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Consider the mass-spring systemwhere dry frictionis present between the block
and the horizontalsurface. The frictionalforce has a constant magnitude
µ
mg
(
is the coefficientoffriction) and alwaysopposes the motion. The differential
equation for the motion of the block can beexpressedas
µ
k
m y
y
y
=−
µ
g
|
y
|
where y ismeasured fromthe positionwhere the spring is unstretched. If the block
is releasedfromrest at y
=
y 0 , verifybynumerical integrationthat thenextpositive
peak valueof y is y 0
4
µ
mg
/
k (this relationship can be derived analytically). Use
80665 m/s 2 and y 0 =
k
=
3000 N/m, m
=
6 kg,
µ =
0
.
5, g
=
9
.
0
.
1m.
18. Integrate the following problemsfrom x
=
0to20 and plot y vs. x :
(a) y +
.
5( y 2
1) y +
=
=
y (0)
=
0
y
0
y (0)
1
0
(b) y =
y (0)
y cos 2 x
y (0)
=
0
=
1
These differentialequations arise in nonlinear vibration analysis.
19.
The solution of the problem
1
x y +
y +
y (0)
y
y (0)
=
1
=
0
is the Bessel function J 0 ( x )
.
Use numerical integration to compute J 0 (5) and com-
pare the result with
0
.
17760, the valuelistedinmathematicaltables. Hint : to
10 12 .
avoid singularityat x
=
0, start the integrationat x
=
20. Consider the initial value problem
y =
y (0)
16
.
81 y
y (0)
=
1
.
0
=−
4
.
1
(a) Derive the analytical solution. (b) Do youanticipate difficulties in numerical
solution of this problem? (c) Try numerical integration from x
=
0to8 to see if
21.
2 R
i 2
R
i 1
R
E ( t )
i 1
L
C
i 2
Kirchoff'sequationsfor the circuit shown are
L di 1
dt
+
Ri 1 +
2 R ( i 1 +
i 2 )
=
E ( t )
(a)
q 2
C +
Ri 2 +
2 R ( i 2 +
i 1 )
=
E ( t )
(b)
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