Mechanics & Vibrations Course Content
Module Title
Dynamics and Vibrations; ME2002
Module Credit
DynA
10
School(s) and Subject Group(s)
Staff Responsible
SEAS- Mechanical & Electrical Engineering
Dr J E T Penny & Prof W T Norris
Learning Outcomes
At the c
B
1
SPECIAL SIGNALS
This has applications in control systems, electronics, communications, mechanical systems,
structures, biophysics, economics, etc.
Signal a stimulus, force, influence, displacement &c, which varies with time: a physical or
symbolic var
LMech 1
1
The Laplace Transform Method
once known as Heaviside's Operational Calculus
(Denbigh , pp 282 299, Howatson, pp 215 - 239: and Maths Texts)
The Laplace transform gives firstly a method of solving ordinary linear differential
equations for values
DynG
Gyroscopes
1
Simple Theory of Gyroscopes
Consider a disc spinning about its axle with angular speed . Then consider its axle
being horizontal and the axle being slowly rotated about a vertical axis - say the z-axis - with
angular speed , called the s
Satellites
DynF
1
Satellites
We consider now a satellite of mass m in free flight in the gravitational field of the
earth. Use cfw_r, polar coordinates in the plane of the motion.
Gravitational attractive force, F
r
Satellite of
mass, m
Earth
If re is th
Rocket propulsion
DynD
1
Rocket Power
Consider a body loaded with fuel ejecting an exhaust and pushing against a
restraining force as shown in the figure.
v
ve
F
m'
m
Let
the restraining force be F,
the mass of vehicle and fuel at time t be m and their ve
DynC
Curvilinear motion and Polar Coordinates
1
Curvilinear Motion
General statement
If the position of a point is
r = r(t)
1
then its velocity is
dr
v = dt
2
and its acceleration
2
' dv d r
a = v = dt = 2
dt
3
The momentum of a particle of mass m is
G =m
Four bar linkages
1
Four Bar Linkages in Two Dimensions
A link has fixed length and is joined to other links and also possibly to a fixed point.
The relative velocity of end B
regard to A is given by
vBA
j
with
VBA = r
B = x+jy
y
= BA k( i x + j y)
= BA (
Tools for Dynamics
DynB
1
Dynamics. Newton Laws
N1L 1. A body continues in uniform rectilinear motion unless
acted on by a force.
Corpus omne persevare in statu suo quiescendi vel movendis uniformiter, nisi
quatenus a viribus impressis cogitur statum illu
Moments of Inertia
DynE
1
Moment of Momentum and Moments of Inertia
Centre of Mass ( C of M) or Centre of Gravity( C of G )
G
i
mi
Let mi be an element of mass in a rigid body and let its vectorial position with regard to
the centre of mass or centre of g
1
Formulae to be learned by heart
Trigonometry
sincfw_A+B) = sinA cosB + cosA sinB
sin(AB) = sinA cosB cosA sinB
cos(A+B) = cosA cosB sinA sinB
cos(AB) = cosA cosB + sinA sinB
tanA + tanB
tan(A+B) = 1 tanA tanB
tanA tanB
tan(AB) = 1+ tanA tanB
cos2A + sin
Dyn H
Mechanical vibrations
1
Mechanical System Resonance
Force,
c
x1 F
M
x2
A force F attempts to move a mass M through a connecting dashpot ( also called a damper
or shock absorber) as shown in the figure
d
F = c dt (length of dashpot)
where c is the da