OPTIMAL SHOCK ISOLATION FOR MULTI-DEGREE-OF-FREEDOM SYSTEMS - Injury Biomechanics and Control: Optimal Protection from Impact - page 146

The peak magnitude of the displacement of the body relative to the base
and the peak magnitude of the force exerted on this body by the spring are
of interest. The solution of Eq. (4.104) subject to the initial conditions of
Eq. (4.105) is given by

cos t k
m

1 .

ma
k

x(t) = − (4.106)

The force exerted on the body is defined by

ma 1 cos t k

m

.

− kx(t) = − (4.107)

Accordingly,

2 ma
k

max

) |

x(t) |= , max

) |

kx(t) |= 2 ma. (4.108)

t ∈ [0 , ∞ t ∈ [0 , ∞

The rigid model of the system under consideration consists of the body
and the base, which are rigidly attached to each other and are moving
together at acceleration a . For this case, the displacement of the body
relative to the base is zero and the force acting on the body is con-
stant and is equal to ma . From the expressions of Eq. (4.108), if follows
that as k

the peak displacement of the body in the elastic model
approaches zero (which is in agreement with the rigid model), whereas the
peak force calculated for the elastic model for any k is twice that of the
rigid model.
If a dashpot with linear characteristic is added to the spring that connects
the body to the base, then the relative motion of the body is governed by
the equation

→∞

m x ¨ + c x ˙ + kx =− ma, (4.109)

where c is the damping coefficient. For this case, the force acting on the
body is expressed as − c x − kx . One might ask if it is possible to approach

rigid attachment of the body with the model of Eq. (4.109) by choosing
sufficiently large values for c and k . This approach is characterized by the
conditions

t ∈ [0 , ∞ ) |

max x(t) | 1 , |

t ∈ [0 , ∞ ) |

max c x ˙ + kx(t) |− ma | 1 . (4.110)