CHAPTER 3
Motion and Dynamics
3.1 Making sense of dynamic equilibrium
The concept of dynamic equilibrium lies behind many types of engineering
analyses and design of rotating equipment. Some key definition points are:
• Formally, an object is in a state of equilibrium when the forces acting on
it are such as to leave it in its state of rest or uniform motion in a straight
line.
• In terms of dynamic equilibrium, this means that it is moving at constant
velocity with zero acceleration (or deceleration).
Figure 3.1 shows the difference between dynamic equilibrium and non-
equilibrium. The concept of dynamic equilibrium is used to design
individual components of rotating equipment.
3.2 Motion equations
Uniformly accelerated motion
Bodies under uniformally accelerated motion follow the general equations
v = u + at
t = time (s)
s = ut + ½at2
a = acceleration (m/s2)
s = distance travelled (m)
u = initial velocity (m/s)
v2 = u2 + 2as
v = final velocity (m/s)
2
u v
s
t
+
=
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66
Angular motion
t = time (s)
θ = angle moved (rad)
α = angular acceleration (rad/s2)
N = angular speed (rev/min)
ω1 = initial angular velocity (rad/s)
ω2 = final angular velocity (rad/s)
Fig. 3.1 Dynamic equilibrium and non-equilibrium
Dynamic equilibrium
Dynamic non-equilibrium
ω2 = ω1 + α
ω22 = ω12 + 2αs
θ = ω1t + ½α2
2
60
N
π
ω =
1
2
2
t
ω ω
θ
−
=
ωb
ωa
ωc
All parts of the
mechanism are
moving with constant
angular velocities
No parts of the
mechanism are
moving with constant
velocity
Accelerating or
decelerating torque
αa=dωα/dt
Motion and Dynamics
67
General motion of a particle in a plane
v = ds/dt
s = distance
a = dv/dt = d2s/dt2 t = time
v = adt
v = velocity
s = vdt
a = acceleration
3.3 Newton’s laws of motion
First law
A body will remain at rest or continue in uniform motion in a
straight line until acted upon by an external force.
Second law When an external force is applied to a body of constant mass
it produces an acceleration that is directly proportional to the
force, i.e. force (F) = mass (m)