Engineering Dynamics

This recitation introduces modal analysis and looks at a double pendulum problem.

This recitation reviews free body diagram strategies, covers equations of motion for multiple degree-of-freedom systems, and addresses a dynamically balanced system.

This recitation covers a direct method of breaking down a problem involving a cart and pendulum.

This recitation covers a Lagrange approach to a problem involving a cart and pendulum.

This recitation includes a concept review for the week, problems with the axis of spin on and not on the principal axis, and a discussion on finding the derivative of a rotating vector. The class concludes with a review of the quiz.

This recitation covers generalized forces in a double pendulum. Questions are also addressed for an upcoming quiz.

This recitation covers a steady state frequency response problem.

This recitation takes an in depth look at modal analysis for a double pendulum system.

This recitation reviews free body diagrams and covers a problem with a torsional spring pendulum followed by a second problem with a rolling pipe on an accelerating truck.

This recitation includes a concept review for the week and covers a problem on velocity and acceleration of a point in a plane using polar coordinates.

Prof. Vandiver goes over analyzing the response of a 2-DOF system to harmonic excitation with transfer functions, using a dynamic absorber to mitigate problem vibration, and does a demonstration of a dynamic absorber using a strobe and a vibrating beam.

Prof. Vandiver goes over wave propagation on a long string, flow-induced vibration of long strings and beams, application of the wave equation to rods, organ pipes, shower stalls with demonstrations, and vibration of beams (dispersion in wave propagation).

Prof. Vandiver begins with an overview then goes over the linearization of a 2-DOF system, free vibration of linear multi-DOF systems, finding natural frequencies and mode shapes of multi-DOF systems, and mode superposition analysis of a 2-DOF system.

Prof. Vandiver goes over the modal expansion theorem, computing mass and stiffness matrices, obtaining uncoupled equations of motion, modal initial conditions, damping in modal analysis, Rayleigh damping, and experimental fitting of damping ratios.

This recitation includes a concept review for the week and covers an amusement park ride problem with velocity in translating and rotating frames. The class also covers questions regarding planar motion problems.

Prof. Vandiver goes over the use of Rayleigh damping to model modal damping ratios, steady state response to harmonic excitation by the method of modal analysis, the direct method for assembling the stiffness of an N DOF system.

Prof. Vandiver goes over the damped response of spring-mass-dashpot system to ICs, the ballistic pendulum example, experimental determination of damping ratio, steady state linear system response to harmonic input, and a beam with a rotating mass shaker.

Prof. Vandiver shows a vibration isolation system with a strobe light and vibrating beam, Hx/F transfer function using complex numbers, vibration isolation system design, predicting natural frequency by SQRT(g/delta), & vibration isolation of the source.

Prof. Gossard goes over obtaining the equations of motion of a 2 DOF system, finding natural frequencies by the characteristic equation, finding mode shapes; he then demonstrates via Matlab simulation and a real 2 DOF system response to initial conditions.

Prof. Vandiver starts with a review of applicable physical laws; he then goes over an example Class 4 problem with moving points of constraint, the tipping box problem, an alternative form of Euler's equation, and ends with a question and answer period.

Prof. Vandiver introduces the single degree of freedom (SDOF) system, finding the EOM with respect to the static equilibrium position, SDOF system response to initial conditions, phase angle in free decay, natural frequencies, and damping ratios.

Prof. Vandiver goes over four classes of rotational problems: (1) rotation about fixed axis through center of mass; (2) fixed axis rotation not through center of mass; (3) unconstrained motion about center of mass; (4) rotation about moving points.

Prof. Vandiver goes over various problems to review for the quiz, such as sticking and sliding in a circular track, a rotating T-bar with an imbalance, a pendulum in an elevator, and other pendulum problems.

Prof. Vandiver introduces Lagrange, going over generalized coordinate definitions, what it means to be complete, independent and holonomic, and some example problems.

Prof. Vandiver goes over the concept questions for the week, the kinematic approach to finding generalized forces, the example of a wheel on moving cart with an incline, and the mass sliding on rod example.

Prof. Vandiver goes over a new formula for computing torque about moving points, the hockey puck problem via direct method and Lagrange, condensing many forces to 1 force and 1 moment at COM, pendulum with Lagrange, Atwood's machine, and falling stick.

Prof. Vandiver goes over the definition of the moment of inertia matrix, principle axes and symmetry rules, example computation of Izz for a disk, and the parallel axis theorem.

Prof. Vandiver goes over the cart and pendulum problem (2 DOF equations of motion), the center of percussion problem, then finally static and dynamic imbalance definitions.

In this class, Prof. Vandiver uses two different notation systems. Please watch this video to see how the different notation systems interrelate.

Prof. Vandiver goes over finding equations of motion and degrees of freedom, the Atwoods machine and rotating mass shaker problems, students' questions about dh/dt and torque, angular momentum for rigid bodies, and the mass moment of inertia matrix, I.

Prof. Vandiver goes over an example problem of a block on a slope, the applications of Newton's 3rd law to rigid bodies, kinematics in rotating and translating reference frames, and the derivative of a rotating vector in cylindrical coordinates.

Prof. Vandiver goes over kinematics (describing the motion of particles and rigid bodies), Newton's three laws of motion, about action and reaction forces, the importance of an inertial reference frames, and the definition of center of mass.

Prof. Vandiver goes over the use of tangential and normal coordinates, a review of linear momentum and impulse, then the definition and derivation of the torque/angular momentum relationship with respect to moving points and rigid bodies.

Prof. Vandiver goes over velocity and acceleration in a translating and rotating coordinate system using polar and cylindrical coordinates, angular momentum of a particle, torque, the Coriolis force, and the definition of normal and tangential coordinates.

Prof. Vandiver first goes over the problem of a body on rollers with an internal rotating mass, then the definition of the mass moment of inertia as a summation, and finally moments and products of inertia.

Prof. Vandiver discusses fictitious forces at length and goes over several problems: the spool problem, the elevator with a cable that breaks, and the cart carrying water on an incline. Finally, he does a rotating mass demonstration.

Prof. Vandiver goes over the time rate of change of linear and angular momentum for a particle, conservation of angular momentum, work equalling the change in kinetic energy, external and internal structural torques, and axis of rotation.

Prof. Vandiver begins the lecture by discussing some concepts students had trouble with, then goes over free body diagrams and degrees of freedom with example problems (hockey puck, elevator, stick against wall), and finally discusses fictitious forces.

Prof. Vandiver introduces key historical thinkers in the study of dynamics. He then derives equations of motion using Newton's laws, gives an introduction to kinematics using reference frames and vectors, and goes over motion in moving reference frames.