⚡ electrical engineering

Frequency-Modulated and Single-Tone Excitation to Reveal Vibro-Acoustic Nonlinearities in Loosened Bolted Joints

This paper proposes a vibro-acoustic method using single-tone and frequency-modulated excitations near a structure's natural frequency to detect bolt loosening in rail-vehicle systems by analyzing the emergence of high-frequency spectral peaks and harmonic band power ratios that distinguish between various preload conditions.

Original authors: Berkay Kullukcu, Robin Pianowski, Dina Hannebauer

Published 2026-05-29

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CC BY 4.0

Original authors: Berkay Kullukcu, Robin Pianowski, Dina Hannebauer

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: The "Loose Bolt" Problem

Imagine a train is a giant, complex Lego structure held together by thousands of screws (bolts). For the train to be safe and quiet, these screws need to be tightened just right. If a screw gets even a tiny bit loose, the parts it holds together stop acting like a solid block and start acting like a rattle.

When a bolt is tight, the connection is stiff and quiet. When it's loose, the metal parts rub against each other, slip, and clatter. This creates extra noise and weakens the structure. The problem is that on a real train, you can't easily unscrew every bolt to check if it's tight, and you can't put a special sensor inside every single screw.

The Solution: "Shaking" the Train to Listen

The researchers at the Technical University of Wildau came up with a clever way to check for loose bolts without taking anything apart. They treated the train part like a musical instrument.

The Setup:

The Shaker: They used a mechanical arm (a shaker) to gently vibrate the metal structure, kind of like tapping a wine glass to hear if it's chipped.
The Ears: They attached a vibration sensor (accelerometer) to the metal and a microphone nearby to listen to the sound.
The Test Subject: They built a metal model of a train part and tested it with one specific bolt tightened to four different levels: completely loose (0%), slightly tight, medium tight, and very tight (80%).

The Experiment: Two Ways to "Sing"

The researchers tried two different ways to vibrate the metal to see how the loose bolt reacted.

1. The Single Note (Single-Tone)
They vibrated the metal at one specific pitch: 130 Hz (a low hum).

The Tight Bolt: When the bolt was tight, the metal hummed cleanly at that one note.
The Loose Bolt: When the bolt was loose, the metal started "singing" extra high notes that weren't supposed to be there. It's like if you hummed a single note, but your loose teeth started chattering, creating a bunch of high-pitched squeaks and rattles on top of your voice. The researchers found these "extra notes" (high-frequency peaks) were a clear sign the bolt was loose.

2. The Sliding Whistle (Frequency-Modulated)
Instead of one fixed note, they made the vibration slide back and forth between 125 Hz and 135 Hz, like a slide whistle or a siren.

Why do this? A loose connection is tricky; it might only start rattling at a very specific moment. By sliding the frequency up and down, they made sure they hit the "sweet spot" where the loose bolt would start acting up.
The Result: This method was even better at spotting the difference. They measured the "loudness" of the main note compared to the "loudness" of the extra high notes (harmonics).
- For a loose bolt, the extra high notes were very loud compared to the main note.
- For a tight bolt, the extra high notes were almost silent.
- The difference was huge: up to 36.5 decibels (which is a massive difference in volume) between the loose and tight states.

The "Why": Friction is the Culprit

Why does a loose bolt make these extra noises?
Think of a tight bolt as two pieces of wood glued together. When you shake them, they move as one.
Think of a loose bolt as two pieces of wood just resting on each other. When you shake them, they stick together for a split second, then slip, then stick again. This "stick-slip" action is chaotic. It acts like a non-linear mixer, taking the low, smooth vibration and turning it into a messy, high-pitched rattle.

What They Found

Loose bolts create "noise pollution" in the vibration: They turn a clean, single vibration into a messy mix of high-frequency squeaks.
Sliding the frequency helps: Moving the vibration slightly up and down (FM) makes these messy squeaks easier to hear and measure than just holding one note still.
It works without sensors inside: You don't need to put a sensor inside the bolt. Just shaking the outside and listening with a microphone is enough to tell if the bolt is loose.

The Bottom Line

This paper proves that you can detect a loose bolt on a train (or similar structure) by simply shaking it and listening to the sound. If the structure starts making weird, high-pitched rattles when you shake it at a specific low pitch, the bolt is likely loose. It's a simple, non-invasive way to keep trains safe and quiet.

Technical Summary: Frequency-Modulated and Single-Tone Excitation to Reveal Vibro-Acoustic Nonlinearities in Loosened Bolted Joints

Problem Statement
Bolted joints are critical in rail vehicle structures, relying on preload (clamp force) to maintain frictional contact and stiffness. Preload loss leads to joint loosening, causing a transition from a stiff, linear state to a soft, nonlinear state characterized by micro-slip, rattling, and unpredictable damping. This degradation increases structure-borne noise, alters vibration transfer, and poses safety risks. While standards like DIN 65151 and ISO 16130 quantify clamp force decay under controlled transverse motion, they often require direct access to preload or axial force, which is impractical for in-service vehicles. Existing monitoring techniques frequently fail to combine controlled shaker tests with the sensing of nonlinear features using minimal, non-invasive sensors.

Methodology
The study employs a rail-vehicle-inspired demonstrator consisting of cantilever beams and metal blocks connected by bolts. The experimental setup utilizes a mechanical shaker (TMS K2007E01) for excitation, a triaxial accelerometer (PCB 356A01) for structural response monitoring, and a microphone positioned 300 mm from the test bolt for acoustic sensing.

The methodology involves the following steps:

Preload Conditions: A specific test bolt (S2) is subjected to four preload states: 0% (0 Nm, loose), 20% (12.5 Nm), 40% (25 Nm), and 80% (50 Nm). Other connecting bolts remain fixed at 80% preload.
Modal Identification: Sine-sweep (1–5000 Hz) and narrow-band white noise (100–350 Hz) excitations are applied to identify the system's dominant natural frequency, determined to be approximately 130 Hz.
Excitation Strategies:
- Single-Tone Excitation: A 130 Hz sinusoidal signal is applied.
- Frequency-Modulated (FM) Excitation: Signals are generated using the equation $x(t) = A \sin (2\pi f_c t + \beta \sin (2\pi f_m t))$ , where the carrier frequency $f_c$ is 130 Hz and the frequency is modulated between 125 Hz and 135 Hz. Modulation frequencies ( $f_m$ ) of 1, 2, 5, 10, and 20 Hz are tested.
Data Analysis: Power Spectral Density (PSD) is analyzed to identify additional spectral peaks and harmonic generation. A harmonic band power ratio ( $R_l$ ) is calculated to quantify the difference between the carrier power (near 130 Hz) and the power of the $l$ -th harmonic band:
$R_l = 10 \log_{10} \left( \frac{P_{out}(130 \text{ Hz})}{P_{out}(l)} \right)$

Key Results

Modal Behavior: The demonstrator exhibits a clear resonance at 130 ± 2 Hz.
Single-Tone Response: Under 130 Hz single-tone excitation, the loose (0% preload) state generates distinct additional high-frequency spectral peaks (e.g., at 1599.5 Hz, 3314.8 Hz, 3919.0 Hz, 5698.9 Hz, and 8451.1 Hz) that are absent or significantly weaker in the tight (80% preload) state. These peaks are attributed to nonlinear energy transfer via frictional micro-slip and intermittent contact.
FM Excitation and Power Ratios: FM excitation between 125 and 135 Hz further distinguishes between preload states. The harmonic band power ratios ( $R_l$ $R_{l}$ ) show a clear separation between the loose state and the 80% preload state.
- For the 2nd harmonic ( $l=2$ ), the difference between the loose and 80% preload states is 17.5 dB.
- For the 6th harmonic ( $l=6$ ), the difference increases to 36.5 dB.
- Table 1 data indicates that as preload increases, the power ratio values become more negative (indicating lower harmonic power relative to the carrier), with the loose state showing the highest relative harmonic content.

Significance and Claims
The paper proposes a proof-of-concept method for detecting bolt loosening using a minimal sensor suite (shaker, accelerometer, microphone) without requiring direct preload measurement. The authors claim that:

Nonlinear Indicators: The presence of additional high-frequency spectral peaks and the magnitude of harmonic band power ratios serve as reliable indicators of joint looseness, driven by the transition from sticking to slipping contact.
FM Advantage: Frequency-modulated excitation offers advantages over single-tone excitation by sweeping through the resonance region (125–135 Hz), which improves the capture of resonance shifts and bandwidth changes caused by varying boundary conditions. Furthermore, the time-varying nature of FM excitation enhances the visibility of nonlinear transfer effects and sidebands.
Practical Application: The method aims to identify reliable indicators in vibration and sound signals that can be obtained with minimal modification to components, addressing the difficulty of retrofitting sensors on in-service rail vehicles.

Limitations and Future Work
The authors explicitly state that the method is currently at the "proof-of-concept" stage. The study is limited to a single bolt, discrete torque levels, and a specific geometry. Future work is identified as necessary to generalize the method to different geometries, multiple bolts, and mounting conditions that better replicate actual rail vehicle body and roof structures. Additionally, the authors note that ideally, future iterations should include direct preload or clamp-force measurements to validate the indirect sensing approach.