:

(Soil Dynamics)

: (Ali Asgari)

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4-5

)Soil Dynamics(

2

Braja M. Das and G.V. Ramana, Principles of SoilDynamics, Second Edition, , Cengage Learning, 2011.

Braja M. Das and Zhe Luo, Principles of Soil Dynamics,third Edition, Cengage Learning, 2016.

Kramer, Steven L. Geotechnical earthquake engineering.Pearson Education India, 1996.

Verruijt, Arnold. An introduction to soil dynamics. Vol.24. Springer Science & Business Media, 2009.

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)Soil Dynamics(

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The animated gif shows the simple harmonic motion of three undampedmass-spring systems, with natural frequencies (from left to right) of ωo,2ωo, and 3ωo. All three systems are initially at rest, but displaced adistance xmfrom equilibrium.

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0 sin ,Q Q t Q

0 sinmz kz Q t

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:

1 sinpz A t

02

sinp

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k m

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sinp

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sin cos sinp h n n

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k m

: 0 0 00 0, 0t

dzz z v

dt

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0

0

2 2sin sin

1n

nn

Q kz t t

0 2 2

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1s n s

n n

z z M t t z Q k M

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0 2 2

1sin sin , ,

1s n s

n n

z z M t t z Q k M

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z z M t t z Q k M

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cos sin ,s n

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dt

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2 2n n nz t t t t

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2

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2

nn

nn

nt n t

mt m t

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: SDOF Kips10

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maxF !!!

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The animation shows response of the masses to the applied forces. The direction andmagnitude of the applied forces are indicated by the arrows. The dashed horizontallines provide a reference to compare magnitudes of resulting steady statedisplacement.

Transient response to an appliedforce: Three identical damped 1-DOF mass-spring oscillators, allwith natural frequency f0=1, areinitially at rest. A time harmonicforce F=F0cos(2 pi f t) is applied toeach of three damped 1-DOF mass-spring oscillators starting attime t=0. The driving frequencies ωof the applied forces are (matchingcolors)f0=0.4, f0=1.01, f0=1.6

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The animation shows the motion of the base and the resulting motion of all threeoscillators together. The horizontal lines indicate the maximum displacement of thebase.

Three simple 1-DOF mass-springoscillators have natural frequencies(from left to right)of f1=1.6, f2=1.0, f3=0.63. At time t=0the base starts moving with sinusoidaldisplacement s(t) = S0sin(t) where thedriving frequency is f=1.001. Thedamping rate for all three oscillators is0.1 so that the initial transient motiondecays and a steady-state is obtained.

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With damping:The animated gif shows two 1-DOF mass-spring systems initially at rest, but displaced from equilibrium by x=xmax . The black mass is undampedand the blue mass is damped (underdamped). After being released from rest the undamped (black) mass exhibits simple harmonic motion while the damped (blue) mass exhibits an oscillatory motion which decays with time.

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g p hz z z

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The 2-DOF system has two natural frequencies, corresponding to the two naturalmodes of vibration for the system. In the lower frequency mode both masses move inthe same direction, in-phase with each other. In the higher frequency mode the twomasses move in opposite direction, 180° out of phase with each other. The animationshows the motion of the 2-DOF system at normalized forcing frequenciesof f.left=0.67 (in-phase mode), fmiddle=1 (undamped classical tuned dynamic absorber),and fright=1.3 (opposite-phase mode). The arrows in the movie represent themagnitude and phase of the force applied to the main mass.

2

1

k

k

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A 1-degree-of-freedom system has1 mode of vibration and 1 naturalfrequency

A 2-degree-of-freedom system has2 modes of vibration and 2 naturalfrequencies

A 3-degree-of-freedom system has 3 modes of vibration and 3 natural frequencies

A 4-degree-of-freedom system has4 modes of vibration and 4 naturalfrequencies