41514 – dynamics of machinery - lecture 2.pdf · (dimensioning & design) (stability analysis...
TRANSCRIPT
![Page 1: 41514 – Dynamics of Machinery - Lecture 2.pdf · (dimensioning & design) (stability analysis and vibrations) (machine elements, strength of materials) (mechanical vibrations) 4](https://reader034.vdocuments.mx/reader034/viewer/2022042111/5e8c6cb5d71c2a4dba0ffbbc/html5/thumbnails/1.jpg)
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)
![Page 2: 41514 – Dynamics of Machinery - Lecture 2.pdf · (dimensioning & design) (stability analysis and vibrations) (machine elements, strength of materials) (mechanical vibrations) 4](https://reader034.vdocuments.mx/reader034/viewer/2022042111/5e8c6cb5d71c2a4dba0ffbbc/html5/thumbnails/2.jpg)
* 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
![Page 3: 41514 – Dynamics of Machinery - Lecture 2.pdf · (dimensioning & design) (stability analysis and vibrations) (machine elements, strength of materials) (mechanical vibrations) 4](https://reader034.vdocuments.mx/reader034/viewer/2022042111/5e8c6cb5d71c2a4dba0ffbbc/html5/thumbnails/3.jpg)
* 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
![Page 6: 41514 – Dynamics of Machinery - Lecture 2.pdf · (dimensioning & design) (stability analysis and vibrations) (machine elements, strength of materials) (mechanical vibrations) 4](https://reader034.vdocuments.mx/reader034/viewer/2022042111/5e8c6cb5d71c2a4dba0ffbbc/html5/thumbnails/6.jpg)
2. Fundamentals: Differentiation (amplitude & direction)
(tangential velocity)
(normal acceleration)
![Page 11: 41514 – Dynamics of Machinery - Lecture 2.pdf · (dimensioning & design) (stability analysis and vibrations) (machine elements, strength of materials) (mechanical vibrations) 4](https://reader034.vdocuments.mx/reader034/viewer/2022042111/5e8c6cb5d71c2a4dba0ffbbc/html5/thumbnails/11.jpg)
Free-Body Diagram fixed arm with a given angle – no angular rotation around point D
3. Example 1 – Particle in 3D (Three Equations)
![Page 12: 41514 – Dynamics of Machinery - Lecture 2.pdf · (dimensioning & design) (stability analysis and vibrations) (machine elements, strength of materials) (mechanical vibrations) 4](https://reader034.vdocuments.mx/reader034/viewer/2022042111/5e8c6cb5d71c2a4dba0ffbbc/html5/thumbnails/12.jpg)
GOAL: Three Dynamic Reaction Forces & No Equation of Motion (dimensioning & design) (machine elements, strength of materials)
3. Example 1 – Particle in 3D (Three Equations)
![Page 13: 41514 – Dynamics of Machinery - Lecture 2.pdf · (dimensioning & design) (stability analysis and vibrations) (machine elements, strength of materials) (mechanical vibrations) 4](https://reader034.vdocuments.mx/reader034/viewer/2022042111/5e8c6cb5d71c2a4dba0ffbbc/html5/thumbnails/13.jpg)
Thank you for your attention!
4. Example 2 – Particle in 3D (Three Equations)
![Page 14: 41514 – Dynamics of Machinery - Lecture 2.pdf · (dimensioning & design) (stability analysis and vibrations) (machine elements, strength of materials) (mechanical vibrations) 4](https://reader034.vdocuments.mx/reader034/viewer/2022042111/5e8c6cb5d71c2a4dba0ffbbc/html5/thumbnails/14.jpg)
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)
![Page 15: 41514 – Dynamics of Machinery - Lecture 2.pdf · (dimensioning & design) (stability analysis and vibrations) (machine elements, strength of materials) (mechanical vibrations) 4](https://reader034.vdocuments.mx/reader034/viewer/2022042111/5e8c6cb5d71c2a4dba0ffbbc/html5/thumbnails/15.jpg)
Thank you for your attention!
5. Example 3 – Particle in 3D (Three Equations)
![Page 16: 41514 – Dynamics of Machinery - Lecture 2.pdf · (dimensioning & design) (stability analysis and vibrations) (machine elements, strength of materials) (mechanical vibrations) 4](https://reader034.vdocuments.mx/reader034/viewer/2022042111/5e8c6cb5d71c2a4dba0ffbbc/html5/thumbnails/16.jpg)
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)
![Page 17: 41514 – Dynamics of Machinery - Lecture 2.pdf · (dimensioning & design) (stability analysis and vibrations) (machine elements, strength of materials) (mechanical vibrations) 4](https://reader034.vdocuments.mx/reader034/viewer/2022042111/5e8c6cb5d71c2a4dba0ffbbc/html5/thumbnails/17.jpg)
6. Example 2 – Particle in 3D (with details)
![Page 18: 41514 – Dynamics of Machinery - Lecture 2.pdf · (dimensioning & design) (stability analysis and vibrations) (machine elements, strength of materials) (mechanical vibrations) 4](https://reader034.vdocuments.mx/reader034/viewer/2022042111/5e8c6cb5d71c2a4dba0ffbbc/html5/thumbnails/18.jpg)
6. Example 2 – Particle in 3D (with details)
![Page 19: 41514 – Dynamics of Machinery - Lecture 2.pdf · (dimensioning & design) (stability analysis and vibrations) (machine elements, strength of materials) (mechanical vibrations) 4](https://reader034.vdocuments.mx/reader034/viewer/2022042111/5e8c6cb5d71c2a4dba0ffbbc/html5/thumbnails/19.jpg)
6. Example 2 – Particle in 3D (with details)
![Page 20: 41514 – Dynamics of Machinery - Lecture 2.pdf · (dimensioning & design) (stability analysis and vibrations) (machine elements, strength of materials) (mechanical vibrations) 4](https://reader034.vdocuments.mx/reader034/viewer/2022042111/5e8c6cb5d71c2a4dba0ffbbc/html5/thumbnails/20.jpg)
6. Example 2 – Particle in 3D (with details)
![Page 21: 41514 – Dynamics of Machinery - Lecture 2.pdf · (dimensioning & design) (stability analysis and vibrations) (machine elements, strength of materials) (mechanical vibrations) 4](https://reader034.vdocuments.mx/reader034/viewer/2022042111/5e8c6cb5d71c2a4dba0ffbbc/html5/thumbnails/21.jpg)
6. Example 2 – Particle in 3D (with details)
![Page 22: 41514 – Dynamics of Machinery - Lecture 2.pdf · (dimensioning & design) (stability analysis and vibrations) (machine elements, strength of materials) (mechanical vibrations) 4](https://reader034.vdocuments.mx/reader034/viewer/2022042111/5e8c6cb5d71c2a4dba0ffbbc/html5/thumbnails/22.jpg)
6. Example 2 – Particle in 3D (with details)
![Page 23: 41514 – Dynamics of Machinery - Lecture 2.pdf · (dimensioning & design) (stability analysis and vibrations) (machine elements, strength of materials) (mechanical vibrations) 4](https://reader034.vdocuments.mx/reader034/viewer/2022042111/5e8c6cb5d71c2a4dba0ffbbc/html5/thumbnails/23.jpg)
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)