Curvilinear Motion: Normal and Tangential Component

Curvilinear Motion: Normal and Tangential Component

◆ Planar motion

Origin of n-t coordinates is located at the particle
Positive t-axis: in the direction of increasing s
Positive n-axis: in the direction from P toward the center of curvature

Center of curvature

ut : unit vector associated with t-axis
un : unit vector associated with n-axis

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Curvilinear Motion: Normal and Tangential Component

◆ Velocity

direction of the velocity is always tangent to the path

v = vut

v = s&

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Curvilinear Motion: Normal and Tangential Component

◆ Acceleration

a = v& = v&ut + vu& t

Vector differentiation

u& t = dut = lim Dut = lim u 't - ut
dt Dt®0 Dt Dt®0 Dt

lim Dut Dq un = q&un
Dt®0 Dt
= lim
Dt®0 Dt

u& t = q&un = s& un = u un ds = rdq
r q& = s&
r
r

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Curvilinear Motion: Normal and Tangential Component

a = atut + anun at ds = vdv

at = v& or

v2

an = r

Magnitude of acceleration

a= a 2 + a2n
t

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Curvilinear Motion: Normal and Tangential Component

Special case

an = 0 rectilinear motion
at = 0 circular motion

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Curvilinear Motion: Normal and Tangential Component

ub = ut ´un
un = ub ´ut

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Curvilinear Motion: Normal and Tangential Component

◆example 1

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Curvilinear Motion: Normal and Tangential Component

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Curvilinear Motion: Normal and Tangential Component

◆ example2

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Curvilinear Motion: Normal and Tangential Component

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Curvilinear Motion: Cylindrical components

◆ Planar motion (Polar coordinates)

r = rur

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Curvilinear Motion: Cylindrical components

◆Velocity

v = r& = r&ur + ru& r

u& r = q&uq

v = vrur + vq uq

vr = r&, vq = rq&

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Curvilinear Motion: Cylindrical components

◆Acceleration u&q = -q&ur

a = v& = d (r&u r + rq&uq )
dt

= &r&ur + r&u& r + r&q&uq + rq&&uq + rq&u&q

u& r = q&uq r&q&uq

a = arur + aquq
ar = &r&- rq&2, aq = rq&&+ 2r&q&

Magnitude of acceleration

a = (&r&- rq&2)2 + (rq&&+ 2r&q&)2

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Curvilinear Motion: Cylindrical components

◆ example

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Curvilinear Motion: Cylindrical components

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Curvilinear Motion: Cylindrical components

◆ 3D motion

v = r&ur + rq&uq + z&uz
a = (&r&- rq&2)ur + (rq&&+ 2r&q&)uq + &z&uz

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Curvilinear Motion: Cylindrical components

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Curvilinear Motion: Cylindrical components

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Absolute Dependent Motion Analysis

The motion of particle A is
dependent on the motion of particle
B by constraint of the lope.

sA + lCD + sB = lT vB = -vA
dsA + dsB = 0 or
dt dt
aB = -aA

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Absolute Dependent Motion Analysis

2sB + h + sA + lred = ltotal
2vB = -vA, 2aB = -aA

2(h - sB ) + h + sA + lred = ltotal
2vB = vA , 2aB = aA

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Absolute Dependent Motion Analysis

◆ example

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Absolute Dependent Motion Analysis

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Relative Motion Analysis: Using Translating Axes

Position

rB = rA + rB
A

Velocity

vB = vA + vB
A

Acceleration

aB = aA + aB
A

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Relative Motion Analysis: Using Translating Axes

◆ example

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Relative Motion Analysis: Using Translating Axes

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