lecture_4_single-degree-of-freedom-system_free-vibration-response_2

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 (Cont..)

Single Degree of Freedom System: Free Vibration Response

Dynamics of Structures

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1

Free Vibration: Viscous Damping

The free vibration response an damped (viscous) system is governed by:

mu&& + cu& + ku = 0

A possible solution of the above equation is: u = Ge λt

Substituting the solution into equation of motion: Gλ2 me λt + cλe λt + Gke λt = 0

λ2 m + cλ + k = 0 ⇒ λ2 +

λ=−

c k λ+ =0 m m

c c2 k ± − 2 2m 4m m Dynamics of Structures

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Free Vibration: Viscous Damping (Cont..)

Based on the value of the discriminant, three different cases arise:

λ=−

c c2 k ± − 2m 4m 2 m

a.

Critically damped system (discriminant is zero)

c2 k − = 0 ⇒ c = ccr = 2 km = 2mω 4m 2 m

b.

Overdamped system (discriminant is positive)

⎛ c2 k c ⎞ ⎟⎟ > 1 − > 0 ⇒ c > (ccr = 2mω ) ⇒ ⎜⎜ ζ = 4m 2 m c cr ⎠ ⎝

c.

Underdamped system (discriminant is negative)

⎛ c2 k c ⎞ ⎟⎟ < 1 − < 0 ⇒ c < (ccr = 2mω ) ⇒ ⎜⎜ ζ = 4m 2 m c cr ⎠ ⎝

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a. Critically Damped System

For critically damped system c2 k − = 0 _ ⇒ _ c = ccr = 2 km = 2mω 4m 2 m

The roots of the differential equation are: λ1 = λ2 = −

c c = − cr = −ω 2m 2m

Therefore the solution of equation of motion is: u = (G1 + G2t )e −ωt

The arbitrary constants are determined from the initial condition: u (0) = uo _ and _ u& (0) = vo

G1 = uo G2 = ωuo + vo Dynamics of Structures

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a. Critically Damped System (Cont..)

The general solution thus becomes: Thus the free vibration of a critically damped system is not oscillatory After the initial disturbance the system will come back to its original position without oscillation.

Examples: Recoiling Gun and Weighing Scale

u = {uo + (vo + ωuo )t}e −ωt 1

Displacement

0.8 0.6 0.4 0.2 0 0

Dynamics of Structures

0.1

0.2

0.3

0.4

0.5

0.6

time

6

3

b. Overdamped System

For overdamped system, damping is greater than critical damping The ratio of damping of a system to its critical damping is called damping ratio and is given by: ζ =

c c = _ ⇒ _ c = 2mωζ ccr 2mω

The roots of characteristic equation may be written as:

λ = −ωζ ± ω ζ 2 − 1 = −ωζ ± ω where _ ω = ω ζ 2 − 1

The general solution of equation of motion is of the form: u = e −ωζt ( A cosh ω t + B sinh ω t )

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b. Overdamped System The arbitrary constants (A and B) are determined from the initial conditions: u (0) = uo _ and _ u& (0) = vo 1.2

A = uo _ and _ B =

vo

ω

+

ζ

uo

ζ 2 −1 v + ωζuo ⎧ ⎫ u = e −ωζt ⎨uo cosh ω t + o sinh ω t ⎬ ω ⎩ ⎭

Displacement

1.0 0.8

Overdamped

0.6

Critically Damped

0.4 0.2 0.0 0

2

4

6

time

8

10

12

14

16

The motion is non-oscillatory

Take more time to come to zero position Examples are: Automatic door closer

Dynamics of Structures

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4

c. Underdamped System

For underdamped system, damping is less than the critical damping, i.e. the damping ratio is less than one The roots of characteristic equation may be written as: λ = −ωζ ± iω 1 − ζ 2 = −ωζ ± ωd where _ ωd = ω 1 − ζ 2

ωd is known as the damped circular frequency The general solution of equation of motion is of the form: u = e −ωζt ( A cos ωd t + B sin ωd t )

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c. Underdamped System

With initial conditions: u (0) = uo _ and _ u& (0) = vo ⎞ ⎛ v + ωζuo u = e −ωζt ⎜⎜ uo cos ωd t + o sin ωd t ⎟⎟ ωd ⎠ ⎝

u = ρe −ωζt sin(ω d t + φ )

1.5 1.0

OR u = ρe −ωζt sin(ωd t + φ ) where _ ρ = tan φ =

u = ρe −ωζt

0.5

(uo )

uoωd vo + uoωζ

2

⎛ v + u ωζ + ⎜⎜ o o ωd ⎝

0.0

⎞ ⎟⎟ _ and ⎠

-0.5 -1.0

0

2

4

6

8

10

12

14

16

u = − ρe −ωζt

-1.5

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5

c. Underdamped System (Cont..)

Underdamped Undamped

1.5 1.0 0.5 0.0 -0.5

0

2

4

6

8

10

12

14

16

-1.0 -1.5

1.5

Overdamped

1.0

Critically Damped

0.5

Underdamped

0.0 -0.5

0

2

4

6

8

10

12

14

16

-1.0

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Phase Plane Diagram: Under-damped u = ρe −ωζt sin(ωd t + φ ) where _ ρ = u&

ωd

⎞

⎛

(uo )2 + ⎜⎜ vo + uoωζ ⎟⎟ _ and _ tan φ = ⎝

ωd

⎠

uoωd vo + uoωζ

⎞ ⎛ ζ sin(ωd t + φ ) ⎟ = ρe −ωζt ⎜ cos(ωd t + φ ) + 2 ⎟ ⎜ 1−ζ ⎠ ⎝

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Logarithmic Decrement

It is the natural log of the ratio of displacement at any time t1 and t1+2π/ωd It represent decay in the magnitude during one cycles Logarithmic Decrement is used to calculate the damping ratio of a system u (t1 ) = ρe −ωζt sin(ωd t1 + φ ) 1

⎛ 2π u ⎜⎜ t1 + ωd ⎝

⎛

2π ⎞

⎛

2π ⎞

−ωζ ⎜⎜ t1 + −ωζ ⎜⎜ t1 + ⎟⎟ ⎟⎟ ⎞ ⎟⎟ = ρe ⎝ ωd ⎠ sin(ωd t1 + 2π + φ ) = ρe ⎝ ωd ⎠ sin(ωd t1 + φ ) ⎠ u (t1 ) e −ωζt1 δ = ln = ln ⎛ 2π ⎞ ⎟ ⎛ −ωζ ⎜⎜ t1 + 2π ⎞ ω ⎟ ⎟⎟ u ⎜⎜ t1 + e ⎝ d⎠ ω d ⎠ ⎝ 2πζ δ= 1− ζ 2

ζ =

δ 4π 2 + δ 2 Dynamics of Structures

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Logarithmic Decrement (Cont..)

For many practical cases:

ζ < 0.2 _ therefore _ 1 − ζ 2 ≈ 1 δ δ = 2πζ ⇒ ζ = 2π

For lightly damped system, the decay of motion is slow, it is desirable to relate amplitudes several cycles apart: u u1 u1 u u u u = 1 2 3 ......... N = e Nδ u2 u3 u4 t u N +1 u2 u3 u4 u N +1

ln

u1 = Nδ = N 2πζ u N +1

ζ =

u 1 ln 1 2πN u N +1

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Logarithmic Decrement (Cont..)

The number of cycles in which the amplitude will decay to half its value at he beginning is given by:

12 10 8

N 50% =

1

δ

ln 2 ≈

1 2πζ

ln 2 ≈

0.11

ζ

4

Since acceleration is easy to measure and also acceleration is proportional to the displacement in case of free vibration, the damping ratio is therefore determined from:

ζ =

6

2 0 0

0.05

0.1

0.15

0.2

u&& 1 ln 1 2πN u&&N +1 Dynamics of Structures

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Free Vibration: Coulomb Damping

Coulomb damping is due to the force of sliding friction The friction force is proportional to the normal force acting on the contact surface: F = µN where µ is the coefficient of friction between the contact surface The friction force is opposite to the direction of motion When the block is moving towards left: mu&& + ku = µN

When the block is moving towards right mu&& + ku = − µN Dynamics of Structures

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Free Vibration: Coulomb Damping (Cont..)

The general solution of equation of motion for block moving towards left is: µN u = A cos ωt + B sin ωt + k And for block moving towards right: u = C cos ωt + D sin ωt −

µN

k A, B, C and D are arbitrary constants to be determined from initial conditions at the start of each half cycle

Now for u (0) = uo _ and _ u& (0 ) = 0,

A = uo −

µN k

− and − B = 0

The solution of equation of motion towards left is: µN ⎞ µN ⎛ u = ⎜ uo − ⎟ cos ωt + k ⎠ k ⎝ Dynamics of Structures

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Free Vibration: Coulomb Damping (Cont..)

At t = π / ω the block will be at the extreme left position and will start moving towards right. Therefore the initial conditions for movement towards right are:

Therefore:

µN ⎞ ⎛π ⎞ ⎛ ⎛π ⎞ u ⎜ ⎟ = −⎜ u o − 2 ⎟ _ and _ u& ⎜ ⎟ = 0 k ⎠ ⎝ω ⎠ ⎝ ⎝ω ⎠ 3µN C = uo −

k

_ and _ D = 0

3µN ⎞ µN ⎛ u = ⎜ uo − ⎟ cos ωt − k ⎠ k ⎝ At the end of one complete cycle t = 2π / ω

µN ⎞ ⎛ 2π ⎞ ⎛ u⎜ ⎟ = ⎜ uo − 4 ⎟ k ⎠ ⎝ω ⎠ ⎝

Thus the amplitude decays by 4 µN / k in one complete cycle and the decay is linear Coulomb damping does not change the frequency of vibration.

At any instant when the mass is at extreme left or right, if the displacement u is equal or less than µN / kthe spring force ku will be equal to or less than the friction force and the block will cease to move Dynamics of Structures

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Free Vibration: Coulomb Damping (Cont..)

Phase plane diagram

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Free Vibration: Hysteretic Damping

The viscous damping force is proportional to the velocity ( f D = cu& ) The velocity in case of free vibration with viscous damping is given by: ⎛ ⎞ ζ = ρ e −ωζt ⎜ cos(ωd t + φ ) + sin(ωd t + φ ) ⎟ 2 ⎜ ⎟ ωd 1− ζ ⎝ ⎠ u&

For small amount of damping, the velocity and hence the damping force is proportional to the frequency of vibration u& ≈ ρωe −ωζt (cos(ωt + φ ) )

This is not true because for most structural system, the damping is either independent of frequency or in some case decrease with increasing frequency

Thus viscous damping is not applicable because major part of damping is from internal friction Dynamics of Structures

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Free Vibration: Hysteretic Damping (Cont..)

The damping resistance occurring from the internal friction is referred to as hysteretic damping, structural damping, or solid damping The loss of energy per cycle is measured from the force deformation hysteresis loop The damping effect may be accounted for using non-linear spring force: mu&& + f s (u ) = 0 Or the effect of hysteretic damping may be considered by using the relation (η is constant): ηk mu&& + u& + ku = 0 ω Since velocity is directly proportional to frequency that makes the damping force independent of frequency. ch =

ηk η c ⇒ ζ h = h = = Const ω ccr 2

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