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14.205 Dynamics Spring 2010

Viscous Damped Free Vibration of SDOF

Systems

GeneralIn reality a vibration without decay in amplitude is never realized. Theprogressively-reduced amplitude of vibration is caused by the presenceof damping forces that dissipate the input energy in a nonconserva-tive system.

Types of damping in dynamics

Viscous damping: Results when a system vibrates in a fluid(e.g., air, oil, water). Examples include shock absorbers, hydraulicdashpots, and sliding of a body on a lubricated surface. For airdamping

fDj = cju2

j (1)

For liquid damping,fDj = cjuj (2)

where cj is the viscous damping coefficient characterizing the damp-ing mechanism of the jth DOF. Notice that uj denotes relativevelocity.

Structural damping (hysteresis damping): Due to internalfriction within the material or at connections between elementsof a structural system. The resulting damping forces are a func-tion of the strain (or deflections) in the structure. For an elasticsystem the jth (or the jth DOF) structural damping force fDjis proportional in magnitude to the internal elastic force fS andopposite in direction to the velocity vector uj .

fDj = igfSj (3)

where i = 1 is the imaginary number and g is a constant. Anequivalent viscous damping coefficient ce for structural dampingcan be defined by

ce =k

(4)

where is a dimensionless structural damping coefficient for thematerial and k is the equivalent stiffness of the system.

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14.205 Dynamics Spring 2010

Coulomb damping (dry friction): Results from the motion

of a body sliding on a dry surface. The resulting damping forceis almost constant, depending on the normal pressure N betweenthe moving body and the surface with a coefficient kinetic friction.

fD = N (5)

An equivalent viscous damping coefficient ce for Coulomb dampingcan be defined by

ce =4fDu

(6)

Negative damping: Results when the nature of the damping

contributes energy to the vibration. This could occur in the aero-dynamics of cables in a bridge.

In most actual physical systems it is very difficult to find the exactexpression for the damping force; viscous damping model is widelychosen for its ease in mathematical treatment.

Other damping mechanisms in physics

Electrical resistance damping: Originates from the delayedresponse of resistance in circuits (Kirchhoffs rule)

Electromagnetic damping: Originates from the delayed re-

sponse of electrons in restoring their original positions. Examplesinclude ballistic galvanometers and eddy current damping.

Collision damping: In plasma vibrations the cooperative behav-ior of the participating electrons is based on the random motion ofthermal agitation. Every electron is associated with a probabilityfunction to collide with another randomly moving particle. Oncecollision occurs, the vibrational energy is reduced in a fractionalrate of energy loss. This happens in ionosphere and in metals.

Viscously damped SDOF systems

Damped: There is damping considered in the systems. Solution to the free vibration problem (ODE) of damped SDOF

systems is a particular solution to the ODE.

Governing equation of a damped SDOF mass-damper-spring sys-tem:

mu + cu + ku = 0 (7)

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Critical damping, cc = 2m

Damping ratio, = ccc

= c2m

Critically-damped systems

u(t) = [u0 + (u0 + u0) t] exp(t) (8)

Undercritically-damped or under-damped systems

u(t) =

u0 cosDt +

u0 + u0D

sinDt

exp(t) (9)

Overcritically-damped or over-damped systems

u(t) = A exp

(+2 1)t

+ B exp

(

2 1)t

(10)

where

A =u0 +

+

2 1

u0

22 1 (11)

B =u0

2 1

u0

22 1

(12)

Effect of initial conditions (I.C.)

u0 = 1 in and u0 = 0 in/s (Figure 1)

u0 = 1 in and u0 = 10 in/s (Figure 2)

u0 = 1 in and u0 = 10 in/s (Figure 3) u0 = 0 in and u0 = 10 in/s (Figure 4)

Effects of damping on free vibration

Reading

[Dynamicsby Hibbeler: Chap. 22, Sec. 4]

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0 5 10 15 20 25

-0.5

0

0.5

1

Time (sec)

Displacement(in)

u0

= 1 in, du0/dt = 0 in/s

[ m=500 lb, k=2000 lb/in, =[0.1 1 4]

Underdamped

Criticaldamped

Overdamped

Figure 1: I.C. case 1: u0 = 1 and u0 = 0

0 5 10 15 20 25-3

-2

-1

0

1

2

3

4

Time (sec)

Displacement(in)

u0

= 1 in, du0/dt = 10 in/s

[ m=500 lb, k=2000 lb/in, =[0.1 1 4]

Underdamped

Criticaldamped

Overdamped

Figure 2: I.C. case 2: u0 = 1 and u0 = 10

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14.205 Dynamics Spring 2010

0 5 10 15 20 25

-4

-3

-2

-1

0

1

2

3

Time (sec)

Displacement(in)

u0

= 1 in, du0/dt = -10 in/s

[ m=500 lb, k=2000 lb/in, =[0.1 1 4]

Underdamped

Criticaldamped

Overdamped

Figure 3: I.C. case 3: u0 = 1 and u0 = 10

0 5 10 15 20 25-3

-2

-1

0

1

2

3

4

Time (sec)

Displacement(in)

u0

= 0 in, du0/dt = 10 in/s

[ m=500 lb, k=2000 lb/in, =[0.1 1 4]

Underdamped

Criticaldamped

Overdamped

Figure 4: I.C. case 4: u0 = 0 and u0 = 10

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Figure 5: Free vibration of damped SDOF systems [Source: A.K. Chopra(2007)]

Figure 6: Free vibration of SDOF with different damping coefficients [Source:A.K. Chopra (2007)]

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