Chapter four - Philadelphia University. acceleration... · Acceleration analysis For a known...

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Chapter four Laith Batarseh Acceleration analysis

Transcript of Chapter four - Philadelphia University. acceleration... · Acceleration analysis For a known...

Page 1: Chapter four - Philadelphia University. acceleration... · Acceleration analysis For a known four-bar mechanism, in a given configuration and known velocities, and a given angular

Chapter four

Laith Batarseh

Acceleration analysis

Page 2: Chapter four - Philadelphia University. acceleration... · Acceleration analysis For a known four-bar mechanism, in a given configuration and known velocities, and a given angular

Acceleration analysis

For a known four-bar mechanism, in a given configuration and known velocities, and a given angular acceleration of the crank, α2 (say CCW), construct the acceleration polygon. Determine α 3 and α 4 .

Page 3: Chapter four - Philadelphia University. acceleration... · Acceleration analysis For a known four-bar mechanism, in a given configuration and known velocities, and a given angular

Acceleration analysis

Solution

For the position vector loop equation:

RAO2+RBA − RBO4 − RO4O2 = 0 --- (1)the velocity equation is

VAO2+VBA − VBO4 = 0 --- (2)

Page 4: Chapter four - Philadelphia University. acceleration... · Acceleration analysis For a known four-bar mechanism, in a given configuration and known velocities, and a given angular

Acceleration analysis

Solution

The acceleration equation is obtained from the time derivative of the

velocity equation as:

AA+ABA = AB

Since RAO2 , RBA , and RBO2 are moving vectors with constant lengths, their

acceleration vectors have normal and tangential components:

Now, ω2, ω3, ω4 and α2 are known, so the components

are known and can be drawn directly

n

B

n

BA

t

A

n

A AandAAA ,,

Page 5: Chapter four - Philadelphia University. acceleration... · Acceleration analysis For a known four-bar mechanism, in a given configuration and known velocities, and a given angular

Acceleration analysis

Solution

Rearrange the loop equation to be seem as follow

Drawing procedures:

Page 6: Chapter four - Philadelphia University. acceleration... · Acceleration analysis For a known four-bar mechanism, in a given configuration and known velocities, and a given angular

Acceleration analysis

Solution

Page 7: Chapter four - Philadelphia University. acceleration... · Acceleration analysis For a known four-bar mechanism, in a given configuration and known velocities, and a given angular

Acceleration analysis

Solution

Page 8: Chapter four - Philadelphia University. acceleration... · Acceleration analysis For a known four-bar mechanism, in a given configuration and known velocities, and a given angular

Acceleration analysis

Solution

Page 9: Chapter four - Philadelphia University. acceleration... · Acceleration analysis For a known four-bar mechanism, in a given configuration and known velocities, and a given angular

Acceleration analysis

Solution

Page 10: Chapter four - Philadelphia University. acceleration... · Acceleration analysis For a known four-bar mechanism, in a given configuration and known velocities, and a given angular

Acceleration analysis

Solution

Page 11: Chapter four - Philadelphia University. acceleration... · Acceleration analysis For a known four-bar mechanism, in a given configuration and known velocities, and a given angular

Acceleration analysis

Using vector algebra

tUtStr

Derive with respect to time

USUStr

UUSUSUSUStr 2

Derive another time with respect to time to find the acceleration

But

Rearrange the terms

USSUSStr 22

tUtU

Page 12: Chapter four - Philadelphia University. acceleration... · Acceleration analysis For a known four-bar mechanism, in a given configuration and known velocities, and a given angular

Acceleration analysis

4-bar mechanism

For the 4-bar mechanism, the length of links are constant and so:

And the equation for acceleration become

As shown in pervious chapters we can find the position and velocity analysis to

4-bar mech. And in this section we will find the acceleration analysis by adding

new input which is α2 and new unknown and they are α3 and α4

0 and SS

USUSUSUStr 22

Page 13: Chapter four - Philadelphia University. acceleration... · Acceleration analysis For a known four-bar mechanism, in a given configuration and known velocities, and a given angular

Acceleration analysis

4-bar mechanism

To apply acceleration analysis on 4-bar mechanism, we derive the loop closure

equation

Dot product both sides by Uθ3 to eliminate α3

443322

2

4444

2

3333

2

2222 UdUdUdUdUdUd

43

2

444344

2

3323

2

222322

cossin

cossin

dd

ddd

Solve for α4:

434

43

2

44

2

3323

2

2223224

sin

coscossin

d

dddd

Page 14: Chapter four - Philadelphia University. acceleration... · Acceleration analysis For a known four-bar mechanism, in a given configuration and known velocities, and a given angular

Acceleration analysis

4-bar mechanism

Dot product both sides by Uθ4 to eliminate α4

Solve for α3:

343

2

4434

2

3324

2

2224223

sin

coscossin

d

dddd

2

44

34

2

33343324

2

222422 cossincossin

d

dddd

Page 15: Chapter four - Philadelphia University. acceleration... · Acceleration analysis For a known four-bar mechanism, in a given configuration and known velocities, and a given angular

Acceleration analysis

Slider crank mechanism

For slider crank mechanism the input is α2 and the outputs will be:

The L.C.E is

Derive twice with respect to time

3 and S

SUaUUdUd 903322

USUdUdUdUd 3322

2

3333

2

2222

Dot product both sides by Uθ3 to eliminate α3

3

2

3323

2

222322 coscossin Sddd

Page 16: Chapter four - Philadelphia University. acceleration... · Acceleration analysis For a known four-bar mechanism, in a given configuration and known velocities, and a given angular

Acceleration analysis

Slider crank mechanism

Solve for

3

2

3323

2

222322

cos

cossin dddS

S

Dot product both sides by U.αto eliminate S

0sincossincos 3

2

333332

2

22222 dddd

33

2223

2

332

2

223

cos

cossinsin

d

ddd