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SD-Lecture05-Review-Vibratory-Motion

# SD-Lecture05-Review-Vibratory-Motion - CEGCEG-5065C Soil...

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CEG CEG-5065C Soil Dynamics 5065C Soil Dynamics Lecture #05 The mathematical language of Vibratory Motion o Types of vibratory motion: periodic versus transient o Simple harmonic motion (SHM) o Trigonometric notation of SHM o Complex notation of SHM o Fourier series analysis of earthquakes Luis Prieto-Portar 2009

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Vibrations propagating through soils and rocks can come from many sources, such as, - earthquakes, - explosions, - large falling weights, unbalanced machinery, impact loads, - heavy traffic, etc. The study of dynamic loads show patterns that can be used to simplify their study. This lecture is a review of vibratory motion, and how they can be expressed in a simple mathematical form . Vibratory motion can be periodic or transient . Periodic motion repeats itself at regular time intervals, called its period T , such as this simple sinusoidal wave,
Periodic motion can be (a) simple harmonic motion , or (b) general periodic motion . Although the latter can be very complex, the use of Fourier series will permit us to express their motion as the sum of many trivial equations of simple harmonic motions.

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A rotating machine that has an unbalanced mass will generate these centrifugal forces upon the foundation.
Dynamic loads vary in their magnitude , direction or position with time. It is possible for more than one type of variation to coexist. Earthquake loads, for example, vary both in magnitude and direction. Thus, they have three orthogonal directions and their corresponding rotation components: a total of six component forces and moments which each vary in magnitude with time. The figure below could be a wheel load rolling over a bridge deck, and is the instance of a force that varies in position with time. This is a periodic load, described as a cycle of motion. The time taken for each cycle is the period. The inverse of the period is the number of cycles per second = the frequency of the load.

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This is the time plot of the intensity of an unbalanced low-speed machine load upon its foundation. It is another example of a simple harmonic motion, albeit a bit more complex to describe mathematically.
The actual plot should include both the static and dynamic loads,

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Similar oscillatory motions occur upon a building’s frame when loaded by steady wind loads and the super-imposed gusts.
Transient motion plots are shown below. Case (c) is the plot of an explosion or an impact load. Case (d) is a typical transient packet from an earthquake. However, even these transient motions can be expressed simply as harmonic motions.

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This plot is the North-South accelerogram of the El Centro, California earthquake that took place on 18 May, 1940. It has been used as a model input for earthquake analysis of the foundations for new buildings during the past six decades.
Another transient motion is single impact of a steel hammer upon a steel plate,

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Contrast the transient of the simple hammer upon a steel plate plot on the previous slide to this plot that shows the vertical acceleration of soil particles close to a pile driving hammer when it hits the pile head-cushion interface.
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SD-Lecture05-Review-Vibratory-Motion - CEGCEG-5065C Soil...

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