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EXAMINATION FOR ME3112E – Mechanics of Machine (Semester I: 2012/2013)

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∑ (

(

(

(

(

(

)

(

(

(

)

(

Based above formula and compare with SolidWorks simulation result, we can see that the theoretical result matches with Simulation quite well, especially when L becomes longer. Table 1: Max Angular Acceleration: Theoretical vs. SolidWorks simulation. (

0.6

0.9

1.2

1.5

1.8

2.1

2.4

4.8

9.6

(

7

10.5

14

17.5

21

24.5

28

56

112

(

9

9

9

9

9

9

9

9

9

Calculated Angular Acc

18.033

12.426

9.571

7.826

6.642

5.783

5.128

2.733

1.434

Simulated Angular Acc

16.405

11.948

9.378

7.732

6.591

5.752

5.108

2.731

1.433

9.93%

4.00%

2.06%

1.21%

0.78%

0.54%

0.38%

0.06%

0.01%

Difference:

Page 1 of 3

I also performed SolidWorks Simulation for Pinned disk case, here are the results: Table 2: Max Angular Acceleration: Theoretical vs. SolidWorks simulation. (

0.6

0.9

1.2

1.5

1.8

2.1

2.4

4.8

9.6

(

7

10.5

14

17.5

21

24.5

28

56

112

(

9

9

9

9

9

9

9

9

9

Calculated Angular Acc 18.033

12.426

9.571

7.826

6.642

5.783

5.128

2.733

1.434

Simulated Angular Acc 16.405 (Welded Disk)

11.948

9.378

7.732

6.591

5.752

5.108

2.731

1.433

Difference: 9.93%

4.00%

2.06%

1.21%

0.78%

0.54%

0.38%

0.06%

0.01%

Simulated Angular Acc 18.033 (Pinned Disk)

12.426

9.571

7.826

6.642

5.783

5.128

2.733

1.434

Difference: 0.001% 0.001% 0.001% 0.001% 0.001% 0.001% 0.001% 0.001% 0.000%

Surprisingly, the simulation result based on pinned disk matches with theoretical calculation perfectly! Which means the previously derived formula for maximum angular acceleration

(

is perfectly applicable for pinned disk case! But what should be the appropriate formula for welded disk case??? SolidWorks Motion Simulation output example:

Displacement vs. Time

Velocity vs. Time

Acceleration vs. Time

Page 2 of 3

For welded disk, the moment of inertia caused by disk should be calculated based on parallel axis theorem:

where: is the moment of inertia of the object about an axis passing through its centre of mass; is the object's mass; is the perpendicular distance between the axis of rotation and the axis that would pass through the centre of mass. Hence the equation of max acceleration will become: (

(

)

(

)

(

Table 3: Max Angular Acceleration: Theoretical vs. SolidWorks Motion simulation. (

0.6

0.9

1.2

1.5

1.8

2.1

2.4

4.8

9.6

(

7

10.5

14

17.5

21

24.5

28

56

112

(

9

9

9

9

9

9

9

9

9

Calculated Angular Acc 16.405 (Use equation 4)

11.948

9.378

7.732

6.591

5.752

5.108

2.731

1.433

Simulated Angular Acc 16.405 (Welded Disk)

11.948

9.378

7.732

6.591

5.752

5.108

2.731

1.433

Difference: -0.002%

0.001%

Calculated Angular Acc 18.033 (Use equation 3)

12.426

9.571

7.826

6.642

5.783

5.128

2.733

1.434

Simulated Angular Acc 18.033 (Pinned Disk)

12.426

9.571

7.826

6.642

5.783

5.128

2.733

1.434

-0.003% 0.002% -0.005% -0.003% 0.007%

0.017% 0.030%

Difference: 0.001% 0.001% 0.001% 0.001% 0.001% 0.001% 0.001% 0.001% 0.000%

This time, the calculated result matches with SolidWorks Motion simulation perfectly! Conclusion: When the disk is welded, we should use parallel axis theorem to calculate its moment of inertia. While when the disk is pinned, it is acting like a point mass at the end of rod. I wish I had figure this out during exam,

.

Page 3 of 3

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(𝒎𝑹

𝒐

𝒎𝑫 𝒈

𝑳𝑪𝑮 𝑰𝜶 𝑳𝑪𝑮

(

∑ (

(

(

(

(

(

)

(

(

(

)

(

Based above formula and compare with SolidWorks simulation result, we can see that the theoretical result matches with Simulation quite well, especially when L becomes longer. Table 1: Max Angular Acceleration: Theoretical vs. SolidWorks simulation. (

0.6

0.9

1.2

1.5

1.8

2.1

2.4

4.8

9.6

(

7

10.5

14

17.5

21

24.5

28

56

112

(

9

9

9

9

9

9

9

9

9

Calculated Angular Acc

18.033

12.426

9.571

7.826

6.642

5.783

5.128

2.733

1.434

Simulated Angular Acc

16.405

11.948

9.378

7.732

6.591

5.752

5.108

2.731

1.433

9.93%

4.00%

2.06%

1.21%

0.78%

0.54%

0.38%

0.06%

0.01%

Difference:

Page 1 of 3

I also performed SolidWorks Simulation for Pinned disk case, here are the results: Table 2: Max Angular Acceleration: Theoretical vs. SolidWorks simulation. (

0.6

0.9

1.2

1.5

1.8

2.1

2.4

4.8

9.6

(

7

10.5

14

17.5

21

24.5

28

56

112

(

9

9

9

9

9

9

9

9

9

Calculated Angular Acc 18.033

12.426

9.571

7.826

6.642

5.783

5.128

2.733

1.434

Simulated Angular Acc 16.405 (Welded Disk)

11.948

9.378

7.732

6.591

5.752

5.108

2.731

1.433

Difference: 9.93%

4.00%

2.06%

1.21%

0.78%

0.54%

0.38%

0.06%

0.01%

Simulated Angular Acc 18.033 (Pinned Disk)

12.426

9.571

7.826

6.642

5.783

5.128

2.733

1.434

Difference: 0.001% 0.001% 0.001% 0.001% 0.001% 0.001% 0.001% 0.001% 0.000%

Surprisingly, the simulation result based on pinned disk matches with theoretical calculation perfectly! Which means the previously derived formula for maximum angular acceleration

(

is perfectly applicable for pinned disk case! But what should be the appropriate formula for welded disk case??? SolidWorks Motion Simulation output example:

Displacement vs. Time

Velocity vs. Time

Acceleration vs. Time

Page 2 of 3

For welded disk, the moment of inertia caused by disk should be calculated based on parallel axis theorem:

where: is the moment of inertia of the object about an axis passing through its centre of mass; is the object's mass; is the perpendicular distance between the axis of rotation and the axis that would pass through the centre of mass. Hence the equation of max acceleration will become: (

(

)

(

)

(

Table 3: Max Angular Acceleration: Theoretical vs. SolidWorks Motion simulation. (

0.6

0.9

1.2

1.5

1.8

2.1

2.4

4.8

9.6

(

7

10.5

14

17.5

21

24.5

28

56

112

(

9

9

9

9

9

9

9

9

9

Calculated Angular Acc 16.405 (Use equation 4)

11.948

9.378

7.732

6.591

5.752

5.108

2.731

1.433

Simulated Angular Acc 16.405 (Welded Disk)

11.948

9.378

7.732

6.591

5.752

5.108

2.731

1.433

Difference: -0.002%

0.001%

Calculated Angular Acc 18.033 (Use equation 3)

12.426

9.571

7.826

6.642

5.783

5.128

2.733

1.434

Simulated Angular Acc 18.033 (Pinned Disk)

12.426

9.571

7.826

6.642

5.783

5.128

2.733

1.434

-0.003% 0.002% -0.005% -0.003% 0.007%

0.017% 0.030%

Difference: 0.001% 0.001% 0.001% 0.001% 0.001% 0.001% 0.001% 0.001% 0.000%

This time, the calculated result matches with SolidWorks Motion simulation perfectly! Conclusion: When the disk is welded, we should use parallel axis theorem to calculate its moment of inertia. While when the disk is pinned, it is acting like a point mass at the end of rod. I wish I had figure this out during exam,

.

Page 3 of 3

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