STUDY NOTES & REFERENCE GUIDE
ALL 12 CHAPTERS COVERED
SOLUTIONS MANUAL
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CONTENTS
Chapter Page
2 ............................................................................................................................. 1
4 9
5 25
6 41
8 ...........................................................................................................................51
9 ...........................................................................................................................61
10 ..........................................................................................................................65
11 ..........................................................................................................................79
12 ..........................................................................................................................91
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StudyNexa ‐ Stuvia | Page 2 of 105
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Chapter 2
2.1 a. Spring constant, k : The change in the force per unit length change of the spring.
b. Coefficient of subgrade reaction, k:
k
Spring constant divided by the foundation contact area, k =
A
c. Undamped natural circular frequency: n = rad/s
W
where m = mass =
g
d. Undamped natural frequency: f n = (in Hz)
2 m
Note: Circular frequency defines the rate of oscillation in term of radians per unit
time; 2π radians being equal to one complete cycle of rotation.
e. Period, T: The time required for the motion to begin repeating itself.
n
f. Resonance: Resonance occurs when =1
g. Critical damping coefficient: cc = 2 k
mW
where k = spring constant; m = mass =
g
h. c c
Damping ratio: D = =
cc 2 km
where c = viscous damping coefficient; cc = critical damping coefficient
i. Damped natural frequency:
d = n 1 − D2
fd = 1 − D2 fn
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2.2 Weight of machine + foundation, W = 400 kN
Spring constant, k = 100,000 kN/m
W 400 kN
Mass of the machine + foundation, m = = = 40.77
g 9.81 m s2
Natural frequency of undamped free vibration is [Eq. (2.19)]
1 k
fn = = = 7.88 Hz
2 m 2 40.77
1 1
From Eq. (2.18), T = = = 0.127 s
fn 7.88
2.3 Weight of machine + foundation, W = 400 kN
Spring constant, k = 100,000 kN/m
Static deflection of foundation is [Eq. (2.2)]
W 400
z = = = 4 10−3 m = 4 mm
s
k 100,000
2.4 External force to which the foundation is subjected, Q = 35.6sint kN
f = 13.33 Hz
Weight of the machine + foundation, W = 178 kN
Spring constant, k = 70,000 kN/m
For this foundation, let time t = 0, z = z0 = 0, zɺ = v0 = 0
W 178 kN
a. Mass of the machine + foundation, m = = = 18.145
g 9.81 m s2
k
n = = = 62.11 rad/s
m
2 2
T= = = 0.101 s
n 62.11
b. The frequency of loading, f = 13.33 Hz
= 2 f = 2 (13.33) = 83.75 rad/s
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© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
StudyNexa ‐ Stuvia | Page 4 of 105
ALL 12 CHAPTERS COVERED
SOLUTIONS MANUAL
StudyNexa ‐ Stuvia | Page 1 of 105
, STUDY NOTES & REFERENCE GUIDE
CONTENTS
Chapter Page
2 ............................................................................................................................. 1
4 9
5 25
6 41
8 ...........................................................................................................................51
9 ...........................................................................................................................61
10 ..........................................................................................................................65
11 ..........................................................................................................................79
12 ..........................................................................................................................91
@
@sseeisismmicicisisoolalatitoionn
StudyNexa ‐ Stuvia | Page 2 of 105
, STUDY NOTES & REFERENCE GUIDE
Chapter 2
2.1 a. Spring constant, k : The change in the force per unit length change of the spring.
b. Coefficient of subgrade reaction, k:
k
Spring constant divided by the foundation contact area, k =
A
c. Undamped natural circular frequency: n = rad/s
W
where m = mass =
g
d. Undamped natural frequency: f n = (in Hz)
2 m
Note: Circular frequency defines the rate of oscillation in term of radians per unit
time; 2π radians being equal to one complete cycle of rotation.
e. Period, T: The time required for the motion to begin repeating itself.
n
f. Resonance: Resonance occurs when =1
g. Critical damping coefficient: cc = 2 k
mW
where k = spring constant; m = mass =
g
h. c c
Damping ratio: D = =
cc 2 km
where c = viscous damping coefficient; cc = critical damping coefficient
i. Damped natural frequency:
d = n 1 − D2
fd = 1 − D2 fn
@ 1
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© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
StudyNexa ‐ Stuvia | Page 3 of 105
, STUDY NOTES & REFERENCE GUIDE
2.2 Weight of machine + foundation, W = 400 kN
Spring constant, k = 100,000 kN/m
W 400 kN
Mass of the machine + foundation, m = = = 40.77
g 9.81 m s2
Natural frequency of undamped free vibration is [Eq. (2.19)]
1 k
fn = = = 7.88 Hz
2 m 2 40.77
1 1
From Eq. (2.18), T = = = 0.127 s
fn 7.88
2.3 Weight of machine + foundation, W = 400 kN
Spring constant, k = 100,000 kN/m
Static deflection of foundation is [Eq. (2.2)]
W 400
z = = = 4 10−3 m = 4 mm
s
k 100,000
2.4 External force to which the foundation is subjected, Q = 35.6sint kN
f = 13.33 Hz
Weight of the machine + foundation, W = 178 kN
Spring constant, k = 70,000 kN/m
For this foundation, let time t = 0, z = z0 = 0, zɺ = v0 = 0
W 178 kN
a. Mass of the machine + foundation, m = = = 18.145
g 9.81 m s2
k
n = = = 62.11 rad/s
m
2 2
T= = = 0.101 s
n 62.11
b. The frequency of loading, f = 13.33 Hz
= 2 f = 2 (13.33) = 83.75 rad/s
@ 2
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© 2017 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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