Virtual Laboratory — Radar Principles: Frequency of a Pendulum

Laboratory Objectives

🎯 Primary Objectives

Upon completion of this virtual laboratory, students will be able to:

Understand Pendulum Dynamics: Derive and verify the relationship between pendulum length, gravitational acceleration, and oscillation frequency for a simple pendulum.
Apply Doppler Radar Principles: Explain how Doppler radar detects periodic motion and measures the frequency of a moving target (pendulum bob).
Analyze Doppler Shift Signals: Interpret the time-varying Doppler frequency shift produced by an oscillating pendulum and relate it to the instantaneous velocity of the bob.
Perform Spectral Analysis: Use FFT-based frequency spectrum analysis to extract the fundamental pendulum frequency from the Doppler radar return signal.
Correlate Theory with Simulation: Compare theoretical predictions of pendulum frequency with simulation results and explain any discrepancies.

🔧 Skills Developed

📐 Mathematical Modeling

Deriving equations of motion
Small-angle approximation
Doppler shift calculations

📡 Radar Signal Processing

Understanding Doppler effect
Signal spectrum analysis
Frequency extraction methods

💻 Simulation & Analysis

Parameter variation studies
Data recording & plotting
Error analysis & comparison

📝 Technical Reporting

Structured lab documentation
Result interpretation
Conclusion formulation

      Prerequisite Knowledge: Basic physics (kinematics, periodic motion), introductory calculus, familiarity with electromagnetic waves, and basic signal processing concepts (FFT, frequency spectrum).
    

Theoretical Background

🕰 1. Simple Pendulum Physics

A simple pendulum consists of a mass (bob) suspended from a fixed point by a massless, inextensible string of length L. When displaced from equilibrium and released, it oscillates under gravity.

Equation of Motion (small-angle approximation, θ < 15°):

d²θ/dt² + (g/L) · θ = 0

where g = 9.81 m/s² (acceleration due to gravity), L = pendulum length (m)

The period and frequency of oscillation are:

T = 2π √(L/g) and f = 1/T = (1/2π) √(g/L)

T = period (s), f = frequency (Hz). Note: f is independent of mass and amplitude (for small angles).

Angular position and velocity as functions of time:

θ(t) = θ&sub0; cos(ωt) and v(t) = −Lωθ&sub0; sin(ωt) = −v_max sin(ωt)

ω = 2πf = √(g/L) (angular frequency), θ&sub0; = maximum angular displacement, v_max = Lωθ&sub0;

Key Insight: The pendulum bob's velocity varies sinusoidally with time, reaching maximum speed at the equilibrium position and zero speed at the turning points. This velocity variation is what creates the time-varying Doppler shift detected by the radar.

📡 2. Doppler Radar Principles

Doppler radar transmits a continuous wave (CW) or pulsed electromagnetic signal toward a target. The reflected signal experiences a frequency shift proportional to the radial velocity of the target relative to the radar.

For a monostatic radar (transmitter and receiver co-located):

f_D = (2v_r / c) · f_c = 2v_r / λ

f_D = Doppler frequency (Hz), v_r = radial velocity (m/s), c = speed of light (~3×10&sup8; m/s), f_c = carrier frequency (Hz), λ = wavelength (m)

For a pendulum, the radial velocity is the horizontal component:

v_r(t) = v(t) · cos(φ) = −v_max sin(ωt) · cos(φ)

φ = angle between radar line-of-sight and direction of motion. For broadside illumination (radar perpendicular to swing plane), cos(φ) ≈ 1.

Therefore, the instantaneous Doppler shift is:

f_D(t) = −(2f_c/c) · v_max sin(ωt) = −f_D,max sin(ωt)

The Doppler shift oscillates sinusoidally at the same frequency as the pendulum, with amplitude f_D,max = 2f_cv_max/c.

        Important: The Doppler shift is positive when the bob moves toward the radar and negative when it moves away. At the turning points (maximum displacement), the velocity is zero, so fD = 0. At the equilibrium position, the velocity is maximum, so |fD| is maximum.
      

📊 3. Frequency Spectrum Analysis

The received radar signal contains the carrier frequency f_c modulated by the Doppler shift. After down-conversion (mixing with a local oscillator at f_c), the baseband signal contains only the Doppler frequency components.

The baseband signal is:

s(t) = A · cos(2π f_D(t) · t) = A · cos(−2π f_D,max sin(ωt) · t)

This is a frequency-modulated (FM) signal. Its spectrum contains sidebands at multiples of the pendulum frequency f.

Using the Jacobi-Anger expansion, the spectrum contains Bessel function sidebands:

s(t) = A · Σ_n=−∞^∞ J_n(β) · e^jnωt

J_n(β) = Bessel function of the first kind, order n. β = f_D,max/f = modulation index. The dominant spectral peak occurs at the pendulum frequency f.

Practical Extraction: In practice, the fundamental pendulum frequency f is extracted by applying the Fast Fourier Transform (FFT) to the Doppler signal and identifying the peak in the magnitude spectrum. For small modulation indices (β < 0.5), the spectrum is dominated by the first-order sideband at frequency f.

🔢 4. Numerical Example

Consider a pendulum with L = 1.0 m, amplitude θ&sub0; = 10° (0.175 rad), and a 10 GHz radar (X-band):

Parameter	Value	Calculation
Period T	2.006 s	2π√(1.0/9.81)
Frequency f	0.498 Hz	1/T
Angular freq. ω	3.132 rad/s	2πf
Max velocity v_max	0.313 m/s	L·ω·θ&sub0;
Max Doppler f_D,max	20.9 Hz	2·v_max·f_c/c
Modulation index β	41.9	f_D,max/f

        Note: With β = 41.9, many Bessel sidebands appear in the spectrum. However, the envelope of the spectrum still peaks at the pendulum frequency f = 0.498 Hz, which can be extracted via FFT analysis.
      

Interactive Simulation

⚙ Parameters

Pendulum Length (L)

1.00 m

Amplitude (θ&sub0;)

10°

Radar Frequency (f_c)

10 GHz

Simulation Speed

1.0×

2.01

Period T (s)

0.50

Freq f (Hz)

20.9

f_D,max (Hz)

📡 Radar & Pendulum Animation

📈 Doppler Signal (Time Domain)

📊 Frequency Spectrum (FFT)

Procedure & Report Guidelines

🔬 Laboratory Procedure

Follow these steps carefully to complete the virtual experiment. Record all data in your lab notebook.

Study the Theory: Review the theoretical background section. Ensure you understand the relationship between pendulum length and frequency, and how Doppler radar measures motion. Write down the key equations in your notebook.
Set Baseline Parameters: Open the simulation. Set the pendulum length to L = 1.00 m, amplitude to θ&sub0; = 10°, and radar frequency to f_c = 10 GHz. Record these values as your baseline configuration.
Record Theoretical Predictions: Before running the simulation, calculate the expected period T, frequency f, and maximum Doppler shift f_D,max using the formulas provided. Show your calculations clearly.
Run Baseline Simulation: Start the simulation and observe the pendulum motion, the Doppler signal waveform, and the FFT spectrum. Let the simulation run for at least 5 complete oscillations to allow the FFT to stabilize.
Extract Measured Frequency: From the FFT spectrum, identify the peak frequency. Compare this with your theoretical prediction. Calculate the percentage error: %Error = |f_theory − f_measured| / f_theory × 100%.
Vary Pendulum Length: Repeat steps 3–5 for at least 5 different lengths: L = 0.25 m, 0.50 m, 1.00 m, 1.50 m, and 2.00 m. Keep amplitude and radar frequency constant. Record all data in a table.
Vary Amplitude: Fix L = 1.00 m and f_c = 10 GHz. Repeat for amplitudes θ&sub0; = 5°, 10°, 15°, and 20°. Verify that the frequency remains approximately constant (isochronism for small angles). Note any deviation at larger angles.
Vary Radar Frequency: Fix L = 1.00 m and θ&sub0; = 10°. Repeat for radar frequencies f_c = 1 GHz, 5 GHz, 10 GHz, and 24 GHz. Observe how f_D,max changes and verify the linear relationship: f_D,max ∝ f_c.
Analyze Doppler Signal Shape: For one configuration, sketch or screenshot the time-domain Doppler signal. Explain why it is sinusoidal and why it crosses zero at the turning points.
Spectral Analysis: Examine the FFT spectrum for different modulation indices (β = f_D,max/f). Note how the spectrum changes with β. For small β, the spectrum should show a single dominant peak.

📊 Data Recording Table (Template)

Use this table format for recording your experimental data:

Trial	L (m)	θ&sub0; (°)	f_c (GHz)
1	0.25	10	10
2	0.50	10	10
3	1.00	10	10
4	1.50	10	10
5	2.00	10	10

📝 Guidelines for Report Writing

Your lab report should be structured, clear, and professionally presented. Include the following sections:

📑 Title Page

Experiment title and course code
Student name, ID, and group number
Date of experiment and submission
Instructor name

🎯 Abstract / Summary

Brief overview (150–200 words)
Objectives, methods, key results
Main conclusions

📚 Theory

Pendulum physics derivation
Doppler radar principles
Relevant equations with definitions
Expected relationships

🔧 Equipment & Setup

Simulation parameters used
Software/tools description
Diagram of setup (if applicable)

📝 Procedure

Step-by-step methodology
Parameter variations
Any deviations from standard

📊 Results

Data tables (neat, labeled)
Graphs: f vs. 1/√L, f_D,max vs. f_c
FFT spectrum screenshots
Sample calculations

🔍 Discussion

Comparison: theory vs. simulation
Sources of error analysis
Physical interpretation
Limitations of small-angle approx.

✅ Conclusion

Key findings summarized
Objectives met? (Yes/No + how)
Recommendations for improvement

📖 References

Textbooks, papers cited
Online resources (with URLs)
IEEE / APA format

📎 Appendices

Raw data sheets
Additional calculations
Simulation screenshots

        Grading Rubric (100 points): Theory & Background (20 pts) | Procedure & Data (20 pts) | Results & Graphs (25 pts) | Discussion & Error Analysis (20 pts) | Presentation & Formatting (15 pts)
      

Report Writing

✍ Tips for a High-Quality Report

Use SI units consistently throughout (meters, seconds, Hz, GHz).
Show all calculations step-by-step; do not just state final answers.
Label all graphs with titles, axis labels (with units), and legends.
Report errors with appropriate significant figures (typically 2–3 sig figs).
Discuss discrepancies honestly — identify whether they are due to simulation limitations, approximations, or measurement uncertainty.
Use proper technical language: "Doppler shift," "radial velocity," "spectral peak," "modulation index," "Fast Fourier Transform."
Include screenshots of the simulation at key moments (e.g., maximum Doppler shift, zero-crossing).
Cite sources for any external equations or data used.

📐 Sample Calculation Format

Given: L = 1.00 m, g = 9.81 m/s², θ&sub0; = 10° = 0.175 rad, f_c = 10 GHz

Step 1 — Theoretical Period:
T = 2π√(L/g) = 2π√(1.00/9.81) = 2π × 0.319 = 2.006 s

Step 2 — Theoretical Frequency:
f = 1/T = 1/2.006 = 0.498 Hz

Step 3 — Maximum Velocity:
ω = √(g/L) = √(9.81/1.00) = 3.132 rad/s
v_max = L·ω·θ&sub0; = 1.00 × 3.132 × 0.175 = 0.313 m/s

Step 4 — Maximum Doppler Shift:
f_D,max = 2·v_max·f_c/c = 2 × 0.313 × 10×10&sup9; / (3×10&sup8;) = 20.9 Hz

Step 5 — Percentage Error:
%Error = |0.498 − 0.500| / 0.498 × 100% = 0.4%

Post-Laboratory Quiz

Answer the following 12 questions. Click "Reveal Answer" to check your response and read the explanation.