Specific Stiffness and Substrate Choice: Deflection and Natural Frequency of Same-Size Discs in Different Materials

For equal disc size the first natural frequency scales as √(E/ρ), and self-weight deflection satisfies w ≈ 1.2g/(2πf₁)², independent of material. Using the exact annular-plate solution this article computes f₁ and sag for six substrates (example Ø200 mm, 2 mm), compares thickness and mass at equal frequency, and gives selection guidance for large-diameter discs.

Once discs get larger and thinner, whether the face sags or shakes matters as much as line accuracy: reflective and small-gap readheads are sensitive to axial face position, so self-weight sag and vibration both become gap changes. A common suggestion is to “switch to a stiffer material”, yet Young’s moduli of glass, sapphire and silicon differ by factors, densities differ too — which quantity should one look at? This article clears deflection and natural frequency with an annular-plate model. Only one quantity matters: specific stiffness E/ρ, and deflection and frequency are two statements of the same thing.

Figure 1 – Six substrate discs Ø200 mm, 2 mm thick: left, side view of the first mode (amplitude exaggerated, spin rate scaled by f1); right, first frequency and self-weight deflection versus rising specific stiffness (example parameters, exact annular-plate solution)

Model: two formulas and one identity

The disc is clamped on a hub. Treat it as a thin annular plate with inner edge (clamp radius a) fixed and outer edge (radius R) free, thickness t. Flexural rigidity

D = E·t³ / [12(1 − ν²)]

First natural frequency

f₁ = λ² / (2πR²) · √(D / ρt) = λ² / (2πR²·√(12(1 − ν²))) · t · √(E/ρ)

where λ² is the frequency coefficient from the exact Bessel-function solution: 4.80 at ν = 0.2, a/R = 0.2, and 5.59 at a/R = 0.25. The lowest mode is a rocking shape with 1 nodal diameter, 2%–6% below the axisymmetric umbrella mode. With the face horizontal under self-weight ρg·t alone, outer-edge sag is

w = α·ρg·t·R⁴ / D = 12α(1 − ν²)·g·R⁴·ρ / (E·t²)

α depends only on geometry: about 0.054 at a/R = 0.2, about 0.043 at 0.25. Comparing the two formulas: f₁ ∝ (t/R²)·√(E/ρ), w ∝ (R⁴/t²)·ρ/E; material enters only through E/ρ (and a near-1 1 − ν² correction). Combining them:

w ≈ κ·g / (2π·f₁)², κ ≈ 1.2–1.4

κ is set by geometry and is material-independent. Self-weight deflection is simply “g over angular frequency squared”: a disc at f₁ = 240 Hz sags about 5.4 μm, at 100 Hz about 31 μm. So do not treat deflection and frequency as two metrics — knowing f₁ gives the self-weight sag (Figure 3, left: all points fall on one line).

Worked example: Ø200 mm, 2 mm thick

Assume outer diameter 200 mm, clamp radius 20 mm, thickness 2 mm, face horizontal. Materials take typical values (sapphire of a-axis order, anisotropy about ±10%; single-crystal silicon the mid-value of 130–169 GPa; alkali-free glass as an order of magnitude):

Material	E (GPa)	ρ (g/cm³)	E/ρ (MJ/kg)	f₁ (Hz)	Outer self-weight deflection (μm)
Soda-lime glass	70	2.50	28.0	239	5.4
Borosilicate glass	63	2.23	28.3	239	5.4
Alkali-free glass	74	2.38	31.1	253	4.8
Fused silica	72	2.20	32.7	256	4.8
Single-crystal silicon	150	2.33	64.4	366	2.3
Sapphire	345	3.98	86.7	427	1.7

Among the four glasses E/ρ differs by only about 20%, f₁ by about 7%; switching soda-lime to fused silica gains almost no stiffness. Sapphire and silicon have E/ρ 2–3 times that of glass, f₁ rises 1.5–1.8× and sag falls to about 0.3–0.4×.

Scaling laws matter more: doubling outer diameter cuts f₁ to 1/4 and multiplies sag by 16; doubling thickness doubles f₁ and cuts sag to 1/4. The same borosilicate disc at Ø200 mm, 1 mm thick has only about 120 Hz and about 22 μm sag; at Ø80 mm, 1 mm thick (clamp radius 10 mm), f₁ is about 870 Hz and sag only 0.44 μm — small discs need no worry.

Equal-frequency design: sapphire gets thinner, but not lighter

Require f₁ ≥ 500 Hz and invert for thickness:

t_min = 2πR²·f₁ / λ² · √(12(1 − ν²)ρ / E)

At Ø200 mm, borosilicate and soda-lime need about 4.2 mm, fused silica about 3.9 mm, silicon about 2.7 mm, sapphire about 2.3 mm (Figure 2). Mass ∝ ρ·t ∝ ρ^1.5/√E; relative to borosilicate: sapphire about 1.00, fused silica 0.92, silicon 0.68, soda-lime 1.12. Sapphire is thinner but denser, so at equal frequency it is no lighter and rotary inertia does not improve; silicon is about three-tenths lighter yet nearly opaque at 850–950 nm near-IR, usable only for reflective or slotted patterns (see another article in this library).

Figure 2 – Minimum thickness versus outer diameter at a fixed first-frequency requirement: t ∝ R²; curves split into glass, silicon and sapphire bands by specific stiffness

Beyond specific stiffness: clamp radius and speed

Clamp radius. Raising a/R from 0.2 to 0.4 lifts λ² from 4.80 to 9.07, nearly doubles f₁, drops α from 0.054 to 0.018 and cuts sag to about one third. Enlarging the hub bond-ring radius is cheaper than thickening the disc or changing material — provided the read tracks are not blocked by the hub.

Rotational speed. Centrifugal stress scales as ρω²R², independent of E. In a free-hole approximation, borosilicate Ø200 mm at 6000 r/min has about 7 MPa hoop stress at the hole edge, about 79 MPa at 20000 r/min (excluding assembly and bond stress). So at low-to-medium speed the selection limit is stiffness and vibration; only at high speed do specific strength and edge defects take over. Also keep rotational frequency and its low harmonics clear of f₁; spinning further splits nodal-diameter modes in the lab frame, which must be checked with a rotating-disc model.

Figure 3 – Left: points of different materials and thicknesses fall on one w–f₁ line; right: relative thickness and mass at a common 500 Hz

Engineering notes

Compute f₁ first, then estimate self-weight sag with w ≈ 1.2g/(2πf₁)²; only when the face is horizontal and gap tolerance is within tens of micrometres is a check needed.
Do not switch among glasses for stiffness: E/ρ differs by only about 20%. Raise f₁ by thickening (f₁ ∝ t), shrinking outer diameter (∝ 1/R²) or enlarging clamp radius.
Actually compute f₁ for large thin glass discs: same borosilicate, Ø150 mm × 3 mm is about 640 Hz, Ø200 mm × 2 mm only about 240 Hz.
Equal-frequency comparisons must look at mass and inertia, not thickness alone: sapphire thinner does not mean lighter.
Sapphire and silicon E vary with crystal direction; table numbers are order-of-magnitude — ask the supplier for orientation and measured modulus before freezing the design.
Real f₁ depends on bond-ring stiffness and is lower than ideal clamp; validate on samples by tap or laser vibrometry.

Summary

For equal-size discs the first natural frequency scales with √(E/ρ) and self-weight deflection with ρ/E, and w ≈ κg/(2πf₁)² joins the two into one fact. Specific-stiffness differences among glasses are small, so material choice brings little change; sapphire and silicon can raise f₁ by about 1.5–1.8×, but sapphire does not reduce mass and silicon is limited by near-IR opacity. Stiffness of large-diameter discs is won mainly from thickness, outer diameter and clamp radius.

Translated from the ENC7 Chinese technical library (article 0572). Read the Chinese original. Need a disc built to your drawing? Request a quote or see our custom encoder discs.