Engineering science

1. Chapter 2- Dynamic Engineering Systems
2.1 Uniform acceleration
• linear and angular acceleration
• Newton’s laws of motion
• mass, moment of inertia and radius of gyration of rotating components
• combined linear and angular motion
• effects of friction
2.2 Energy transfer
• gravitational potential energy
• linear and angular kinetic energy
• strain energy
• principle of conservation of energy
• work-energy transfer in systems with combine linear and angular motion
• effects of impact loading
2.3 Oscillating mechanical systems
• simple harmonic motion
• linear and transverse systems;
• qualitative description of the effects of forcing and damping

2. Review Hook’s law
relaxed position
FX = 0
x
x=0
relaxed position
FX = -kx > 0
x
x<0
x=0

3. Simple Harmonic Motion (SHM)
• Definition:
The motion that occurs when an
object is accelerated towards a mid-
point. The size of the acceleration is
dependent upon the distance of the
object from the mid-point.
Very common type of motion
eg. sea waves, pendulums, springs

4. Simple harmonic motion
X=0
X=A; v=0; a=-amax
X=0; v=-vmax; a=0 F=-kx
F=ma
X=-A; v=0; a=amax ma = -kx
a= -kx/m
X=0; v=vmax; a=-0 x = -ma/k
X=A; v=0; a=-amax
X=-A X=A

5. Properties

9. SHM plotted on a paper

10. Basic definitions

11. Transverse and longitudinal waves

12. Relation between SHM & Circular motion

14. Simple Harmonic Motion
x(t) = [A]cos(ωt)
v(t) = -[Aω]sin(ωt)
a(t) = -[Aω2]cos(ωt)
Maximum value
xmax = A Angular frequency
vmax = Aω ω = 2πf = 2π/T
amax = Aω2

15. Pendulum
• A mass, called a bob,
suspended from a fixed point
so that it can swing in an arc
determined by its momentum
and the force of gravity.
• The length of a pendulum is
the distance from the point of
suspension to the center of -Lmg sinθ =Iα
gravity of the bob. α = -(Lmg/I) θ and
α= - ω 2 θ
Lmg
∴ ω=
I

16. • A clock has a pendulum that performs one
full swing every 1.0 sec. The object at the
end of the string weights 10.0 N. What is
the length of the pendulum?

19. Damped oscillations
• Only ideal systems oscillate indefinitely
• In real systems, friction retards the motion
• Friction reduces the total energy of the system and the
oscillation is said to be damped
• Example: Shock absorber:
With a low viscosity fluid, the
vibrating motion is preserved,
but the amplitude of vibration
decreases: This is known as
underdamped oscillation

20. More Types of Damping
• With a higher viscosity, the object returns
rapidly to equilibrium after it is released and
does not oscillate
– The system is said to be critically damped
• With an even higher viscosity, the piston
returns to equilibrium without passing through
the equilibrium position, but the time required
is longer
– This is said to be over damped
Plot a : under damped
Plot b : critically damped
Plot c : over damped

21. Combination of springs

22. Questions
1. A mass spring system has m=5kg and k=2000N/m,
A=50cm. Find the velocity and acceleration when
x=30cm.
2. A mass spring system has m=5kg and k=1600N/m,
A=2m. Find
1. Total energy
2. Kinetic energy to potential energy ratio when x=1m
3. A k=800N/m spring is stretched by 200N force on a
horizontal surface with a 4kg object connected to the one
end of it and released. Find the
1. Amplitude
2. Maximum acceleration and velocity
3. Find the velocity when x=30cm and x=-30cm

23. • An object performs simple harmonic motion, its position
given by x(t)=20cos(31.4t) in cm. Find the
– Amplitude
– Frequency and period of oscillation
• An object performs SHM, its position is given by
x(t)=50cos(20πt) in cm. Find the
– Maximum value of displacement, velocity and acceleration
– Number of oscillations in 5 sec.
• An object makes 10 complete oscillations in 5 seconds
between two points 40cm apart.
– Write an expressions for displacement and velocity
– Find the position and speed at t=0.125s
• An oscillations having an amplitude of 10cm and a period
of 4s starts from Xmax at t=0
– Find x at t=1s
– Find x at t=0.5s