Lecture 174 of 223Vibration Analysis by Mode Superposition1 h 22 min
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
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.
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- 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