15
The Modeling of Two-dof Mechanical Systems
The Modeling of Two-dof
Mechanical Systems
Sistemi vibranti 2 g.d.l. - Oscillazioni libere 2 / 16
Examples of modeling of two-dof of
mechanical systems.
1k
1x
1m
2k
2m
2x
x
m
1k 2k
1 2
Free vibrations
Sistemi vibranti 2 g.d.l. - Oscillazioni libere 3 / 16
The Modeling of Two-dof Mechanical Systems
1k
1 1 1, ,x x x
1m
2k 3k
2m
2 2 2, ,x x x
Referring to Figure choose the two independent linear coordinates x1 and x2
from the static equilibrium position of the two masses m1 and m2.
Sistemi vibranti 2 g.d.l. - Oscillazioni libere 4 / 16
2m1m
The Modeling of Two-dof Mechanical Systems
1 1 1, ,x x x 2 2 2, ,x x x
1eF
2eF
3eF
2 2m x1 1m x
We write the equations of dynamic equilibrium for the two masses::
1 2 1 1e eF F m x
1 1 2 1 2 1 1k x k x x m x
2 3 2 2e eF F m x
2 2 1 3 2 2 2k x x k x m x
Mass 1 Mass 2
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The Modeling of Two-dof Mechanical Systems
The system of equation of motion becomes:
1 1 1 1 2 1 2
2 2 2 2 1 3 2
0
0
m x k x k x x
m x k x x k x
and:
1 1 1 2 1 2 2
2 2 2 1 2 3 2
0
0
m x k k x k x
m x k x k k x
i.e. a system of homogeneous linear differential equations of second order with
constant coefficients, coupled to the term k2.
Sistemi vibranti 2 g.d.l. - Oscillazioni libere 6 / 16
The Modeling of Two-dof Mechanical Systems
In matrix notation:
1 2 21 1 1
2 2 32 2 2
0 0
0 0
k k km x x
k k km x x
2x2 2x1 2x2 2x1 2x1
M x K x 0
And in compact form:
Sistemi vibranti 2 g.d.l. - Oscillazioni libere 7 / 16
Sistema a due gradi di libertà libero
The system accepts solutions of the type:
t
1 1
t
2 2
z
z
x X e
x X e
t
1 1
t
2 2
z
z
x z X e
x z X e
2 t
1 1
2 t
2 2
z
z
x z X e
x z X e
2
1 1 1 2 2 2
2
1 2 2 2 2 3
0
0
X m z k k X k
X k X m z k k
Substituting in the system has:
Sistemi vibranti 2 g.d.l. - Oscillazioni libere 8 / 16
Sistema a due gradi di libertà libero
The algebraic system admits solutions different from the banal if and only if the
determinant of the matrix of coefficients is equal to zero, ie:
obtaining the following equation in biquadratic z:
2
1 1 2 2
2
2 2 2 3
0m z k k k
k m z k k
4 2 2
1 2 1 2 3 2 1 2 1 2 2 3 2m m z m k k m k k z k k k k k
Sistemi vibranti 2 g.d.l. - Oscillazioni libere 9 / 16
Sistema a due gradi di libertà libero
in general:
4 2
1 2 3 0a z a z a
whose solutions are valid:
22 2 2 1 31 2
1
4
2
a a a az
a
It can be shown that the radicand is always positive and that its root is always
less than a2, then it follows that the roots are both negative, and then you have
four imaginary roots, two by two conjugated.
2
1 2z
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Sistema a due gradi di libertà libero
At the two solutions found, the system is:
having determinant zero, is reduced to a single equation of the two present,
being a linear combination of the other.
2
1 1 1 2 2 2
2
1 2 2 2 2 3
0
0
X m z k k X k
X k X m z k k
It is not possible to determine the constants X1 e X2, but only and only their
relationship.
1 1 2
1 1 122 1 1 1 1 2
X z kz z z
X z m z k k
1 2 2
2 2 222 2 1 2 1 2
X z kz z z
X z m z k k
for
for
1 1 1 2 1X z X z
1 2 2 2 2X z X z
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Sistema a due gradi di libertà libero
Therefore, the general solution of the system of differential equations is a linear
combination of the two solutions corresponding to z = z1 ed a z = z2 , so we can,
wirte:
1 2
1 2
t t
1 1 1 1 2
t t
2 2 1 2 2
t
t
z z
z z
x X z e X z e
x X z e X z e
1 2
1 2
t t
1 1 2 1 2 2 2
t t
2 2 1 2 2
t
t
z z
z z
x X z e X z e
x X z e X z e
Sistemi vibranti 2 g.d.l. - Oscillazioni libere 12 / 16
Sistema a due gradi di libertà libero
To determine the four arbitrary
constants A, B, y1 e y2 must impose
four initial conditions, for t = 0:
0
0
0
0
1 1
1 1
2 2
2 2
0
0
0
0
x x
x v
x x
x v
being the imaginary can be put :2
1 2z
1 1z i 2 2z i
by the Euler equations, we can write the integral in the following form:
1 1 1 1 2 2 2
2 1 1 2 2
t sen t ψ sen t ψ
t sen t ψ sen t ψ
x A B
x A B
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Sistema a due gradi di libertà libero
By imposing that the initial conditions satisfy the following relationship:
0 0
0 0
1 1
1
2 2
0x v
Bx v
1 1 1 1
2 1 1
t sen t ψ
t sen t ψ
x A
x A
Then the system with two degrees of freedom, begins to vibrate sinusoidally with
pulsation 1 as if he had a single degree of freedom, and this vibration, pulsation
with 1 and the amplitude ratio 1 constant, is called the first mode of vibration
of the system.
the general integral becomes:
Sistemi vibranti 2 g.d.l. - Oscillazioni libere 14 / 16
Sistema a due gradi di libertà libero
By imposing that the initial conditions satisfy the following relationship:
0 0
0 0
1 1
2
2 2
0x v
Ax v
1 2 2 2
2 2 2
t sen t ψ
t sen t ψ
x B
x B
Then the system with two degrees of freedom, begins to vibrate sinusoidally with
pulsation 2 as if he had a single degree of freedom, and this vibration, pulsation
with 2 and the amplitude ratio constant 2, is defined according to its own way
of vibrating system second mode of vibration of the system.
the general integral becomes:
Sistemi vibranti 2 g.d.l. - Oscillazioni libere 15 / 16
Sistema a due gradi di libertà libero
If the initial conditions are generic system will vibrate according to the sum of two
sinusoids, as shown in figure:
I modo II modo moto effettivo
t
x
