68
I.-B. Dragna et al.
There exist numerous examples of vibrational systems with collisions. We
deal only with the fundamental characteristics of mechanical vibrations [13, 18]
and consider only the case of harmonic excitations. Elimination of the Lagrange
multipliers is presented in [30].
2 The Mechanical System
The mechanical system is presented in Fig. 1. It consists in the rigid bodies denoted
by 0, 1, 2, …, n that move without friction on horizontal bars. The rigid body 0 has
an oscillatory motion in the form
x 0 = X 0 + A 0 sin(ωt),
(1)
where X 0 is its initial position.
The rigid bodies are linked one to another by springs of stiffness k 1 , k 2 , …, k n
and non-deformed lengths equal to the corresponding distances between horizontal
bars, that is
l
0
p = y p .
(2)
On the n bars, there are situated some fixed stoppers situated (for a generic rigid
body denoted by p, 1 ≤ p ≤ n) at the distances b ps and b pd , respectively, from the
ends of the bars.
One considers as known the masses m p , 1 ≤ p ≤ n, of the rigid bodies, and the
coefficients of restitution r p between the rigid body p and the stoppers situated on
its own bar. We will denote by x p the position of the rigid body p.
For an arbitrary spring p, its length at a moment of time is
l p =
y 2
p +
x p+1 − x p
2 ,
(3)
wherefrom one gets its elongation
l p = l p − l
0
p =
y 2
p +
x p+1 − x p
2 − y p .
(4)
The elastic force in the spring p has the expression
F p = k p l p = k p
y 2
p +
x p+1 − x p
2 − y p
.
(5)
By isolating one generic rigid body p < n, the elastic forces F p and F p+1 read
I.-B. Dragna et al.
There exist numerous examples of vibrational systems with collisions. We
deal only with the fundamental characteristics of mechanical vibrations [13, 18]
and consider only the case of harmonic excitations. Elimination of the Lagrange
multipliers is presented in [30].
2 The Mechanical System
The mechanical system is presented in Fig. 1. It consists in the rigid bodies denoted
by 0, 1, 2, …, n that move without friction on horizontal bars. The rigid body 0 has
an oscillatory motion in the form
x 0 = X 0 + A 0 sin(ωt),
(1)
where X 0 is its initial position.
The rigid bodies are linked one to another by springs of stiffness k 1 , k 2 , …, k n
and non-deformed lengths equal to the corresponding distances between horizontal
bars, that is
l
0
p = y p .
(2)
On the n bars, there are situated some fixed stoppers situated (for a generic rigid
body denoted by p, 1 ≤ p ≤ n) at the distances b ps and b pd , respectively, from the
ends of the bars.
One considers as known the masses m p , 1 ≤ p ≤ n, of the rigid bodies, and the
coefficients of restitution r p between the rigid body p and the stoppers situated on
its own bar. We will denote by x p the position of the rigid body p.
For an arbitrary spring p, its length at a moment of time is
l p =
y 2
p +
x p+1 − x p
2 ,
(3)
wherefrom one gets its elongation
l p = l p − l
0
p =
y 2
p +
x p+1 − x p
2 − y p .
(4)
The elastic force in the spring p has the expression
F p = k p l p = k p
y 2
p +
x p+1 − x p
2 − y p
.
(5)
By isolating one generic rigid body p < n, the elastic forces F p and F p+1 read