Concepts and Content Learning

• Understand the importance of data preprocessing in real-world data analysis, including handling missing data and noise.
• Calculate and interpret key time-domain features (mean, RMS, peak-to-peak) as basic indicators of vibration intensity.
• Apply the Fast Fourier Transform (FFT) to sensor data to transition from the time domain to the frequency domain.
• Correctly identify dominant frequencies from a frequency spectrum and relate them to the structure's natural frequencies.
• Gain a foundational understanding of how sensor readings are combined to approximate a vibration mode shape.
• Relate analytical results (frequencies, amplitudes) to the structural health and performance assessment (e.g., susceptibility to resonance).

Utilization of MATLAB and Impact on Learning

• MATLAB is utilized as the primary computational tool for efficient handling of large time-series data and application of complex algorithms like the FFT and digital filters (Butterworth). This approach improves student learning by:
• Bridging Theory and Practice: Students directly apply the theoretical concepts of FFT, filtering, and modal analysis that they learn in lectures to realistic data.
• Rapid Iteration and Visualization: MATLAB's powerful plotting and numerical capabilities allow students to quickly visualize preprocessing results, frequency spectra, and mode shapes, providing immediate feedback and improving conceptual understanding.

Specific Skill Development:

• Critical Thinking: Interpreting spectra and diagnosing structural issues.
• Data Analysis: Selecting appropriate preprocessing methods and calculating features.
• Synthesis of Ideas: Connecting time-domain, frequency-domain, and mode shape results.
• Computation: Programming signal processing algorithms.

Other Skills:

Technical Report Writing: Documenting analysis steps, interpreting results, and providing recommendations in the conclusion.

This activity is ideal for an upper-level undergraduate or early graduate course in Structural Dynamics, Vibration Engineering, or a general Data Analysis/Signal Processing course with a civil/mechanical engineering focus.

Educational Level: Junior/Senior Undergraduate or First-Year Graduate Engineering.

Class Size/Institution: Adaptable to all sizes, from small seminars to large lectures with lab sections. Suitable for any institution type.

Activity Type/Time: A structured lab activity or longer homework assignment. It requires approximately 2-3 hours for students to complete all steps, analysis, and write the discussion.

Prerequisite Disciplinary Skills: Students should have a solid understanding of single degree-of-freedom (SDOF) and multi degree-of-freedom (MDOF) dynamic systems, concepts of natural frequency and damping, and the theory behind the Fourier Transform (understanding the transition from the time domain to the frequency domain).

Course Placement:This activity is best situated after lectures on structural dynamics theory and before or concurrently with topics on modal analysis or structural health monitoring.

Adaptability: It is highly adaptable. The core signal processing steps are standard. It could easily be modified for use in a mechanical engineering context (e.g., machinery vibration) or with different software (e.g., Python/NumPy/SciPy).

Required MATLAB Skills Students must be able to:

•  Load and manipulate data (readtable, indexing).
• Perform basic matrix/vector operations (mean, std, max, min).
• Create and interpret 2D plots (plot, xlabel, title).
• Apply signal processing functions (fillmissing, butter for filter design, filter, fft, findpeaks).

This activity requires students to use the provided MATLAB code template and a data file (vibration_data.csv) to conduct a comprehensive structural vibration analysis. The activity is structured as a step-by-step procedure:

• Load and Preprocess Data: Students load the CSV file and practice data hygiene by using functions like fillmissing() and considering basic outlier detection, which prepares them for handling real-world data imperfections.
• Time-Domain Analysis: They calculate basic statistical measures (mean, std) and key vibration metrics (RMS, peak-to-peak), providing an initial quantitative assessment of vibration intensity. They then implement a low-pass filter (butter and filter) to understand how to remove high-frequency noise.
• Frequency-Domain Analysis: This is the core of the activity. Students apply the fft() function, calculate the correct frequency vector, and plot the resulting spectrum. They then use findpeaks() to programmatically identify the dominant frequencies, which are essential for identifying the structure's natural frequencies.
• Vibration Mode Analysis: A simplified approach is used where the "mode shape" is represented by an average of the sensor signals. This introduces the concept of modal coordinates, which is then discussed relative to theoretical mode shapes.
• Conclusion & Discussion: Students synthesize their findings, discuss the implications of the identified frequencies and amplitudes for structural health, and propose potential engineering solutions (e.g., adding damping).

Why MATLAB? While this activity could be performed with other tools (e.g., Python with NumPy/SciPy), MATLAB is chosen because of its ubiquitous use in engineering curricula, its robust built-in functions for signal processing (Signal Processing Toolbox), and its highly effective environment for engineering-focused data visualization and rapid prototyping. The syntax is also very clear for engineers, making the complex concepts of FFT and filtering more accessible.

CSV Sensor Data (Comma Separated Values 486bytes Sep28 25)
Problem Live Script (MATLAB Live Script 247kB Sep28 25) 
Script to create Sensor data (Matlab File 874bytes Sep28 25) 

Tips for Instructors and Common Student Confusion

FFT Interpretation: Students often struggle with correctly calculating the frequency axis (f) for the FFT plot, particularly with handling the sampling frequency (Fs) and the two-sided spectrum. Emphasize the relationship f=(0:N−1)∗(1/(N⋅T)) where T is the time step, and the need to plot only the first half of the resulting array (positive frequencies).

Filter Design: The low-pass filter step can be challenging. Clearly explain the role of the cutoff frequency (cutoff) and the order of the filter (butter(4, ...)). If students use the wrong sampling frequency (Fs), the filter results will be incorrect. Stress the importance of knowing the actual data acquisition parameters.

Mode Shape Simplification: Be transparent with students that calculating the mode shape as a simple average is a major simplification. This step is intended to introduce the concept of a mode shape (the relative displacement/acceleration of different parts of the structure) rather than a rigorous modal analysis, which would require more advanced techniques (like the Peak Picking Method or Modal Identification).

Data File Necessity: The activity is impossible without the vibration_data.csv file. Ensure this file is available and contains sufficient dynamic content (i.e., clear peaks in the frequency domain) for students to find meaningful results. A good dataset would include a mix of low-frequency ambient vibrations and distinct peaks representing natural frequencies.

Code Review: Encourage students to review the code line-by-line, not just run it. Ask them to explain what the butter() function does or why they use abs(Y_pos) when plotting the spectrum.

Student understanding is assessed via a written technical report or a filled-out template that includes:

1. Screenshots of the key plots: time-domain vibration patterns, filtered signals, and the final frequency spectrum.

2. Tabulated Results of the time-domain metrics (mean, RMS, peak-to-peak) and the identified dominant frequencies.

3. Discussion and Conclusion: A detailed section addressing the implications of the results:

* What are the structure's primary natural frequencies?

* Do the RMS values indicate high or low vibration severity?

* How does the observed mode shape relate to typical theoretical mode shapes (e.g., first bending mode)?

If a known excitation frequency (e.g., traffic) were present, would the structure be at risk of resonance?

Resources for Structural Vibration Sensor Analysis Activity

Resource: MATLAB Documentation: fft
Description: The official MathWorks documentation page for the Fast Fourier Transform function.
Relevance: Essential for understanding how to correctly implement the FFT and interpret its output, especially regarding the frequency vector calculation.

Resource: Chopra, A. K. (2017). Dynamics of Structures: Theory and Applications to Earthquake Engineering
Description: A standard textbook in structural dynamics. Chapters on modal analysis and frequency response.
Relevance: Provides the theoretical foundation for natural frequency, mode shapes, and damping concepts.

Resource: Signal Processing with MATLAB (Online Tutorial)
Description: Various online tutorials from MathWorks or academic sites dedicated to signal processing in MATLAB.
Relevance: Helps students troubleshoot issues with filtering (butter, filter) and understanding how to transition from the discrete-time domain to the frequency domain.