Dynamics of Rigid Bodies

5. Plane Kinematics of Rigid Bodies

Reference : Engineering Mechanics : Dynamics (J.L.Meriam, L.G.Kraige), 7th Edition, pp.353, E.5.9

Type of Analysis : Plane Kinematics of Rigid Bodies, Relative Velocity

Type of Element : Rigid Body (one part)

Comparison of results

Object Value Theory RecurDyn Error

𝑣𝐺 [m/s] 19.23 19.24 0.01

𝜔𝐴𝐵 [rad/s] 29.5 29.5 0.00

Note

1. Theoretical Solution

1) Basic Conditions

𝜔𝑂𝐵 = 1500 𝑟𝑒𝑣/𝑚 = 1500×2𝜋

60𝑟𝑎𝑑/𝑠 = 50𝜋 𝑟𝑎𝑑/𝑠

𝑙1 = 0.25 𝑚 , 𝑙2 = 0.1 𝑚 , 𝑟 = 0.125 𝑚 , θ = 60°

2) Geometrical Constraint

𝑟

𝑠𝑖𝑛𝛽=

𝑙

𝑠𝑖𝑛𝜃

𝛽 𝜃

B

A O

G 𝑙1 𝑟

𝑙2 𝜔𝑂𝐵

𝑣𝐵

𝑣𝐴

𝑣𝐴/𝐵

𝛾

𝜔𝐴𝐵

𝜔𝐴𝐵

β = 𝑠𝑖𝑛−1 (𝑟

𝑙∙ 𝑠𝑖𝑛𝜃) = 𝑠𝑖𝑛−1 (

0.125

0.35∙ 𝑠𝑖𝑛60) = 18.02°

∴ γ = 71.98°

𝑣𝐵 = 𝑟 ∙ 𝜔𝑂𝐵 = 0.125×50𝜋 = 19.63 𝑚/𝑠

3) The Equation of Relative Velocity

𝑣𝐴⃗⃗⃗⃗ = 𝑣𝐵⃗⃗ ⃗⃗ + 𝑣𝐴/𝐵⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗

𝑣𝐴

𝑠𝑖𝑛78.02=

𝑣𝐵

𝑠𝑖𝑛𝛾

𝑣𝐴 =𝑣𝐵

𝑠𝑖𝑛71.98 ∙ 𝑠𝑖𝑛78.02 = 20.19 𝑚/𝑠

𝑣𝐴/𝐵

𝑠𝑖𝑛30=

𝑣𝐵

𝑠𝑖𝑛𝛾

𝑣𝐴/𝐵 =𝑣𝐵

𝑠𝑖𝑛71.98 ∙ 𝑠𝑖𝑛30 = 10.32 𝑚/𝑠

𝜔𝐴𝐵 =𝑣𝐴/𝐵

𝑙 = 29.49 𝑟𝑎𝑑/𝑠

𝑣𝐺⃗⃗ ⃗⃗ = 𝑣𝐴⃗⃗⃗⃗ + 𝑣𝐺/𝐴⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗

= 20.19𝑖 +29.49�⃗� ×(0.25𝑐𝑜𝑠18.02𝑖 + 0.25𝑠𝑖𝑛18.02𝑗 ) = 20.19𝑖 + (7.01𝑗 − 2.28𝑖 ) = 17.91𝑖 + 7.01𝑗

𝑣𝐺 = √(17.912 + 7.012) = 19.23 𝑚/𝑠

𝑣𝐵 𝑣𝐴/𝐵

𝑣𝐴

𝛾 30° 78.02°

2. Numerical Solution – Using RecurDyn

1) Create New Model

- Set the model name : PE5_9

- 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.

Point : -395.3381739, 0.0, 0.0

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

(2) Create a “Link”, “Body1”

Point1 : -395.3381739, 0.0, 0.0

Point2 : -45.3381739,0,0, 0.0

- 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 two markers on the two points of the “Link1”

Point : -395.3381739, 0.0, 0.0

Point : -45.3381739,0,0, 0.0

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

- Modify the properties of “Body1”

- Modify the point “CM” of “Body1”.

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

(4) Rotate “Body1”

- Rotate “Body1” 18.01673623° to the left (counter clockwise) by using the “Basic Object Control”

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

(5) Create a “Cylinder”, “Body2”

Point1 : -425.0, 0,0, 0.0

Point2 : -365.0, 0,0, 0.0

Radius : 40

- 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.

(6) Create a “Link”, “Body3”

- 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 “Link1” as shown below.

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

(7) Change the names of the bodies : Rename

3) Create “Joint”

1) Revolute Joint

2) Translational Joint

4) Create “Motion”

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

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

the “Name” and −50 ∗ pi [𝑟𝑎𝑑 𝑠⁄ ] as the “Expression”.

5) Analysis

- Execute the Dynamic/Kinematic icon to simulate (this example problem is a kinematic

problem because the number of degrees of freedom is “0”).

- 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”

(1) The velocity on X axis and the velocity on Y axis of the “CM” of the “CRod”

- The velocity of the “CM” of the “CRod” : 19240.12 mm/s

(2) The angular velocity on Z axis of the “CRod” : 29.5 rad/s

3. Problems to Consider

1) What can you do to calculate the velocity of an arbitrary point on “CRod” in the RecurDyn model?

2) In this problem, calculate the velocity and the acceleration of the end point “B” on “CRod”.