lecture_3_single-degree-of-freedom-system_free-vibration-response

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.

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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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