SPINAL INJURY CONTROL - Injury Biomechanics and Control: Optimal Protection from Impact - page 188

The system with two isolators belongs to the class of three-component
models considered in Section 4.2. The airframe, the seat pan, and the
occupant (modeled by two masses connected by a spring-and-dashpot
element) play the roles of the base, container, and object, respectively.
Equations (5.48) - (5.51) can be reduced to the form of Eq. (4.157), and
Problem 5.3 can be treated as a particular case of Problem 4.8. Represent
Eqs. (5.48) - (5.51) as

m 1 ( x ¨ +¨ z) + C( x ˙ −˙ y) + K(x − y) = 0 , (5.59)

m 1 ( x ¨ +¨ z) + m 2 ( y ¨ +¨ z) = F, (5.60)

m s ( x s ¨ +¨ z) = F 1 − F, (5.61)

z ¨ =− v(t), (5.62)

where Eq. (5.49) has been replaced by the sum of Eqs. (5.48) and (5.49),
and in the resulting system let

x = x 2 − z + y 1 , y = x 2 − z, z = x 0 , x s = x 1 − x 0 ,

m s = M, m 1 = μ 1 , m 2 = m, F = F 2 , v =− σ, (5.63)

C( y ˙ −˙ x) + K(y − x) = f 1 .

This leads to the set of equations

μ 1 ( y 1 ¨ +¨ x 2 ) = f 1 (y 1 , y 1 ), ˙

m x 2 + μ 1 ( x 2 + y 1 ) = F 2 ,

(5.64)

M x 1 = F 1 − F 2 ,

x 0 = σ(t),

which coincides with Eq. (4.157) for n 1. The coordinates x 0 , x 1 ,and x 2

measure the displacements of the airframe, the seat pan, and the occupant's
lower torso relative to the ground, while y 1 designates the displacement
of the upper torso relative to the lower torso. By using the transformation
of Eq. (4.159) this system can be reduced to the form of Eq. (4.158).
The initial conditions for the variables x 0 , x 1 , x 2 ,and y 1 are given by
Eq. (4.160).

=