Engineering Science 3 - MECH 226 - Home Page

1. Introduction.
These notes are designed to help you to learn how to solve a range of problems in engineering science. For the majority of problems in statics it can be assumed that superposition applies, this requires that only small strains and small deflections are occurring. In such cases the various loads and moments acting on a body can be separated and the effects of each calculated as though the one being considered is the only force or moment acting. The total resulting stresses (and strains) can then be found by summing the individual contributions of all the forces and moments.

In the early days of engineering, machinery was slow moving and the main forces present were those due to gravity acting on the machine masses. However a key aspect of engineering progress has been the requirement for machinery to operate at ever increasing speeds. It is now very common for dynamic (or inertia) load effects to dominate and gravity forces are commonly insignificant.

A systematic approach is important - the procedure below is applicable to many problems.

The consideration given to the effects of friction is often different in static and dynamic problems. Frequently in statics problems friction plays a vital role - eg the forces acting on a ladder leaning against a wall - and in these cases, friction must be considered. However in dynamics problems it is common for friction effects to be quite small and consequently ignored. This is partly due to the fact that in most dynamic systems the efficiency has to be quite high, e.g. better than 95% for a good gearbox, so the error that occurs in ignoring friction is normally quite small and it greatly simplifies the initial problem solution. Where a detailed design is required for a critical high powered system, then the friction effects would be considered, mainly because a great deal of heat may need to be dissipated.

2. Units
It is normally easier to convert all values into SI units - m, kg, s, N, radians etc. - prior to starting calculations.

3. For all problems it is vital that you understand the configuration.
The following is a useful first step for most problems:

  • draw a layout sketch showing all the features and all the external forces and moments that may be acting
  • If they are not already included, put your own xyz axes on the diagram, this will simplify your explanation.
  • For 'determinant' systems, the values of the external forces and moments may be evaluated by summing forces and taking moments. For statically indeterminant systems, where deflections have to be considered, a more complex approach making use of the stiffnesses and deflections of components must be used

  • 4. For statics problems it is usual to resolve forces and moments so their components are parallel or perpendicular to the axes of the component parts (and xyz axes).
    Statics problems that are determinant can now be solved by considering the effects of the components of the forces and moments and summing the individual results - the problems considered in this module involve only small strains and small displacements so superposition can be used.

    5. For dynamics problems:

  • Draw separate free diagrams for every individual part in the system. All forces and moments that may be present must be shown (if some of them work out to be zero, that is ok)
  • The system equations for each component can then be written down (there may be up to 6 equations - 3 translation and 3 rotation)
  • When solving any problem that involves translation and rotation it is much better to consider the translation of the centre of mass and the rotation about the centre of mass. Attempting to solve using rotation about some other point will often lead to errors (unless it is rotation about a fixed axis).
  • Write down any equations of restraint. (as an example, for a road wheel, this will link the rotation of the wheel to its translation)
  • It should now be possible to solve the problem by working from one 'end' to another, usually substituting for variables as you go, until you are have an equation in only 1 variable, which can then be solved. You then work back, substituting the the variables as they are evaluated.
  • At this point you should have values of forces and moments that are acting on each component. It is then possible to evaluate the stresses and their distribution in each component which result from the external loads and the self mass effects.
  • Think! Do the results look reasonable?

  • 6. Energy Methods
  • An approach using energy methods may be useful in some statics and in some dynamics problems, particularly those where several components are linked together.

  • 7. Vibration Problems
    The following topics are of interest:
  • Free vibration
  • Steady state response of a system subject sinusoidal excitation - specifically:
  • i) Isolating a delicate device from external vibration
  • ii) Preventing vibration generated by a machine reaching other adjacent equipment
  • Transient response of a system to a shock load
  • NB: It will not be possible in this module to cover all these topics.

    Links to Notes for this Module: Not all of these topics will be covered, some are included for completeness and provide supplementary information. Topics marked '*' are of primary importance.

    Module Course Work outline information. *
    Thin cylinder experiment

  • Module data sheet *
  • Module nomenclature *
  • Statics - Stress and Strain

  • Strain in 2 dimensions: Part 1; Part 2; Part 3 *
  • Stresses and strains in thin cylinders *
  • Thin cylinder tutorial questions
  • Thin cylinder tutorial questions solns. Question 1a Q 1b; Q 2; Q 3a; Q 3b; Q 4;
  • Deflections and slopes in beams Part 1; Part 2 *
  • Beams tutorial questions
  • Beams tutorial questions solns. Question 1a; Q 1b; Q 2a; Q 2b; Q 3a; Q 3b; Q 4a; Q 4b; Q 4c; Q 5a; Q 5b;
  • Stress in 2 dimensions: Part 1; Part 2; Part 3 - worked eg.; Part 4 - worked eg.; Part 5 *
  • Two dimensional stress and strain tutorial questions *
  • Solution to Tutorial Question 1a; Q 1b; Q 1c; Q 2a; Q 2b; Q 2c; Q 3a; Q 3b; Q 4a; Q 4b; Q 5a; Q 5b; *
  • Dynamics

  • Moment of inertia *
  • Linear and rotary motion *
  • Acceleration of geared systems *
  • Vibrations
  • Free undamped vibration notes and examples, 47 kB .pdf file *
  • Applet to simulate free vibration of a second order system *
  • Rotary oscillations *
  • Compound pendulum *
  • Free damped vibration notes and examples, 73 kB .pdf file *
  • Tutorial questions vibration tutorial questions and solutions *
  • Forced undamped vibration notes and examples, 63 kB .pdf file
  • Additional questions and worked examples.
    Beams 1
    Beams 2
    Stress and strain in two dimensions.
    Solutions for Beams 2
    Solutions for 2D stress and strain questions:
    1.a, 1.a circle, 1.b, 1.b circle, 1.c, 1.c circle, 2.a, 2.b, 2 circle, 3, 3 circle

    Previous Exam Questions - from June 2003:

    Questions 1 and 2 Q1 soln part 1 and Q1 soln part 2 and Q1 soln part 3
      Q2 soln part 1 and Q2 soln part 2 and Q2 soln part 3
    Questions 3 and 4 Q3 soln part 1 and Q3 soln part 2
      Q4 soln part 1 and Q4 soln part 2
    Questions 5 Q5 soln part 1 and Q5 soln part 2

    1. 'Mechanics and Materials for Design', by N H Cook, McGraw-Hill, Int. student Ed., 1985, ISBN: 0-07-Y66157-X.
    2. 'Mechanics of Materials', Gere and Timoshenko, Brooks/Cole, 2nd ed. 1984, ISBN: 0-534-03099-8.

    There are several books with titles 'Mechanics of Materials' or 'Strength of Materials' or similar which are suitable for the 'statics' part of the course contents.

    3. 'Principles of Engineering Mechanics', H R Harrison and T Nettleton, Arnold, 1978, ISBN: 0-7131-3378-3 (deals with dynamics rather than statics).

    There are several American textbooks on Dynamics, but they tend to have a lot more than is needed for this course on dynamics and barely enough on vibration:

    4. 'Mechanics for Engineers - Dynamics', F P Beer and E R Johnson, McGraw-Hill, Int. Ed., 4th Ed. 1987, ISBN: 0-07-100135-2.

    Return to Index of Online Documentation.

    David J Grieve, 26th November 2003.