Mechanical Engineering Projects B03

The purpose of these projects is to relate the concepts of linear systems to mechanical vibrations, especially resonance. 

The following problems incorporate differential equations, but they do not assume any knowledge of differential equations on your part, only basic calculus and physics. For each of these projects, turn in

a paper with a cover page listing

all group members,

the group number,

project number and title.

An introduction telling what you intend to discuss and show in this paper.

The body may be a mix of text and hand printing of equations. Please be careful to have correct spelling, complete sentences, neatly written mathematics and easily understandable concepts.

A conclusion summarizing what you learned and reported.

List any resources (books, web, people).

 

Project #1 What is resonance? due date: Monday, Nov 17.

Newton's Second Law governs mechanical systems in the following differential form:

F(t)   =    mx’’ + cx' + kx

            ...where x stands for displacement from rest and t is time. The external force is F(t), the mass is m, c is a damping coefficient, and the spring constant is k. The ‘ indicates a derivative with respect to time. Even if you don't understand this equation yet, accept that it is true!

We want to be able to discuss mechanical systems (machines, bridges, suspension components) in the terms of that equation. First we need to work on some basics:

1. In your study of mechanical systems, you will learn that mechanical systems store energy.  So what is an energy storing system? What types of energy do they store? Can you relate this to material properties? (i.e. stress, strain, modulus of elasticity)

2. Pick your favorite textbook (be it math, physics, or mechanical systems) and write down the definition of resonance. Now, how can you relate this definition to the energy storage in mechanical systems? What causes resonance?

In the simplest terms, in what physical forms do we often observe resonance? Feel free to refer to every day examples. Cars, washing machines, stereo systems, airplanes and bridges all contain mechanical subsystems which we experience regularly.

3. Grab that textbook again, and define natural frequency. How do you find the natural frequency? What is driving frequency? Using your book, comment on the interaction of natural frequency and driving frequency, 

4. What is damping? Is it a natural phenomenon, or applied by the system designer? Can an undamped system exist in the real world? 

5. Take another look at the equation above. Can you relate any of the four concepts you just worked on to the different terms of that equation? (bonus)

 

Questions? send email me:   goulet@wpi.edu

 

Project #2
Resonance in Spring-Mass Systems

 

Let's recap Project 1:

Most machine components are made from "elastic" materials. Some are more elastic than others, but even structural steel can be compressed and allowed to rebound slightly - just not as much as a rubber band (see Young's Modulus). This allows materials to store kinetic and potential energy. In short, nearly any system with moving components may be subject to resonance.

Resonance is a harmonic condition that can occur in systems that store energy which manifests itself in the form of unwanted and potentially damaging forces and displacements. Most often, the physical phenomenon we observe is vibration.

The machine elements described above possess one or more natural frequencies, which are a function of mass and material characteristics. If the machine experiences a driving force whose frequency is the same as the natural frequency of the system, resonance will occur. Due to the destructive nature of many resonance conditions, engineers often refer to the natural frequency as the critical frequency or critical speed.

So to simplify things, we could say that what we are really talking about is vibration, usually undesirable vibration. Vibration is just another term for oscillating physical displacement. 

Let's go back to Newton's Law....

F(t) = mx’’ +  cx’ +  kx

                            

                          ...and let's apply it to a spring-mass system. We'll look at two different cases, and explore the importance of damping. 

1. What are the two types of vibration? What term or terms of the above equation change from one type to another?

2. Assume F(t) = 0 (no external force), and c = 0 (no damping). Rewrite the equation. Explain it words what each term represents.

So if you grab the mass and pull it downward to a displacement x, and then let go, it will snap back higher than it started, and then recoil downward again, and back up again, etc. Will Newton's Law will tell us how exactly how that displacement x will affect a mass "m" and a spring "k", and after how many cycles the system will come to rest? Explain why the term "decay" is important to this type of cyclic motion.

3. In physics and mechanics, what is the Principle of Superposition? 

4. Assume F(t) = 0, but now there is damping.

How does the decay differ? Can we control the decay? (hint: This is a broad question. Don't be shy about finding examples!)

5. Grab a textbook if necessary, but broadly explain the value of damping in mechanical systems. Do the same principles apply to other types of motion, such as torsion?

6. A homogeneous example, assuming no external force, F(t):

                                                mx’’  +  kx = 0

                        Assuming no forcing function,show (i.e. do work!) that solutions to the governing equation  are  cos(w0t)  and  sin(w0t)    

                                        where w0 =sqrt(k/m)  (i.e. the natural frequency)

                        Show that this problem is linear by showing that for any constant K  that

                                    1)     Kcos(w0t)   and  Ksin(w0t)    are also solutions   

 

                                    2)      K1 cos(w0t)   +  K2 sin(w0t)      is also a solution

hint: think about what it means to be a “solution”…

(in the context of the course, this makes the set of all solutions a 2 dimensional vector space)


7. Look back at what you wrote for project 1, and answer this question again: Can an undamped system exist in the real world? If so, give an example. If not, why should we bother modeling it mathematically? (bonus)

Project #3

Real examples and applications ; theory behind it.

A. Look up the Tacoma Narrows Bridge. Do we know how for certain how it failed? Explain. How did the design of its replacement differ?

B. Download the following two videos and watch them carefully.

            video 1:  goalposts

            video 2:  an externally driven mass-spring system

What type of vibration are you seeing in each one, free or forced? Explain in detail.

What can you say about the forcing function in each case? 

Note:  if you find the videos are quite small, hit the button marked “undock player”. You

can then maximize the window.

C. Look again at the goalpost video... how could the motion of the goalposts be damped? Think about what machine components you would use, and how and where you would mount them. (hint: ask for help! You aren't expected to answer this like an expert vibration control engineer, but that doesn't mean you can't ask someone who has more experience than you! Cite all sources! Sketches are a plus!)

 

D.  You may have asked yourself the question "what does all this have to do with Linear Algebra?"  Your final job is to address exactly that.  If we start with the presumption that resonance is an important concept in engineering, then we will look at how resonance turns out to be a fascinating interaction between particular and homogeneous solutions under certain circumstances.  We do that as follows:

 

            1) we know  sin(w0t) and cos(w0t)  are homogeneous solutions to

 

                                    my''  +   k y  =    F0 cos(wt)            where  w0 = sqrt(k/m) and also assume w0 ≠ w

                                    y(0) = 0     y'(0) = 0                         ( initially at rest, at the origin)

 

            2)  verify that

                                    yp(t)  =   - F0    cos(wt)               is a particular solution   (substitute!!)

                                                  m( w2 – w02)

 

            3)  show that    y(t) =    F0      (cos(w0t)  -  cos(wt) )

                                                  m( w2 – w02)

                        satisfies the initial conditions   and is thus the general solution

 

            The phenomenon of resonance occurs when  w approaches w0.  First plot this solution with Maple or Matlab using the following  sample data:

                                    F0 = 1               m = 1          w0 = 2   and w = 2.1

                        and graph the solution for t between 0 and 20 Pi

Next, if w is exactly equal to w0 then we would be dividing by 0, which is meaningless.  One may apply LHopital's rule and get the solution under this circumstance (no, you don't have to do it!).  It turns out to be

 

                  y(t) = t sin(w0t)/(2w0)

plot this for the same time frame and describe how you see the amplitude behaving.  Relate this to your prior understanding of  resonance. Ask questions; come by and discuss it!!