How Does a 10% Reduction in Titanium Wall Thickness Due to Uniform Corrosion Change the Heater's First Mode Vibration Frequency and Amplitude?

The natural vibration frequency in the first mode for the process engineer monitoring an aging titanium immersion heater, which has suffered uniform wall thinning of 10% (for example from 1.5 mm to 1.35 mm) due to corrosion, decreases approximately 5-10% and the vibration amplitude under the same excitation force increases 15-25%. The natural frequency f_n is proportional to sqrt(I/m), where I is the area moment of inertia (∝ t³ for thin walls) and m is mass per unit length (∝ t). A 10% reduction in t decreases I by around 27% (from 0.93 to 0.73) and m by 10%. The overall effect on f_n is sqrt(0.73/0.90)=sqrt(0.81)=0.90. So f_n is down 10%. The amplitude of the vibration A under resonant condition is inversely proportional to the damping ratio ζ which is not significantly impacted by uniform wall thinning. However, for off-resonant condition, with constant excitation force, the amplitude increases by about 1/(f_n²), which is a 23% amplitude increase (1/0.90² = 1/0.81 = 1.23). This greater amplitude causes fatigue and fretting wear on the support points.

Influence of wall thickness on vibrational characteristics
The first mode natural frequency of a cantilevered titanium heater tube is given by f_n = (λ² / 2πL²) × sqrt(EI/m) where λ = 1.875 for the first bending mode, L is length, E is elastic modulus, I = π/64 × (D⁴ - d⁴) is the area moment of inertia and m is mass per unit length. For a thin-walled tube (t << D) I~D^3*t and m~D*t. Uniform reduction in t reduces both I and m proportionate to t 1 and t 1 correspondingly but I has a further t 2 dependence from the (D 4 - d 4 ) expansion. A 10% drop of t to 1.35 mm for a 25 mm OD tube with 1.5 mm wall reduces I by 27% and m by 10%, and f_n by 10%. The amplitude of vibration $A$ for a harmonic force $F$ at a frequency $\omega$ is $A$ = $F$ / ($m$ * $\sqrt{(\omega_{n}^2 - \omega^2)^2 + (2\zeta\omega_{n}\omega)^2}$). The reaction increases because the near to resonance increases as f_n lowers.

Quantification of Changes in Vibration Characteristics with 10% Wall Thinning
Wall t (mm) Thinned Wall t (mm) Reduction (%)I Reduction (%) m Reduction (%) f_n Reduction (%) Amplitude Increase at Constant Excit. (%)
1.5 1.35 10 27 10 9.5 22 2.0 1.80 10 26 10 9.0 21 2.5 2.25 10 25 10 8.5 20 1.0 0.90 10 28 10 10.0 23 1.5 (seamless) 1.35 10 27 10 9.5 22 1.5 (welded) 1.35 10 27 10 9.5 (but damping lower) 28 (lower damping)
Guidance for Vibration Assessment of Thinned Heaters: Scenario Based
Heater Installation and Operating ConditionsOriginal Wall (mm) Thinned Wall (mm) f_n Shift (Hz) Resonance RiskSuggested Action
Vertical, L=800 mm, low flow velocity (0.3 m/s) 1.5 1.35 18 → 16.3 Low (vortex shedding frequency far below f_n) No action needed
Horizontal, L=1,000 mm, cross-flow 0.8 m/sec 1.5 1.35 5.2 → 4.7 Moderate (0.8× f_n before, now 0.9× f_n vortex shedding)Check for help. If possible reduce flow.
Horizontal, L=1,000 mm, cross flow 0.9 m/s 1.5 1.35 5.2 → 4.7 High (vortex shedding may now also coincide with f_n)Replace heater or install more support.
Any heater with resonant state before thinning 1.5 1.35 Any Very high (f_n shift may move to resonance)Need to add immediate assistance .
Heater w/ numerous supports (short spans) 1.5 1.35 > 30 → > 27 Low (f n too high for vortex shedding) No change needed.
Heater in high viscosity fluid (damping high) 1.5 1.35 18 → 16.3 Low (viscous damping suppresses vibration) No intervention required.
Life Assessment Engineering Considerations
Ultrasonic thickness testing to detect uniform wall thinning of 10% should be part of the remaining life evaluation to measure the change in vibration characteristics. This reduction in f_n (usually 5 to 10 percent) brings the heater closer to any resonant frequencies in the system such as vortex shedding from cross flow, impeller blade pass frequencies or flow driven pulsations. If the original design contained a 20% safety margin between f_n and the closest excitation frequency, then a 10% drop in f_n reduces the safety margin to 10% -- still adequate. If the original margin was only 10% thinning the heater may now put it at resonance. The fretting wear at support points is accelerated by the higher vibration amplitude (15-25%). PTFE or rubber support liners that were once adequate may be worn through now, permitting metal-to-metal contact and galvanic corrosion. For heaters with 10% thinning working close to resonant conditions, the initial vibration margins can be restored by lowering the flow velocity by 10-15% or by installing an intermediate support.

Conclusion: 10% Wall Thinning Reduces f_n by ~10% and Increases Amplitude by ~20%
Uniform corrosion of the titanium heater wall thickness by 10% leads to a drop of the first mode natural vibration frequency by about 9-10% and an increase in vibration amplitude under constant excitation by 20-25%. The decrease in frequency is attributed to a 25-28% decrease in area moment of inertia I largely countered by a 10% decrease in mass per unit length. The increase in amplitude is due to the inverse square connection between amplitude and frequency at resonance. For heaters close to resonance conditions (vortex shedding, impeller frequencies) 10% thinning can induce the heater to resonate, leading to premature fatigue failure. If the thickness monitoring indicates a 10% uniform wall loss, a vibration study should be conducted to determine whether the frequency shift has reduced the safety margins to unacceptable levels. It is suggested to install intermediate supports, reduce flow velocity or change the heater in the case of resonance.