UTPA STEM/CBI Courses/Engineering Analysis/Second Order Differential Equations with Applications in Engineering

Course Title: Engineering Analysis II MECE 3450

Lecture Topic: Second Order Differential Equations with Applications in Engineering

Instructor: Dumitru Caruntu

Institution: UTPA

Backwards Design

Course Objectives

• Primary Objectives- By the next class period students will be able to:
  • Model mathematically mechanical engineering (ME) challenge problems
  • Solve the differential equations and interpret the results

• Sub Objectives- The objectives will require that students be able to:
  • Integrate second order differential equations, homogeneous and nonhomogeneous
  • Grasp the physical meaning of the mathematical concepts

• Difficulties- Students may have difficulty:
  • Connect the real world problems to mathematical concepts - modeling
  • Understand the meaning or what systems ODEs model – see physical meaning
  • Solve and discuss the results in terms of engineering concepts
  • Checking their work
• Real-World Contexts- There are many ways that students can use this material in the real-world, such as:
  • Very specific mechanical engineering design problems, from simple structural elements to very complex systems and structures
  • Make engineering decisions based on modeling, simulations, and predictions

Model of Knowledge

• Concept Map
  • Homogeneous 2nd Order Ordinary Differential Equations (ODE)
    • Linear, constant coefficients, homogeneous
    • Characteristic equation
    • General solution
    • ME applications – free oscillations of mass-spring system
  • Nonhomogeneous 2nd Order ODE
    • General solution
    • Method of undetermined coefficients
    • Method of variation of parameters
    • ME applications – Forced oscillations in mechanical systems, and electrical circuits

• Content Priorities
  • Enduring Understanding
    • Engineering meaning of first and second derivatives and their coefficients
    • Finding the general solution of 2nd order ODEs by using characteristic equation
    • Finding particular solutions of nonhomogeneous ODEs
  • Important to Do and Know
    • Write the mathematical model of a real ME problem
    • Know the meaning of nonhomogeneous terms in the ODEs
    • The importance of initial conditions for ODE and their meaning for ME problems
  • Worth Being Familiar with
    • Mathematical modeling of electromechanical systems
    • Mechanical vibrations of systems and structures
    • Dynamics of systems

Assessment of Learning

• Formative Assessment
  • In Class
    • Find the general solution of a differential equation describing a mass-spring system.
    • Find the solution of an initial value problem associated to this equation. Specific initial values are given.
    • Discuss engineering concepts.
  • Homework (individual)
    • Write a report in which possible areas of research for solving a specific engineering problem are identified.
    • Specifically indicate the what engineering concepts have to be used in the modeling.
• Summative Assessment
  • Decide on the relevant areas of research for the given challenge.
  • Make recommendations based on the assessment.
  • Justify the recommendations.

Legacy Cycle

OBJECTIVE

By the next class period, students will be able to:

• Link real world problems to engineering and mathematical concepts.
• Understand the meaning of concepts such as acceleration, mass, damping, elastic constant, resonance, differential equations, and initial conditions.
• Understand the importance of decision making based on technical assessments.

The objectives will require that students be able to:

• Conduct a thorough review of the problem at hand.
• Model real world problems using technical concepts.
• Formulate engineering assessments based on modeling, simulation, and experimental research.

THE CHALLENGE “… the problem of subway induced vibrations on line 4 of Beijing metro, which is currently under construction and is planned to pass in close proximity of the Physics Laboratory of Beijing University. The laboratory has a lot of equipment that is very sensitive to traffic induced vibrations and future operation of metro line 4 is a matter of concern. Hence, it is important to study the influence of subway induced vibrations inside the laboratory and to propose a viable solution to mitigate the vibrations. ….” S. Gupta, W.F. Liu, G. Degrande, G. Lombaert, W.N. Liu, Prediction of vibrations induced by underground railway traffic in Beijing, Journal of Sound And Vibration 310 (2008) 608-630. The challenge consists of you making recommendations to mitigate the induced vibrations or reduce them to an acceptable level. The Board of Directors of the company will spend hundreds of millions of dollars for further research based on your recommendation. Build a simple model (ODE) of the vibration problem, produce or show simulations, make your recommendations, and justify them.

GENERATE IDEAS

• Students should discuss what criteria should be used to compare different options: When it comes to vibrations is frequency important? What about damping? Is mass going to play a significant role? How about elasticity of structures? Are the soil and concrete structures between subway and the lab important? Is this a mechanical or civil engineering problem?
• How do engineers model the motion? What are the engineering concepts in modeling vibrations? What are the constants that describe the characteristics of such elements? Are the realistic values of these constants important?
• In general, these discussions can be summarized into a conclusion that engineering modeling of vibrations is based on 2nd order ODEs to include concepts such as mass, damping, elasticity, acceleration, and velocity. Students can discuss what factors they want to choose for further investigation.

MULTIPLE PERSPECTIVES

• By this point, students should have generated a list of relevant ideas for this problem.
• Students should discuss data is needed and what sources they need to refer to obtain these data.
• Websites are to be visited (e.g. journal article, magazine articles, success stories, other engineering sources)
• After an initial discussion led by the students, instructor can provide some printed materials. Students should have further discussions.
• The discussions should identify most important data, and modeling tools to solve the problem.

RESEARCH & REVISE

• By this point, students should have collected all relevant data to further their research.
• Students are to analyze the information at hand in order to write a report in which they suggest the directions of research.
• Their recommendations have to be supported by evidence, and justified (why these recommendations).
• A number of students present their findings. All students will go back to revise their recommendations, showing what is different from the initial report, and what they learned.

TEST YOUR METTLE

Students will take a quiz in which a problem modeling the problem at hand has to be numerically solved. They will indicate based on this problem what are the directions of research for the given challenge based on their experience at that point. They need to know to solve systems of ODEs.

GO PUBLIC

Students will have a power point presentation, in which they simulate a technical presentation for the Board of Directors that have to decide how to direct $300 Million for research in order to solve the problem of induced vibrations by the subway system. This work can be eventually showcased on the website.

Pre-Lesson Quiz

Various material can be tested here, depending on time constraints.

Test Your Mettle Quiz

Various material can be tested here, depending on time constraints.