52
2 Single Degree of Freedom (SDOF) Systems
a(t) =
n
k=1
[a k cos(ν k t) + b k sin(ν k t)] , 0 ≤ t ≤ τ,
where n = 100, ν k = 2 π k/τ, τ = 10 s, a k = sin(π/k), and b k = sin
3 π/(2 k)
.
Calculate and plot the Fourier transforms of the ground acceleration and the
displacement x(t) of the system. Compare your results with those of Example 2.14.
Consider as in this example only the particular solution.
Problem 2.9 Repeat the calculation in the previous problem and assume that
oscillator is damped with damping ratio ζ = 0.05. Consider only the steady-state
solution. Compare the Fourier transforms of the damped and undamped oscillators.
Problem 2.10 Consider an undamped SDOF system with stiffness k = 10 N/m
and mass m = 10 kg which is subjected to the harmonic force
f (t) =
q cos(ν t), 0 ≤ t ≤ t 1
0, t 1 < t,
where q = 10 N, and t 1 = 100 s.
Determine and plot the motions of mass m for (a) ν = 0.4 rad/sec, (b) ν =
0.999 rad/sec, and (c) ν = 0.9 rad/sec. Assume zero initial conditions. Comment on
your results.
Problem 2.11 Using Duhamel’s integral, determine the response of the system in
Problem 2.10 subjected to the forcing function
f (t) =
q 1 , 0 ≤ t ≤ t 1
q 2 , t 1 < t,
where q 1 = 10 N, q 2 = 15 N, t 1 = 10 s.
Problem 2.12 Determine the motion of an undamped SDOF system subjected to
the forcing function f (t) = q
exp(−at) − exp(−bt)
, a, b > 0. The oscillator is
at rest at the initial time. Plot the oscillator displacement for a/ω = 0.05, 0.1, and
0.5. Assume ω = 1, stiffness k = 1, b = 0.1, and q = 1.
Problem 2.13 Repeat the analysis of the previous problem and assume that the
SDOF is damped with ζ = 0.05.
Problem 2.14 Use the FD method to find the displacement of the SDOF system in
Example 2.15 under the forcing function f (t) = sin(ν t), where ν = ω.
Problem 2.15 Find the displacement function of the SDOF system in Problem 2.8
by the Fourier series method of Sect. 2.6.2.
2 Single Degree of Freedom (SDOF) Systems
a(t) =
n
k=1
[a k cos(ν k t) + b k sin(ν k t)] , 0 ≤ t ≤ τ,
where n = 100, ν k = 2 π k/τ, τ = 10 s, a k = sin(π/k), and b k = sin
3 π/(2 k)
.
Calculate and plot the Fourier transforms of the ground acceleration and the
displacement x(t) of the system. Compare your results with those of Example 2.14.
Consider as in this example only the particular solution.
Problem 2.9 Repeat the calculation in the previous problem and assume that
oscillator is damped with damping ratio ζ = 0.05. Consider only the steady-state
solution. Compare the Fourier transforms of the damped and undamped oscillators.
Problem 2.10 Consider an undamped SDOF system with stiffness k = 10 N/m
and mass m = 10 kg which is subjected to the harmonic force
f (t) =
q cos(ν t), 0 ≤ t ≤ t 1
0, t 1 < t,
where q = 10 N, and t 1 = 100 s.
Determine and plot the motions of mass m for (a) ν = 0.4 rad/sec, (b) ν =
0.999 rad/sec, and (c) ν = 0.9 rad/sec. Assume zero initial conditions. Comment on
your results.
Problem 2.11 Using Duhamel’s integral, determine the response of the system in
Problem 2.10 subjected to the forcing function
f (t) =
q 1 , 0 ≤ t ≤ t 1
q 2 , t 1 < t,
where q 1 = 10 N, q 2 = 15 N, t 1 = 10 s.
Problem 2.12 Determine the motion of an undamped SDOF system subjected to
the forcing function f (t) = q
exp(−at) − exp(−bt)
, a, b > 0. The oscillator is
at rest at the initial time. Plot the oscillator displacement for a/ω = 0.05, 0.1, and
0.5. Assume ω = 1, stiffness k = 1, b = 0.1, and q = 1.
Problem 2.13 Repeat the analysis of the previous problem and assume that the
SDOF is damped with ζ = 0.05.
Problem 2.14 Use the FD method to find the displacement of the SDOF system in
Example 2.15 under the forcing function f (t) = sin(ν t), where ν = ω.
Problem 2.15 Find the displacement function of the SDOF system in Problem 2.8
by the Fourier series method of Sect. 2.6.2.