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Intermediate module

Dynamics and Vibrations

Analyze particles and rigid bodies in motion, oscillations, natural frequency, damping, and resonance.

Course outline only for now. Full chapter-level lessons are still in progress. Use this page for readiness, concepts, worked-example format, practice, review, and portfolio direction. Complete course contents are live today for Math, Physics, and Statics.

01

Readiness check

Before starting, confirm the prerequisite habits.

  • Draw FBDs for moving bodies.
  • Use Newton second law with signs.
  • Understand rad/s versus Hz.
  • Recognize static equilibrium position.
0 or 1 weak itemContinue, but slow down at the worked example.
2 weak itemsReview the foundation page linked in the roadmap before solving practice problems.
3 or more weak itemsStep back to prerequisites; this module depends on them.
02

The core idea

Build equations of motion and predict acceleration or vibration response.

Dynamics extends statics by keeping the m a term: you still draw a free body, but now net force or moment equals mass times acceleration, and oscillation emerges when a restoring force opposes displacement.

omega = sqrt(k/m)
Works when: you draw the FBD, choose a coordinate, and write Sum F = m a (or Sum M = I alpha) with a consistent positive direction before solving.
Breaks down when: you treat a rotating or accelerating frame as inertial, or drop the inertia term in a system that is actually accelerating.
Figure 1. Concept model for Dynamics and Vibrations. The figure names inputs, computed variables, geometry, and result.
input/load result/constraint computed variable dimension/model geometry
03

The method

1Model

Make the physical situation visible.

2Relate

Translate the model into symbols.

3Solve

Calculate only after the model is clear.

4Check

Use units, scale, and limiting cases.

04

Worked example

Figure 2. Worked problem setup: A 4 kg machine component is mounted on a spring with stiffness 900 N/m. Ignore damping and find natural frequency in Hz.
Figure 3. Calculation model. The result follows from the model, units, and reasonableness check.

A 4 kg machine component is mounted on a spring with stiffness 900 N/m. Ignore damping and find natural frequency in Hz.

  1. Problem A 4 kg machine component is mounted on a spring with stiffness 900 N/m. Ignore damping and find natural frequency in Hz.
  2. Given and find m = 4 kg, k = 900 N/m. Find: Natural frequency f_n.
  3. Assumptions Idealized model, consistent units, and no hidden effects outside the stated scope.
  4. Step omega_n = sqrt(k/m) = sqrt(900/4) = 15 rad/s.
  5. Step f_n = omega_n / (2 pi) = 2.39 Hz.
  6. Step Increasing stiffness raises frequency and increasing mass lowers it.
  7. Step State that damping is ignored.
  8. Conclusion fn = 2.39 Hz. Carry this result into the design decision, not just into the answer box.
05

Misconceptions and diagnostics

MistakeSymptomDiagnostic questionCorrection
Forgetting the inertia termSolves a moving system as if staticIs anything accelerating here?Add m a (or I alpha) when acceleration is nonzero.
Wrong natural frequencyomega comes out with mass and stiffness swappedDoes stiffer or heavier raise the frequency?omega = sqrt(k/m): stiffness up raises omega, mass up lowers it.
Mixing mass and weightUses W where m belongs in m aDid you divide weight by g to get mass?Keep m in kg; convert weight W = m g explicitly.
06

Practice ladder

Level 1: direct skill

Redo the worked example with one changed input. Predict the trend before calculating.

Check yourself

The trend must match the governing relation: omega = sqrt(k/m).

Level 2: mixed concept

Draw the model from memory, label knowns and unknowns, then write the first equation without looking.

Check yourself

Your first equation should connect the model to fn.

Level 3: independent problem

Create a similar problem from a real object near you. State assumptions, solve it, and include a reasonableness check.

Check yourself

A valid solution has a sketch, given/find list, governing relation, units, and a conclusion.

Level 4: transfer task

Turn the result into a design decision: what would you change if the output missed its target by 25 percent?

Check yourself

Name the design variable with the strongest influence and justify it from the equation.

07

Working with AI, and proving it yourself

Useful AI role

Ask for a critique of assumptions, units, diagram labels, and missing checks after you have attempted the solution.

Do not outsource

Do not paste the problem and accept a final answer. Your evidence is the model, the checks, and the explanation.

08

Retrieval and spaced review

Closed-notes prompts: draw the FBD of the moving body, choose a coordinate and positive sense, write Sum F = m a or Sum M = I alpha, and identify the system's natural frequency.

TodayRedo the worked example from a blank page.
+1 daySolve Level 1 without notes.
+3 daysSolve Level 2 with changed numbers.
+7 daysConnect this module to another course.
+30 daysAdd a portfolio artifact.
09

Mapping and portfolio task

Course mapping

Dynamics and vibrations turn statics into motion: it feeds controls (the plant model), machine design (fatigue from vibration), and robotics (manipulator dynamics).

First-pass focus: definitions, model setup, units, and worked examples. Save edge cases for the second pass.

Portfolio task

Create a one-page natural-frequency note for a real mounted mass: sketch, assumptions, equations, result, reasonableness check, limitation, and recommendation.