Lecture 88 of 223Finding Equations of Motion for Rigid Body Rotation1 h 4 min
Tipping box problem-alternative form of Eulers equation
Prof. J. Kim Vandiver and Prof. David Gossard · Massachusetts Institute of Technology
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- Engineering Dynamics, Massachusetts Institute of Technology (at MIT OpenCourseWare)
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- CC BY-NC-SA 4.0, as MIT OpenCourseWare states it. Non-commercial use only.
- Attribution
- Prof. J. Kim Vandiver, Prof. David Gossard. Engineering Dynamics. Fall 2011. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu/. License: Creative Commons BY-NC-SA.
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All the lectures
- Lecture 1: History of Dynamics; Motion in Moving Reference Frames
- Modeling in dynamics
- History of the study of dynamics
- An example deriving equations of motion using Newton's laws
- Introduction to kinematics using reference frames and vectors
- Moving frames: translation and rotation
- Notation Systems
- Lecture 2: Newton's Laws and Describing the Kinematics of Particles
- Purpose of concept questions
- Kinematics: describing the motion of particles and rigid bodies
- A review of Newton's three laws of motion
- The importance of inertial reference frames
- Newton's 3rd Law action and reaction forces
- The definition of center of mass
- The definition of center of rod
- Lecture 3: Motion of Center of Mass; Acceleration in Rotating Ref. Frames
- Problem solving with an example of a block on a slope
- Newton's 3rd law recap and application to rigid bodies
- Kinematics: in translating and rotating frames
- Kinematics: derivative of a rotating vector in cylindrical coordinates
- Lecture 4: Movement of a Particle in Circular Motion w/ Polar Coordinates
- Vector form of velocity and acceleration in a translating and rotating coordinate system and the expression of them in polar and cylindrical coordinates
- Two examples using polar coordinates
- Angular momentum of a particle, torque as the time rate of change of angular momentum, and the appearance of the coriolis force
- Candy shooter example of coriolis force done in polar coordinates
- Definition of normal and tangential coordinates
- Notation Systems
- Recitation 2: Velocity and Acceleration in Translating and Rotating Frames
- Lecture 5: Impulse, Torque, and Angular Momentum for a System of Particles
- Lecture outline and Use of tangential and normal coordinates with an example
- A review of the principle of linear momentum and impulse
- The definition of the angular momentum of a particle with respect to a moving reference point
- Example of the torque/angular momentum relationship to a carnival ride
- Lecture 6: Torque and the Time Rate of Change of Angular Momentum
- Intro to the time rate of change of linear and angular momentum for a particlecoordinates
- Example 1: Rotating mass on a string; dP/dt = tension
- Example 2a-Pulling the string from a rotating mass through a hole at the axis of rotation. Conservation of angular momentum, h
- Example 2b-work done equals the change in kinetic energy
- Example 3b: Computing external and internal structural torques using dh / dt; angular momentum vector not aligned with the axis of rotation
- Question and answer session
- Recitation 3: Motion in Moving Reference Frames
- Lecture 7: Degrees of Freedom, Free Body Diagrams, and Fictitious Forces
- Muddy card discussion
- Degrees of freedom and free body diagrams
- Example 2: Hockey puck
- Example 3: Rod leaning on a wall
- Free body diagrams with two bodies
- Fictitious force in the elevator
- Fictitious centrifugal force
- Lecture 8: Fictitious Forces and Rotating Mass
- Reading assignment from Williams on kinematics
- Concept questions intro showing problems
- Concept question discussion-spool and string etc
- Fictitious force: elevator cable breaks
- Demo of cart on incline with water in container
- Complete fictitious force discussion
- Rotating mass demo related to homework 1
- Recitation 4: Free Body Diagrams
- Lecture 9: Rotating Imbalance
- Body on rollers with an internal rotating mass
- Mass moment of inertia definition as a summation
- Moments and Products of Inertia
- Lecture 10: Equations of Motion, Torque, Angular Momentum of Rigid Bodies
- Equations of motion (EOM) and degrees of freedom (DOF)
- Example 1: Finding EOMs and DOFs
- Example 2: EOM for Atwoods machine
- Example 3: Muddy card issue about dh/dt and torques
- Example 4: Rotating mass shaker and effect of gravity on torque
- Angular momentum for rigid bodies and the mass moment of inertia matrix, [I]
- Recitation 5: Equations of Motion
- Lecture 11: Mass Moment of Inertia of Rigid Bodies
- Definition of moment of inertia matrix
- Principle axes and symmetry rules
- Example computation of Izz for a disk
- Parallel axis theorem
- Lecture 12: Problem Solving Methods for Rotating Rigid Bodies
- Four classes of rotational problems
- Class 1: Rotation about fixed axis through center of mass
- Class 2: Fixed axis rotation not through center of mass-pendulum example
- Class 3: Unconstrained motion about center of mass-hockey puck example
- Equivalent forces and moments acting only upon the center of mass
- Class 4: Rotation about moving points; H/A alternative formulation in terms of H/G
- Class 4 example of the pipe rolling on a truck bed
- Recitation 6: Angular Momentum and Torque
- Lecture 13: Four Classes of Problems With Rotational Motion
- Review of applicable physical laws
- Example 4: Problem with moving points of constraint
- Tipping box problem-alternative form of Eulers equation
- Question and answer period
- Lecture 14: More Complex Rotational Problems and Their Equations of Motion
- Cart and pendulum 2 DOF equations of motion
- Center of percussion problem
- Static and dynamic imbalance definitions
- Recitation 7: Cart and Pendulum, Direct Method
- Notation Systems
- Lecture 15: Introduction to Lagrange With Examples
- Generalized coordinate definitions
- Complete, independent and holonomic
- Example problems
- Recitation 8: Cart and Pendulum, Lagrange Method
- Lecture 16: Kinematic Approach to Finding Generalized Forces
- Concept question discussion
- Kinematic approach to finding generalized forces Example of a wheel on moving cart with an incline
- Mass sliding on rod example
- Lecture 17: Practice Finding EOM Using Lagrange Equations
- New formula for computing torque about moving points
- Example of hockey puck with a pull string-direct method
- Hockey puck by Lagrange
- Condensing many forces to one force and one moment at center of mass
- Generic pendulum by Lagrange
- Examples: Atwood's machine and falling stick
- Recitation 9: Generalized Forces
- Lecture 18: Quiz Review From Optional Problem Set 8
- Problem 1-Sticking sliding in circular track
- Problems 2 and 3-Rotating T-bar with imbalance
- Problems 4 and 5-Pendulum in elevator
- Problem 6 and 7-more pendula
- Lecture 19: Introduction to Mechanical Vibration
- Introduction to the single degree of freedom (SDOF) system
- Finding the equation of motion with respect to the static equilibrium position
- Response of the SDOF system to initial conditions
- Meaning of phase angle in free decay
- Engineers approach to the solution of the 2nd order differential equation with initial conditions
- Estimation of natural frequencies and damping ratios from measured response: the logarithmic decrement
- Lecture 20: Linear System Modeling a Single Degree of Freedom Oscillator
- Concept question discussion of Problem set 9-keep
- Damped response of spring-mass-dashpot system to initial conditions
- Finding the response of a pendulum to initial conditions
- Example: Ballistic pendulum
- Experimental determination of damping ratio
- Steady state linear system response to harmonic input
- Demonstration of a beam with a rotating mass shaker
- Lecture 21: Vibration Isolation
- Demonstration of a vibration isolation system-strobe light and vibrating beam
- Euler's formula for complex numbers
- Hx/F transfer function by complex number approach
- Base motion Hx/y transfer function-vibration isolation system design
- Predicting natural frequency by SQRT(g/delta)
- Vibration isolation of the source-H-Ft/Fo transferfunction-with strobe light demo
- Lecture 22: Finding Natural Frequencies and Mode Shapes of a 2 DOF System
- Obtain the equations of motion of a 2 DOF system
- Finding the natural frequencies by the characteristic equation
- Finding the modes shapes of the 2 DOF system
- Matlab simulation of the response of the 2 DOF system to initial conditions
- Physical demonstration of the motion of a real 2 DOF system
- Recitation 10: Steady State Dynamics
- Lecture 23: Vibration by Mode Superposition
- Review of the concept questions with Homework 10
- Overview of the vibration material to be covered in the course
- Linearization of a 2-DOF system
- Free vibration of linear multi-DOF systems-intro to mode superposition
- Finding natural frequencies and mode shapes of multi-dof systems
- Mode superposition analysis of a 2-DOF system
- Lecture 24: Modal Analysis: Orthogonality, Mass Stiffness, Damping Matrix
- The modal expansion theorem
- Vibrating string demonstration
- Demonstration of modes with the 2 DOF mechanical model
- Orthogonality and the computation of the mass and stiffness matrices
- Response to initial conditions using modal analysis
- Obtaining uncoupled equations of motion and determination of modal initial conditions
- Dealing with damping in modal analysis
- Rayleigh damping and experimental fitting of damping ratios
- Recitation 11: Double Pendulum System
- Lecture 25: Modal Analysis: Response to IC's and to Harmonic Forces
- Outline of the final three lectures
- Response to initial conditions and the use of Rayleigh damping to model modal damping ratios
- Steady state response to harmonic excitation by the method of modal analysis
- Demonstration of the response of a string to harmonic excitation
- Demonstration of the response of a string to initial deflection
- Direct method for assembling the stiffness of an N DOF system
- Lecture 26: Response of 2-DOF Systems by the Use of Transfer Functions
- Discussion of problem set concept questions
- Review of the use of a transfer function to analyze the response of a single DOF system to harmonic input
- Analysis of the response of a 2-DOF system to harmonic excitation by means of transfer functions and the introduction of the application of the dynamic absorber to mitigate problem vibration
- Demonstration of a dynamic absorber using a strobe and a vibrating beam
- Questions and answers: discussion of helicopter video and the 300 ton dynamic absorber in the Hancock building
- Lecture 27: Vibration of Continuous Structures: Strings, Beams, Rods, etc.
- Demonstration of wave propagation on a long string
- My personal research area: Flow-induced vibration of long strings and beams in fluid flow
- Derivation of the "wave equation"
- Natural frequencies and mode shapes of the string under tension-with demonstration
- Application of the wave equation to rods, organ pipes, shower stalls with demonstrations
- The vibration of beams: natural frequencies as a function of boundary conditions, dispersion in wave propagation
- Recitation 12: Modal Analysis of a Double Pendulum System
- Lecture 1: History of Dynamics; Motion in Moving Reference Frames
- Lecture 2: Newton's Laws and Describing the Kinematics of Particles
- Lecture 3: Motion of Center of Mass; Acceleration in Rotating Ref. Frames
- Lecture 4: Movement of a Particle in Circular Motion w/ Polar Coordinates
- Notation Systems
- Recitation 2: Velocity and Acceleration in Translating and Rotating Frames
- Lecture 5: Impulse, Torque, and Angular Momentum for a System of Particles
- Lecture 6: Torque and the Time Rate of Change of Angular Momentum
- Recitation 3: Motion in Moving Reference Frames
- Lecture 7: Degrees of Freedom, Free Body Diagrams, and Fictitious Forces
- Lecture 8: Fictitious Forces and Rotating Mass
- Recitation 4: Free Body Diagrams
- Lecture 9: Rotating Imbalance
- Lecture 10: Equations of Motion, Torque, Angular Momentum of Rigid Bodies
- Recitation 5: Equations of Motion
- Lecture 11: Mass Moment of Inertia of Rigid Bodies
- Lecture 12: Problem Solving Methods for Rotating Rigid Bodies
- Recitation 6: Angular Momentum and Torque
- Lecture 13: Four Classes of Problems With Rotational Motion
- Lecture 14: More Complex Rotational Problems and Their Equations of Motion
- Recitation 7: Cart and Pendulum, Direct Method
- Lecture 15: Introduction to Lagrange With Examples
- Recitation 8: Cart and Pendulum, Lagrange Method
- Lecture 16: Kinematic Approach to Finding Generalized Forces
- Lecture 17: Practice Finding EOM Using Lagrange Equations
- Recitation 9: Generalized Forces
- Lecture 18: Quiz Review From Optional Problem Set 8
- Lecture 19: Introduction to Mechanical Vibration
- Lecture 20: Linear System Modeling a Single Degree of Freedom Oscillator
- Lecture 21: Vibration Isolation
- Lecture 22: Finding Natural Frequencies and Mode Shapes of a 2 DOF System
- Recitation 10: Steady State Dynamics
- Lecture 23: Vibration by Mode Superposition
- Lecture 24: Modal Analysis: Orthogonality, Mass Stiffness, Damping Matrix
- Recitation 11: Double Pendulum System
- Lecture 25: Modal Analysis: Response to IC's and to Harmonic Forces
- Lecture 26: Response of 2-DOF Systems by the Use of Transfer Functions
- Lecture 27: Vibration of Continuous Structures: Strings, Beams, Rods, etc.
- Recitation 12: Modal Analysis of a Double Pendulum System