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