Single Degree of Freedom (SDOF) system υ(τ)
1 0 0 1 0 1 0 1 0 1
k m F(t)
Figure 1: Undamped SDOF system 1111111 0000000 u(t) 0000000 1111111 0000000 1111111 M 0000000 1111111 1111 0000 1111 0000 0000000 1111111 0000000 1111111 0000000 1111111 F(t)
00 11 0 1 00 111 0 011 1 00 11 0 1 00 00 000 1 011 1 00 11 0011 11 0 1 00 11 00 11
Figure 2: Example of overhead water tank that can be modeled as SDOF system 1. Equation of motion (EOM) Mathematical expression defining the dynamic displacements of a structural system. Solution of the expression gives a complete description of the response of the structure as a function of time Derivation of EOM 1. Dynamic Equilibrium (Using D’Alembert’s principle) 2. Principle of Virtual Work 3. Hamilton’s principle (Using Lagrange’s equation) Dynamic Equilibrium D’Alembert’s principle states that a mass develops an inertial force proportional to its acceleration and opposing its motion. (See Figure 3) m¨ u + ku = F (t) Equation of Motion
(1)
for F (t) = 0, the response is termed as free vibration and occurs due to initial excitation. 1
..
mu
m ku
F(t)
Figure 3: Dynamic force equilibrium Free Vibration m¨ u + ku = 0 linear,homogeneous second order differential equation k ⇒ u¨ + u = 0 m s k k ⇒ u¨ + ωn2 u = 0 ωn2 = , ωn = ωn = natural frequency m m
(2)
Solution of Equation 2 will be, u(t) = C1 eıωn t + C2 e−ıωn t = C1 (cos ωn t + ı sin ωn t) + C2 (cos ωn t − ı sin ωn t) = (C1 + C2 ) cos ωn t + ı(C1 − C2 ) sin ωn t
(3)
Applying the initial conditions, u(t)|@t=0 = u0 = C1 + C2 u(t)| ˙ ˙ 0 = ıωn (C1 − C2 ) @t=0 = u
(4)
Substituting Equation 4 into Equation 3, we get, u(t) = u0 cos ωn t +
u˙ 0 sin ωn t ωn
(5)
Again, substituting, u0 = A cos φ u˙ 0 = A sin φ ωn
(6)
into Equation 5, we get, u(t) = A cos φ cos ωn t + A sin φ sin ωn t = A cos(ωn t − φ)
2
(7)
where, A is the amplitude and φ is the phase angle v u µ ¶ u u˙ 0 2 A = tu20 +
Ã
and φ = tan
ωn
−1
u˙ 0 /ωn u0
!
(8)
Free vibration of damped SDOF system Modeling of damping is perhaps one of the most difficult task in structural dynamics. It is still a topic of research in advanced structural dynamics and is derived mostly experimentally. Viscous Damping The most common form of damping is viscous damping. Equation of Motion ..
u
111 000 11 00 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11
k F(t)
m
11111 00000
c
Figure 4: SDOF with viscous damping m¨ u + cu˙ + ku = 0 c k ⇒ u¨ + u˙ + u = 0 m m ⇒ u¨ + 2ξωn u˙ + ωn2 u = 0
(9)
c where, ξ = 2mω is the viscous damping factor. Assuming a solution u(t) = Cest and n substituting in Equation 9, we get,
s2 + 2ξωn s + ωn2 = 0 ⇒s=
−2ξωn ±
µ
= −ξ ±
q
4ξ 2 ωn2 − 4ωn2
¶2
q
ξ 2 − 1 ωn
(10)
Depending on the value of ξ, the nature of s and correspondingly u(t) will be determined, √ √ (−ξ− ξ 2 −1)ωn t (−ξ+ ξ 2 −1)ωn t + C2 e u(t) = C1 e ¸ · √ 2 √ 2 (11) = C1 e (ξ −1)ωn t + C2 e− (ξ −1)ωn t e−ξωn t
3
Case I Under-damped system, 0 < ξ < 1 For ξ < 1, s1 , s2 are complex numbers and given as, µ
¶
q
|ξ 2
s1 s2 = −ξ ± ı
− 1| ωn (12)
Therefore,
µ ı
u(t) = C1 e Considering
√
|ξ 2 −1|ωn t
−ı
√
+ C2 e
¶ |ξ 2 −1|ωn t
e−ξωn t
(13)
q
|ξ 2 − 1|ωn = ωd , Equation 13 can be written as, ³
´
u(t) = C1 eıωd t + C2 e−ıωd t e−ξωn t [(C1 + C2 ) cos ωd t + ı(C1 − C2 ) sin ωd t] e−ξωn t
(14)
where, ωd is referred as damped natural frequency. Substituting (C1 + C2 ) = A cos φ and ı(C1 − C2 ) = A sin φ into Equation 14, we get, u(t) = A cos(ωd t − φ)e−ξωn t
(15)
Applying initial conditions as, u(t)|@t=0 = u0 and u(t)| ˙ ˙ 0 , we get, @t=0 = u "
C1 + C2 = u0
u˙ 0 u0 ξ and ı(C1 − C2 ) = +√ ωd 1 − ξ2
#
Thus for these initial conditions, the response can be written as, Ã
"
#
!
u˙ 0 u0 ξ +√ sin ωd t e−ξωn t u(t) = u0 cos ωd t + 2 ωd 1−ξ
(16)
Case II Critically-damped system, ξ = 1 Critical damping is the minimum damping required to stop the oscillations. s1 , s2 = −ωn The solution is of the form, u(t) = (C1 + C2 t)e−ωn t
(17)
Even here, C1 and C2 can be obtained from the initial conditions given. Case III Over-damped system, ξ > 1 There is no oscillatory motion in an over-damped system. u(t) = (C1 eωd t + C2 e−ωd t )e−ξωn t
(18)
For a over-damped system, higher the values of ξ, the slower the rate of the decay (See Figure 5). 4
Figure 5: Free vibration of under-damped, critically damped and over-damped system
5