Free vibration of damped structures (Case 3: 𝐜𝟐 − 𝟒𝐤𝐌 < 𝟎)
In this case, the motion is said to be underdamped, or simply damped. The values of m1 and
m2 in equation 1.9 are the conjugate complex numbers
𝑚 =−
𝑐
1
(4𝑘𝑀 − 𝑐 ) 𝑤ℎ𝑒𝑟𝑒 𝑖 = √−1
∓𝑖
2𝑀
2𝑀
𝑚 =−
𝑐
∓ 𝑖𝜔 … … 1.13
2𝑀
Where ωd is sometimes called the circular natural frequency with damping and is expressed in
radians per second. The equation of motion now becomes
𝑣 = 𝑒𝑥𝑝[−(𝑐 ⁄2𝑀 )𝑡][𝐴 𝑒𝑥𝑝(𝑖𝜔 𝑡) + 𝐴 𝑒𝑥𝑝(−𝜔 𝑡)] … … 1.14
Expressing the terms in the equation in trigonometric functions and simplifying, we have
𝑣 = 𝐶 𝑒𝑥𝑝(− 𝑐𝑡⁄2𝑀 )𝑠𝑖𝑛(𝜔 𝑡 + 𝛼) … … 1.15
Or in the form
𝑣 = 𝐷 𝑒𝑥𝑝(− 𝑐𝑡 ⁄2𝑀 )𝑐𝑜𝑠(𝜔 𝑡 + 𝛽) … … 1.16
Where C, D, α and β are arbitrary constants to be determined from the initial conditions. The
motion described by any of these equations is known as a damped free vibration and a typical
displacement-time plot is shown in figure 1.5. It is of interest to note that the underdamped
system oscillates about the neutral position, with a constant circular frequency ωd.
Figure 1.5: Damped free vibration
Note that the motion is not truly periodic, because the multiplying factor 𝑒𝑥𝑝(− 𝑐𝑡⁄2𝑀) is
continuously decreasing. Figure 1.5 shows that the displacement x is zero at intervals, i.e. the
mass M passes through the position of static equilibrium at intervals. From equation 1.15, x is
zero whenever 𝑠𝑖𝑛(𝜔 𝑡 + 𝛼) is zero, i.e. whenever
𝜔 𝑡 + 𝛼 = 𝜋 + 𝑛𝜋
Or
1
𝑡=
1
𝜋
(𝜋 − 𝛼) + 𝑛
… … 1.17
𝜔
𝜔
That is, x is zero at regular intervals of π/ωd. The period Td of the damped free vibration is twice
this interval, i.e.
𝑇 =
2𝜋
… … 1.18
𝜔
The frequency fd of the vibration is the reciprocal of Td
𝑓 =
1
𝜔
=
… … 1.19
𝑇
2𝜋
And the free-vibration frequency of the damped system may be given as
𝜔 =𝜔 1−𝜀
The natural period of damped vibration, Td is related to the natural period Tn without damping
by
𝑇 =
𝑇
√1 − 𝜀
Referring to figure 1.5, the amplitudes at points M, N, P… occur at successive intervals of Td.
From equations 1.15 or 1.16 the ratios of the amplitudes are constant:
𝑥
𝑥
𝑒𝑥𝑝(− 𝑐𝑡⁄2𝑀)
=
=⋯=
= 𝑒𝑥𝑝(𝑐𝑇 ⁄2𝑀)
𝑥
𝑋
𝑒𝑥𝑝[−𝑐(𝑡 + 𝑇 )⁄2𝑀 ]
A convenient way to determine the amount of damping present in the systems is to measure
the rate of decay of free vibration. The larger the damping, the greater will be the rate of decay.
Considering two successive peaks such as 𝑣 and 𝑣
(𝑛 + 1)
which occur at times 𝑛
and
, respectively, then the ratio of these two successive values is given by
𝑣
𝑣
= 𝑒𝑥𝑝(2𝜋𝜀 𝜔 ⁄𝜔𝑑)
Where the logarithmic decrement of damping, δ, will be given as
𝛿 = 𝑙𝑛
𝑣
𝑣
=
2𝜋𝜀
√1 − 𝜀
Further, if we know the amplitudes of two successive cycles, we can find the amplitude after p
cycles from
𝑣
=
𝑣
𝑣
2
𝑣
Alternatively, the natural logarithm of the ratio is called the logarithmic decrement, i.e.
𝑙𝑜𝑔𝑎𝑟𝑖𝑡ℎ𝑚𝑖𝑐 𝑑𝑒𝑐𝑟𝑒𝑚𝑒𝑛𝑡 𝛿 =
𝑐𝑇
… 1.20
2𝑀
The logarithmic decrement is often used as an indication of the damping capacity of a structure.
Free vibration of undamped structures (Case 4: 𝐜 = 𝟎)
If c = 0, the system is undamped, i.e. there is no damping on the SDOF system. From equation
1.9
𝑚 , 𝑚 = ∓𝑖
𝑘
= ∓𝑖𝜔 … 1.21
𝑚
So that the equation of motion becomes
𝑢(𝑡) = 𝐴 𝑒
+𝐴 𝑒
… 1.22
Or in terms of trigonometric functions,
𝑢(𝑡) = 𝐴 sin 𝜔𝑡 + 𝐵 cos 𝜔𝑡 … 1.23
Equation 1.23 gives the displacement of the mass at any time t.
Since
𝐴 sin 𝜔𝑡 + 𝐵 cos 𝜔𝑡 = 𝐴 𝑠𝑖𝑛(𝜔𝑡 + 2𝜋) + 𝐵 𝑐𝑜𝑠(𝜔𝑡 + 2𝜋)
= 𝐴 sin 𝜔 𝑡 +
2𝜋
2𝜋
+ 𝐵 cos 𝜔 𝑡 +
𝜔
𝜔
The displacement 𝑢(𝑡) of equation 1.23 has the same value at time t as at time 𝑡 +
. We
therefore say the equation is periodic (Figure 1.6). The period, commonly given in seconds, is
denoted by T. thus
𝑇=
2𝜋
… 1.24
𝜔
Where
𝜔=
𝑔
𝑚𝑔
, 𝑎𝑛𝑑 𝛿 =
𝛿
𝑘
The reciprocal of T is the natural frequency,
𝑓=
1
𝜔
1 𝑘
=
=
… 1.25
𝑇 2𝜋 2𝜋 𝑚
3
Form equation 1.25 shows that the period and frequency depend only on mass m and spring
stiffness k. The displacement at a particular time t and the amplitude of the vibration depend
on the initial displacement and velocity, but these initial quantities do not affect the period and
frequency.
The quantity ω is called the natural angular frequency.
The two constants A and B in the general solution (equation 1.23) can be determined from the
given initial conditions of motion. For example, the initial conditions 𝑢(0) = 𝑢 𝑎𝑛𝑑 𝑢̇ (0) =
𝑢̇ will give 𝐵 = 𝑢 𝑎𝑛𝑑 𝐴 = 𝑢̇ ⁄𝜔 .
Replacing the values of A and B in equation 1.23, we have the equation of motion as
𝑢(𝑡) =
𝑢̇
sin 𝜔𝑡 + 𝑢 cos 𝜔𝑡
𝜔
The general equation may be written in an alternative form if we use the identity as
sin 𝛼 cos 𝛽 + cos 𝛼 sin 𝛽 = sin(𝛼 + 𝛽)
The solution will therefore be
𝑢(𝑡) = 𝐶 sin(𝜔𝑡 + 𝛼) … 1.26
Where the constants
𝐶=
𝐴 +𝐵
𝛼 = tan
𝐴
𝐵
Using the values of A and B, we have:
𝑢(𝑡) = 𝜌 cos(𝜔𝑡 + 𝛼) … … 1.27
Where ρ is the amplitude of displacement and α is the phase angle given by:
𝜌=
𝑢 +
𝑢̇
𝜔
Where 𝑢 and 𝑢̇ are the initial displacement and velocity of the system respectively.
The natural vibration properties ω, T and f depend only on the mass and stiffness of the
structure. Stiffer of two SDF systems having the same mass will have the higher natural
frequency and the shorter natural period. Similarly, the heavier (more mass) of two structures
having the same stiffness will have the lower natural frequency and the longer natural period.
The qualifier natural is used to emphasize the fact that these are natural properties of the system
when it is allowed to vibrate freely without any external excitation. Because the system is
linear, these vibration properties are independent of the initial displacement and velocity.
4
The phase angle determines the amount by which u(t) lags behind the function cos ωt. The
general appearance of the undamped free vibration is shown in Figure 1.6.
Figure 1.6: Undamped free vibration
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