130
5 Continuous Systems
v(x, t) =
∞
n=1
sin
(2 n − 1) π
2 l
x
2
l
l
0
v 0 (x) ϕ r (x) dx
cos(ω n t)
+
2
ω n l
l
0
˙
v 0 (x) ϕ r (x) dx
sin(ω n t)
.
(5.41)
We conclude with the observation that the infinite series representations of the
solutions v(x, t), e.g., the series of Eqs. 5.20, 5.32, and 5.38, have to be truncated
for numerical calculations. The accuracy of the resulting calculations depends on
the numbers of terms retained from the infinite series of v(x, t) and the rate of
convergence of these series.
5.3 Problems
Problem 5.1 The continuous beam in Example 5.1 is subjected to the initial
conditions v 0 (x) = ϕ n (x) = sin
n π x/l
and ˙
v 0 (x) = 0. Find the free vibration
solution of the beam.
Problem 5.2 The continuous beam in Example 5.1 is subjected to the harmonic
forcing function f (x, t) = q sin(ν t) with frequency ν > 0 and amplitude q > 0.
Find the forced vibration solution of the beam.
Problem 5.3 Consider a cantilever of length l with constant stiffness E I and
constant mass per unit length m. Find the first five modal frequencies and shapes
and plot the modal shapes. View the cantilever as a flexural beam. Assume l = 1,
EI = 1, and m = 1.
Hint: The following steps can be used for solution: (1) Use the general expression
of ϕ(x) in Eq. 5.8. (2) Impose the boundary conditions at the fixed end, and use
resulting relationships to eliminate two constants from the expression of ϕ(x).
(3) Impose the boundary conditions at the free end to obtain a linear homogeneous
system of equations in the remaining two constants in the expression of ϕ(x). Since
the trivial solution is not possible, set the determinant of this system zero. The
resulting equation is
cos(β l) cosh(β l) = −1, or cos(β l) +
1
cosh(β l)
= 0,
where β is defined in Eq. 5.8. The first five roots of the equation give the first five
modal frequencies. (4) Use any of the 2 equations imposing the boundary conditions
at the free end to construct modal shapes.
Problem 5.4 A simple support is added to the cantilever of the previous problem.
Find the first five modal frequencies and shapes. Plot the modal shapes.
5 Continuous Systems
v(x, t) =
∞
n=1
sin
(2 n − 1) π
2 l
x
2
l
l
0
v 0 (x) ϕ r (x) dx
cos(ω n t)
+
2
ω n l
l
0
˙
v 0 (x) ϕ r (x) dx
sin(ω n t)
.
(5.41)
We conclude with the observation that the infinite series representations of the
solutions v(x, t), e.g., the series of Eqs. 5.20, 5.32, and 5.38, have to be truncated
for numerical calculations. The accuracy of the resulting calculations depends on
the numbers of terms retained from the infinite series of v(x, t) and the rate of
convergence of these series.
5.3 Problems
Problem 5.1 The continuous beam in Example 5.1 is subjected to the initial
conditions v 0 (x) = ϕ n (x) = sin
n π x/l
and ˙
v 0 (x) = 0. Find the free vibration
solution of the beam.
Problem 5.2 The continuous beam in Example 5.1 is subjected to the harmonic
forcing function f (x, t) = q sin(ν t) with frequency ν > 0 and amplitude q > 0.
Find the forced vibration solution of the beam.
Problem 5.3 Consider a cantilever of length l with constant stiffness E I and
constant mass per unit length m. Find the first five modal frequencies and shapes
and plot the modal shapes. View the cantilever as a flexural beam. Assume l = 1,
EI = 1, and m = 1.
Hint: The following steps can be used for solution: (1) Use the general expression
of ϕ(x) in Eq. 5.8. (2) Impose the boundary conditions at the fixed end, and use
resulting relationships to eliminate two constants from the expression of ϕ(x).
(3) Impose the boundary conditions at the free end to obtain a linear homogeneous
system of equations in the remaining two constants in the expression of ϕ(x). Since
the trivial solution is not possible, set the determinant of this system zero. The
resulting equation is
cos(β l) cosh(β l) = −1, or cos(β l) +
1
cosh(β l)
= 0,
where β is defined in Eq. 5.8. The first five roots of the equation give the first five
modal frequencies. (4) Use any of the 2 equations imposing the boundary conditions
at the free end to construct modal shapes.
Problem 5.4 A simple support is added to the cantilever of the previous problem.
Find the first five modal frequencies and shapes. Plot the modal shapes.