Christopher Sullivan
Explanation of States of Loading
P12031
States of Loading
To simulate Richard swinging across the deck in a worst case scenario we will assume Richard behaves
much like a frictionless solid arm pendulum, and that gravity is off axis by 30 deg. The system of
equations for the second order differential equations for a pendulum is shown in equation 7 and 8
where 𝜃̈ represents the angular acceleration, 𝜃̇ represents the angular velocity, and 𝜃 represents the
angle . With the known initial conditions of equation 6 it is possible to get the rotational velocity
when 𝜃 = 0 using a simple ODE solver, we are solving for the velocity so that we can use a change in
momentum to solve for the force put into the system. Figure 2 shows an animation of the pendulum
moving across the system.
𝜃̈ =
𝜃
𝜋 − .0001
[ ̇] = [
]
0
𝜃
(6)
𝜃̇ = 𝜃̇
(7)
−𝑔 sin 30 sin 𝜃
(𝑅 sin 𝜃)2 + (𝑅 cos 𝜃)2
(8)
1
0.8
0.6
0.4
R
0.2
0
-0.2
θ
-0.4
-0.6
-0.8
-1
-1
-0.5
0
0.5
1
Figure 2.
Figure 3 shows the plot of angular velocity vs time, this data was generated from the movement of the
pendulum.
Christopher Sullivan
Explanation of States of Loading
P12031
Angular Velocity vs Time
0
Angular Velocity (rad/s)
-1
-2
-3
-4
-5
-6
1
1.5
2
2.5
3
3.5
Time (s)
Figure 3.
As you can see the maximum angular velocity that will be reached is at the vertex of the swing and is
5.4 𝑟𝑎𝑑⁄𝑠 converting this can be converted to meters per second and mph respectively by multiply by
the radius of the track.
𝑣𝑅 = 3.7 𝑚⁄𝑠
(9)
𝑣𝑅 = −8.2 𝑚𝑝ℎ
(10)
Once we know the velocity that Richard will be traveling when he hits the track stop we can by using
conservation of energy. The impact force felt by the track stopper will be the difference in kinetic energy
divided by the distance after the collision that it took Richard to come to rest. To make it easier we are
assuming that Richard bounces back about 5 in or .127 m. Using equation 9 where 𝑣𝑟 is the velocity of
Richard, 𝑚 is the mass of Richard and the chair, and 𝑑 will be the linear distance Richard bounces back
off the rubber.
1 2
𝑣𝑟 𝑚⁄
𝐹𝑐 = 2
𝑑
The force of the impact is roughly equivalent to 6000N or 1348 lb. This force will later be applied to the
model of our track platform.
(11)
Christopher Sullivan
Explanation of States of Loading
P12031
To account for the adverse weather condition that Richard will be facing on the boat some initial
background about waves needs to be gathered first so that we might make some basic assumptions.
The first basic assumption is defining a “bad” wave, essentially the sizing of the waves that Richard
might face. Based on figure 4 an average height of 7m with sustained winds of 40knots and also from
figure 4 a period of 16 seconds will be assumed to be a worst case scenario. From figure 5 the
wavelength of a wave with a period of 16 sec is known to be about 400m now that relative to the boat is
quite long and therefore the boat will generally follow the angle of the wave as it floats long its surface.
Now that we have a handle on the height, and speed of these waves, we can start to calculate the
significant accelerations that our system needs to be able to handle. Acceleration of Richard will also
cause a significant internal reaction forces between Richard, his chair, and the track platform.
Figure 4. (Anthoni, 2000)
Figure 5. (Anthoni, 2000)
Christopher Sullivan
Explanation of States of Loading
P12031
To calculate these accelerations a wave path must be created, that can be written as a simple
mathematical equation. Assuming the wave is in the shape of a sin wave the equation of the wave can
be show as equation 10. Where Ymax is the maximum height of the wave and P is the period of the
waves. Figure 6 shows the high of the wave as a function of time.
𝑌𝑤 (𝑡) =
𝑦𝑚𝑎𝑥
2𝜋𝑡 𝜋
𝑦𝑚𝑎𝑥
sin (
− )+
2
𝑃
2
2
7
Wave
6
5
Hight
4
3
2
1
0
0
Figure 6.
2
4
6
8
Time (s)
10
12
14
16
Because Richard is not in the dead center of the boat and because the boat generally follows the
angle of the wave as it travels along its surface there will be relative motion. To find this the first thing
we will do is find the angle that the boat is at and how this changes with time. To do this we take the
equation 9 and differentiate it. Now you have an equation that represent the slope of the boat with
respect to time, and to get it into a usable angle you simply need to take the arctangent of the slope.
Figure 7 represents the set of angles that the boat will be at with respect to time while the wave moves
under it.
Christopher Sullivan
Explanation of States of Loading
P12031
1
Angle Of the Boat
0.8
0.6
Angle (rad)
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
2
4
6
Figure 7.
8
Time (s)
10
12
14
16
Now that we have the angle we can use it to calculate the position of Richard with respect to
the center of the boat. With this in mind you can differentiate these equations twice to get the
acceleration of Richard, and the system. The resulting Accelerations are shown in figure 8.
0.8
Acc Y
Acc X
0.6
Acceleration (m/s 2)
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
0
Figure 8.
2
4
6
8
Time (s)
10
12
14
16
Christopher Sullivan
Explanation of States of Loading
P12031
As you can see there is a spike in the acceleration in the X direction right as the boat would be
rounding the top of the wave which is exactly what we expected to see. Now that we have this
information we will be able to calculate the body forces of the individual pieces along with the reactions
that they may cause to their interlocking parts. Assuming Richard and his chair weigh approximately
136kg using simple force = mass*acceleration and adding in gravitational effects. At their peak
acceleration he is applying 1425 newtons down on the track, and 24 newtons towards the middle of the
boat. This information can be fed into an ANSYS model of the track platform to find the stress
distribution of the platform.
Christopher Sullivan
Explanation of States of Loading
P12031
To simulate a crash with another boat there are just too many variables to take into
consideration. So instead of finding out exactly how much acceleration there is in a single kind of crash
table 1 lists out the magnitude of different kinds of common acceleration.
Table 1. (http://en.wikipedia.org/wiki/G-force)
For our experiments with ANSYS we will slowly ramp up the acceleration until the maximum Von Mises
Stress exceeds these percussionary limits (at around 1 FOS) and this will be the amount of acceleration
that the system will be rated to handle. There will be five cases all include Richards internal forces added
to the situation at the unsupported area of the track, pure port acceleration, a pure stern acceleration,
pure bow acceleration, an equal combination of port and bow, and port and stern. Once we know these
maximum accelerations in each direction it can be converted to Gs to compare with other kinds of high
G activities.
Christopher Sullivan
Explanation of States of Loading
P12031
Works Cited
Anthoni, D. J. (2000). Oceanography: waves theory and principles of waves, how they work and what
causes them. Retrieved 10 27, 2011, from seafriends:
http://www.seafriends.org.nz/oceano/waves.htm
MatWeb. (n.d.). Retrieved 11 3, 2011, from Aluminum 2011-T6:
http://www.matweb.com/search/DataSheet.aspx?MatGUID=66a81429bea54053bbdc39cfce0f2
407&ckck=1
The Engineering ToolBox. (n.d.). Retrieved 11 3, 2011, from Modulus or Rigidity:
http://www.engineeringtoolbox.com/modulus-rigidity-d_946.html