3. Rotational Dynamics Flashcards
Rotational Reference Frames
-R denotes a rotational reference frame, e.g. a room rotating around a central axis -I indicates an inertial reference frame
Rotation Vector
-the rotation vector is Ω_ -this is generally set to Ω_ = Ωez^ -where Ω is the rate of rotation, and Ω>0 indicates anticlockwise rotation
Period of Rotation
τ = 2π/Ω
Centrifugal Force Description
-proportional to distance from the axis of rotation -e.g. when you are in a car and it goes around a bend, your momentum is in the tangential direction so you experience a radial force
Coriolis Force Description
-associated with motion -if a person in a rotational reference frame throws a ball, the ball deflects to follow the path of rotation
Rotaional Reference Frame
Position
to describe position x_(t), we choose a basis {ej_} for j=1,2,3
-in R, choose {ej~_(t)} fixed in the room:
ej~_ = P[Ωt] ei_
-so that:
x_(t) = xj~(t) ej~_(t)
Inertial Reference
Position
- to describe position x_(t), we choose a basis {ej_} for j=1,2,3
- in I, {ej_} is fixed in time:
x_(t) = xj(t) ej_
Relationship Between Position in Inertial and Rotational Reference Frames
-the relationship between the two sets of coordinates is: xj(t) ej_ = xj~(t) ej~(t)
Rotation Matrix
-P[Ωt] is a 3x3 matrix describing rotation around the z-axis by Ωt -with entries cos(Ωt), -sin(Ωt), 0 in the first row, sin(Ωt), cos(Ωt), 0 in the second row and 0,0,1 in the third row
Rotational Basis Vectors
e1~_ = cosθ e1_ - sinθ e2_ e2~_ = sinθ e1_ + cosθe2_ e3~_ = e3_
Inertial Basis Vectors
e1_ = cosθ e1~_ + sinθ e2~_ e2_ = -sinθ e1~_ + cosθ e2~_ e3_ = e3~_
Inertial Reference Frame
Velocity
dx_/dt |I = dxj/dt ej_
Rotational Reference Frame
Velocity
dx_/dt |R = dxj~/dt ej~_
-the apparent velocity
Relationship Between Velocity in Inertial and Rotational Reference Frames
dx_/dt|I = dx_/dt|R + Ω_ x x_
Relationship Between Acceleration in Inertial and Rotational Reference Frames
d²x_/dt²|I = d²x_\dt²|R + 2Ω_xdx_/dt|R + Ω_x(Ω_xx_)
Equation of Motion in the Inertial Reference Frame
m d²x_/dt²|I = F_
Equation of Motion in the Rotational Reference Frame
m d²x_/dt²|R = F_ - 2m Ω_ x dx_/dt|R - mΩ_ x (Ω_ x x_)
-the secon term on the RHS represents the Coriolis force and the third term the Centrifugal force
Ficticious Forces in the Rotational Reference Frame
-in moving to the rotational reference frame, ficticious forces arise which keep track of the fact that we are in a rotating reference frame
Properties of the Centrifugal Force
-if Ω_ = Ω ez^ and x_ = r er^ + z ez^
Fu_ = mΩ²r er^
- this is always positive, as expercted since the centrifugal force points outwards
- a ‘radial gravity’
- a conservative force
Fu_ = -∇(1/2 mΩ²r²)
Coriolis Force Properties
Fc_ . x_ = 0
- so Fc_ is perpendicular to the direction of motion
- no work done
- deflecction is energetically free
Coriolis Dominated Dynamics
|Fu_|/|Fc_| ~ Ωr/U << 1
- Fu_ negligible and Fc_ dominant
- Fu_ can also be balanced out in the governing equation:
–by gravity in a planetary context
–by the reaction force from the outer wall in a tank
Coriolis Dominated
Equation
Fu_ = 0
=>
mx’‘_ = -2mΩ_ x x’_ + F_
Coriolis Dominated
Solutions
F_ = (0,F)
=>
x = a/4Ω² [2Ωt - sin(2Ωt)]
y = a/4Ω² [1 - cos(2Ωt)]