398
10 Dynamic Direct Stiffness Method
Condition given by Eq. (10.2b) yields
C
cos λL 1
sin λL 1
−
cosh λL 1
sinh λL 1
= D
−
cos λL 2
sin λL 2
+
cosh λL 2
sinh λL 2
(10.17)
Combination of Eqs. (10.16) and (10.17) leads to the frequency equation as
cot λL 1 − coth λL 1 = coth λL 2 − cot λL 2
(10.18)
When both spans are equal, that is L 1 = L 2 , Eq. (10.18) becomes
tan λL = tanh λL
(10.19)
If C = D and L 1 = L 2 = L, it can be shown from Eq. (10.17) that
sin λL sinh λL = 0
or
sin λL = 0
(10.20)
Therefore, for a two-span beam having equal span length, there are two frequency
equations given by Eqs. (10.19) and (10.20), respectively. The lowest frequency is
corresponding to the lowest root of Eq. (10.20), that is, λL = π. Usually, the odd
modes will correspond to the mode shape given by Eq. (10.20), that is, same as that
of simply supported beam, and the even modes will correspond to Eq. (10.19). First
four mode shapes have been shown in Fig. 10.2.
10.3 Methods Analogous to Classical Methods in Statical
Analysis
Free vibration analysis of indeterminate frames can be done by methods, which
are similar to classical methods adopted in statical analysis. Hence, these methods
have been given identical names as the classical methods, such as three-moment
theorem, moment distribution method and so on. Here, we limit our discussion to
the three-moment theorem only.
Let us consider a uniform simply supported beam of span L n , mass m n per unit
length and flexural rigidity E n I n . Supports at two ends are indicated as (n −1) and
n as shown in Fig. 10.3. The end conditions for the beam are
y(0, t) = y(L n , t) = M(0, t) = 0
(10.21)
10 Dynamic Direct Stiffness Method
Condition given by Eq. (10.2b) yields
C
cos λL 1
sin λL 1
−
cosh λL 1
sinh λL 1
= D
−
cos λL 2
sin λL 2
+
cosh λL 2
sinh λL 2
(10.17)
Combination of Eqs. (10.16) and (10.17) leads to the frequency equation as
cot λL 1 − coth λL 1 = coth λL 2 − cot λL 2
(10.18)
When both spans are equal, that is L 1 = L 2 , Eq. (10.18) becomes
tan λL = tanh λL
(10.19)
If C = D and L 1 = L 2 = L, it can be shown from Eq. (10.17) that
sin λL sinh λL = 0
or
sin λL = 0
(10.20)
Therefore, for a two-span beam having equal span length, there are two frequency
equations given by Eqs. (10.19) and (10.20), respectively. The lowest frequency is
corresponding to the lowest root of Eq. (10.20), that is, λL = π. Usually, the odd
modes will correspond to the mode shape given by Eq. (10.20), that is, same as that
of simply supported beam, and the even modes will correspond to Eq. (10.19). First
four mode shapes have been shown in Fig. 10.2.
10.3 Methods Analogous to Classical Methods in Statical
Analysis
Free vibration analysis of indeterminate frames can be done by methods, which
are similar to classical methods adopted in statical analysis. Hence, these methods
have been given identical names as the classical methods, such as three-moment
theorem, moment distribution method and so on. Here, we limit our discussion to
the three-moment theorem only.
Let us consider a uniform simply supported beam of span L n , mass m n per unit
length and flexural rigidity E n I n . Supports at two ends are indicated as (n −1) and
n as shown in Fig. 10.3. The end conditions for the beam are
y(0, t) = y(L n , t) = M(0, t) = 0
(10.21)