UTPA STEM/CBI Courses/Dynamics/The Big Slip
Course Title: Dynamics
Lecture Topic: The Big Slip
Instructor: R.A. Freeman
Institution: UTPA
Backwards Design
[edit | edit source]Course Objectives
- To master the fundamental concepts planar particle and rigid-body kinematic and dynamics analysis.
- To understand that the mathematical models used for analysis are also used for design.
- To be introduced to math and engineering software for dynamic analysis, simulation, and design of free and constrained planar particle and rigid-body systems.
- Primary Objectives- By the end of the course students will be able to:
- Solve for the one-dimensional kinematics of a particle using the basic relationships between position, velocity, acceleration, and time.
- Solve problems involving the kinematics of one-dimensional constrained motion of particles, with specific application to pulley systems.
- Describe two-dimensional motion of a particle using cartesian, polar, and path coordinate systems.
- Convert from one coordinate system to another and solve problems involving a mixture of coordinate systems.
- Apply the concepts of angular velocity and acceleration, along with the notions of relative velocity and acceleration and the vector cross product, to determine the velocity and acceleration of points fixed in rigid-bodies.
- Apply the concept of an instant center to determine the velocities of points in single rigid-bodies and of the bodies themselves in constrained rigid-body systems/gear trains/linkages.
- Determine the position, velocity, and acceleration of points fixed in rigid-bodies that are rolling without slip.
- Describe the positions, velocities, and accelerations of all members of constrained planar-rigid-body linkages and gear-trains.
- Determine the relative and absolute velocities and accelerations of independent particles, particles moving within rotating discs, and linkages containing prismatic joints in terms of the application of rotating-reference-frames and the notion of coincident points.
- Apply Newton’s 2nd Law to obtain the differential equations of motion of free and constrained particles and rigid-bodies in one- and two-dimensions and to solve for forces, moments, and/or accelerations, including situations involving friction.
- Apply the Work-Energy relationship to obtain the integral velocity-displacement equations of free and constrained particles and rigid-bodies in one- and two-dimensions and to solve for forces, moments, displacements, and/or velocities, including situations involving friction.
- Apply the Linear and Angular Impulse and Momentum relationships, including Linear Impact, to obtain the integral velocity-time equations of free and constrained particles and rigid-bodies in one- and two-dimensions and to solve for forces, moments, velocities, and/or time, including situations involving friction.
- Determine if a problem is solvable in terms of the relationship between the number of independent kinematics and kinetics equations available and the number of unknowns.
- Sub Objectives- The objectives will require that students be able to:
- Generate position equations using basic geometric and trigonometric relationships
- Take derivatives of polynomials and trigonometric functions, including applying the chain rule.
- Integrate polynomials and trigonometric functions.
- Describe physical quantities (pos., vel., force, torque/moment) in terms of vectors.
- Determine the magnitude and direction of vectors.
- Apply the vector dot and cross products.
- Solve systems of simultaneous linear equations (at least 2 equations in 2 unknowns)
- Draw Free-Body-Diagrams
- Difficulties- Students may have difficulty:
- Drawing correct FBD’s
- Decomposition of vectors
- Simple differentiation and integration
- Simultaneous use of multiple coordinate systems
- Kinematic description of particles in terms of rotating reference frames
- Description and determination of work done by forces
- Determination of velocity as a function of position via work-energy due to integration difficulties
- Understanding of the definition of angular momentum as the moment of the linear momentum about some point
- Real-World Contexts- There are many ways that students can use this material in the real-world, such as:
- Thrill Rides: Roller-Coasters, Bungee Cords, Skydiving
- Automotive Applications:
- Particles: Race cars, passenger cars, road design, friction
- Rigid-Bodies: Drive train / Transmission (Gearing), Steering (Linkages), Suspension (Shock absorbers and natural frequency)
- Measurement devices: Accelerometers
- Control devices: Attitude gyroscope
- Power Tools (Gearing: force-velocity duality or virtual work or power balance)
- Robotics
- Biomechanics
- Pulleys
Model of Knowledge
- Concept Map
-
Dynamics Concept Map
- Content Priorities
- Enduring Understanding
- Basic Kinematic relationships between pos., vel., accel., and time (s, v, a, t)
- Free-Body-Diagram (FBD): Generation and Interpretation
- Relative position, velocity, and acceleration (unconstrained and constrained)
- Coordinate systems: Rectangular/Cartesian, Path/Normal-Tangential, & Polar (Cyl.)
- Trigonometry (sines, cosines, and tangents)
- Newton’s 2nd Law
- Euler’s Equation(s) of Motion
- Friction
- Enduring Understanding
- Important to Do and Know
- Work – Energy
- Impulse – Momentum
- Impact
- Instant Centers
- Power Balance: Force-Velocity Duality
- Algebra: parametric descriptions
- Linear Algebra: solution of simultaneous equations
- Basic Differential and Integral Calculus
- Important to Do and Know
- Worth Being Familiar with
- Geometry
- Numerical integration
- Worth Being Familiar with
Assessment of Learning
- Formative Assessment
- In Class (groups)
- Computer simulations using both math and engineering software
- Challenge solution presentations
- Homework (groups)
- Standard text book homework assigned
- Challenge solution reports
- In Class (groups)
- Summative Assessment
- Exams formulated to assess both Knowledge/Efficiency & Innovation/Transfer
- Include problems identical to those addressed in class (just change the numbers) as well as those requiring both "near" (similar problem switching knowns and unknowns) and "far" (a totally different question addressable using the same theory) transfer
- Exams formulated to assess both Knowledge/Efficiency & Innovation/Transfer
Legacy Cycle
[edit | edit source]OBJECTIVE
By the end of the module (2 or 3 class periods), students will be able to:
- Describe the trajectory of a particle in "free" flight. This will include being able to determine, based on initial conditions, if, when, and how a target can be reached, and the corresponding "terminal" velocity.
- Use Newton’s 2nd Law to determine the forces acting on a particle sliding down a 2 dimensional path.
- Identify which forces acting on the particle do work and which do not.
- Determine the velocity of a particle after it has traveled along a constrained path, including simple paths involving friction, using work-energy.
The objectives will require that students be able to:
- Describe two-dimensional motion of a particle using cartesian coordinates.
- Apply Newton’s 2nd Law to obtain the differential equations of motion of free and constrained particles in two-dimensions and to solve for forces, and/or accelerations, including situations involving friction.
- Apply the Work-Energy relationship to obtain the integral velocity-displacement equations of free and constrained particles in one- and two-dimensions and to solve for forces, displacements, and/or velocities, including situations involving friction.
THE CHALLENGE
Do you believe the big slip video is real? Make an argument in support of your conclusion.
GENERATE IDEAS
The class is asked what they believe is important to know or do in able to answer the challenge. It is expected that they will list:
- Slide geometry
- Slide friction
- Ramp angle
- Height and distance from ramp to pool
- Velocity when leaving the ramp
- Persons weight
MULTIPLE PERSPECTIVES
One could get other expert opinions but I usually serve as the expert. I generally point out that the problem can be broken into two phases of motion; one on the slide analyzed primarily using Work-Energy (especially if friction is assumed negligible), and one in the air analyzed as a particle ballistics problem. Various computer simulations are used to illustrate the effect of the different parameters including;Slide geometry, slide friction, ramp angle, height and distance from ramp to pool,velocity when leaving the ramp, persons weight, etc.
RESEARCH & REVISE
The students are provided with "lecture notes" covering Newton's 2nd law, Work-Energy, and 1-D Friction. They are also provided with numerous solved problems illustrating the covered concepts.
R&R Notes, Examples,& Simulations
TEST YOUR METTLE
They are given textbook problems in the area of work-energy of particles, ballistics of particles, and mixed work-energy/ballistics problems. A nice example of the mixed problem involves a carnival game where a spring operated plunger shots a nickel up an incline, into the air, and (hopefully) into a cup. A computer simulation of this problem is also provided. TYM Simulations
GO PUBLIC
Each group (typically 2 students) is required to write a report addressing the challenge that is turned in to the instrutor. They are also required to create a presentation of their investigation that they may have to give to the class. I generally only have time for a couple of groups per challenge to present. The following is a simple grade sheet that I use for the challenge investigation/solution.
Big Slip Grade Sheet
- Need estimates of:
- Slide Geometry (Δh)
- Slide friction
- Ramp angle
- Ballistic Geometry (Δx and Δy)
- Flight time Δt (I measured it to be 2 seconds)
- Need to calculate:
- Vo from Work-Energy
- Δx and Δy
- Need to compare assumptions and calculations and conclude.