2.8 Problems
51
by arguments similar to those used to establish Eq. 2.93. Since x 0 − μ g/ω 2 > 0 by
assumption, the velocity is negative so that the motion continues from right to left
till time t ∗ = π/ω at which the velocity vanishes. The motion in the time interval
[0, t ∗ ] is given by Eq. 2.95. It starts at x 0 and ends at x(t ∗ ) = −(x 0 − 2 μ g/ω 2 ).
The absolute value of the displacement at time t ∗ is smaller than x 0 since |x(t ∗ )| =
|(−x 0 +μ g/ω 2 )+μ g/ω 2 | ≤ μ g/ω 2 ≤ x 0 since −x 0 +μ g/ω 2 ≤ 0 by assumption
and μ g/ω 2 > 0. This observation is consistent with the fact that friction dissipates
energy. The next cycle can be treated in the same manner. The initial displacement
and velocity for this cycle are x(t ∗ ) = −(x 0 − 2 μ g/ω 2 ) and ˙
x(t ∗ ) = 0.
2.8 Problems
Problem 2.1 Repeat the calculations in Example 2.1 by using the coordinate y
whose origin is at the top of the undeformed spring and is pointing upward.
Problem 2.2 The displacement x(t) of a single degree of freedom system is the
solution of
2 ¨
x(t) + 1.6 π ˙
x(t) + 128 x(t) = 10 cos(3t − 2.34), t ≥ 0.
Find the natural frequency ω and damping ratio ζ of the system.
Problem 2.3 Find the particular solution of the undamped SDOF system in
Eq. 2.36 for the forcing function f (t) = q cos(ν t), ν = ω.
Problem 2.4 Find the constants c 1 and c 2 in the expression of the homogeneous
solution x h (t) in Eq. 2.19 from the initial conditions (x 0 , ˙
x 0 ).
Problem 2.5 Complete the calculations following Eq. 2.46 to find the coefficients
α and β in the expression of the particular solution x p (t).
Problem 2.6 Show that Eq. 2.45 is an alternative form of Eq. 2.44.
Hint: Write x p (t) of Eq. 2.44 in the form
x p (t) = x st r d (ν)
− 2 ζ (ν/ω) r d (ν) cos(ν t) +
1 − (ν/ω)
2
r d (ν) sin(ν t)
,
use the notations sin(ϕ) = 2 ζ (ν/ω) r d (ν) and cos(ϕ) =
1 − (ν/ω) 2
r d (ν),
convince yourself that they are admissible, and complete the analysis by employing
a trigonometric identity.
Problem 2.7 Repeat the calculations in Problem 2.3 for the forcing function
f (t) = q cos(ν t) and show that the particular solution has the expression x p (t) =
x st r d (ν) cos(ν t − ϕ) with the notations of Eqs. 2.44 and 2.45.
Problem 2.8 An undamped single degree of freedom system with natural frequency ω = 3 π/2 rad/sec is subjected to a seismic ground acceleration
51
by arguments similar to those used to establish Eq. 2.93. Since x 0 − μ g/ω 2 > 0 by
assumption, the velocity is negative so that the motion continues from right to left
till time t ∗ = π/ω at which the velocity vanishes. The motion in the time interval
[0, t ∗ ] is given by Eq. 2.95. It starts at x 0 and ends at x(t ∗ ) = −(x 0 − 2 μ g/ω 2 ).
The absolute value of the displacement at time t ∗ is smaller than x 0 since |x(t ∗ )| =
|(−x 0 +μ g/ω 2 )+μ g/ω 2 | ≤ μ g/ω 2 ≤ x 0 since −x 0 +μ g/ω 2 ≤ 0 by assumption
and μ g/ω 2 > 0. This observation is consistent with the fact that friction dissipates
energy. The next cycle can be treated in the same manner. The initial displacement
and velocity for this cycle are x(t ∗ ) = −(x 0 − 2 μ g/ω 2 ) and ˙
x(t ∗ ) = 0.
2.8 Problems
Problem 2.1 Repeat the calculations in Example 2.1 by using the coordinate y
whose origin is at the top of the undeformed spring and is pointing upward.
Problem 2.2 The displacement x(t) of a single degree of freedom system is the
solution of
2 ¨
x(t) + 1.6 π ˙
x(t) + 128 x(t) = 10 cos(3t − 2.34), t ≥ 0.
Find the natural frequency ω and damping ratio ζ of the system.
Problem 2.3 Find the particular solution of the undamped SDOF system in
Eq. 2.36 for the forcing function f (t) = q cos(ν t), ν = ω.
Problem 2.4 Find the constants c 1 and c 2 in the expression of the homogeneous
solution x h (t) in Eq. 2.19 from the initial conditions (x 0 , ˙
x 0 ).
Problem 2.5 Complete the calculations following Eq. 2.46 to find the coefficients
α and β in the expression of the particular solution x p (t).
Problem 2.6 Show that Eq. 2.45 is an alternative form of Eq. 2.44.
Hint: Write x p (t) of Eq. 2.44 in the form
x p (t) = x st r d (ν)
− 2 ζ (ν/ω) r d (ν) cos(ν t) +
1 − (ν/ω)
2
r d (ν) sin(ν t)
,
use the notations sin(ϕ) = 2 ζ (ν/ω) r d (ν) and cos(ϕ) =
1 − (ν/ω) 2
r d (ν),
convince yourself that they are admissible, and complete the analysis by employing
a trigonometric identity.
Problem 2.7 Repeat the calculations in Problem 2.3 for the forcing function
f (t) = q cos(ν t) and show that the particular solution has the expression x p (t) =
x st r d (ν) cos(ν t − ϕ) with the notations of Eqs. 2.44 and 2.45.
Problem 2.8 An undamped single degree of freedom system with natural frequency ω = 3 π/2 rad/sec is subjected to a seismic ground acceleration