Summary of Dynamics of Rigid Bodies

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Summary of Dynamics of Rigid Bodies
If O is fixed
Kinematics relations between points on a rigid
body
Note formulas are for ? positive in a CCW (right
hand rule) sense
Now we will solve some problems
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Alternative approach use moment equation for
fixed-axis rotation
Reaction forces (two of the unknowns) do not
appear in the moment equation
One EOM for ?(t), and all the rest follows from
other equations
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EOM angular acceleration is a function only of
angle
A bit of trickier using differentiation
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Suppose we start the body closer to ?3?/2
Define ? ? - 3?/2
For ? ltlt 1, sin? ? , so we get
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Analysis of a Car Crash
A 2500 kg car crashes into a fixed barrier wall
at speed vo5 m/s and is observed to rebound with
a velocity vr2 m/s. The total impact time is
0.5s
Average deceleration of the car is a (2 m/s
(-5m/s))/(0.5s) 14 m/s2

• Now examine the passenger
• Treat as a rigid body IG1.2mL2
• Wearing a seatbelt, so body rotates around O
• Center of gravity of torso shown
• Distance to head shown
• At what velocity does head strike dashboard?

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Set up coordinates x, y
Motion of G in terms of coords
Point O is not fixed
What is motion of O?
Motion of Car
So, we have
Draw Free Body Diagram
Set Fma and MI?
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Dont really care about G so eliminate using
force equations
Solve set of equations for ?(t), R(t), N(t), xO,
yO
Stop calculation when ?5?/6
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Solve set of equations for ?(t), R(t), N(t), xO,
yO
Put in matrix form to solve in MATLAB
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Go to the MATLAB.
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Now lets try to save the passenger by adding an
air bag
Model airbag as force acting on head Pressure x
(Area of Head/Torso hitting airbag)
Acts over angles ? gt 7?/12
What response time and pressure (Pmax, tab) are
needed to prevent passengers head from hitting
the dashboard?
?/12
GT
L
O
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Force due to airbag modifies the Equations of
Motion
Putting in accelerations and writing in matrix
form, we have
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Total crash takes 0.5s so what kind of tab do
we need?
Want sufficient airbag force to stop the head,
but not too much
We have not designed actual contact of airbag
and passenger we assume force acts on the head.
Proper design requires more detail. Children
should not sit in the front seat because the
airbag acting only on their heads could lead to
death itself.
Go to MATLAB
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Response time tab0.2s Pmax 900N just keeps
head from hitting dashboard
? (rad)
xO (m)
Time (s)
Time (s)
yh (m)
? (rad/s)
Airbag at 63 pressure
xh (m)
Time (s)
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Dynamics of Rigid Bodies
What is the equivalent of F ma ? How does the
angular velocity change ? What are momentum,
kinetic energy, potential energies, angular
momentum of a rigid body ? - Rigid body has both
translation and rotation What is the rotational
analog of mass? i.e. what is resistance to
angular acceleration? ? Moment of Inertia
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Dynamics of Rigid Bodies
Forces are applied at different locations on the
body Forces friction, normal reaction, gravity,
other applied forces
Forces causes the body to both translate and
rotate
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For now, just one quantity Angular momentum
Sum of angular momenta of masses mi at ri moving
at Vi
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Recall Rate of change of h is equal to total sum
of all moments acting on the body
Sum of the moments around the center of mass is
proportional to the angular acceleration
Coefficient of proportionality is the Moment of
Inertia IG
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Suppose point O is FIXED in space?
Fixed-Axis Rotational Motion
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Linear momentum
Velocity at G rotation of i around G
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Gravitational Potential Energy
Just like a point particle of mass m at G
yi
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Kinetic Energy
Sum of KE of all blocks of mass
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Moment of Inertia around G, geometric property of
the body
New Quantity
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Point-particle Motion
Rigid-body Motion
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Summary of Dynamics of Rigid Bodies
If O is fixed
Gravitational Potential Energy
Kinetic Energy
Linear Momentum
Angular Momentum
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Energy methods good to relate initial position,
velocity to final position, velocity
No friction Reaction forces do no work
Initially starts at rest, with yG0 ? Initial
energy is zero
At any angle ?, we then have
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To get reaction forces, recall that we had
At bottom position, ?3?/2, we thus have
As found via analyzing full MATLAB solution
and