2.1 The following terms :a. Spring constantSpring Constant or
force constant is defined as the applied force if the displacement
in the spring is unityb. Coefficient of subgrade reactionDefinition
:1. The ratio of load per unit area on soil to the corresponding
deformation2. Load per unit area of horizontal surface of a mass of
soil, to corresponding settlement of the surfacec. Undamped natural
circular frequencyd. Undamped natural frequencye. Periodf.
ResonanceThe concept in which two or more equivalent dot formulas
for the same arrangement of atoms (resonance structures) are
necessary to describe the bonding in a molecule or ion.g. Critical
damping coefficienth. Damping ratioThe damping ratio is a
dimensionless measure describing how oscillations in a system decay
after a disturbance.i. Damped natural frequencyThe natural
frequency is the rate at which an object vibrates when it is not
disturbed by an outside force
2.2 A machine foundation can be idealized to a mass-spring
system, as shown in Figure 2.4.GivenWeight of machine + foundation
= 400 kNSpring constant = 100,000 kN/mDetermine the natural
frequency of undamped free vibration of this foundation and the
natural period.Solution : the natural frequency of undamped free
vibration :
Natural period :
2.3 Refer to Problem 2.2, What would be the static deflection zs
of this foundation?Solution :
2.4 Refer to Example 2.3. For this foundation let time t = 0, z
= z0 = 0.z = u0 = 0.a. Determine the natural period T of the
foundation.b. Plot the dynamic force on the subgrade of the
foundation due to the forced part of the response for time t =0 to
t = 2T.c. Plot the dynamic force on the subgrade of the foundation
due to the free part of the response for t = 0 to 2T.d. Plot the
total dynamic force on the subgrade [that is, the algebraic sum of
(b) and (c)]. Hint: Refer to Eq. (2.33).
Example 2.3A machine foundation can be idealized as a
mass-spring system. This foundation can be subjected to a force
that can be given as Q (kN) = 35.6 sin t .Given f= 13.33 Hz Weight
of the machine + foundation = 178 kNSpring constant= 70,000
kN/mDetermine the maximum and minimum force transmitted to the
subgrade.
Solution :
But
Thus
Maximum force on the subgrade = 178 + 102.18 = 280.18 kNMinimum
force on the subgrade = 178 102.18 = 75.82 kN
a. The natural period T of the foundation.
b. Plot the dynamic force on the subgrade of the foundation due
to the forced part of the response for time t =0 to t = 2T.
c. Plot the dynamic force on the subgrade of the foundation due
to the free part of the response for t = 0 to 2T.
d. Plot the total dynamic force on the subgrade [that is, the
algebraic sum of (b) and (c)]. Hint: Refer to Eq. (2.33).
2.5 A foundation of mass m is supported by two springs attached
in series. (See Figure P2.5). Determine the natural frequency of
the undamped free vibration.
Figure P2.5Solution :
The natural frequency of the undamped free vibration :
