Figure 12 : Dynamic model of the system
The mathematical model for the system shown in Figure 12 is:
(1.1)
In which m represents the total mass of the user and k represents the spring constant of the shoe insole.
The solution of equation 1.1 gives position function x(t). Taking the Laplace transform of both sides gives:
(1.2) This can be rewritten as:
(1.3)
Solving for X(s) gives:
(1.4)
The inverse Laplace transform of equation 1.4 gives:
(1.5)
The following initial values were assumed and used to plug into the final acceleration equation.
Differentiating equation 1.5 twice gives the following two equations:
(1.6)
(1.7)
Plugging the initial values into equation 1.7 allows us to solve for the acceleration of the foot as it hits the ground:
(1.8)
The maximum acceleration occurs at approximately t = 0.0344s. This time represents the actual contact time for the foot on the ground as the user is running. The maximum acceleration then becomes 31.9m/s 2 . To find the maximum possible force acting on the user's foot at the time of impact, Newton 's equation can be used:
(1.9)
A value of 2893 Newton 's corresponds to a force of approximately 650 lbs. This value of F represents the maximum force acting on the foot while the person is running. One constraint of the force sensitive resistors being used is that they can only withstand pressures up to 175 psi. To determine whether this pressure will be exceeded by a 200 lb man running, the previously found force was divided between portions of the foot. The percentages used are from a study done by the American Fitness Association. The load was distributed for sprinting (see figure 13), running (see figure 14), and standing (see figure 15). The loads were then divided by the different areas of the foot to find the pressure on each portion of the foot (see figure 16). Figure 17 shows a diagram of where these regions are located on the foot. All pressures were found to be under the 175psi threshold.
Sprinting |
650 |
lbs |
|
|
|
Relative loads (%) |
Load(lbs) |
Area(in 2 ) |
Pressure(psi) |
Medial heel |
0.015 |
9.75 |
2.02 |
4.83 |
Lateral heel |
0.018 |
11.7 |
2.02 |
5.79 |
Medial midfoot |
0.005 |
3.25 |
3.75 |
0.87 |
Lateral midfoot |
0.035 |
22.75 |
3.75 |
6.07 |
Medial forefoot |
0.277 |
180.05 |
3 |
60.02 |
Central forefoot |
0.192 |
124.8 |
3 |
41.60 |
Lateral forefoot |
0.176 |
114.4 |
3 |
38.13 |
Hallux |
0.147 |
95.55 |
2.25 |
42.47 |
Second toe |
0.081 |
52.65 |
1 |
52.65 |
Lateral toes |
0.054 |
35.1 |
3 |
11.70 |
Figure 13 : Load and pressure distribution for sprinting
Running |
650 |
lbs |
|
|
|
Relative loads (%) |
Load(lbs) |
Area(in 2 ) |
Pressure(psi) |
Medial heel |
0.078 |
50.7 |
2.02 |
25.10 |
Lateral heel |
0.088 |
57.2 |
2.02 |
28.32 |
Medial midfoot |
0.024 |
15.6 |
3.75 |
4.16 |
Lateral midfoot |
0.098 |
63.7 |
3.75 |
16.99 |
Medial forefoot |
0.187 |
121.55 |
3 |
40.52 |
Central forefoot |
0.156 |
101.4 |
3 |
33.80 |
Lateral forefoot |
0.182 |
118.3 |
3 |
39.43 |
Hallux |
0.096 |
62.4 |
2.25 |
27.73 |
Second toe |
0.049 |
31.85 |
1 |
31.85 |
Lateral toes |
0.042 |
27.3 |
3 |
9.10 |
Figure 14 : Load and pressure distribution for running
Standing |
200 |
lbs |
|
|
|
Relative loads (%) |
Load(lbs) |
Area(in 2 ) |
Pressure(psi) |
Medial heel |
0.3 |
60 |
2.02 |
29.70 |
Lateral heel |
0.3 |
60 |
2.02 |
29.70 |
Medial midfoot |
0.065 |
13 |
3.75 |
3.47 |
Lateral midfoot |
0.015 |
3 |
3.75 |
0.80 |
Medial forefoot |
0.12 |
24 |
3 |
8.00 |
Central forefoot |
0.04 |
8 |
3 |
2.67 |
Lateral forefoot |
0.12 |
24 |
3 |
8.00 |
Hallux |
0.03 |
6 |
2.25 |
2.67 |
Second toe |
0.007 |
1.4 |
1 |
1.40 |
Lateral toes |
0.003 |
0.6 |
3 |
0.20 |
Figure 15 : Load and pressure distribution for standing
|
Minimum Load (lbs) |
Maximum Load (lbs) |
Minimum Pressure (psi) |
Maximum Pressure (psi) |
Medial heel |
9.75 |
60 |
4.83 |
29.70 |
Lateral heel |
11.7 |
60 |
5.79 |
29.70 |
Medial midfoot |
3.25 |
15.6 |
0.87 |
4.16 |
Lateral midfoot |
3 |
63.7 |
0.80 |
16.99 |
Medial forefoot |
24 |
180.05 |
8.00 |
60.02 |
Central forefoot |
8 |
124.8 |
2.67 |
41.60 |
Lateral forefoot |
24 |
118.3 |
8.00 |
39.43 |
Hallux |
6 |
95.55 |
2.67 |
42.47 |
Second toe |
1.4 |
52.65 |
1.40 |
52.65 |
Lateral toes |
0.6 |
35.1 |
0.20 |
11.70 |
Figure 16 : Max and min loads on the foot
Figure 17 : Foot Diagram
The mathematical representation of the system is only an estimate. After concluding the dynamic analysis the group found that the human body was very difficult to model as a dynamic system. In our case, the foot proved difficult to model due to the inability to predict the effect of the ligaments on the model. They serve as a damper and make very precise analysis very difficult to perform.