Introduction to the Methodology of Mechanics

Many years ago when I was a PhD student, being short of cash (as anything changed?), I undertook to metricate Schaum’s Strength of Materials, authored by William Nash.  The Schaum’s series are tutorial books and contain hundreds of solved examples.  Metricating a book such as this is no small task, as the problems all have to be reworked with sensible (integer) values of variables.  Also there were some 750 problems for which no solutions existed, and so they had to be worked out from scratch. In all I had to do 1750 mechanics problems in a year, i.e. about 35 per week.  There is no better training in mechanics if you can afford the time!!!

In going through these problems it eventually dawned upon me that I was using a standard process to solve the problems.  This process is actually implicit in the original text, and is also there in most mechanics textbooks, although again hidden and never explicitly stated.  I therefore compiled the method I was using and called it a ""methodology, which is shown below.

The importance of the methodology is that it is a way of thinking about problem solving at a Meta (Above)  level; it is about thinking how to think.  This to me is the very essence of education.  Students shoul  be taught at the Meta level – how to obtain methodologies to solve whole classes of problems - rather than endless specific examples in the hope that some understanding may emerge.  This latter type of teaching is what I experienced, and what most students still experience.

I have used this methodology for many years, at all levels from Foundation Year to the Masters year, and all the student groups exposed to it are highly appreciative.  It gives them a clear framework within which they can apply the methodology to every problem, and they feel ‘safe’.  When I have introduced the methodology to students who have been taught by someone else in earlier years, they frequently ask the question “why weren’t we taught this at the start?”.  This is the impetus to put the methodology on my web page.

MECHANICS

Mechanics      - Science (study) of MOTION of objects

                        - motion needs ENERGY

STEP 1: Choose axis system (Generally but not restrictively 2D)

- Rectangular Co-ordinates (Cartesian)

- Normal &Tangential (N&T) Co-ordinates

STEP 2: Overall Equilibrium

ΣF_v = ΣF_h = ΣM = 0

Equilibrium Satisfied            Equilibrium Not Satisfied

STATIC                                  DYNAMIC

STEP 3: Method of Sections / Free Body Analysis

Free Body Equilibrium:

        ΣF_v = ΣF_h = ΣM = 0             STATIC               

        ΣF_v = ΣF_h = ΣM  not = 0      DYNAMIC

                                                                                         = MASS x ACCⁿ

-        “Equations of Motion”

-        “Degrees of Freedom”

STEP 4: Determinate or Indeterminate??

Three equations of equilibrium

< 3 unknowns - determinate

> 3 unknowns – indeterminate

Statically Determinate                  Dynamically Determinate

         SOLVE                                                    SOLVE

STEP 5: Indeterminate Systems

Need more Independent equations – COMPATIBILITY

For a structure to move in a Compatible manner the movement of adjacent parts should be the same.

  Statically Indeterminate         Dynamically Indeterminate

Displacement Compatibility       Kinematic Compatibility

            Δ₁=Δ₂                        V₁=V₂

            SOLVE                                   SOLVE

NOTE:  This ^“Methodology” or “Paradigm” will specify all mechanics problems. Non-linear problems require a further Linearisation step and a Minimum Energy criterion.

The "Free Bodies" are FINITE therefore when combined with Matrix Methods and solved on a computer this methodology is known as FINITE ELEMENT ANALYSIS