1. Modeling & Simulation of Dynamic Systems
Lecture-5
Modeling Examples
Dr. Imtiaz Hussain
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2. Rotational and Translational Motion (Example-1)
Consider the system shown below.
The disc is of radius ‘R’ and has a moment of inertia ‘J’. There is friction between the disc and the block of mass ‘m’.

3. Rotational and Translational Motion
Free body diagram
𝑚 𝑥
𝐹=𝑚 𝑥 +𝑘𝑥+𝑐 𝑥 + 𝐹 𝑓 (1)

4. Rotational and Translational Motion
Free body diagram of disc
Since 𝑥=𝑅𝜃
𝑇=𝐽 𝜃
𝐹 𝑓 𝑅=𝐽 𝜃
𝐹 𝑓 = 𝐽 𝑅 2 𝑥

5. Rotational and Translational Motion
Substituting the value of 𝐹 𝑓 in equation (1)
Taking Laplace Transform of the equation
𝐹=𝑚 𝑥 +𝑘𝑥+𝑐 𝑥 + 𝐽 𝑅 2 𝑥
𝐹 𝑠 =𝑚 𝑠 2 𝑋(𝑠) +𝑘𝑋(𝑠)+𝑐𝑠𝑋(𝑠) + 𝐽 𝑅 2 𝑠 2 𝑋(𝑠)
𝑋(𝑠) 𝐹(𝑠) = 1 𝑚+ 𝐽 𝑅 2 𝑠 2 +𝑐𝑠+𝑘

6. Example-2 Consider the system shown below (Assume no friction).
The relation between the rotation of the disc and the linear displacement moved by the disc is given by x = Rθ

7. Example-2
Free body diagram of the disc
Where 𝑥=𝑅𝜃
0=𝑚 𝑥 +2𝑘𝑥

8. Example-3

9. Example-3
Free Body diagram of Mass m
𝑚 𝑦
𝑚 𝑦 +𝑐 𝑦 −𝑅 𝜃 +𝑘 𝑦− 𝑅 2 𝜃 =0

10. Example-3 Free body diagram of drum
𝑇=𝐽 𝜃 +𝑐 𝑅 𝜃 − 𝑦 𝑅+𝑘 𝑅 2 𝜃−𝑦 𝑅 2 +𝑘𝜃𝑅 𝑅 +𝑐 𝑅 2 𝜃 ( 𝑅 2 )

11. Example-4: Elevator

12. Example-4: Elevator
The cage of an elevator is hoisted by a long cable wound over a drum driven through a gear-set by an electric motor. The motor is relay-operated (i.e., either on or off)

13. Example-4: Elevator 𝒏 𝒈𝒆𝒂𝒓 𝜽 𝒓𝒊𝒎 𝒓 𝒅𝒓𝒖𝒎 𝜽 𝒎𝒐𝒕𝒐𝒓 𝒆 𝒎𝒐𝒕𝒐𝒓 𝒌 𝒄𝒂𝒃𝒍𝒆 𝒙 𝒄𝒂𝒈𝒆
𝒎 𝒄𝒂𝒈𝒆

14. Example-4: Elevator Differential Equation
𝒎 𝒄𝒂𝒈𝒆
𝑭 𝒎 𝒄𝒂𝒈𝒆
𝑭 𝒌 𝒄𝒂𝒃𝒍𝒆
𝑭 𝒈
𝑚 𝑐𝑎𝑔𝑒 𝑥 𝑐𝑎𝑔𝑒+ 𝑘 𝑐𝑎𝑏𝑙𝑒 𝑥 𝑐𝑎𝑔𝑒 − 𝑥 𝑟𝑖𝑚 + 𝑚 𝑐𝑎𝑔𝑒 𝑔=0

15. Example-4: Elevator Transmission 𝑥 𝑟𝑖𝑚 =𝑟 𝑑𝑟𝑢𝑚 𝜃 𝑑𝑟𝑢𝑚
𝑥 𝑟𝑖𝑚 =𝑟 𝑑𝑟𝑢𝑚 𝜃 𝑑𝑟𝑢𝑚
𝜃 𝑑𝑟𝑢𝑚 =𝑛 𝑔𝑒𝑎𝑟 𝜃 𝑚𝑜𝑡𝑜𝑟
𝜃 𝑚𝑜𝑡𝑜𝑟 =𝑒 𝑚𝑜𝑡𝑜𝑟 𝐾 𝑡 / 𝑅 𝑎 𝑠[𝐽𝑠+(𝐵+ 𝐾 𝑡 𝐾 𝑏 / 𝑅 𝑎 )]

16. Example-5: Tap Drive

17. Example-5: Tap Drive

18. Example-5: Tap Drive

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