Dynamics of Rigid Bodies

5. Plane Kinematics of Rigid Bodies

References : Engineering Mechanics : Dynamics (J.L.Meriam, L.G.Kraige), 7th Edition, pp.346, P.5.54

Type of Analysis : Plane Kinematics of Rigid Bodies, Absolute Motion

Type of Element : Rigid Body (one part)

Comparison of results

Object Value Theory RecurDyn Error

𝜔𝐴𝐵 [rad/s] 0.572 0.572 0.0

Note

1. Theoretical Solution

1) Basic Conditions

θ = 45°, 𝜔𝑂𝐴 = �̇� = 3 rad/s , 𝑙 = 0.2 𝑚 , 𝑟 = 0.1 𝑚

2) Geometrical Constraint

With the trigonometric function formula,

𝑙

𝑠𝑖𝑛𝛼=

𝑟

𝑠𝑖𝑛𝛽

𝑙

sin (𝜋 − (𝜃 + 𝛽))=

𝑟

𝑠𝑖𝑛𝛽

𝑙

sin (𝜃 + 𝛽)=

𝑟

𝑠𝑖𝑛𝛽

𝑟 ∙ 𝑠𝑖𝑛(𝜃 + 𝛽) = 𝑙 ∙ 𝑠𝑖𝑛𝛽 (1)

Differentiate both sides of this equation,

𝑟 ∙ (�̇� + �̇�) ∙ 𝑐𝑜𝑠(𝜃 + 𝛽) = 𝑙 ∙ �̇� ∙ 𝑐𝑜𝑠𝛽

Where, 𝜔𝑂𝐴 = �̇�, 𝜔𝐴𝐵 = 𝜔 = �̇�

(𝑙 ∙ 𝑐𝑜𝑠𝛽 − 𝑟 ∙ 𝑐𝑜𝑠(𝜃 + 𝛽)) �̇� = 𝑟 ∙ �̇� ∙ 𝑐𝑜𝑠(𝜃 + 𝛽)

𝜔 = �̇� =𝑟 ∙ �̇� ∙ 𝑐𝑜𝑠(𝜃 + 𝛽)

(𝑙 ∙ 𝑐𝑜𝑠𝛽 − 𝑟 ∙ 𝑐𝑜𝑠(𝜃 + 𝛽))

With the equation (1),

𝑟 ∙ 𝑠𝑖𝑛(𝜃 + 𝛽) = 𝑙 ∙ 𝑠𝑖𝑛𝛽

𝑟 ∙ (𝑠𝑖𝑛𝜃 ∙ 𝑐𝑜𝑠𝛽 + 𝑐𝑜𝑠𝜃 ∙ 𝑠𝑖𝑛𝛽) = 𝑙 ∙ 𝑠𝑖𝑛𝛽

Multiply both sides of this equation by 1

𝑐𝑜𝑠𝛽,

𝑟 ∙ (𝑠𝑖𝑛𝜃 + 𝑐𝑜𝑠𝜃 ∙ 𝑡𝑎𝑛𝛽) = 𝑙 ∙ 𝑡𝑎𝑛𝛽

(𝑙 − 𝑟 ∙ 𝑐𝑜𝑠𝜃) ∙ 𝑡𝑎𝑛𝛽 = 𝑟 ∙ 𝑠𝑖𝑛𝜃

𝑡𝑎𝑛𝛽 =𝑟 ∙ 𝑠𝑖𝑛𝜃

(𝑙 − 𝑟 ∙ 𝑐𝑜𝑠𝜃)

𝛽 = 𝑡𝑎𝑛−1𝑟 ∙ 𝑠𝑖𝑛𝜃

𝑙 − 𝑟 ∙ 𝑐𝑜𝑠𝜃

If, θ = 45°, 𝜔𝑂𝐴 = �̇� = 3 rad/s , 𝑙 = 0.2 𝑚 , 𝑟 = 0.1 𝑚,

𝛽 = 𝑡𝑎𝑛−10.1 ∙ 𝑠𝑖𝑛45

0.2 − 0.1 ∙ 𝑐𝑜𝑠45= 28.68°

𝜔 = �̇� =0.1 ∙ 3 ∙ 𝑐𝑜𝑠(45 + 28.68)

(0.2 ∙ 𝑐𝑜𝑠28.68 − 0.1 ∙ 𝑐𝑜𝑠(45 + 28.68))= 0.572 𝑟𝑎𝑑/𝑠

2. Numerical Solution – Using RecurDyn

1) Create New Model

- Set the model name : P5_54

- Set the “Unit” to “MMKS”

- Set the “Gravity” to “-Y”

2) Create an Object Shape

(1) Create a marker after clicking the “Ground” or selecting“Ground – Edit” in the “Database”

window.

[Geometry] – [Solid and Marker] – [Marker]

Point : 200.0, 0.0, 0.0

- Click the “Exit” icon to exit from the “Ground Edit” mode.

(2) Create a “Link”, “Body1”

- Double click “Body1” in the working window or select “Body1 – Edit” (Click the right mouse

button after selecting “Body1” in the “Database” window, then, select “Edit” in the pop up box.)

in the “Database” window on the right to enter the “Body Edit” mode.

- Modify the properties of “Link1” as shown below.

- Create a marker on a point of “Link1”

[Geometry] – [Solid and Marker] – [Marker]

Point : 100.0, 0.0, 0.0

- Click the “Exit” icon to exit from the “Body Edit” mode.

(3) Rotate “Body1”

- Rotate “Body1” 45° to the left (counter clockwise) by using the “Basic Object Control”

(Rotate) with the “Inertial Marker” as the “Reference Frame” as shown below.

(4) Create a “Cylinder”, “Body2”

- Click the point (200, 0, 0) and the end point of “Body1” to create a cylinder with radius of 2mm

as shown below.

- Double click “Body2” in the working window or select “Body2 – Edit” (Click the right mouse

button after selecting “Body2” in the “Database” window, then, select“Edit” in the pop up box.) in

the “Database” window on the right to enter the “Body Edit” mode.

- Modify the properties of “Cylinder1” as shown below.

- Click the “Exit” icon to exit from the “Body Edit” mode.

(5) Create a “Cylinder”, “Body3”

- Click the point “CM” of “Body2” and the end point of “Body2” to create a cylinder with radius of

6mm as shown below.

- Double click “Body3” in the working window or select “Body3 – Edit” (Click the right mouse

button after selecting “Body3” in the “Database” window, then, select “Edit” in the pop up box.)

in the “Database” window on the right to enter the “Body Edit” mode.

- Modify the properties of “Cylinder1” as shown below.

(6) Create a “Cylinder”, “Cylinder2” in the “Body Edit” mode

(7) Use the “Boolean – Subtract” tool to remove “Cylinder2” from “Cylinder1”. After selecting the

“Subtract” tool, select “Cylinder1” and then select “Cylinder2”.

(8) Create a “Cylinder”, “Cylinder3” in the “Body Edit” mode

(9) Use the “Boolean – Subtract” tool to remove “Cylinder3” from “Subtract1”. After selecting the

“Subtract” tool, select “Subtract1” and then select “Cylinder3”.

- Click the “Exit” icon to exit from the “Body Edit” mode.

(10) Adjust the “Icon Size” and the “Marker Size” by Using the “Icon Control” Tool

11) Change the names of the bodies : Rename

3) Create “Joint”

(1) Revolute Joint

- Ground, OA, (0, 0, 0)

- OA, AB, (70.7106781186548, 70.7106781186547, 0)

- Ground, C, (200, 0, 0)

(2) Translational Joint

- C, AB, (183.838834764832, 8.83883476483174, 0)

4) Create “Motion”

- Select “RevJoint1” and click the right mouse button.

- Select “Property” – “Include Motion” – “Velocity” (angular) – “EL” button – “Create” – Enter

the “Name” and 3 [𝑟𝑎𝑑 𝑠⁄ ] as the “Expression”.

5) Analysis

- Execute the Dynamic/Kinematic icon to simulate.

- Set the “End Time” and the “Step” as shown below.

- Set the “Maximum Time Step” to “0.001” and the “Integrator Type” to “DDASSL”.

6) Execute the “Plot”

[View] - [TraceData]

- The angular velocity on Z axis of the link “AB” : -0.572

3. Problems to Consider

1) The masses of the bodies were not set in this problem. If the “User Input” is used to change

them, does the result of the simulation change? Verify it through a simulation.

2) If there is no difference, consider what the reason is.