CE-5113: DYNAMICS OF STRUCTURES By: Dr. Mohammad Ashraf (engineerashraf@yahoo.com) Office: CE: B109
Department of Civil Engineering, University of Engineering and Technology, Peshawar
Module-3 Single Degree of Freedom System: Free Vibration Response
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Solution of Equation of Motion
For a single degree of freedom system the equation of motion is a linear second order differential equation:
mu&& + cu& + ku = p(t )
Various methods for the solution of equation of motion are: z z z z
Classical Solution Duhamel’s Integral Frequency Domain Method Numerical Methods
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Free Vibration
The vibration of a system excided by initial disturbances (initial displacement or velocity) without any external force is called free vibration
mu&& + cu& + ku = 0
The equation of motion becomes a homogenous 2nd order linear differential equation The study of free vibration response is important because: z z
Many practical systems are excited by initial disturbances The complete solution of force vibration also include free vibration component. Dynamics of Structures
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Undamped Free Vibration
The free vibration response an undamped system is governed by:
mu&& + ku = 0
In reality there is no existence of undamped system, however the its response gives an insight into the nature of damped system. A possible solution of the above 2nd order ordinary differential equation is: λt
u = Ge
Substituting the solution into equation of motion: Gλ2 me λt + Gke λt = 0 ⇒ λ2 m + k = 0 ⇒ λ2 + ω 2 = 0 ⇒ λ = ±iω where _ ω =
k m
called circular frequency of system
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Undamped Free Vibration
The general solution of the equation of motion now becomes:
u = G1e iωt + G2 e − iωt = A cos ωt + B sin ωt
The velocity is obtained by differentiating the above equation:
u& = − Aω sin ω t + Bω cos ωt
The arbitrary constants A and B are determined from the initial conditions i.e. At t = 0, u (0) = uo _ and _ u& (0) = vo v A = uo _ and _ B = o
The solution of undamped free vibration response of single degree of freedom system is, therefore: v u = uo cos ωt + o sin ωt ω
ω
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Undamped Free Vibration
The solution may be written as: u = ρ sin (ωt + φ ) uω ⎛v ⎞ where _ ρ = uo2 + ⎜ o ⎟ _ and _ tan φ = o vo ⎝ω ⎠ 2
ρ is known as the amplitude of vibration and φ is called the phase angle
Both amplitude and phase angle depend upon the initial condition and natural circular frequency of the system
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Undamped Free Vibration
⎛ It can be shown that: u (t1 ) = u⎜ t1 + ⎝
2π ⎞ ⎟ ω ⎠
u (t1 ) = ρ sin (ωt1 + φ ) ⎧ ⎛ 2π ⎞ 2π ⎞ ⎫ ⎛ u ⎜ t1 + ⎟ +φ⎬ ⎟ = ρ sin ⎨ω ⎜ t1 + ω ⎠ ω ⎠ ⎭ ⎝ ⎩ ⎝ 2π ⎞ ⎛ u ⎜ t1 + ⎟ = ρ sin{(ωt1 + φ ) + 2π } ω ⎠ ⎝ 2π ⎞ ⎛ u ⎜ t1 + ⎟ = ρ sin (ωt1 + φ ) ω ⎠ ⎝
Thus the vibration repeats itself after T = 2π and is the natural period of ω system ω The reciprocal of T is call the natural frequency of system; f = 2π
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Natural Frequency of a Frame
The natural frequency of a 2D frame with fixed base and rigid beam of mass m, neglecting mass of columns, is given by:
k m 24 EI k= 3 L 24 EI ω= mL3
ω=
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Natural Frequency of a Suspended Mass
W = k∆ st ⇒ mg = k∆ st ⇒ k =
ω= T= f =
mg ∆ st
k mg / ∆ st g = = m m ∆ st 2π
ω 1 2π
= 2π
∆ st g
g ∆ st
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Phase Plan Diagram
u = ρ sin (ωt + φ ) u& = ρ cos(ωt + φ )
ω
(u )2 + ⎛⎜ u ⎞⎟ &
⎝ω ⎠
2
= ρ2
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Phase Plan Diagram (Initial Displacement)
u (0 ) = u0 , _ u& (0 ) = 0 2
⎛ vo ⎞ ⎟ = u0 ⎝ω ⎠
ρ = uo2 + ⎜ tan φ =
uoω π = ∞ ⇒φ = vo 2
π⎞ ⎛ u = uo sin ⎜ ωt + ⎟ = uo cos ωt 2⎠ ⎝ u& π⎞ ⎛ = uo cos⎜ ωt + ⎟ = uo sin ωt ω 2⎠ ⎝ Dynamics of Structures
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Phase Plan Diagram (Initial Velocity)
u (0 ) = 0, _ u& (0 ) = vo 2
v ⎛ vo ⎞ ⎟ = o ω ⎝ω ⎠
ρ = uo2 + ⎜ tan φ = u= u&
ω
vo
ω
=
uoω = 0⇒φ = 0 vo
sin (ωt )
vo
ω
cos(ωt )
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Phase Plan Diagram (Impulse)
uω ⎛ vo ⎞ ⎟ _ and _ tan φ1 = o vo ⎝ω ⎠ 2
ρ1 = uo2 + ⎜
ut1ω ⎛ u&t1 + I / m ⎞ ⎟ _ and _ tan φ1 = ω u&t1 + I / m ⎝ ⎠ u = ρ1 sin (ωt + φ1 ) − − for − 0 ≤ t ≤ t1 2
ρ 2 = ut21 + ⎜
u = ρ 2 sin (ω (t − t1 ) + φ2 ) − − for − t ≥ t1
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Phase Plan Diagram (Ground Motion)
⎛0⎞
2
ρ = u g2 + ⎜ ⎟ = u g ⎝ω ⎠ 3π tan φ = −∞ _ φ =
2 3π ⎞ ⎛ u = u g + u g sin ⎜ ωt + ⎟ 2 ⎠ ⎝ u = u g (1 − cos ωt )
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