Drone Propeller Gyroscopic Moment and Shaft Bearing Loads

Blade geometry is integrated at 2000 radial stations. With $w(r)=c(r)t(r)$ and total blade mass $m_b$, the normalized mass element is $dm=m_b w(r)dr/\int w(r)dr$. Rectangular sections give $I_{p,b}=\int(r^2+c^2/12)dm$ and $\Delta I=I_u-I_v=\int(c^2/12-r^2)dm$ for two opposite blades. The hub has $I_{p,h}=m_hR_h^2/2$. Equally spaced identical blades with count at least three cancel $\Delta I$. The measured option takes $\Delta I=-\eta I_{p,prop}$; the low-inertia transverse principal axis defines rotor phase zero. Bell inertia is axisymmetric and enters $I_p$ only.

For cant $\beta$, $\Omega_x=p\cos\beta-r\sin\beta$ and $\Omega_y=q$. Signed spin $\omega$ is positive CCW viewed from above. The transmitted moment is the reaction to the torque accelerating rotor angular momentum:

Thus the mean-vector magnitude is $G=I_p|\omega|\Omega_\perp$ and the circular harmonic amplitude is $A=|\Delta I\omega|\Omega_\perp$. This is the asymmetric-gyroscope treatment discussed by Den Hartog; Bock, NACA TM 1099, printed pp. 4–5 (attached PDF pp. 5–6), provides an open derivation and numerical example.

Take shaft z from far bearing A at zero to near bearing B at L and the load plane at L+a. With hub radial force $(F_x,F_y)$, $R_{Ax}=(aF_x+C_y)/L$, $R_{Ay}=(aF_y-C_x)/L$ and $\mathbf R_B=-\mathbf F-\mathbf R_A$. At B the bending vector is $(C_x-aF_y,C_y+aF_x)$. The bending vector varies linearly between section endpoints, so the global peak is the larger of the two endpoint mean magnitudes, plus A. Radial reaction and displacement extrema use the same exact circular-harmonic envelope, rather than maxima from sampled plots.

Timoshenko beam integration imposes zero translation at A and B with free support rotations. For an isotropic solid circular section, $G_s=E/[2(1+\nu)]$, effective shear area $A_s=0.9A_{shaft}$ and shear rigidity $K_s=G_sA_s$. The 0.9 shear-area convention follows the OpenSees Elastic Timoshenko Beam documentation. Including both bending and shear, the tip compliances are
$c_F=\frac{a^2(L+a)}{3EI}+\frac{a(L+a)}{LK_s},\quad c_M=\frac{aL/3+a^2/2}{EI}+\frac{a}{LK_s},$
$c_\phi=\frac{L/3+a}{EI}+\frac{1}{LK_s}.$
Thus $(\delta_x,\delta_y)=c_F(F_x,F_y)+c_M(C_y,-C_x)$ and hub section rotation is $(\phi_x,\phi_y)=c_M(F_x,F_y)+c_\phi(C_y,-C_x)$. Centreline tip slope is $\mathbf v'=\boldsymbol\phi+\mathbf F/K_s$, which differs from section rotation when a radial force is present.

For either bending plane, put $Q=C_y$ for x displacement and $Q=-C_x$ for y displacement, with $B=Fa+Q$. With $s=z-L$, the displacement profile is
$v(z)=\frac{B(z^3-L^2z)}{6EIL}\quad(0\leq z\leq L),$
$v(L+s)=\frac{BLs}{3EI}+\frac{Bs^2}{2EI}-\frac{Fs^3}{6EI}+\frac{s}{K_s}\left(\frac BL+F\right)\quad(0\leq s\leq a).$
The section rotation equals the bending-only profile derivative plus $B/(LK_s)$ throughout the shaft. This also retains the shear-induced support rotation under a pure end couple. Bending-only contributions remain available to compare with the original Euler–Bernoulli FE benchmark; total displacement, rotation, profile endpoints and centreline slopes are checked against three independent 100-element Timoshenko solutions.

Compression is a validity screen, not a buckling prediction. Conservatively take the maximum entered compression $P_c=\max(0,-F_{ax})$ along the full hub-to-far-bearing-A load path of length $S=L+a$. This includes the case in which the axial reaction actually enters at B. A sufficient elastic stability lower bound for the two fixed translational supports is
$P_b=\left[\frac{4S^2}{\pi^2EI}+\frac{S}{LK_s}\right]^{-1}.$
This bound is derived here from the strain-energy form, using the support condition $v(0)=v(L)=0$. Writing shear strain $\gamma=v'-\phi$, subtract the mean section rotation over the bearing span. The one-zero Poincaré inequality and the norm $\sqrt{S/L}$ of the shear mean-subtraction operator give $\int v'^2dz\leq[4S^2/(\pi^2EI)+S/(LK_s)]\int(EI\phi'^2+K_s\gamma^2)dz$. The allowable screen $P_c/P_b\leq0.05$ limits destabilizing geometric stiffness to 5% of the elastic energy form; larger compression is rejected for beam-column analysis. MIT 16.20 Handout 5 provides the Euler/beam-column stability framework; this conservative support-specific bound is an original energy estimate.

For a circular shaft, $Z=\pi d^3/32$. Rotating the moment components into material coordinates gives fully reversed bending at one cycle per revolution. Maximizing over material fibres and both critical endpoints gives $\sigma_a=K_f M_{peak}/Z$, with $K_f=1+q(K_t-1)$. Shigley’s Mechanical Engineering Design, 10th edition, Chapters 6–7, supplies the Marin and distortion-energy/modified-Goodman framework. Surface factors use the MPa-based ground or machined correlations, capped at one; the polished option uses one. For 2.79–50 mm shafts, $k_b=1.24[d/(1\,\mathrm{mm})]^{-0.107}$; below 2.79 mm use one. Bending load factor is one and $S_e=k_a k_b k_e k_{other}\min(0.5S_{ut},700\,\mathrm{MPa})$.

Steady concentrated axial and torsional stresses form $\sigma_{m,eq}=\sqrt{\sigma_{ax}^2+3\tau_m^2}$. Goodman design utilization is $n_d(\sigma_a/S_e+\sigma_{m,eq}/S_{ut})$. The separate elastic yield utilization is $n_d\sqrt{(K_t\sigma_b+|\sigma_{ax}|)^2+3\tau_m^2}/S_y$. Both must be at most one for the displayed stress-margin verdict.

The play input is rotation progress from 0° to 360°. The physical rotor angle is this progress for CCW and its negative for CW. Blades, material-fibre marker, instantaneous moment, plotted waveforms and spin-direction indicator use that same signed angle.

Identical balanced blades with linear projected chord and axial thickness, uniform effective density and a centered rectangular section; no blade sweep or twist is resolved. Total blade mass excludes the separate hub. A measured inertia ratio describes two opposite blades with known transverse principal axes. Bell and hub inertia are axisymmetric.

Body angular rates and shaft speed are constant over a revolution, with spin much faster than body rotation. The retained terms are first order in manoeuvre rate. The shaft is a uniform isotropic solid circular Timoshenko beam on two perfectly rigid translational supports; section rotations at the bearings are free, and loads enter at the propeller plane. Concentration factors conservatively apply at whichever endpoint gives the greater bending amplitude. Radial hub force is fixed in the motor frame; axial thrust and drive torque are steady. The material follows a steel endurance-limit model.

Use this first-order Timoshenko beam response away from bending resonances. Displacements and hub section rotations include shear compliance with G = E/[2(1 + ν)] and effective circular shear area 0.9A. Hub section rotation differs from centreline tangent by the radial shear strain; both enter the small-motion condition. The two-blade applied moment has a 2/rev harmonic; the rotating-fibre bending stress has a 1/rev harmonic. Spin-up torque, transverse angular acceleration, quadratic body-rate terms, flexible propeller dynamics, bearing/housing compliance and assembly vibration amplification are excluded. Body-rate/spin ratio above 0.1 and large beam slopes or deflections trigger input-dependent messages.

The geometry option approximates solid, unswept blades using projected chord and axial thickness. For twisted, hollow or nonuniform-density parts, use measured/CAD polar inertia and transverse inertia difference. Increasing blade count at fixed total blade mass keeps the same radial mass distribution; it does not add blade mass automatically. The measured two-blade option assumes the low-inertia transverse principal axis is phase zero. Symmetric three-or-more-blade cancellation does not cover damaged or unequal blades.

Marin correlations follow Shigley’s 10th edition for steels. Size correction is used over 2.79–50 mm; below that range a factor of one avoids extrapolation. Temperature and environmental effects must be supplied through the other-modifiers input. No beneficial compressive mean-stress credit is taken. Elastic concentration factors and notch sensitivity require actual material and notch data. The fatigue criterion screens repeated high-cycle demand; it does not predict finite life, rainflow damage, crack growth, fretting or corrosion fatigue. Axial bearing load sharing and bearing catalog life are outside the model.

Both stress utilizations include the requested design factor; the verdict addresses only those checks. Deflection and bearing reactions need comparison with the actual motor’s clearances and bearing limits. The rotation-progress animation is slowed for viewing and follows the selected spin direction. Numerical integration uses 2000 stations and curves use 721 phase samples and 401 beam stations; engineering peak values are analytic envelopes, not sample-limited maxima.

Axial stability and beam-column response are excluded from the reported transverse response. Compression is accepted only below 5% of a conservative elastic stability lower bound for the full hub-to-far-bearing-A load path. This deliberately assumes compression throughout both the bearing span and overhang, even if the actual axial reaction is carried at B. The threshold limits the omitted destabilizing geometric stiffness to 5% in the elastic energy form (up to about 5.3% amplification in the elastic energy norm). Larger compression blocks results and requires a beam-column/buckling assessment with the actual axial load route, end restraint and imperfections. Positive tension does not activate this compression screen; its geometric stiffening is omitted.