1. Impact simulation the helmet bounces upward after hitting the ground.
the helmet bounces upward after hitting the ground
90° z x g O
an elastic collision take place between the helmet moving upward with speed
and the head moving downward with speed

2. Impact simulation From: conservation of momentum
conservation of kinetic energy
We can calculate:

3. Impact simulation from calculation: from FE simulation:
the FE simulation stops before the head reaches the maximum height

4. Influence of material propertiesx1 k M x0 xf
by Newton's Second Law:
Hooke's Law:
The sum of the forces on the mass generates the equation of motion:
Where
and

5. Influence of material properties
From Hooke’s law:
E is the Young's modulus (modulus of elasticity)
S0 is the original cross-sectional area through which the force is applied
L0 is the original length of the spring
we know:
mhead= 4.57kg, L0 = 0.26m, khelmet= 3113 N/m, h0 = 1.4m
we can calculate:
no plastic deformation take place, thus:

6. Influence of material properties
but
for
from calculation:
from FE simulation:
thus
tmax is the time needed by helmet-head to fall from 1.4m on a rigid surface and coming back to the initial position
For different material properties if E ↓ → k helmet ↓
Thus when k helmet ↓ → ∆x ↑ but ω ↓ and a ↓

7. Influence of material propertiesx0 k M c x1 xf
due to plastic deformation
thus
for elastic-plastic conditions

8. Cross-sectional area influence
Helmet with vents sliding
Helmet without vents sliding
For a helmet with vents S0 ↓ → k helmet ↓
Thus when k helmet ↓ → ∆x ↑ but ω ↓ and a ↓ →for absorbing the same energy the helmet needs to deform more

9. Cross-sectional area influence
increasing the deformation the stress in the helmet will increase

10. Force exerted on the head with helmet
Impact time ≈ 0.002s
Helmet+head v1=5.22m/s
Impact duration = 0.002s
mhead=4.57kg
mhelmet=0.233kg
Pi=mi x vi = 4.803kg x 5.22m/s = 25.07kgm/s
Pf = mf x vf = 4.803kg x 0m/s = 0kgm/s
I=force x time
F = ∆P/t = Pf – Pi = kgm/s /0.002s = N
∆v = a x t
a = ∆v/t = 5.22m/s/0.002s = 2610 m/s2 ≈ 261g
from FE simulation aFEM = 252g

11. Force exerted on the head without helmet
Impact time ≈ s
Head v1=5.22m/s
Impact duration = s
mhead=4.57kg
Pi=mi x vi = 4.57kg x 5.22m/s = 23.86kgm/s
Pf = mf x vf = 4.57kg x 0m/s = 0kgm/s
I=force x time
F = ∆P/t = Pf – Pi = kgm/s / s = N
∆v = a x t
a = ∆v/t = 5.22m/s/ s = m/s2 ≈ 1214g
from FE simulation aFEM = 1002g