Certainly! Here’s a step-by-step example project you could use as a template. I'll use a simple, accessible object: **a metal ruler clamped to a table so it acts like a cantilever beam**. This system is easy to model and fits the requirements. --- ## **Project Summary** ### **1. Object Selection** **Object:** A standard 30 cm metal ruler, clamped at one end to a desk, with the other end free to vibrate (acting as a cantilever beam). **Why this object?** - Easy to set up at home or in a lab. - Fundamental (lowest) frequency is typically under 60 Hz for a length of 20–25 cm overhanging. - Easily excited and measured. --- ### **2. Modeling the System** **Physical Properties:** - Length of overhanging part: L = 0.25 m - Width: w = 0.025 m - Thickness: t = 0.002 m - Material: Steel (density ≈ 7800 kg/m³, Young’s modulus E ≈ 200 GPa) **Mass Estimation:** - Volume = L × w × t = 0.25 × 0.025 × 0.002 = 1.25 × 10⁻⁵ m³ - Mass (m) = Volume × Density = 1.25 × 10⁻⁵ × 7800 ≈ 0.098 kg **Model:** Treat as a **single degree-of-freedom (1DOF)** system using the equivalent mass and stiffness for a cantilever beam. - **Equivalent mass (m_eq)** at tip ≈ 0.23 × total mass ≈ 0.023 kg - **Equivalent stiffness (k):** \( k = \frac{3EI}{L^3} \) Where \( I = \frac{wt^3}{12} \) (second moment of area) Plugging in values: - \( I = \frac{0.025 × (0.002)^3}{12} = 1.67 × 10^{-10} \, m^4 \) - \( k = \frac{3 × 200 × 10^9 × 1.67 × 10^{-10}}{(0.25)^3} ≈ 5.1 \, N/m \) --- ### **3. Experimental Measurement** **Excitation:** - Pull the free end down by ~1 cm and release (initial condition). - Record the motion using a smartphone camera (slo-mo mode if possible). **Video Analysis:** - Use software (e.g., Tracker, or even frame-by-frame in VLC or YouTube) to track the tip position over time. - Measure time between peaks to find **frequency**. - Measure the decay in peak amplitudes for **damping ratio**. **Example Results:** - Period between peaks = 0.25 s ⇒ Frequency = 4 Hz - Amplitude decays from 10 mm to 6 mm in 5 cycles: - Logarithmic decrement: \( \delta = \frac{1}{5} \ln\left(\frac{10}{6}\right) ≈ 0.102 \) - Damping ratio \( \zeta = \frac{\delta}{2\pi} ≈ 0.016 \) (underdamped) --- ### **4. Analysis of Inputs** #### **A. Initial Condition (as above)** - Simulate response using \( x(t) = X_0 e^{-\zeta\omega_n t} \cos(\omega_d t) \) - Compare with measured data. #### **B. Harmonic Input (Optional Extension)** - Tap the tip repeatedly at a set frequency (e.g., with a pencil), observe resonance. --- ### **5. Model Validation** - Compare predicted frequency from model: - \( f_n = \frac{1}{2\pi} \sqrt{\frac{k}{m_{eq}}} ≈ \frac{1}{2\pi} \sqrt{\frac{5.1}{0.023}} ≈ 2.37 \) Hz - Measured: 4 Hz - Discuss discrepancies: - Possible errors in estimating mass, overhang, or stiffness. - Real boundary conditions differ from ideal clamp. - Camera measurement error. --- ### **6. Conclusion** - The ruler behaves as an underdamped SDOF oscillator. - The model predicts the order of magnitude well, but exact frequency differs – likely due to approximations and real-world deviations. - Damping is low, as expected for metal. --- ## **PowerPoint Outline** **Slide 1: Introduction** - Brief overview of the project and its goals. **Slide 2: Object & Setup** - Photo of the ruler setup. - Description of physical properties. **Slide 3: Modeling** - Diagram of the SDOF model. - Equations for mass and stiffness. **Slide 4: Experimental Measurement** - Screenshot from video tracking. - Table of measured frequencies/damping. **Slide 5: Analysis & Comparison** - Graph: Measured vs. model response. - Discussion of accuracy and sources of error. **Slide 6: Conclusion** - Key findings. - What worked well, what to improve. --- **If you want the MATLAB code for plotting the response, or a sample data table, let me know!** You can adapt this template to any vibrating object you choose (e.g., a pendulum, a phone vibrating on a table, etc.). Just follow the same steps: measure, model, analyze, compare.