41514 – Dynamics of Machinery - Lecture 2.pdf · (dimensioning & design) (stability analysis...

41514 – Dynamics of Machinery – Theory, Experiment, Phenomenology and Industrial Applications –

Ilmar Ferreira Santos

1. Recapitulation – Mathematical Modeling & Steps

2. Fundamentals

3. Example 1

4. Example 2

5. Example 3

6. Example 2 (with details)

* Mass Elements M Particle Rigid Body Distributed * Spring Elements K Elasticity Theory & Material Magnetism Fluid Mechanics * Damping Elements D Fluid Mechanics Contact Mechanics (friction)

Mathematical Model

Mechanical Model

Physical System

Assumptions (simplifications)

Newton, Euler, D‘Alembert, Lagrange, Hamilton, Jourdain

(principles & axioms)

Static Equilibrium Position (Linearization) (structure)

(machine)

Solution:

(eigenvalues)

(eigenvectors)

1. Recapitulation – Mathematical Modeling & Steps

* Mass Elements M Particle Rigid Body Distributed * Spring Elements K Elasticity Theory & Material Magnetism Fluid Mechanics * Damping Elements D Fluid Mechanics Contact Mechanics (friction)

Mathematical Model

Mechanical Model

Physical System

Assumptions (simplifications)

Newton, Euler, D‘Alembert, Lagrange, Hamilton, Jourdain

(principles & axioms)

• Kinematics 1. Reference Frames (systems of coordinates) 2. Transformation Matrices 3. Position Vectors 4. Velocity Vectors (linear and angular) 5. Acceleration Vectors (linear and angular)

• Dynamics 6. Mass Properties (mass center, moments of inertia) 7. Force and Moment Vectors 8. Dynamic Equilibrium: (Newton, Euler, Lagrange …) 9. Equilibrium Poition (Linearization, Vibration Analysis)

1. Recapitulation – Mathematical Modeling & Steps

2. Fundamentals: Vectors & Reference Frames

2. Fundamentals: Differentiation & Reference Frames

2. Fundamentals: Differentiation (amplitude & direction)

(tangential velocity)

(normal acceleration)

2. Fundamentals: Vectors & Reference Frames

2. Fundamentals: Absolute & Relative Velocities

2. Fundamentals: Absolute & Relative Velocities

2. Fundamentals: Absolute and Relative Accelerations

Free-Body Diagram fixed arm with a given angle – no angular rotation around point D

3. Example 1 – Particle in 3D (Three Equations)

GOAL: Three Dynamic Reaction Forces & No Equation of Motion (dimensioning & design) (machine elements, strength of materials)

3. Example 1 – Particle in 3D (Three Equations)

Thank you for your attention!

4. Example 2 – Particle in 3D (Three Equations)

GOAL: Two Dynamic Reaction Forces & One Equations of Motion (dimensioning & design) (stability analysis and vibrations) (machine elements, strength of materials) (mechanical vibrations)

4. Example 2 – Particle in 3D (Three Equations)

Thank you for your attention!

5. Example 3 – Particle in 3D (Three Equations)

GOAL: One Dynamic Reaction Force & Two Equations of Motion (dimensioning & design) (stability analysis and vibrations) (machine elements, strength of materials) (mechanical vibrations)

5. Example 3 – Particle in 3D (Three Equations)

6. Example 2 – Particle in 3D (with details)

6. Example 2 – Particle in 3D (with details)

6. Example 2 – Particle in 3D (with details)

6. Example 2 – Particle in 3D (with details)

6. Example 2 – Particle in 3D (with details)

6. Example 2 – Particle in 3D (with details)

GOAL: Two Dynamic Reaction Forces & One Equations of Motion (dimensioning & design) (stability analysis and vibrations) (machine elements, strength of materials) (mechanical vibrations)

6. Example 2 – Particle in 3D (with details)