Scribd
Upload
42 views2 pages

Equation Sheet

This document provides equations and concepts related to advanced dynamics, including: - Position, velocity, and acceleration equations in Cartesian, cylindrical, and spherical coordinate systems - Rotation matrices for rotations about x, y, z axes and sequences of rotations - Kinematics equations relating displacement, velocity, and acceleration between different reference frames - Equations of motion including Newton-Euler equations, angular momentum, Lagrange's equations, and virtual work - Key equations for kinetic and potential energy, parallel axis transformation of moments of inertia, and products of inertia.

Uploaded by

Steven
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
42 views2 pages

Equation Sheet

This document provides equations and concepts related to advanced dynamics, including: - Position, velocity, and acceleration equations in Cartesian, cylindrical, and spherical coordinate systems - Rotation matrices for rotations about x, y, z axes and sequences of rotations - Kinematics equations relating displacement, velocity, and acceleration between different reference frames - Equations of motion including Newton-Euler equations, angular momentum, Lagrange's equations, and virtual work - Key equations for kinetic and potential energy, parallel axis transformation of moments of inertia, and products of inertia.

Uploaded by

Steven
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
You are on page 1/ 2

EGH413 Advanced Dynamics Equation Sheet

General
𝑑𝑟𝑃⁄𝑂 𝑑𝑣̅ 𝑃 𝜕𝐴
𝑣̅ 𝑃 = 𝑎̅𝑃 = 𝐴̇ ̅ = ̅ × 𝐴̅
+𝜔
𝑑𝑡 𝑑𝑡 𝜕𝑡
2
𝑛𝑐
𝑇2 + 𝑉2 = 𝑇1 + 𝑉1 + 𝑊1→2 𝑊1→2 = ∮ ∑𝐹̅ (𝑟̅ ) ⋅ 𝑑𝑟̅ 𝐻𝑂 = ∑(𝑟̅𝑖 × 𝑚𝑖 𝑣̅𝑖 )
1

Position, velocity, and acceleration of a point 𝑷

System 𝒓̅𝑷/𝑶 ̅𝑷
𝒗 ̅𝑷
𝒂

Cartesian 𝑥𝑖̂ + 𝑦𝑗̂ + 𝑧𝑘̂ 𝑥̇ 𝑖̂ + 𝑦̇ 𝑗̂ + 𝑧̇ 𝑘̂ 𝑥̈ 𝑖̂ + 𝑦̈ 𝑗̂ + 𝑧̈ 𝑘̂

Cylindrical 𝑅𝑒̅𝑅 + 𝑧𝑒̅𝑧 𝑅̇ 𝑒̅𝑅 + 𝑅𝜃̇𝑒̅𝜃 + 𝑧̇ 𝑒̅𝑧 (𝑅̈ − 𝑅𝜃̇ 2 )𝑒̅𝑅 + (𝑅𝜃̈ + 2𝑅̇ 𝜃̇)𝑒̅𝜃 + 𝑧̈ 𝑒̅𝑧

[𝑟̈ − 𝑟𝜙̇ 2 − 𝑟𝜃̇ 2 sin2 𝜙]𝑒̅𝑟


𝑟̇ 𝑒̅𝑟 + 𝑟𝜃̇ sin 𝜙 𝑒̅𝜃
Spherical 𝑟𝑒̅𝑟 +[𝑟𝜙̈ + 2𝑟̇ 𝜙̇ − 𝑟𝜃̇ 2 sin 𝜙 cos 𝜙]𝑒̅𝜙
+ 𝑟𝜙̇ 𝑒̅𝜙
+[𝑟𝜃̈ sin 𝜙 + 2𝑟̇ 𝜃̇ sin 𝜙 + 2𝑟𝜙̇𝜃̇ cos 𝜙]𝑒̅𝜃

Rotation Matrices

Rotation Type Rotation Matrix Visualization

1 0 0
𝑥-axis [𝑅𝑥 ] = [0 cos 𝜃𝑥 sin 𝜃𝑥 ]
0 − sin 𝜃𝑥 cos 𝜃𝑥

cos 𝜃𝑦 0 − sin 𝜃𝑦
𝑦-axis [𝑅𝑦 ] = [ 0 1 0 ]
sin 𝜃𝑦 0 cos 𝜃𝑦

cos 𝜃𝑧 sin 𝜃𝑧 0
𝑧-axis [𝑅𝑧 ] = [− sin 𝜃𝑧 cos 𝜃𝑧 0]
0 0 1

Sequence of Body
[𝑅] = [𝑅𝑛 ] … [𝑅2 ][𝑅1 ]
Fixed Rotations

Sequence of Space
[𝑅] = [𝑅1 ] … [𝑅𝑛−1 ][𝑅𝑛 ]
Fixed Rotations
Kinematics an intermediate reference
(𝑥𝑃 )𝑜
Δ𝑟̅𝑃 = Δ𝑟̅𝑂′ + [𝑅]𝑇𝑓 (Δ𝑟̅𝑃 )𝑥𝑦𝑧 + ([𝑅]𝑇𝑓 − [𝑅]𝑇𝑜 ) [(𝑦𝑃 )𝑜 ]
Displacement
(𝑧𝑃 )𝑜

Velocity 𝑣̅ 𝑃 = 𝑣̅ 𝑂′ + (𝑣̅ 𝑃 )𝑥𝑦𝑧 + 𝜔


̅ × 𝑟̅𝑃/𝑂′

Acceleration 𝑎̅𝑃 = 𝑎𝑂′ + (𝑎̅𝑃 )𝑥𝑦𝑧 + 𝛼̅ × 𝑟̅𝑃/𝑂′ + 𝜔


̅ × (𝜔 ̅ × (𝑣̅ 𝑃 )𝑥𝑦𝑧
̅ × 𝑟̅𝑃/𝑂′ ) + 2𝜔

Equations of Motion (and helpful equations)


Newton-Euler (when 𝐴 is 𝑑𝑃̅ 𝑑𝐻̅𝐴
an allowable point) ∑𝐹𝐺 = = 𝑚𝑎̅𝐺 ∑𝑀𝐴 =
𝑑𝑡 𝑑𝑡
̅𝐴 = (𝐼𝑥𝑥 𝜔𝑥 − 𝐼𝑥𝑦 𝜔𝑦 − 𝐼𝑥𝑧 𝜔𝑧 )𝑖̂ + (𝐼𝑦𝑦 𝜔𝑦 − 𝐼𝑦𝑥 𝜔𝑥 − 𝐼𝑦𝑧 𝜔𝑧 )𝑗̂
𝐻
Angular Momentum
+ (𝐼𝑧𝑧 𝜔𝑧 − 𝐼𝑧𝑥 𝜔𝑥 − 𝐼𝑧𝑦 𝜔𝑦 )𝑘̂
𝑑 𝜕ℒ 𝜕ℒ
Lagrange’s Equations ( )− = 𝑄𝑗 ℒ = 𝑇−𝑉
𝑑𝑡 𝜕𝑞̇ 𝑗 𝜕𝑞𝑗
𝑛
Virtual Work by non-
𝛿𝑊 𝑛𝑐
= ∑ 𝐹̅𝑖 ⋅ 𝛿𝑟̅𝑖 = ∑ 𝑄𝑗 𝛿𝑞𝑗
conservative forces
𝑖 𝑗
1 1
𝑇 = 𝑚𝑣̅ 𝐺 ⋅ 𝑣̅ 𝐺 + 𝜔 ̅𝐺
̅⋅𝐻 Centre of Mass
2 2
Kinetic Energy
1
𝑇= 𝜔 ̅𝑂
̅⋅𝐻 Pure rotation about point 𝑂
2

𝑉𝑔𝑟𝑎𝑣𝑖𝑡𝑦 = 𝑚𝑔𝑍 Gravity potential energy (𝑍 is


height above datum)
Potential Energy
1 Spring potential energy (Δℓ is
𝑉𝑠𝑝𝑟𝑖𝑛𝑔 = 𝑘𝛥ℓ2 change from free length)
2

Parallel Axis Transformation

𝐼𝑥 ′ 𝑥 ′ = 𝐼𝑥𝑥 + 𝑚(𝑦𝐵2 + 𝑧𝐵2 )


Moments of inertia 𝐼𝑦′ 𝑦′ = 𝐼𝑦𝑦 + 𝑚(𝑥𝐵2 + 𝑧𝐵2 )
𝐼𝑧 ′ 𝑧 ′ = 𝐼𝑧𝑧 + 𝑚(𝑥𝐵2 + 𝑦𝐵2 )

𝐼𝑥 ′ 𝑦′ = 𝐼𝑥𝑦 + 𝑚𝑥𝐵 𝑦𝐵
Products of inertia 𝐼𝑥 ′ 𝑧 ′ = 𝐼𝑥𝑧 + 𝑚𝑥𝐵 𝑧𝐵
𝐼𝑦′ 𝑧 ′ = 𝐼𝑦𝑧 + 𝑚𝑦𝐵 𝑧𝐵

You might also like