Spindle Vibration & Deflection Reduction

Introduction

Peddinghaus reported excessive spindle vibration while drilling weight-reduction holes in O’Hare Airport I-beams, linking it to long-term damage to the electronics box.

Our capstone used FEA to study how repositioning bearings changed spindle support, motor-end overhang, bending modes, and natural frequencies.

Can changing where the drill’s shaft is supported reduce shaking and help protect the machine?

Simplified Model

In September 2026, I revisited the motor-end overhang using a stepped-beam model with varying geometry, materials, and distributed mass.

The gear near the support did not dominate modal inertia; the tail supplied bending flexibility. Frequency estimates resembled the simulation, but assumed geometry limits comparison.

Retrospective stepped-beam approximation with an idealized root clamp, five equivalent sections, a C-ring point mass, and estimated axial stations — Retrospective stepped-beam approximation of the motor-end overhang. Dimensions were estimated from report figures. Rev A, page 3.

The baseline is a linear, undamped Euler–Bernoulli beam with an ideal root clamp and free tail. The axial coordinate \(s\) runs from the root to \(L\); the stepped sections have rigidity \(E(s)I(s)\) and line density \(\mu(s)=\rho(s)A(s)\). The C-ring contributes a point mass \(m_r\) at \(s_r\).

Quantity	Symbol	Model value
Tip stiffness	\(k_L\)	\(3.08262016\times10^6~\mathrm{N/m}\)
Translational modal mass	\(m_{\mathrm{eff}}\)	\(0.688433~\mathrm{kg}\)
Rotary-inertia contribution	\(m_\theta\)	\(0.00971902~\mathrm{kg}\)
Total modal mass	\(m\)	\(0.69815157~\mathrm{kg}\)
Gyroscopic coefficient	\(g\)	\(0.01943803~\mathrm{kg}\)
Spin speed	\(N\)	\(3000~\mathrm{rpm}\)

Nonrotating Rayleigh model

The static tip-load deflection \(v(s)\) defines the normalized trial shape; \(q(t)\) is tail displacement. Primes denote derivatives with respect to \(s\).

\[ \phi(s)=\frac{v(s)}{v(L)},\qquad w(s,t)=\phi(s)q(t),\qquad \phi(0)=\phi'(0)=0 \]

\[ \begin{aligned} k_L^{-1}&=\int_0^L\frac{(L-s)^2}{E(s)I(s)}\,ds\\ m_{\mathrm{eff}}&=\int_0^L\mu(s)\phi(s)^2\,ds+m_r\phi(s_r)^2 \end{aligned} \]

\[ \begin{gathered} m_{\mathrm{eff}}\ddot q+k_Lq=0\\ \boxed{f_R=\frac{1}{2\pi}\sqrt{\frac{k_L}{m_{\mathrm{eff}}}}\approx336.8~\mathrm{Hz}} \end{gathered} \]

Rotating extension

The extension retains the same shape and stiffness in two transverse planes, adding rotary inertia and gyroscopic coupling. The cross-sectional area moments \(I\) and \(J_p\) satisfy \(J_p=2I\) for the circular equivalents.

\[ w_x=\phi q_x,\qquad w_y=\phi q_y \]

\[ \begin{aligned} m_\theta&=\int_0^L\rho I(\phi')^2\,ds\\ g&=\int_0^L\rho J_p(\phi')^2\,ds\\ m&=m_{\mathrm{eff}}+m_\theta \end{aligned} \]

For constant positive spin about \(+s\), the linear equations in right-handed, stationary axes \((x,y,s)\) are:

\[ \begin{aligned} m\ddot q_x+g\Omega\dot q_y+k_Lq_x&=0\\ m\ddot q_y-g\Omega\dot q_x+k_Lq_y&=0 \end{aligned} \]

\[ \begin{aligned} \Omega&=\frac{2\pi N}{60}\approx314.15927~\mathrm{rad/s}\\ f_0&=\frac{1}{2\pi}\sqrt{\frac{k_L}{m}}\\ b&=\frac{g\Omega}{4\pi m}\approx0.69605~\mathrm{Hz}\\ f_\pm&=\sqrt{f_0^2+b^2}\pm b \end{aligned} \]

Here \(f_0\) includes rotary inertia at zero spin; \(f_-\) and \(f_+\) are the backward- and forward-whirl branches at 3000 rpm.

\[ \boxed{ \begin{aligned} f_0&\approx334.43~\mathrm{Hz}\\ f_-&\approx333.73~\mathrm{Hz}\\ f_+&\approx335.13~\mathrm{Hz} \end{aligned} } \]

Rotary inertia lowers the zero-spin frequency; gyroscopic coupling splits the stationary-frame branches without damping. These are model predictions, not measurements. Results use full-precision coefficients; the backward branch’s proximity to the independent Rayleigh–Ritz estimate is coincidental.

Frequency Comparison

The baseline FEA lies near the observed ≈310 Hz band; the proposed bearing change raises the predicted first mode to 383.27 Hz. Local overhang estimates cluster near 334–337 Hz.

Local-model detail

Rayleigh and its rotary-inertia and rotating extensions on an expanded frequency scale.

On smaller screens, scroll the table horizontally to read all columns.

Test or model	Frequency (Hz)	Condition
Test and full-spindle FEA · 2025
Unloaded test	≈310	Y-direction band, 250–3000 rpm; mode shape unverified.
Baseline FEA	307.17	First bending mode; original bearings, 3000 rpm.
Modified “2x” FEA	383.27	Predicted first mode; proposed bearing change and 22-mm shift.
Local overhang models · September 2026
Rayleigh	336.8	Nonrotating; one bending shape, no rotary inertia.
Rayleigh–Ritz	333.7	Nonrotating first mode; two station coordinates.
Rotary inertia · zero spin	334.43	One shape in two planes; whirl branches coincide.
Backward whirl	333.73	Rotating extension; 3000 rpm, stationary frame.
Forward whirl	335.13	Rotating extension; 3000 rpm, stationary frame.

Initial Testing

Our on-site campaign combined impact tests, unloaded free-spinning tests, and loaded cutting tests. A tri-axial accelerometer mounted on the spindle housing recorded the response.

Annotated test setup showing an impact hammer beside the spindle housing and mounted accelerometer on the left, the custom torsion bar at upper right, and the end mill cutting a workpiece at lower right — Test setup: housing-mounted accelerometer and impact hammer (left), custom torsion bar (upper right), and cutting operation (lower right). Open the image for the full-resolution view.

Impact tests

Transverse x/y impacts and torsional impacts using a custom torsion bar did not establish quantitative modal conclusions.

Unloaded tests

Twelve 10-second free-spinning trials covered 250–3000 rpm in 250-rpm steps. Wavelet analysis showed a persistent band near 310 Hz in the y-direction, not an experimentally established mode shape.

Loaded tests

Cutting tests used short and long tools at multiple speeds. Long-tool bands varied with speed, with some accelerometer saturation and no single frequency for unloaded-model comparison.

Loaded test: an end mill cutting a steel workpiece, as shown in the final presentation. This footage illustrates the cutting operation; the approximately 310 Hz reference comes from the separate unloaded tests.

Model references over the unloaded wavelet

Y-direction data from the final report, Figure 9. Twelve separate 10-second tests at 250, 500, …, 3000 rpm are stitched along the time axis; this is not a continuous speed ramp. The original raster is preserved, with axes redrawn for readability. View the original X/Y/Z plot at full resolution.

Finite Element Analysis

Refined against unloaded tests, ANSYS placed the first mode near the observed band, with bending at the motor-end overhang.

It used grounded outer bearing faces, cylindrical joints, and Coriolis effects at 3000 rpm, with bearing stacks and the rotary union nonrotating. Unloaded-test correlation did not independently validate loaded cutting.

Original spindle CAD assembly with detailed gear teeth and bearing features above a simplified assembly of stepped cylindrical sections, labeled Original Model and Approximate Model — Original spindle CAD (top) and the approximate geometry used for the finite element model (bottom). Open the image for the full-resolution comparison.

Bearing Modification

We recommended modifying the floating-bearing stack and shifting it 22 mm toward the drive. The preferred configuration raised the first simulated frequency from approximately 307 Hz to 383 Hz, an increase of about 25%. This was a predicted frequency shift, not a measured reduction in operating vibration.

Side-by-side spindle CAD cross-sections labeled Current and New, with red outlines highlighting the existing floating-bearing arrangement on the left and the proposed raised bearing stack on the right — Existing floating-bearing arrangement (left) and proposed configuration (right). The highlighted region shows the support change near the drive end. Open the image for the full-resolution comparison.

Outcome

The proposed bearing modification increased the first ANSYS bending-mode frequency from 307.17 to 383.27 Hz (+24.8%), moving it away from the observed ≈310 Hz response band to reduce resonance risk.

The project linked measured vibration, a simulated bending mechanism, and a targeted bearing recommendation. The retrospective calculations added a clearer explanation of why support placement and distributed inertia mattered. Confirming the proposed modification’s operating benefit would require testing the modified spindle under representative cutting conditions.

Observed band and model estimates

The highlighted final row shows the proposed FE configuration at 383.27 Hz, a 24.8% increase over the 307.17 Hz baseline. The dashed reference marks the reported unloaded band at approximately 310 Hz. Squares represent full-spindle FE configurations; circles represent the observation and local mathematical estimates. The proposed configuration is not a measured post-modification result.