Engineering Science
By
Mr. Isaac Kuma Yeboah.
What is Engineering Science?
• Engineering science is a broad discipline that
encompasses many different scientific principles and
associated mathematics that underlie engineering.
• Engineering science integrates engineering, biological,
chemical, mathematical, and physical sciences with
the arts, humanities, social sciences, and the
professions to tackle the most demanding challenges
and advance the well-being of global society.
What do Engineering Scientists do?
• The unique knowledge and interdisciplinary skill set of
engineering scientists allows
them to merge
multidisciplinary resources to propose and develop
innovative, enduring solutions and transform the latest
scientific discoveries into enabling new technologies.
• Where do engineering scientists work? Because of the
breath of their training, engineering scientists are well
prepared to lead national and international interdisciplinary
teams in a diverse array of science and engineering
endeavours, including the legal profession, medicine,
business, and government service.
SI Units
• Units: The system of units used in engineering and
science is the Systeme Internationale d’ Unites
(International System of Units), usually abbreviated to
SI units and is based on the metric system.
• The SI unite was introduce in 1960 and is now
adopted by the majority of countries as the official
system of measurement.
• Prefixes: SI units may be made larger or smaller by
using prefixes that denote multiplication or division by
a particular amount.
Force, Mass and Acceleration
• When an object is pushed or pulled, a force is applied to the object, this
force is measured in Newtons (N). The effects of pushing or pulling an
object are:
i. To cause a change in the motion of the object .
ii. To cause a change in the shape of the object.
• If a change occurs in the motion of the object, that is its velocity changes
from u to v, then the object accelerate. Thus, it follows that acceleration
results from a force being applied to an object.
• If a force is applied to an object and it does not move, then the object
change in shape, that is, deformation of the object takes place.
• Usually the change in shape is so small that it cannot be detected by just
watching the object, however when very sensitive measuring instrument s
are used very small changes in dimensions can be detected.
Newton’s Laws of Motion
• To make a stationary object move or to change the direction in which
the object is moving requires a force to be applied externally to the
object.
• This concept is known as Newton’s first law of motion and may be
stated as “An object remains in a state of rest , or in continues in a
state of uniform motion in a straight line, unless it is acted on by an
externally applied force”.
• Since a force is necessary to produce a change of motion, an object
must have some resistance to a change in its motion.
• The resistance to a change in motion is called the inertia of an object
and the amount of inertia depends on the mass of the object.
Reference Frame
• Newtonian (Inertial) frame of Reference have the following
characteristics:
1. Unaccelerated Cartesian frame, that is the origin referenced to
inertial (non-moving ) frame.
2. Right-hand rule , that is origin can translate at constant linear
velocity.
3. Frame cannot rotate with respect to inertial origin.
Position in 3 dimensions of a non-moving
frame.
Velocity and Momentum.
Newton’s Laws of Motion: Dynamics of a
Particle
Translation in a Linear motion.
Centripetal Acceleration
• When an object moves in a circular path at constant speed, its
direction of motion is continually changing and hence its velocity
(which depends on both magnitude and direction) is also continually
changing.
• Since acceleration is the change in velocity divided by time taken,
the object has an acceleration.
• Let the object with a constant angular velocity of ω and a tangential
velocity of magnitude v and let the change of velocity for a small
change of angle of θ = ωt be V.
• Then, v2 – v1 = V.
Centripetal Acceleration
A Guide to Solving Physics Problems.
• When faced with a new question, we employ a strategy to break the
problem down into a series of 5 steps to digest and analyse it.
• These 5 key steps will help you develop a logical, structured method
and universal approach.
1. Keywords
2. Diagram
3. Concepts
4. Symbols
5. Dimensions and Numbers
5 Key steps to problem solving
• Step 1: Keywords are there words in the question that contain
additional information about the problem? Frictionless, light, uniform.
• Highlight these words so that they stand out-they will help to simplify
the solution and allow us to neglect concept that we don’t need to
consider.
• Step 2: Draw a diagram, the action of drawing a diagram helps us to
digest and summarise the information in a question.
• Drawing a diagram will save time and effort later and is the key to
finding a solution.
• When drawing the diagram, label the quantities that are given with
symbols (e.g. u for a velocity, d for a displacement); staying in symbols
rather than using numbers is vital.
5 Key steps to problem solving
• Step 3: Key concepts and mathematics, fluency with mathematical
rearrangements is essential but we need to be sure that we are logical
with our approach and consider the physical concepts that we need
before throwing algebra at the problem.
• Identify which concepts are relevant to the problem, write down the
relevant physical principles and equations that might be useful
particularly if they connect the quantities that are given in the
question.
• Step 4: Stay in symbols even if the question gives numerical values,
represent each of them with a symbol.
• This may appear to overcomplicate the problem, but it really helps
when checking the solution or trying to find a mistake.
5 Key steps to problem solving
• Step 5: Check dimensions, then put in number, check if reasonable before putting
numbers into an algebraic expression, experienced physicists check whether their
answer has the correct dimensions.
• If we are trying to find a quantity of time, each term in our expression must have
dimensions of only time no length, or mass or charge.
Dynamics: Pop-up Toy
Force and Work Done
• Using a force for the pop-up toy is more challenging because the
force due to the spring is not constant throughout the motion.
However, analysing the problem through forces should give us the
same answer.
• How does the force exerted by the spring on the mass cause the toy
leap off the ground?
• Solution: Hooke’s Law tells us that the magnitude of the force, F,
needed to compress a spring by an amount x, is given by F = kx.
• To compress the spring by an additional small amount δx we need to
do work on the spring.
• The amount of work done is δW = Fδx as δx become very small (tends
to zero), δW tends to dW.
Force and Work Done
• The work done to compress the spring from stage 1 to stage 2 can be
find by integrating with respect to dx between the limits of the
compression at stage 1 to the compression at stage 2.
Force and Work Done
Step 5: Dimensions and Numbers
Statics Prism
• A prism has a cross section that is an isosceles triangle. It has a
unique angle of 30, and a mass of m = 100g. You wish to lift it by
touching the upper two faces only. If the coefficient of friction
between the prisms surface and your skin is µ = 0.400, what is the
minimum normal force you need to apply to each face in order to
support the prism?
• Solution: To add the frictional force on the prism we think about the
direction that the prism wants to fall.
• We know that friction will oppose this motion, using Newton’s third
law, the frictional force on our fingers must be equal in magnitude but
opposite in direction to that on the prism.
Centre of Gravity and Equilibrium
• The centre of gravity of an object is a point where the resultant
gravitational force acting on the body may be taken to act. For objects
of uniform thickness lying in a horizontal plane, the centre of gravity
is vertically in line with the point of balance of the object.
• A thin uniform rod the point of balance and hence the centre of
gravity is halfway along the rod as shown in figure (a).
• A thin flat sheet of a material of uniform thickness is a lamina and the
centre of gravity of a rectangular lamina lies at the point of
intersection of its diagonals, as shown in figure (b).
• The centre of gravity of a circular lamina is at the centre of the circle,
as shown in figure ( c ).
Equilibrium
• An object is in equilibrium when the forces acting on the object are such
that there is no tendency for the object to move. The state of equilibrium
of an object can be divided into three groups.
I. If an object is in stable equilibrium and it is slightly disturbed by pushing
or pulling (i.e. a disturbing force is applied), the centre of gravity is raised
and when the disturbing force is removed, the object returns to its
original position. Thus a ball bearing in a hemispherical cup is in stable
equilibrium as shown in figure (a).
II. An object is in unstable equilibrium if, when a disturbing force is applied,
the centre of gravity is lowered and the object moves away from its
original position. Thus, a ball bearing balanced on top of a hemispherical
cup is in unstable equilibrium, as shown in figure (b).
III. When an object in neutral equilibrium has a disturbing force applied, the
centre of gravity remains at the same height and the object does not
move when the disturbing force is removed. Thus, a ball bearing on a flat
horizontal surface is in neutral equilibrium, as shown in figure ( c ).
The Resultant of Two Coplanar Forces.
• When forces are all acting in the same plane, they are called coplanar.
For two forces acting at a point, there are three possibilities, these
are;
a) For forces acting in the same direction and having the same line of
action, the angle force having the same effect as both of the force,
called the resultant force or just the resultant, is the arithmetic sum
of the separate forces. Forces of F1 and F2 acting at point P, as
shown in figures below have exactly the same effect on point P as
force F.
Triangle of Forces Method
• A simple procedure for the triangle of forces method of
vector addition is as follows:
I. Draw a vector representing one of the forces, using an
appropriate scale and in the direction of its line of action.
II. From the nose of this vector and using the same scale,
draw a vector representing the second force in the
direction of its line of action.
III. The resultant vector is represented in both magnitude and
direction by the vector drawn from the tail of the first
vector to the nose of the second vector.
The Parallelogram of Forces Method
• A simple procedure for the parallelogram of forces method of vector
addition is as follows:
I. Draw a vector representing one of the forces, using an appropriate
scale and in the direction of its line of action.
II. Form the tail of this vector and using the same scale draw a vector
representing the second force in the direction of its line of action.
III. Complete the parallelogram using the two vectors drawn in (i) and
(ii) as two sides of the parallelogram.
IV. The resultant force is represented in both magnitude and direction
by the vector corresponding to the diagonal of the parallelogram
drawn from the tail of the vectors in (i) and (ii).
Resultant of Coplanar Forces by Calculation
• An alternative to the graphical methods of determining the resultant
of two coplanar forces is by calculation.
• This can be achieved by trigonometry using the cosine rule and the
sine rule, or by resolution of forces.
Resultant of more than Two Coplanar Forces
• For the three coplanar forces F1, F2 and F3 acting at a point as shown in
figure below, the vector diagram is drawn using the nose-to tail method.
The procedure is:
I. Draw oa to scale to represent force F1 in both magnitude and direction.
II. From the nose of oa, draw ab to represent force F2.
III. From the nose of ab, draw bc to represent force F3.
IV. The resultant vector is given by length oc in the figure. The direction of
resultant oc is from where we started. i.e. point o, to where we finished,
i.e. point c.
• The resultant force, given by oc, has the same effect on the point as forces
F1, F2 and F3 have when acting together. The resulting vector diagram of
figure is called the polygon of forces.
Coplanar Forces in Equilibrium
Resolution of Forces
• A vector quantity may be expressed in terms of its horizontal and
vertical components.
Summary
The Concept of Differentiation.
• Suppose we wished to measure the rate of propane gas flow through a
hose to a torch.
• Flowmeters appropriate for measuring low flow rates of any gas are
typically very expensive, making it impractical to directly measure the flow
rate of propane fuel gas consumed by this torch at any given moment.
• We could, however, indirectly measure the flow rate of propane by placing
the tank on a scale where its mass (m) could be monitored over time.
• By taking measurements of mass over short time periods (Δt), we could
calculate the corresponding differences in mass (Δm), then calculate the
ratio of mass lost over time to calculate average mass flow rate
Graphed as a function over time.
Graphed as a function over time.
• We should recall from basic geometry that the slope of a line or line
segment is defined as its rise (vertical height) divided by it run
(horizontal width).
• Thus, the average mass flow rate calculated within each time period
may be represented as the pitch (slope) of the line segments
connecting dots, since mass flow rate is defined as a change in mass
per (divided by) change in time.
• Periods of high propane flow (large flame from the torch) show up on
the graph as steeply-pitched line segments.
• Periods of no propane flow reveal themselves as flat portions on the
graph (no rise or fall over time).
• Flow is calculated just the same as before, a quotient of mass and time
difference, except here the differences are infinitesimal in magnitude.
• The unit of flow measurement reflects this process of division, just as
before, with mass flow rate expressed in unites of kilograms per minute.
• The rate of flow is graphically represented by the slope of the graph,
steeply-sloped points on the graph represent moments of high flow rate,
while shallow-sloped point on the graph represent moments of low flow
rate.
• Such a ration of differential quantities is called a derivative in calculus.
Derivatives-especially time-based derivatives such as flow rate find many
applications in instrumentation as well as the general sciences.
• Some of the most common time – based derivative functions include the
relationships between position (x), velocity (v), and acceleration (a).
The Engineering realm concept of gain.
• Gain is generally defined as the ratio of output change to input
change.
• An electronic amplifier with an input signal of 2 volts (peak-to-peak)
and an output signal of 8.6 volts (peak-to-peak), would be said to
have a gain of 4.3, since the change in output measured in peak-topeak volts is 4.3 times larger than the corresponding change in input
voltage.
• This gain may be expressed as a quotient of difference, or it may be
expressed as a derivative instead.
The concept of Integration.
• Suppose we wished to measure the consumption of propane over
time for a large propane storage tank supplying a building with
heating fuel, because the tank lacked a level indicator to show how
much fuel was left at any given time.
• The flow rate is sufficiently large, and the task sufficiently important,
to justify the installation of a mass flowmeter, which registers flow
rate at an indicator inside the building.
• By measuring true mass flow rate, it should be possible to indirectly
measure how much propane has been used at any time following the
most recent filling of the tank.
• In order to accurately calculate the amount of propane mass
consumed by the building over time, we must treat each period of
constant flow a its own propane quantity, calculating the mass lost
during each period.
• Then summing those mass differences to arrive at a total mass loss
for the entire time interval covered by the graph.
• Since we know the difference (loss) in mass over a time period is
equal to the average flow rate for that period multiplied by the
period’s duration (Δm = WΔt).
Free-Body Diagrams
• A free-body diagram is a sketch of the system to which you will apply the
laws of mechanics, and all the non-negligible external forces and couples
which act on it.
• The diagram indicates what material is in the system, what is and what is
not, known about the forces.
• Generally there is a force or moment component associated with any
connection that causes or prevents a motion.
• Conversely, there is no force or moment component associated with
motions that are freely allowed.
• Mechanics reasoning entirely rests on free body diagrams and may student
errors in problem solving are due to problems with their free body
diagrams.
A free-body diagram (FBD)
• In mechanics a system is often called a body and when it is conceptually
isolated it is free, as in free from its surroundings.
• A free-body diagram (FBD) is a sketch of an isolated system and the
external forces which act on it.
• Closed systems in thermodynamics; A system usually a fluid, is isolated
with rigid walls that allow no heat, motion or material to pass.
• Then, bit by bit, as the subject is developed, the response of the system to
certain interactions across the boundaries is allowed.
• The gas expanding in a refrigerator follows the same rules of heat-flow and
work as when it was expanded in its isolated container.
• The laws of mechanics are applied using the forces shown on a free body
diagram and not using any other forces.
Some basic mechanics assumptions
• The free-body is a closed system; As in elementary thermodynamics
we will only be concerned with so-called closed systems.
• The mechanics of open systems, where material crosses the system
boundaries, is important in fluid mechanics.
• Such open fluid systems are first seen in some elementary dynamics
problems like rockets, where material is allowed to cross the system
boundaries.
• The equations governing these open systems are deduced from
careful application of the more fundamental governing mechanics
equations of closed systems.
How to draw a free-body diagram
Kinematics of Wheeled Mobile Robots
• Kinematic model describes geometric relationships that are present in the
system.
• The relationship between input (control) parameters and behavior of a
system velocities and is presented by a set of differential first – order
equations.
• Dynamic models describe a system motion when forces are applied to the
system.
• This models include physics of motion where force, energies, system mass,
inertia, and velocity parameters are used.
• Descriptions of dynamic models are given by differential equations of the
second order.
• In wheeled mobile robotics, usually kinematic models are sufficient to
design locomotion strategies, while for the other systems such as robots in
space, air, or walking robots, dynamic modelling is also needed.
Internal Kinematics.
• Internal kinematics explains the relation between system internal
variables e.g., wheel rotation and robot motion.
Differential Drive
• Differential drive is a very simple driving mechanism that is quite often
used in practice, especially for smaller mobile robots.
• Robots with this drive usually have one or more castor wheels to support
the vehicle and prevent tilting, both main wheels are placed on a common
axis.
• The velocity of each wheel is controlled by a separate motor.
• The input (control) variables are the velocity of the right wheel Vr(t) and
the velocity of the left wheel Vl(t).
• The meaning of other variables are as follows: r is the wheel radius; L is
the distance between the vehicle centre (middle point between the
wheels) and (ICR point).
• In each instance of time both wheels have the same angular velocity w(t)
around the ICR.
Forward and Inverse Kinematics