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Introduction
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Basic Concepts
1.1
Coordinates, Motion, displacement, Velocity, Acceleration
Origin or reference point (0, 0, 0).
Cartesian rectangular coordinates (x, y, z) and time t.
Displacement of a point referred to an origin (x, y, z).
Velocity ( dx
, dy , dz ).
dt dt dt
2
2
2
Acceleration ( ddt2x , ddt2y , ddt2z ).
1.2
Simple Harmonic Motion of a Point in 1D Cartesian Coordinates
x = X sin(ωt) = Im(X exp(iωt))
dx
= ωX cos(ωt) = Im(iωX exp(iωt))
dt
d2 x
= −ω 2 X sin(ωt) = Im(−ω 2X exp(iωt))
2
dt
Euler equation
exp(iθ) = cos θ + i sin θ
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1.3
Mass, Stiffness, Damping
Free body diagrams, Newton’s Laws and D’Alembert’s principle.
Force applied to a point particle F .
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Inertia force m ddt2x .
F =m
d2x
dt2
Stiffness (or a streched linear elastic rod)
k=
dF
A
=E
dx
L
Stiffness (or a bent linear elastic beam)
k=
dF
I
= 3E 3
dy
L
Stiffness (or a twisted linear elastic rod)
k=
J
dF
=G
dψ
L
Damping Force
F =c
1.4
dx
dt
Lagrange Equations
Kinetic Energy T , Potential Energy U.
Generalized Displacements qi , Generalized Forces Qi .
1.5
Units
Absolute and Gravitational Systems.
SI
Mass (kg); Force (N); Length (m); Time (s)
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