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2. Kinematics of the material point in orthogonal curvilinear coordinates
Curvilinear coordinates (q1, q2, q3) are a coordinate system for Euclidean
space in which the coordinate lines may be curved.
Examples: polar, rectangular, spherical, and cylindrical coordinate systems.
Lecture 2. Curvilinear coordinates 1
Theoretical Mechanics
(2.3)
We assume the existence of a vector function 𝑟 : Ω ⊂ 𝑅3 → 𝐷 ⊂ 𝑅3 such that
Lecture 2. Curvilinear coordinates 2
Theoretical Mechanics
(2.1)
(2.2)
This function associates to each triplet 𝑞1, 𝑞2, 𝑞3 ∋ Ω an unique position
M 𝑥, 𝑦, 𝑧 ∈ R3 and vice versa.
(2.3)
Thus, the application should be is a diffeomorphism of class C2 and in order to exist
is necessary that:
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(2.4)
(2.5)
In this case the equation of motion are
Lecture 2. Curvilinear coordinates
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The curves:
Γ1:
Γ2:
Γ3:
are called curves of coordinates. In a similar way the surfaces of coordinates are
obtained.
Lecture 2. Curvilinear coordinates
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The elementary displacement on
is:
and the length of the curve arc covered by
particle M is given by:
Lecture 2. Curvilinear coordinates
𝑀 𝑑𝑟 2
𝑞1
𝑞2
𝑞3
𝑑𝑟 1 𝑑𝑟 3
𝑒 1
𝑒 2
𝑒 3
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Theoretical Mechanics
Taking into account that 𝑑𝑟 is tangent to the curve Γ1 then
Generally, we have:
where 𝐻𝑖 is the coefficient of Lamé corresponding to the curve Γ1 (i = 1,2,3).
However, for a general motion the elementary displacement on the trajectory is:
𝑑𝑟 = 𝑑𝑟 𝑖
3
𝑖=1
= 𝜕𝑟
𝜕𝑞𝑖
3
𝑖=1
𝑑𝑞𝑖 = 𝐻𝑖𝑑𝑞𝑖𝑒 𝑖
3
𝑖=1
𝑑𝑠 = 𝑑𝑟 = 𝐻𝑖𝑑𝑞𝑖2
3
𝑖=1
; 𝑑𝑠𝑖 = 𝐻𝑖𝑞𝑖 , 𝑖 = 1,2,3
(2.6)
(2.7)
(2.8)
Lecture 2. Curvilinear coordinates
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Theoretical Mechanics
Velocity in curvilinear coordinates
Taking into account that and
one obtain:
where
𝑣𝑖 = 𝐻𝑖𝑞𝑖 , (𝑖 = 1,2,3)
are the components of the velocity in the frame 𝑒 1, 𝑒 2, 𝑒 3 .
or
𝑣 = 𝐻𝑖𝑞𝑖 𝑒 𝑖
3
𝑖=1
= 𝑣𝑖𝑒 𝑖
3
𝑖=1
(2.9)
(2.8)
Lecture 2. Curvilinear coordinates
Lecture 1. Introduction 8
Theoretical Mechanics
Acceleration in curvilinear coordinates
Consider the components of the acceleration:
We have
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Finally, one obtain
(2.10)
Remark: If the curve of coordinate is a straight line or a circle of radius R then we
have 𝐻line = 1 and 𝐻circle = 𝑅.
Indeed, for Γ1 line we have 𝑑𝑞1 = 𝑑𝑠1
Lecture 2. Curvilinear coordinates
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For Γ1 circle we have 𝑑𝑠1 = 𝑅𝑑𝜃, and thus
Lecture 2. Curvilinear coordinates
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Particular curvilinear coordinates –cylindrical coordinates
(q1, q2, q3) :
Lecture 2. Curvilinear coordinates
Equations of the motion are:
Functional relations and the Jacobian:
(2.11)
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The curves:
Γ𝑟 is a line with the versor e𝑟 𝐻𝑟 = 1
Γ𝜃 is a circle with the versor e𝜃 𝐻𝜃 = 𝑟
Γ𝑧 is a line with the versor e𝑧 𝐻𝑧 = 1
Lecture 2. Curvilinear coordinates
The curves of coordinates (𝑟, 𝜃, 𝑧) are orthogonal.
(orthogonality condition)
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Thus, the velocity and acceleration components are:
(2.12)
Lecture 2. Curvilinear coordinates
𝑑𝑟 = 𝑑𝑟 𝑟 + 𝑑𝑟 𝜃 + 𝑑𝑟 𝑧 = 𝑑𝑟 𝑒 𝑟 + 𝑟𝑑𝜃 𝑒 𝜃 + 𝑑𝑧 𝑒 𝑟
𝑑𝑠 = 𝑑𝑟 = 𝑑𝑟 2 + 𝑟𝑑𝜃 2 + 𝑑𝑧 2
Displacement:
𝑑𝑠𝑟 = 𝐻𝑟𝑑𝑟 = 𝑑𝑟; 𝑑𝑠𝜃 = 𝐻𝜃𝑑𝜃 = 𝑟𝑑𝜃; 𝑑𝑠𝑧 = 𝐻𝑧𝑑𝑧 = 𝑑𝑧
𝑑𝑟 𝑟 = 𝑑𝑠𝑟𝑒 𝑟 = 𝑑𝑟𝑒 𝑟; 𝑑𝑟 𝜃 = 𝑑𝑠𝜃𝑒 𝜃 = 𝑟𝑑𝜃𝑒 𝜃; 𝑑𝑟 𝑧 = 𝑑𝑠𝑧𝑒 𝑟 = 𝑑𝑧𝑒 𝑟;
(2.13)
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Theoretical Mechanics
(2.14)
Lecture 2. Curvilinear coordinates
𝑎 = 𝑎𝑟𝑒 𝑟 + 𝑎𝜃𝑒 𝜃 + 𝑎𝑧𝑒 𝑧 = 𝑟 − 𝑟𝜃 2 𝑒 𝑟 +1
𝑟
𝑑
𝑑𝑡𝑟2𝜃 𝑒 𝜃 + 𝑧 𝑒 𝑧
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Theoretical Mechanics
(2.15)
Remark: For z=0, the motion in polar coordinates is obtained.
Equations of the motion are:
The velocity and acceleration components are
𝑣𝑟 = 𝑟 is the radial velocity
𝑣𝜃 = 𝑟𝜃 is the transversal (circumferential) velocity
(2.16)
Lecture 2. Curvilinear coordinates
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Theoretical Mechanics
Remark: It is also possible to calculate directly, using (2.6) the values for the
Lame’s coefficients in cylindrical coordinates.
(2.17)
Lecture 2. Curvilinear coordinates
𝐻𝑟 =𝑑𝑥
𝑑𝑟
2
+𝑑𝑦
𝑑𝑟
2
+𝑑𝑧
𝑑𝑟
2
= cos2 𝜃 + sin2 𝜃 + 02 = 1
𝐻𝜃 =𝑑𝑥
𝑑𝜃
2
+𝑑𝑦
𝑑𝜃
2
+𝑑𝑧
𝑑𝜃
2
= r2cos2 𝜃 + 𝑟2sin2 𝜃 + 02 = 𝑟
𝐻𝑧 =𝑑𝑥
𝑑𝑧
2
+𝑑𝑦
𝑑𝑧
2
+𝑑𝑧
𝑑𝑧
2
= 02 + 02 + 12 = 1
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Theoretical Mechanics
Lecture 2. Curvilinear coordinates
The angular velocity describes the rate of
change of angular position, and the
instantaneous axis of rotation. In this case
(counter-clockwise rotation) the vector points up.
In this case the normal and
tangent versors and the polar
coordinate versors have the
same directions.
https://ocw.mit.edu/courses/aeronautics-and-astronautics/16-07-dynamics-fall-2009/lecture-
notes/MIT16_07F09_Lec05.pdf
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How to find the sense of the angular velocity:
Lecture 2. Curvilinear coordinates
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Lecture 2. Curvilinear coordinates
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Lecture 2. Curvilinear coordinates
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Lecture 2. Curvilinear coordinates
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Lecture 2. Curvilinear coordinates
Particular curvilinear coordinates –spherical coordinates
The curves:
Γ𝑟 is a line with the versor e𝑟 𝑟 𝐻𝑟 = 1
Γ𝜃 is a circle of radius 𝑟 with the versor e𝜃 𝜃 𝐻𝜃 = 𝑟
Γ𝜙 is a circle of radius 𝑟 sin 𝜃 with the versor e𝜙 𝜙 𝐻𝜙 = 𝑟 sin 𝜃
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Theoretical Mechanics
(2.18)
Lecture 2. Curvilinear coordinates
𝑑𝑟 = 𝑑𝑟 𝑟 + 𝑑𝑟 𝜃 + 𝑑𝑟 𝜙 = 𝑑𝑟 𝑒 𝑟 + 𝑟𝑑𝜃 𝑒 𝜃 + 𝑟 sin 𝜃 𝑑𝜙𝑒 𝜙
𝑑𝑠 = 𝑑𝑟 = 𝑑𝑟 2 + 𝑟𝑑𝜃 2 + 𝑟 sin 𝜃 𝑑𝜙 2
Displacement:
𝑑𝑠𝑟 = 𝐻𝑟𝑑𝑟 = 𝑑𝑟; 𝑑𝑠𝜃 = 𝐻𝜃𝑑𝜃 = 𝑟𝑑𝜃; 𝑑𝑠𝜙 = 𝐻𝜙𝑑𝜙 = 𝑟 sin 𝜃 𝑑𝜙
𝑑𝑟 𝑟 = 𝑑𝑟𝑒 𝑟; 𝑑𝑟 𝜃 = 𝑟𝑑𝜃𝑒 𝜃; 𝑑𝑟 𝜙 = 𝑟 sin 𝜃 𝑑𝜙𝑒 𝜙;
(2.19)
Velocity: 𝑣 = 𝑣𝑟𝑒 𝑟 + 𝑣𝜃𝑒 𝜃 + 𝑣𝜙𝑒 𝜙
𝑣𝑟 = 𝑟
𝑣𝜃 = 𝑟 𝜃
𝑣𝜙 = 𝑟 sin 𝜃 𝜙
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Theoretical Mechanics
(2.20)
Lecture 2. Curvilinear coordinates
Acceleration:
Other choice :
(2/16)
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Theoretical Mechanics
Lecture 2. Curvilinear coordinates
Example: J. L. Meriam, L. G. Kraige, J. N. Bolton, Engineering Mechanics:
Dynamics, 8th Ed, Wiley, 2015
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Theoretical Mechanics
Lecture 2. Curvilinear coordinates
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Theoretical Mechanics
Lecture 2. Curvilinear coordinates
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Theoretical Mechanics
Lecture 2. Curvilinear coordinates
Example: J. L. Meriam, L. G. Kraige, J. N. Bolton, Engineering Mechanics:
Dynamics, 8th Ed, Wiley, 2015
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Theoretical Mechanics
Lecture 2. Curvilinear coordinates
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Theoretical Mechanics
Lecture 2. Curvilinear coordinates
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Theoretical Mechanics
Lecture 2. Curvilinear coordinates