mechanism building creating of kinematic chain · 2015-11-12 · creating of kinematic chain....

Ing. Šimon Kovář, Ph.D.

Mechanism building

CREATING OF KINEMATIC CHAIN

GEOMETRICAL PRECISION OF MECHANISMS

Slider crank mechanism

manufacturing tolerance - Δr

error in the position of the slider - Δs=?

Analytical determine slider position error.

crank

slider

coupler

frame

Mechanism Building| 12.11.2015

Slider crank mechanism

For small dimensions is possible: 𝑑𝑟 = Δ𝑟

𝑑𝑠 = Δ𝑠

Mechanism Building| 12.11.2015

Swinging mechanism

For small dimensions is possible:

𝑑𝐻 = Δ𝐻

𝑑𝜓 = Δ𝜓

Mechanism Building| 12.11.2015

Swinging mechanism

manufacturing tolerance - ΔH

error in the position of the

slider - ΔΨ = ?

Analytical determine rocker

position error.

crank

slider

rocker

frame

Mechanism Building| 12.11.2015

Swinging mechanism

Δ𝜓 = −Δ𝐻 ∗ 𝑟 ∗ 𝑠𝑖𝑛𝜑

(𝐻 − 𝑟 ∗ 𝑐𝑜𝑠𝜑)2+𝑟2 ∗ 𝑠𝑖𝑛2𝜑

𝑐𝑜𝑠2𝜓 =(𝐻 − 𝑟 ∗ 𝑐𝑜𝑠𝜑)2

(𝐻 − 𝑟 ∗ 𝑐𝑜𝑠𝜑)2+𝑟2 ∗ 𝑠𝑖𝑛2𝜑

Mechanism Building| 12.11.2015

4 bar mechanism

Transfer function

manufacturing tolerance - Δb

error in the position of the

rocker - Δ=?

Analytical determine slider

position error.

Mechanism Building| 12.11.2015

4 bar mechanism

For small dimensions is possible:

𝑑𝑏 = Δ𝑏

𝑑𝜓 = Δ𝜓

Δ𝜓𝐶 = Δ𝜓𝑎 + Δ𝜓𝑏 + Δ𝜓𝑐 + Δ𝜓𝑑For finish error valid:

Mechanism Building| 12.11.2015

TRNASFORMATION OF MECHANISMS

Definition for transformation angle

The angle between the direction

of absolute motion and direction

of the relative motion of the

point in which the driven

member gets the motion impulse

Mechanism Building| 12.11.2015

4 bar mechanism

Mechanism Building| 12.11.2015

4 bar mechanism

Important limits for analysis: maximal and minimal values of transformation angle

𝑎𝑝𝑝𝑜𝑖𝑛𝑡 𝜓1 = 0° 𝑎𝑛𝑑 𝜓2 = 180° to

Mechanism Building| 12.11.2015

4 bar mechanism

Ideal transfer between the driving and the driven parts is 90°. Optimal transformation angle is 90°.

Mechanism Building| 12.11.2015

4 bar mechanisms

Very important milestone is transformation in reversal points of mechanism.

Mechanism Building| 12.11.2015

4 bar mechanisms - example

𝜇 1,2 =?

a

d

b

c

Mechanism Building| 12.11.2015

4 bar mechanisms - example

𝜇 1,2 =?

Mechanism Building| 12.11.2015

4 bar mechanisms - example

Results: 𝜇1 = 𝜇𝑚𝑖𝑛 = 104°00´

𝜇2 = 𝜇𝑚𝑎𝑥 = 81°50´

𝜓𝑂𝑃𝑇1 = 105°20´

𝜓𝑂𝑃𝑇2 = 254°40´

𝜇𝑈1 = 87°10´

𝜇𝑈2 = 98°40´

𝜓𝑈1 = 121°40´

𝜓𝑈2 = 302°00´

Mechanism Building| 12.11.2015

Slider – crank mechanism

Mechanism Building| 12.11.2015

Slider – crank mechanism

Valid for maximum and

minimum

reversal points

Mechanism Building| 12.11.2015

Slider – crank mechanism

example

r= 13 mml= 39 mme= 3 mm

𝜇 1,2 =?e

l

r

Mechanism Building| 12.11.2015

Slider – crank mechanism - example

r= 13 mml= 39 mme= 3 mm

𝜇 1,2 =?

Mechanism Building| 12.11.2015

Slider – crank mechanism - example

Results: 𝜇1 = 𝜇𝑚𝑖𝑛 = 64°40´

𝜇2 = 𝜇𝑚𝑎𝑥 = 104°50´

𝜓𝑂𝑃𝑇1 = 193°20´

𝜓𝑂𝑃𝑇2 = 346°40´

𝜇𝑈1 = 86°40´

𝜇𝑈2 = 83°20´

𝜓𝑈1 = 183°20´

𝜓𝑈2 = 6°40´

Mechanism Building| 12.11.2015