VIETNAMESE - GERMAN UNIVERSITY
MECHANICS 1 STATICS FORMULA SHEET
Authors:
Hoang Minh Khoi, MEN2021
Khuong Khanh Phuong, MEN2021
Supervisor:
Dr. Tran Tuan Minh
This document is allowed in the exam room
NO modification of this document is allowed
1
Equilibrium Conditions
1.1
Vector Appendix:
⎛𝑎 ⎞
⎛𝑎 ⎞
1
⎜𝑏⎟
• Unit vector: 𝒗 = ⎜𝑏⎟ → Unit vector: 𝒖 = √
⎜ ⎟
2
2
2
𝑎 + 𝑏 + 𝑐 ⎜⎝𝑐 ⎟⎠
⎝𝑐 ⎠
𝑐
𝑏
|𝑐| = |𝑎||𝑏| sin 𝜃
𝜃
𝑎
• Vector product:
| 𝒆𝑥 𝒆𝑦 𝒆𝑧 |
|
| |𝑎 𝑎 |
|𝑎 𝑎 |
|𝑎 𝑎 |
– 𝒄 = 𝒂 × 𝒃 = ||𝑎𝑥 𝑎𝑦 𝑎𝑧 || = || 𝑦 𝑧 || 𝒆𝑥 − || 𝑥 𝑧 || 𝒆𝑦 + || 𝑥 𝑦 || 𝒆𝑧
| 𝑏 𝑏 𝑏 | | 𝑏𝑦 𝑏𝑧 |
| 𝑏𝑥 𝑏𝑧 |
| 𝑏𝑥 𝑏𝑦 |
| 𝑥 𝑦 𝑧|
– 𝒄 =𝒂×
{𝒃 = |𝑎| |𝑏| sin 𝜃 𝒆𝒊
𝒆𝒊 ∶ the unit vector perpendicular to vector 𝒂 and 𝒃
where
𝜃 ∶ the angle between vector 𝒂 and 𝒃
𝑏
𝜃
• Dot product:
𝑎
𝑏 cos 𝜃
⎛𝑎1 ⎞ ⎛𝑏1 ⎞
– 𝒂 ⋅ 𝒃 = ⎜𝑎2 ⎟ ⋅ ⎜𝑏2 ⎟ = 𝑎1 𝑏1 + 𝑎2 𝑏2 + 𝑎3 𝑏3
⎜ ⎟ ⎜ ⎟
⎝𝑎3 ⎠ ⎝𝑏3 ⎠
– 𝒂 ⋅ 𝒃 = |𝑎||𝑏| cos 𝜃
1.2
Specified formulas:
• Force: 𝑭 = 𝑭𝑥 + 𝑭𝑦 + 𝑭𝑧 = 𝐹𝑥 𝒆𝑥 + 𝐹𝑦 𝒆𝑦 + 𝐹𝑧 𝒆𝑧
where 𝒆𝑖 is the unit vector of 𝑭𝑖 (𝑖 = 𝑥, 𝑦, 𝑧)
• Moment:
1. 𝑀 = ℎ.𝐹
{
ℎ ∶ perpendicular distance from the point of application to the line of force
where
𝐹 ∶ magnitude of applied force
| 𝒆𝒙 𝒆𝒚 𝒆𝒛 |
|
|
2. 𝑴 = 𝒓 × 𝑭 = || 𝑟1 𝑟2 𝑟3 || = 𝑀𝑥 𝒆𝑥 + 𝑀𝑦 𝒆𝑦 + 𝑀𝑧 𝒆𝑧
|𝐹 𝐹 𝐹 |
| 1 2 3|
{
𝒓 ∶ the position vector pointing from point O to an arbitrary point on the action line of 𝑭
where
𝑭 ∶ the applied force
1
2
Center of Gravity, Center of Mass, Centroids
2.1
Centroid:
• Line
– Non-linear:𝑥 =
∫ 𝑥 𝑑𝑠
𝑦=
,
∫ 𝑦 𝑑𝑠
∫ 𝑑𝑠
∫ 𝑑𝑠
∑
∑
𝑥𝑙
𝑦𝑙
– Linear: 𝑥 = ∑ 𝑖 𝑖 , 𝑦 = ∑ 𝑖 𝑖
𝑙𝑖
𝑙𝑖
• Area
– Non-linear:𝑥 =
∫ 𝑥 𝑑𝐴
𝑦=
,
∫ 𝑦 𝑑𝐴
∫ 𝑑𝐴
∫ 𝑑𝐴
∑
∑
𝑥𝐴
𝑦𝐴
– Linear: 𝑥 = ∑ 𝑖 𝑖 , 𝑦 = ∑ 𝑖 𝑖
𝐴𝑖
𝐴𝑖
• Volume
– Non-linear: 𝑥 =
∫ 𝑥 𝑑𝑉
,
𝑦=
∫ 𝑦 𝑑𝑉
,
𝑧=
∫ 𝑧 𝑑𝑉
∫ 𝑑𝑉
∫ 𝑑𝑉
∫ 𝑑𝑉
∑
∑
∑
𝑥𝑉
𝑦𝑉
𝑧𝑉
– Linear: 𝑥 = ∑ 𝑖 𝑖 , 𝑦 = ∑ 𝑖 𝑖 , 𝑧 = ∑ 𝑖 𝑖
𝑉𝑖
𝑉𝑖
𝑉𝑖
2.2
Center of mass:
• Non-linear: 𝑥 =
∫ 𝑥𝜌 𝑑𝑉
∫ 𝜌𝑑𝑉
∑
𝑥𝜌𝑉
• Linear: 𝑥 = ∑ 𝑖 𝑖 𝑖 ,
𝜌𝑖 𝑉𝑖
2.3
,
𝑦=
∫ 𝑦𝜌 𝑑𝑉
∫ 𝜌𝑑𝑉
∑
𝑦𝜌𝑉
𝑦 = ∑ 𝑖 𝑖 𝑖,
𝜌𝑖 𝑉𝑖
,
𝑧=
∫ 𝑧𝜌 𝑑𝑉
∫ 𝜌𝑑𝑉
∑
𝑧𝜌𝑉
𝑧= ∑𝑖 𝑖 𝑖
𝜌𝑖 𝑉𝑖
Possible Centroids
2
3
Support Reactions
4
Trusses
Zero-force members cases:
3
𝑆1 = 0 , 𝑆2 = 0
5
Beams
5.1
Quantities relationships
𝑆1 = 𝐹 , 𝑆2 = 0
𝑑𝑀
=𝑉
𝑑𝑥
𝑑𝑉
= −𝑞,
𝑑𝑥
5.2
𝑆1 = 𝑆2 , 𝑆 3 = 0
Relationships of V, M and loads
Types of loads
V diagram
M diagram
𝑞=0
const
linear
𝑞 = const
linear
parabola
𝑞 = linear
parabola
cubic
concentrated
force
jump
kink
moment couple
continuous
4
jump
5.3
Relationships of lines:
𝒇 ′′
𝒇′
𝒇
(+)
(+)
(−)
(+)
(+)
(−)
(−)
(+)
(+)
(−)
(−)
(+)
(−)
(−)
6
Static friction
6.1
Basic formula:
|𝐻| < 𝐻0 ,
𝐻0 = 𝜇0 𝑁
⎧𝐻 ∶ static friction force
⎪
⎪𝐻0 ∶ limiting friction force
⎨
⎪𝜇0 ∶ coefficient of static friction
⎪𝑁 ∶ normal force
⎩
6.2
Problem types:
1. Static friction (stays at rest): 𝐻 < 𝜇0 𝑁
2. Limiting friction (at rest an on the verge of moving): 𝐻 = 𝜇0 𝑁
3. Kinetic friction (the body slips): 𝑅 = 𝜇𝑁
5
6.3
Belt friction
𝑆1 ≤ 𝑆2 𝑒𝜇0 𝜙
The system is in equilibrium if
𝑆1 𝑒−𝜇0 𝜙 ≤ 𝑆2 ≤ 𝑆1 𝑒𝜇0 𝜙
7
Virtual Work and Stability
7.1
Principle of Virtual Work
∑
∑
𝛿𝑈 =
𝐹𝑖 𝛿𝑑𝑖 +
𝑀𝑖 𝛿𝜙𝑖 = 0
{
𝛿𝑑𝑖 ∶ virtual linear displacement
where
𝛿𝜙𝑖 ∶ virtual angular displacement
7.2
Stability of Equilibrium States
• Equation of Stability State: 𝛿𝑉 = −𝛿𝑈 =
𝑑𝑉
=0
𝑑𝑥
• Types of potential energy:
– Gravitational: 𝑉 = 𝑚𝑔ℎ
– Linear Spring: 𝑉 = 12 𝑐 (Δ𝑥)2
– Torsion Spring: 𝑉 = 12 𝑐𝑇 (Δ𝛼)2
• Stability criterion:
1. 1st derivative:
– 𝑉′ =0
– 𝑉′ >0
– 𝑉′ <0
→
→
→
neutral
stable
unstable
2. 2nd derivative:
– 𝑉 ′′ (𝑥0 ) = 0
– 𝑉 ′′ (𝑥0 ) > 0
– 𝑉 ′′ (𝑥0 ) < 0
→
→
→
required further investigations
minimum → stable
maximum → unstable
6