Lecture (3)
Modeling of SDOF
Systems
Dr. Ahmed Hegazy
Mechanical Model
Physical Model
Dr. Ahmed Hegazy
Email: a.hegazy@eaeat.edu.eg
1
Simplified Mechanical Model
Dr. Ahmed Hegazy
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Simplified Mechanical Model
Dr. Ahmed Hegazy
Email: a.hegazy@eaeat.edu.eg
3
• The mass or inertia element is a mechanical element stores both
kinetic and potential energy. The inertia force and inertia torque can be
expressed as:
The Inertia Force
σ 𝑭𝒙 = 𝒎𝒙ሷ
for translational motion in x direction
The Inertia moment
σ 𝑴𝒐 = 𝑰𝒐 𝜽ሷ
For rotational motion about axis O
σ 𝑴𝒄 = 𝑰𝒄 𝜽ሷ
c is the instantaneous center of rotation
Dr. Ahmed Hegazy
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Dr. Ahmed Hegazy
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Parallel-Axes Theorem:
𝐼 = 𝐼𝐺
2
+ 𝑚𝑑
2
7
2
2
𝐼 = 𝑀𝑅 + 𝑀𝑅 = 𝑀𝑅 2
5
5
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Dr. Ahmed Hegazy
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σ 𝐹𝑥 = 𝑚𝑥ሷ
−𝑘𝑥 − 𝑐𝑥ሶ = 𝑚𝑥ሷ
𝑚𝑥ሷ + 𝑐 𝑥ሶ + 𝑘𝑥 = 0
÷𝒎
𝑥ሷ + 2ζ𝜔𝑛 𝑥ሶ + 𝜔𝑛2 𝑥 = 0
𝜔𝑛 =
ζ=
𝑐
𝑐𝑜
𝑘
𝑓𝑛 =
𝑚
=
𝑐
2𝑚𝜔𝑛
=
𝑐
2 𝑘𝑚
Equation of Motion
𝜔𝑛
2𝜋
Natural frequency
Damping Factor
Damping Ratio
𝑐𝑜 = 2𝑚𝜔𝑛 = 2 𝑘𝑚
Critical Damping Coefficient
Dr. Ahmed Hegazy
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8
System
Rectilinear
Rotational
Equation of Motion
𝑚𝑥ሷ + 𝑐 𝑥ሶ + 𝑘𝑥 = 𝐹(𝑡)
𝐼 𝜃ሷ + 𝑐𝑇 𝜃ሶ + 𝑘 𝑇 𝜃 = 𝑇(𝑡)
Natural Frequency
Damping Factor
Dr. Ahmed Hegazy
𝜔𝑛 =
𝑘
𝑚
𝑐
𝑐
𝑐
ζ= =
=
𝑐𝑜 2𝑚𝜔𝑛 2 𝑘𝑚
Email: a.hegazy@eaeat.edu.eg
𝜔𝑛 =
𝑘𝑇
𝐼𝑜
𝑐𝑇
𝑐𝑇
𝑐𝑇
ζ=
=
=
𝑐𝑇 𝑜 2𝐼𝑜 𝜔𝑛 2 𝑘 𝑇 𝐼𝑜
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Produce the equation of motion for the system shown, and calculate the
equivalent mass moment of inertia 𝑰𝒆𝒒 , and the equivalent stiffness 𝑲𝒆𝒒 .
σ 𝑀𝑜 = 𝐼 𝜃ሷ
𝑥 = 𝑟 sin 𝜃
𝐹 𝑡 𝑟 − 𝐾 2𝑟 𝜃 = 𝑚𝑟 + 𝐼 𝜃ሷ
2
2
𝑥 = 𝑟𝜃
𝑚𝑟 2 + 𝐼 𝜃ሷ + 4𝑘𝑟 2 𝜃 = 𝐹 𝑡 𝑟
2
𝐼𝑒𝑞 = 𝑚𝑟 + 𝐼
For small 𝜃
𝑥ሶ = 𝑟𝜃ሶ
𝑘𝑒𝑞 = 4𝑘𝑟
2
𝑥ሷ = 𝑟𝜃ሷ
𝑥ሷ
𝜃ሷ =
𝑟
Dr. Ahmed Hegazy
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For the system shown, produce the equation of motion. And find the
natural frequency.
σ 𝑀𝑜 = 𝐼 𝜃ሷ
−𝑘𝑟 2 𝜃 − 𝑘 2𝑟 2 𝜃 = 𝐼 + 𝑚1 2𝑟 2 + 𝑚2 𝑟 2 𝜃ሷ
𝐼 + 4𝑚1 𝑟 2 + 𝑚2 𝑟 2 𝜃ሷ + 5𝑘𝑟 2 𝜃 = 0
𝐼𝑒𝑞 = 𝐼 + 4𝑚1 𝑟 2 + 𝑚2 𝑟 2
𝜔𝑛 =
𝐾𝑒𝑞
𝐼𝑒𝑞
Dr. Ahmed Hegazy
=
𝑘𝑒𝑞 = 5𝑘𝑟 2
5𝑘𝑟 2
4𝑚1 𝑟 2 +𝑚2 𝑟 2 +𝐼
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For the system shown, produce the equation of motion.
σ 𝑀𝑜 = 𝐼 𝜃ሷ
−𝐾𝑟 𝜃 − 𝑐𝑟 𝜃ሶ =
2
3
2
3
2
2
𝑚𝑟 2
2
+ 𝑚𝑟 2 𝜃ሷ
𝑚𝑟 2 𝜃ሷ + 𝑐𝑟 2 𝜃ሶ + 𝑘𝑟 2 𝜃 = 0
𝑚𝜃ሷ + 𝑐𝜃ሶ + 𝑘𝜃 = 0
Dr. Ahmed Hegazy
Email: a.hegazy@eaeat.edu.eg
12
For the system shown, produce the equation of motion. And find the
natural frequency.
σ 𝑀𝑜 = 𝐼 𝜃ሷ
−𝑘
𝐿2
9
𝐿 2
3
𝑚𝜃ሷ +
𝜃 − 2𝑘
4𝐿2
9
𝑐 𝜃ሶ +
𝐿 2
3
𝐿2
3
𝜃−𝑐
3
𝜔𝑛 =
𝑚𝐿2
𝜃ሶ =
𝑘𝜃 = 0
𝑚𝜃ሷ + 4𝑐𝜃ሶ + 3𝑘𝜃 = 0
Dr. Ahmed Hegazy
2𝐿 2
12
÷
𝐾𝑒𝑞
𝐼𝑒𝑞
=
+𝑚
𝐿 2
6
𝜃ሷ
𝑳𝟐
𝟗
3𝑘
𝑚
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Example (5):
For the system shown, produce the equation of motion. Parameters for the
suspension system may be m=300 kg, c=1200 N.s/m, and k=12000 N/m.
σ 𝐹 = 𝑚𝑥ሷ
−𝐾(𝑥 − 𝑦) − 𝑐(𝑥ሶ − 𝑦)
ሶ = 𝑚𝑥ሷ
𝑚𝑥ሷ + 𝑐 𝑥ሶ + 𝑘𝑥 = 𝑐𝑦ሶ + 𝑘𝑦
300𝑥ሷ + 1200𝑥ሶ + 12000𝑥 = 1200𝑦ሶ + 12000𝑦
÷ 𝟑𝟎𝟎
𝑥ሷ + 4𝑥ሶ + 40𝑥 = 4𝑦ሶ + 40𝑦
Dr. Ahmed Hegazy
Email: a.hegazy@eaeat.edu.eg
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