Principles-of-Soil-Dynamics-3rd-Edition-Das-Solutions-Manual.pdf

Principles of Soil Dynamics 3rd Edition Das SOLUTIONS MANUAL Full clear download (no formatting errors) at: https://testbankreal.com/download/principles-soi https://testbankreal.com/dow nload/principles-soil-dynamics-3rd-e l-dynamics-3rd-editionditiondas-solutions-manual/

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 : Spring constant divided by the foundation contact area, k

c.

Undamped natural circular frequency fr equency:: where m = mass =

d.

k

n

m

rad/s

W g

Undamped natural frequency: frequency: f n

k

1 2

(in Hz)

k A

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i.

Damped natural frequency fr equency:: d

n

1 D 2

f d 1 D 2 f

n









2.2

Weight of machine + foundation, W = 400 kN Spring constant, k = 100,000 kN/m 400 W kN Mass of the machine + foundation, m = = = 40.77 g

9.81

Natural Natural frequenc frequency y of of undampe undamped d free vibration vibration is [Eq. [Eq. (2.19)] (2.19)] k 1 1 100,000 = 7.88 Hz f n = 2 40.77 m 2

From Eq. (2.18), T =

1

f n

=

1

= 0.127 s

7.88

m s

2









Force due to forced part, F k 1

Q k 0

1

2

(70, 000)

sin t 2

35.6 35.670, 000 2

sin(83.75t) 2

1 83.75 62.11 43.51sin(83.75 t ) kN See the plot below for F for F 1 vs. t









The plot of variation of the dynamic force on the subgrade of the foundation f oundation due to (a) forced part, (b) free part, and (c) total of the response for time t = 0 to t = 2T 2T is shown in the figure above (Part b).









2.5

The natural frequency of the undamped free vibration of the spring mass system is given by f n

k eq

1 2

m

where k eq = equivalent stiffness of the spring system

For springs attached in series, the equivalent stiffness is given by k 1 k 2 1 1 1 1 , or k eq k 1 k 2 k eq k 1 k 2

The natural frequency of the given undamped free vibration spring mass system is k 1k 2 1

f

n

2π

k 1 k 2

m











Now, the given given spring system can be reduced reduced to three springs springs in series.





















a.

Damped natural frequency [Eq. (2.67)]

f d 1 D 2f n

D

c cc

c 2 km

11.67

= 0.031

2 2600 13.76











Z 0 Z 5

e 0.974 = 2.649;

Z 5

25 2.649

= 9.44 mm

After 5 cycles of disturbance, the amplitude amplitude of vibration = 9.44 mm











2.10

Q0 = 6.7 kN = 3100 rad/min = 51.67 rad/s Weight of machine + foundation, W = 290 kN Spring constant, k = 875 MN/m = 875,000 kN/m ω

Natural Natural angular angular frequency, frequency, n

k

875,000 103

m

290 10 9.81

3

= 172.04 rad/s

































Principles of Soil Dynamics, 3E

Das/Luo














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2.7Free Rotating-Mass-Type Excitation 2.3 Vibration of a Spring-Mass ∗

∗

In many cases of foundation equipment, vertical vibration of foundation is produced by counter-rotating masses. Since horizontal forces on the foundation at any instant cancel, the net vibrating force on the foundation can be determined to be equal to 2meeω2sinωt (where me = mass of each counter-rotating element, e = eccentricity, and ω = angular frequency of the masses). In such cases, the equation of motion with viscous damping can be modified to the form:












Das/Luo

2.72.3 Rotating-Mass-T Rotating-MassType Excitation Cont’d Free Vibration of a Spring-Mass

∗

The angular resonant frequency for rotating-mass-type rotating-mass-typ e excitation can be obtained as:












Das/Luo

2.8 Ratio Rati o 2.3 Determination Free Vibration of Damping a Spring-Mass ∗

The damping ratio D can be determined from free and forced vibration tests on a system. In a free vibration test, the system is displaced from its equilibrium position, after which the amplitudes of displacement are recorded with time:

∗

If D is small, then

∗

Where Z n = the peak amplitude of the nth cycle. Thus,












Das/Luo

2.9 Instrument 2.3 Vibration-Measuring Free Vibration of a Spring-Mass ∗

Based on the theories of vibration presented in the preceding sections, it is now possible to study the principles of a vibration-m vibration-measuring easuring instrument, as shown below. ∗

∗

The instrument consists of a spring-massdashpot system. It is mounted on a vibrating base. The relative motion of the mass m with respect to the vibrating base is monitored. Neglecting the transients let the absolute motion of the mass be given as:












Das/Luo

2.9 Instrument 2.3 Vibration-Measuring Free Vibration of a Spring-Mass Cont’d ∗

∗

So, the equation of motion for the mass can be written as:

If the natural frequency of the instrument ωn is small and ω / ωn is large, then for practically all values of D, the magnitude of Z /Z′ is about 1. Hence the instrument works as a velocity pickup.














Das/Luo

System with Two Degrees of Freedom 2.3 Free Vibration of a Spring-Mass 2.10 Vibration of a Mass-Spring System ∗

A mass-spring with two degrees of freedom is shown on the right. If the masses m and m are displaced












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2.10 of a Mass-Spring System 2.3Vibration Free Vibration of a Spring-Mass Cont’d ∗

Let












Das/Luo

2.11 Translation Rotation of 2.3Coupled Free Vibration of a and Spring-Mass a Mass-Spring System (Free Vibration) ∗

The figure below shows a mass-spring system that will undergo translation translatio n and rotation. The equations of motion of the mass m can be given as












Das/Luo

2.11 Translation Rotation of 2.3Coupled Free Vibration of a and Spring-Mass a Mass-Spring System (Free Vibration) Cont’d ∗

The general equations of motion can be given as:












Das/Luo

2.3 Free Vibration of a Spring-Mass Principles of Soil Dynamics 3rd Edition Das SOLUTIONS MANUAL Full clear download (no formatting errors) at: https://testbankreal.com/dow https://testba nkreal.com/download/princi nload/principles-soil-dynamic ples-soil-dynamics-3rd-edition-das-so s-3rd-edition-das-solutions-manua lutions-manual/ l/ principles of soil dynamics by braja m. das pdf principles of soil dynamics 3rd edition pdf principles of foundation engineering engineering 7th edition solution solution manual

Principles-of-Soil-Dynamics-3rd-Edition-Das-Solutions-Manual.pdf

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