Rigid Body Analysis in a Plane: The Most ... - MAELabs UCSD

Translation
Rigid Body Analysis in a Plane: The Most Common Machine Design Analysis
General Case: Dynamic Translation and Rotation Special Case 1: Quasi-Static Translation and Rotation
Key Assumption: Accelerations are low enough to be neglected, and thus
a r and θ & are negligible.
y
F1
Translation is evaluated at the
Center of Mass (CM)
Vector Notation
r r
ΣF
= ma
CM
Scalar Notation in x and y
ma ΣF =
x
ma ΣF =
y
CM, x
CM, y
x
F2
F3
M
Center of Mass (CM)
Governing Equations Governing Equations
Rotation
Translation
Rotation
The moments are calculated at
the CM
ΣMCM = ICMθ&
&
The ΣM includes moments from
all forces and also applied pure
moments.
Comments: This is the general case, thus these equations are
always valid for rigid body analysis in a plane. In many cases,
assumptions can be made that simplify the above equations. But it
never hurts to use these equations to double check special case
analysis.
y
F1
Vector Notation
r
ΣF ≈ 0
Scalar Notation
ΣFx ≈ 0
ΣFy ≈ 0
x
F2
F3
M
A
Arbitrary Point
on Rigid Body
The moments can be calculated
about any point.
ΣM A ≈
0
The point “A” is any point on the rigid
body. Since dynamic forces are
negligible, the Center of Mass does
not enter the governing equations.
The proof of this is left as an exercise
for the reader.
Comments: This analysis is similar to static analysis, but can be
applied to moving parts. Quasi-static motion applies to any case
where accelerations are low. This can be impending motion where
motion is slowly beginning to occur or constant velocity motion.
However, it does not apply to motion with frequent starts and
stops, since accelerations can be high. Inequalities can be used to
evaluate whether impending motion will occur; for example, if
ΣFx>0 horizontal motion will occur.

Special Case 2: Dynamic Rotation about a Pivot Special Case 3: Quasi-Static Rotation about a Pivot
Key Assumption: Accelerations are low enough to be neglected, and thus
a r and θ & are negligible.
y
F1
x
Rotation θ
F2
P (pivot)
F3
M
Governing Equations
The moments and inertia are calculated about the pivot
= I &θ
&
ΣM P P
ΙP = Moment of Inertia about pivot point P.
Comments:
• The direction of positive rotation is shown as counterclockwise
to be consistent with the right-hand rule.
• The moment of inertia of a machine element can be difficult to
calculated, but often can be estimated by summing key
masses on the object multiplied by the distance to the pivot
squared.
y
F1
x
Rotation θ
F2
P (pivot)
F3
M
Governing Equations
The moments are calculated about the pivot
ΣM P ≈ 0
Comments:
• Impending motion can be calculated with an inequality, for
example ΣMP>0 indicates that counterclockwise motion will
occur.
• Special case 3 is a subset of special case 1. The force balance
equation are still valid, and can be used to calculate the
reaction forces at the pivot.