BASIC OPTIMAL SHOCK ISOLATION: SINGLE-DEGREE-OF-FREEDOM SYSTEMS - Injury Biomechanics and Control: Optimal Protection from Impact - page 50

3.1 BASIC PROBLEMS

3.1.1 Mechanical Model: Equation of Motion

The system shown in Fig. 3.1 consists of a rigid body (base) that can move
translationally along a straight line, designated by the coordinate z that is
measured from a fixed point of an inertial reference frame, and another rigid
body (object) attached to the base by a device (shock isolator) so that the
object can move relative to the base along the same line. The acceleration
of the base z (shock acceleration pulse) is given as a function of time, that

is, the base is subject to a kinematic shock disturbance. Then the motion
of the object relative to the base is governed by the differential equation

¨

m x ¨ = f − m z(t), ¨ (3.1)

where m is the mass of the object, x is the coordinate measuring the dis-
placement of the object relative to the base, and f is the force (control
force) exerted by the isolator on the object. Divide both sides of Eq. (3.1)
by m and denote

f
m ,

u = v(t) =− z(t) (3.2)

so that Eq. (3.1) becomes

x ¨ = u + v(t). (3.3)

The notation of Eq. (3.3) conforms to that commonly used in control the-
ory: u is the control variable that should be chosen to provide the desired
properties for the system behavior and v is an uncontrolled excitation which

Object

Control
force

u m

Base

x

z

FIGURE 3.1 Single-degree-of-freedom model.