Structural Dynamics Introduction 

On motion • Aristotle struggled with the idea of how motion actually works, evident from below: “What about the fact that when things are thrown they continue to move when the thrower is no longer touching them? It would be impossible to explain why something which has been set in motion should stop anywhere: why should it stop here rather than there?” • The conceptual difficulty — defining motion — remained up to and beyond Newton’s time. 2 

A brief history: On motion (contd.) • Aristotle defined circular motion, and shape of the heaven (celestial bodies) accurately: “Circular movement is the primary kind of movement. Only circular motion can be continuous and eternal.” “The shape of the heaven is of necessity spherical; for that is the shape most appropriate to its substance and also by nature primary.” 3 

Library of Alexandria — 300 B.C. 4 

Alexandria • There were two schools of thought: • Geocentric model — Ptolemy, the author of Almagest (200 AD) • Heliocentric model — Aristarchus of Samoa (300 AD) • The Church later supported the geocentric model, as it was popular, emphasized Earth’s privilege. 5 

A brief history: On motion (contd.) • Ptolemy followed the rule that heavenly motions are perfect and so must be circular. • He realized that simple model of Sun and planets moving on circular paths centered on the Earth could never give accurate results. The solution was a combination of circles. 6 

A brief history: after Greeks • Before 14 century, motion was not understood. • 14 century brought Renaissance — Art led to geometry, mathematics, and eventually, physics. • Renaissance renewed the curiosity about everything, including “heavenly” bodies. • However, discussing heavenly bodies (i.e., planets) was a religious matter. 7 

Renaissance • Change it brought, began challenging age-old conventions & thoughts. • Among them was studying “heavenly” bodies. • Copernicus (1473—1543) was the first to realize ours was a heliocentric model: “We revolve around the Sun like any other planet.” 8 

— Johann Wolfgang von Goethe “Of all discoveries and opinions, none may have exerted a greater effect on the human spirit than the doctrine of Copernicus. The world has scarcely become known as round and complete in itself when it was asked to waive the tremendous privilege of being the center of the universe.” 9 

Inertia • In 16th century, Galileo invented the telescope — the rest is history, as they say. • His discoveries overturned 1400 years of scientific wisdom. • He was not alone: Tycho Brahe made similar breakthroughs. Kepler exchanged notes with Galileo. • Based on his observations, Galileo first defined Inertia, as we know today. 10 

— Galileo Galilei “If an object is left alone, is not disturbed, it continues to move with a constant velocity in a straight line if it was originally moving, or it continues to stand still if it was just standing still.” 11 

Kepler’s laws of planetary motion 1. The orbit of each planet is in shape of an ellipse with the Sun at one focus. 2. In any equal time intervals, a line from the planet to the Sun will sweep out equal areas. 3. The total orbit times for planet 1 & planet 2 have a ratio: a 3 2 1 : a 3 2 2 12 

— Johannes Kepler My goal is to show that the heavenly machine is not a kind of divine living being but similar to a clockwork insofar as almost all the manifold motions are taken care of by one single absolutely simple magnetic bodily force, as in a clockwork all motion is taken care of by a simple weight. And indeed I also show how this physical representation can be presented by calculation and geometrically. 13 

Isaac Newton 1642 — 1727 14 

— Albert Einstein “The whole evolution of our ideas about the processes of nature might be regarded as an organic development of Newton’s ideas.” 15 

— Einstein’s comment when writing about his theory of relativity “No one must think that Newton’s creation can be overthrown in any real sense by this or any other theory. His clear and wide ideas will forever retain their significance as the foundation on which our modern conceptions of physics have been built.” 16 

Life of Newton • Joined the University of Cambridge, first as a student, then as a fellow, and as a professor spending 35 years there. • He graduated with a bachelor of arts degree in 1665. • Not an inspiring lecturer, he lectured to an empty room. • Began working on optics, and invented reflecting telescope. 17 

The Principia Mathematica • Newton’s mathematical capability was unmatched, and he put it to great use. • In response to Edmund Halley’s challenge (in 1686), Newton produced a nine page paper proving that the force (F) moving a planet was inversely proportional to the square of planet’s distance (d) from the Sun. • Amazed by the paper, Halley encouraged Newton to write in depth, resulting in his tour de force — Philosophiae Naturalis Principia Mathematica. F / 1 d2 18 

Newton’s three laws • Newton was a close follower of Galileo — read all his notes, commentaries, and books. • Newton adopted Galileo’s definition of inertia — defined for heavenly bodies — to earthly bodies as his first law. • The next thing needed a rule to find how an object changes its speed if something is affecting it —> equating momentum to force became the second law. • Lastly, conservation of momentum, i.e., F = -F, became the third law. 19 

Second law: Momentum & Force • Newton’s second law provided a method of determining how velocity changes under different influences called forces. • This is the founding equation in Structural Dynamics. F = m dv dt = ma F = d dt (mv) Note the slight difference between the two equations. The second one assumes constant mass — for simplicity. 20 

Mechanics • Quantum (new): subatomic behavior — no further discussion on this here. • Classical: • Newtonian mechanics — kinematics (the original theory of motion), and dynamics (forces). • Statics — semi rigid bodies in mechanical equilibrium. • Solid mechanics, elasticity — properties of deformable bodies. • Fluid — motion of fluids. • Continuum — mechanics of continuation (both solids and fluids). • Hydraulics — mechanical properties of fluids. • Soil — mechanical behavior of soils. 21 

Statics, Dynamics & Linearity • Statics — Time invariant • Dynamics — Time variant • Dynamic systems can be linear if: 1. Amplitudes of displacement & load are proportional 2. Stress is proportional to strain (damping acceptable) 3. Mass needs to be constant 22 

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Effective stiffness Stiffnesses in parallel get added. Examples: two legs of a portal frame. Stiffnesses in series are calculated for effectiveness. Examples: haunched beams, soil layers. 1 ke = 1 k1 + 1 k2 ke = k1 + k2 

Static analysis 26 F = k ⇥ Dynamic analysis F = m ⇥ a k ⇥ = m ⇥ a 

A word about notations, conventions You may find these are used interchangeably — usual math convention short form. ¨ x = d 2 x dt 2 ˙ x = dx dt 27 

At any time t, the corresponding displacement is x, then the restoring force will be When in motion, ! Solving this above quadratic equation for x = xo @ t = to: fk = Kx 28 M ¨ x = fk = Kx M ¨ x + Kx = 0 x = x0 cos r K M (t t0) ! for t > t0 Oscillation of an SDOF: No forcing & no damping. 

The solution, x 1. Response is harmonic. 2. Deflection amplitude is xo 3. Oscillation (natural) frequency is 29 !n = r K M 

Acceleration, velocity, spring force Acceleration Velocity Spring force Kx = Kx0 cos ( !n( t t0) Undamped — free to vibrate (no resistance) 30 ¨ x = ! 2 nx0 cos ( !n( t t0) ˙ x = !nx0 sin ( !n( t t0) 

–Johnny Appleseed “Type a quote here.” Damping as a force (resistance to unrestricted motion), fd may be written as fd = C ˙ x Steady state oscillation of an SDOF with forcing & viscous damping 31 

Applied cyclic load in the presence of damping, where f is a sinusoidally varying function of t at frequency The equation of motion thus becomes M ¨ x + C ˙ x + Kx = f0 cos ( !t ) f ( !t ) = f0 cos ( !t ) ! 32 

Amplitude of the motion (from steady state, and algebraic substitutions): x0 = f0 p ( K M! 2)2 + C 2 ! 2 Phase lag angle between applied force and resulting motion: = tan 1 ✓ C! K M!2 ◆ 33 

Dynamic Amplification Factor, Q Q = amplitude of displacement equivalent static displacement = x0 f0/K Q = 1 p (1 ⌦2)2 + (2⇣⌦)2 34 Damping ratio: ⇣ = C 2M!n = C 2 p MK Critical damping Frequency ratio: ⌦ = ! !n Natural frequency Forcing frequency At resonance: Q = 1 2⇣ (⌦ = 1) 

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Application: when to use what 36 

Additional reading/viewing • Justin Pollard’s book, The Rise & Fall of Alexandria • Alejandro Amenabar’s film, Agora • Colin Pask’s book, Magnificent Principia • Caltech’s book: Feynman lectures on Physics • Barltrop and Adams: Dynamics of Fixed Marine Structures 37 

Thank you. Questions? 38