Non dimensional analysis of axially polarized passive magnetic bearings

Research Article Non dimensional analysis of axially polarized passive magnetic bearings Mintu Karmakar1 · Susenjit Sarkar1 Received: 9 October 2019 / Accepted: 22 April 2020 © Springer Nature Switzerland AG 2020 Abstract Ranges of investigation on passive magnetic bearing exist where radial and axial forces and variations of a few coefficients of radial and axial stiffnesses are shown for a particular set of chosen parameters and dimensions. The paper presents a non-dimensional approach for the same with additional parameters like all coefficients of stiffnesses and natural frequency. The model is considered for static analysis of axially polarised ring shaped magnets for both stator and rotor part in three degrees of freedom with three linear translations in 3D Cartesian coordinate system. The significance of this non-dimensional method is more generalised to the designer where boundary parameters like maximum force on the rotor of passive magnetic bearing, natural frequency, etc. can be easily estimated by simple conversion without doing separate numerical simulation for different parameters every time. The analysis of additional parameters like non-dimensional natural frequency can be the input for non-dimensional dynamic analysis. This paper also provides a way for maximization of radial force with the optimized solution of physical dimensions of passive magnetic bearing as inner radius of stator and outer radius of rotor magnet while keeping others as input parameters. The proposed model is validated with data available for radial and axial forces as well as for radial and axial stiffnesses that found in existing literature for similar kind of problem. Keywords Non-dimensional · Passive magnetic bearing · Radial force maximization · Natural frequency · Axially polarised Abbreviations Fsr Magnetic force between ( stator and rotor (N) ) H or Tm 𝜇0 Absolute permeability 4𝜋 × 10−7 AN2 or m A 𝜋 3.1415926535 (Non-dimensional) qs Magnetic pole strength of stator magnet (Wb or Am) qr Magnetic pole strength of rotor magnet (Wb or Am) rsr Distance between stator and rotor magnetic pole (m) 𝜎s Stator magnetic surface charge density (T) 𝜎r Rotor magnetic surface charge density (T) Ss Stator magnetic pole surface area per pole (m2) Sr Rotor magnetic pole surface area per pole (m2) Br1 Residual magnetism of rotor magnetic material (T) Br2 Residual magnetism of stator magnetic material (T) r1 Radial distance of rotor small elemental area dSr from rotor axis (m) r2 Radial distance of stator small elemental area dSs from stator axis (m) α Angular displacement of stator small elemental area dSs about stator axis measured from x axis (radian) β Angular displacement of rotor small elemental area dSr about rotor axis measured from x axis (radian) * Mintu Karmakar, ; Susenjit Sarkar, | 1Department of Mechanical Engineering, Jadavpur University, Kolkata, West Bengal 700032, India. SN Applied Sciences (2020) 2:987 | https://doi.org/10.1007/s42452-020-2809-x Vol.:(0123456789) Research Article SN Applied Sciences (2020) 2:987 x Radial displacement of rotor centroid from stator centroid (global origin) along x axis (m) y Radial displacement of rotor centroid from stator centroid (global origin) along y axis (m) z Axial displacement of rotor centroid from stator centroid (global origin) along z axis (m) u Radial displacement of rotor small elemental area dSr from stator small elemental area dSs along x axis (m) v Radial displacement of rotor small elemental area dSr from stator small elemental area dSs along y axis (m) z1 Displacement of small elemental area dSr of rotor’s South Pole from small elemental area dSs of stator’s North Pole along z axis (m) z2 Displacement of small elemental area dSr of rotor’s North Pole from small elemental area dSs of stator’s South Pole along z axis (m) z3 Displacement of small elemental area dSr of rotor’s North Pole from small elemental area dSs of stator’s North Pole along z axis (m) z4 Displacement of small elemental area dSr of rotor’s South Pole from small elemental area dSs of stator’s South Pole along z axis (m) Ls Stator length along stator z axis (m) Lr Rotor length along stator z axis (m) δ Maximum radial clearance between rotor and stator cylindrical axis (m) Fref Reference force of rotor and stator magnets (N) Fx Magnetic force along x axis (N) F0 External radial force per bearing applied on journal having rotor (N) R1 Inner radius of rotor magnet (m) R2 Outer radius of rotor magnet (m) R3 Inner radius of stator magnet (m) R4 Outer radius of stator ( )magnet (m) N k Position stiffness m kxx Position stiffness when variation( of)force and N displacement both along x axis m kyy Position stiffness when variation( of)force and N displacement both along y axis m kzz Position stiffness when variation( of)force and N displacement both along z axis m kxy Position stiffness when variation of force along x axis ( and)variation of displacement along y direcN tion m kyx Position stiffness when variation of force along y axis ( and)variation of displacement along x direcN tion m Vol:.(1234567890) | https://doi.org/10.1007/s42452-020-2809-x kxz Position stiffness when variation of force along x axis ( and)variation of displacement along z direcN tion m kzx Position stiffness when variation of force along z axis ( and)variation of displacement along x direcN tion m kyz Position stiffness when variation of force along y axis ( and)variation of displacement along z direcN tion m kzy Position stiffness when variation of force along z axis ( and)variation of displacement along y direcN tion m ω Frequency (rad/s) m Mass (kg) φx Amplitude of displacement along x axis (m) φy Amplitude of displacement along y axis (m) φz Amplitude of displacement along z axis (m) e 2.71828 (Non-dimensional) t Time (s) ωref Reference natural frequency (rad/s) S Force ratio (Non-dimensional) ar Frequency that governs growth of vibration (rad/s) ωr Natural frequency (rad/s) Nx Number of space discretisation along x axis (Non-dimensional) Ny Number of space discretisation along y axis (Non-dimensional) Nz Number of space discretisation along z axis (Non-dimensional) 1 Introduction In recent past, there are less numbers of available research papers on passive magnetic bearings in comparison to active magnetic bearings. Lower magnetic strength and instability of passive magnetic bearings, kept the researchers in abeyance for further development. Research on passive magnetic bearings was started using basic study of permanent magnets for radial and axial forces [1]. Yung et al. [2] derived the equation of force between two small magnetic dipoles smaller than their separation using Taylor expansion for the first non-zero term by vector differential and path integral derivation approaches. Simon et al. [3] in their paper, discussed about spin stabilization of magnetic levitating top on magnetic base. Gyrosco (...truncated)