. 10

Flow chart of the design optimization.

Formulation of the optimization problem

Design optimization of the integrated starter generator was carried out to minimize its mass. The design parameters included bearing span, shaft thickness, distances between components (left end bearing to main generator, main generator to exciter, exciter to PMG), and housing stiffness. The objective function is the minimization of the shaft’s mass. The optimization problem was formulated as follows:

$${\text{Minimize}}\,\,Obj.\left( {L,{d_{outer}},{t_{shaft}},{\rho _{shaft}}} \right)={\rho _{shaft}}L\pi \left[ {\frac{{{d_{outer}}^{2}}}{4} - \frac{{{{\left( {{d_{outer}} - 2{t_{shaft}}} \right)}^2}}}{4}} \right]$$

(19)

$$\begin{gathered} {\text{S}}{\text{.T}}{\text{.}}\,\,{G_1}=1 - 0.8\frac{{{\omega _{\hbox{min} }}}}{{{\omega _n}}} \leqslant 0\,\,||\,\,1 - 1.2\frac{{{\omega _n}}}{{{\omega _{\hbox{max} }}}} \leqslant 0 \hfill \\ \,\,\,\,\,\,\,\,\,\,\,{G_2}=\frac{{{\sigma _b}}}{{{S_{yield}}}}{n_s} - 1 \leqslant 0 \hfill \\ \,\,\,\,\,\,\,\,\,\,\,0 \leqslant {L^*} \leqslant 1 \hfill \\ \,\,\,\,\,\,\,\,\,\,\,0 \leqslant {t_{shaft}}^{*} \leqslant 1 \hfill \\ \,\,\,\,\,\,\,\,\,\,\,0 \leqslant {L_1}^{*} \leqslant 1 \hfill \\ \,\,\,\,\,\,\,\,\,\,\,0 \leqslant {L_2}^{*} \leqslant 1 \hfill \\ \,\,\,\,\,\,\,\,\,\,\,0 \leqslant {L_3}^{*} \leqslant 1 \hfill \\ \,\,\,\,\,\,\,\,\,\,\,30,367 \leqslant {k_H}[N/mm] \leqslant 738,752\, \hfill \\ \end{gathered}$$

where $\:{\omega\:}_{\text{m}\text{i}\text{n}}$ and $\:{\omega\:}_{\text{m}\text{a}\text{x}}$ denote the minimum and maximum operating speeds of the integrated starter generator, respectively, and $\:{\omega\:}_{n}$ denotes the critical speed associated with the forward modes. Since the integrated starter generator is configured with a single shaft, backward modes do not induce significant stress in the shaft; therefore, only forward modes are considered in the design. To prevent excitation due to shaft imbalance, eccentricity, misalignment, or flatness, engine order was considered up to 2X^45,46. Additionally, a 20% margin is maintained for critical speeds. d_shaft, t_shaft, and ρ_shaft represent the outer diameter, thickness, and material density of the shaft, respectively, and the superscript * indicates normalized dimensionless variables. The distance between the PMG and the right end bearing is calculated as the difference between bearing span and the lengths and distances between components. S_yield is the yield strength of the shaft material, and the safety factor, n_s, is set to 1.5. σ_b represents the bending stress of the shaft and is formulated as shown in the equation below.

$${\sigma _b}= \mp \frac{{{M_{\hbox{max} }}{d_{shaft}}}}{{2{I_{shaft}}}}$$

(20)

where I_shaft represents the moment of inertia of the shaft, and M_max is the maximum bending moment due to unbalancing load. The maximum bending moment is calculated under the condition that the unbalancing loads of the main generator, exciter, and PMG act in the same direction. The balancing quality grade was set to G 6.3 ⁴⁷. The required design conditions pertain to maintaining structural integrity under centrifugal forces, unbalanced loads, and thermal loads, as well as avoiding critical speeds during the operation of the integrated starter generator.

Optimization results and discussion

Figure 11 presents the convergence histories of the objective function. The results indicate that the mass of the shaft converges to 0.3 kg as the genetic algorithm evolves over successive generations, along with the design parameters. Table 4 provides a summary of the optimized parameters, including the mass, critical speed, and bending stress. The geometric parameters are normalized as follows: length values are normalized based on the bearing span, mass values are normalized by the mass of the main generator, and moments of inertia are normalized according to those of the main generator. Additionally, the critical speed is normalized with respect to the maximum operating speed. These optimized values satisfy the specified strength and operating speed requirements.

Fig. 11

Objective function history.

Table 4 Optimization result.

Verification by the 3D finite element analysis

In this section, the vibration and structural analyses of the optimized integrated starter generator were conducted using ANSYS (structural analysis and modal analysis) as the solver, while the thermo-structural analyses of the main generator, exciter, and PMG were performed using ABAQUS (general, coupled temperature-displacement). The finite element models were developed utilizing both software packages, as illustrated in Figs. 12 and 13. For the 3D structural and vibration analyses, the finite element model consisted of over 77,973 linear solid elements, and the accuracy was verified through convergence studies. The thermo-structural model comprised more than 15,000 linear solid elements. In the structural and vibration analyses of the integrated starter generator, the load and boundary conditions were defined in accordance with those described in Sect.  3 to ensure consistency across analyses. For the thermo-structural analysis of the main generator, exciter, and PMG, a quarter-symmetric model was employed to enhance computational efficiency, with symmetry boundary conditions applied on the model’s symmetry plane. The boundary conditions involved fixing the radial displacement at the connection point with the shaft, while the load conditions included the maximum centrifugal force and the maximum operational temperature. The material properties used in all analyses were consistent with those specified in Sect.  3 of the study. Figure 14 presents the results of the 3D structural and vibration analyses of the optimized integrated starter generator. The Campbell diagrams indicate the critical speed, which maintains a margin of over 20% relative to the operational rotational speed, ensuring safe operation. The first natural mode corresponds to the first bending mode of the shaft, while the second mode corresponds to the second bending mode. Furthermore, the maximum stress due to unbalance load in the shaft is 60.2 MPa, localized at the main generator connection, with a safety margin of 12.7. Additionally, Fig. 15 also shows the results of the 3D thermo-structural analysis of the main generator, exciter, and PMG. In the main generator, the maximum stress is 256.5 MPa, occurring in the damper bar, with a minimum safety margin of 0.12. In the exciter, the maximum stress reaches 526.7 MPa at the hub, with a safety margin of 0.57. For the PMG, the maximum stress is 149.7 MPa, occurring in the magnet, with a safety margin of 0.87. Overall, the results demonstrate that both the optimized integrated starter generator and its components maintain adequate thermo-mechanical stability and vibrational performance under typical operating conditions, confirming their suitability for practical application.

Fig. 12

Finite element models of the optimized integrated starter generator.

Fig. 13

Finite element models of the main generator, exciter, and PMG.

Fig. 14

Result of the structural and vibration analyses of the optimized integrated starter generator.

Fig. 15

Result of the thermo-structural analyses of the main generator, exciter, and PMG.

Conclusion

This study proposes a computational framework for the optimization and vibration analysis of an integrated starter generator (ISG) based on a stiffness-equivalent method. The proposed approach is formulated on the principle of minimum total potential energy, enabling the equivalent representation and stiffness compensation of auxiliary structures, including the main generator, exciter, and permanent magnet generator (PMG). The accuracy of the proposed method was validated through a direct comparison between two three-dimensional finite element models: a detailed fine model and a stiffness-equivalent model. The modal characteristics obtained from both models showed identical mode shapes, where the first and second modes correspond to the first and second bending modes of the shaft, respectively. The discrepancies in the first and second critical speeds between the two models were within 8.8%. Furthermore, it was confirmed that the proposed stiffness-equivalent model maintains a comparable level of accuracy across different material cases, indicating limited sensitivity to material property variations. In terms of computational efficiency, the stiffness-equivalent model required only 66 s of computation time, representing a substantial reduction compared to the 1,128 s required for the fine-model finite element analysis under identical CPU conditions. An optimization procedure was subsequently performed using the proposed stiffness-equivalent method in conjunction with PyAnsys to obtain a lightweight ISG design that satisfies all operational constraints. This optimization framework enables efficient exploration of design variables while achieving a minimum-weight configuration. The performance of the optimized ISG was further verified through three-dimensional thermo-structural and vibration analyses. The results demonstrated that the critical speeds maintain margins exceeding 20% relative to the operational rotational speed, thereby ensuring safe operation. The maximum stress induced by unbalanced loading was 60.1 MPa, localized at the main generator–shaft connection, corresponding to a safety margin of 12.7. In addition, the minimum safety margins of the main generator, exciter, and PMG were confirmed to be greater than 0.12. These results confirm that both the optimized ISG and its constituent components exhibit sufficient thermo-mechanical integrity and vibrational stability under representative operating conditions, demonstrating their suitability for practical applications. Since the ISG design process is initiated based on electromagnetic performance considerations, the proposed computationally efficient framework allows the systematic construction of a database of structural and vibrational properties of integrated starter generators with respect to key design variables. This database can be directly applied within the baseline geometry and allowable design-variable ranges derived from electromagnetic design, thereby facilitating rapid design evaluation and optimization. Overall, the validation results demonstrate that the proposed method is well suited for integrated starter generator design and optimization, owing to its high computational efficiency, structural fidelity, and practical applicability.