Dynamics of Structures, 5th Edition Solution Manual

Name: Dynamics of Structures, 5th Edition Solution Manual
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Author: Benjamin White

Dynamics of Structures, 5th Edition Solution Manual is your ultimate textbook solutions guide, providing answers to the most difficult questions.

Benjamin White

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1
CHAPTER 1
Problem 1.1
Starting from the basic definition of stiffness, determine
the effective stiffness of the combined spring and write the
equation of motion for the spring–mass systems shown in
Fig. P1.1.
Figure P1.1
Solution:
If k e is the effective stiffness,
f kS e u=
f S f Sk u2
k u1
k1
k2
u
Equilibrium of forces: f k k uS = +( 1 )2
Effective stiffness: + 2k f u k ke S= = 1
Equation of motion: mu k u p te ( )+ =

Problem 1.2
2
Starting from the basic definition of stiffness, determine
the effective stiffness of the combined spring and write the
equation of motion for the spring–mass systems shown in
Fig. P1.2.
Figure P1.2
Solution:
If k e is the effective stiffness,
f k u= (S e a)
f S
k1 k2
u
If the elongations of the two springs are 1u and ,2u
u u u= + (b1 2 )
Because the force in each spring is fS ,
f k uS = 1 1 f k uS = 2 2 (c)
Solving for u1 and u2 and substituting in Eq. (b) gives
f 1 1 1f f
k k k
S
e
S S
= +
1 2
⇒ 1 2k k ke
= + ⇒
k k k
k k
e = +
1 2
1 2
Equation of motion: mu + ke u p(t= ) .

3
Problem 1.3
Starting from the basic definition of stiffness, determine
the effective stiffness of the combined spring and write the
equation of motion for the spring–mass systems shown in
Fig. P1.3.
Figure P1.3
Solution:
k1
k2
k 3
m
⇓
Figure P1.3a
k1 k 3k2+1
m
⇓ Figure P1.3b
u
m
k e
Figure P1.3c
This problem can be solved either by starting from the
definition of stiffness or by using the results of Problems
P1.1 and P1.2. We adopt the latter approach to illustrate
the procedure of reducing a system with several springs to
a single equivalent spring.
First, using Problem 1.1, the parallel arrangement of
k1 and k2 is replaced by a single spring, as shown in Fig.
1.3b. Second, using the result of Problem 1.2, the series
arrangement of springs in Fig. 1.3b is replaced by a single
spring, as shown in Fig. 1.3c:
1 1 1
1 2 3k k k ke
= + +
Therefore the effective stiffness is
k k k k
k k k
e = +
+ +
( )1 2 3
1 2 3
The equation of motion is mu + =ke u p(t ) .

4
Problem 1.4
Derive the equation governing the free motion of a simple
pendulum that consists of a rigid massless rod pivoted at
point O with a mass m attached at the tip (Fig. P1.4).
Linearize the equation, for small oscillations, and
determine the natural frequency of oscillation.
Figure P1.4
Solution:
1. Draw a free body diagram of the mass.
m
O
θ
L
T
mg sin
θ mg cos
θ
2. Write equation of motion in tangential direction.
Method 1: By Newton’s law.
- =m mg sin
θ a
- =m mg s in
θ L
θ (a)
mL
θ
θ+ =mg sin 0
This nonlinear differential equation governs the motion for
any rotation
θ.
Method 2: Equilibrium of moments about O yields
2 g sinmL m L
θ
θ= -
or
g sin 0mL m
θ +
θ =
3. Linearize for small
θ.
For small
θ, sin
θ ≈
θ , and Eq. (a) becomes
gmL m
θ + 0
θ =
g 0
L
θ
θ+ =
( )
| │
( )
 (b)
4. Determine natural frequency.
g
ωn = L

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