ME 6590 Multibody Dynamics
Homework #1
1. The system shown has three components, a
vertical tower T , a horizontal arm M , and
a body B . Body B is a disk with radius r
and is positioned relative to M by the
angle θ 3 . Arm M has length L and is
positioned relative to T with the angle θ 2 .
Tower T is positioned relative to the fixed
frame XYZ with the angle θ1 . (In the
position shown, θ1 and θ 2 are both zero.)
Hence, B is positioned relative to XYZ
with a 2-1-3 body-fixed rotation sequence.
Using the matrix-vector notation discussed
in class, complete the following.
Note: In each case, find expressions that are correct at any general position where
θ1 ≠ 0, θ 2 ≠ 0, and θ 3 ≠ 0 .
a) Find the XYZ components of r P the position vector of P relative to O in terms of L ,
r , and the three angles θ1 , θ 2 , and θ 3 . Express your results as a matrix-vector equation.
b) Find the XYZ components of v P the velocity of P relative to the fixed frame XYZ in
terms of L , r , the angles θ1 , θ 2 , and θ 3 , and the body-fixed angular velocity
components. Express your results as a matrix-vector equation. Use the “two-point”
formula for velocity to help you formulate the equation.
c) Find the XYZ components of a P the acceleration of P relative to the fixed frame XYZ
in terms of L , r , the angles θ1 , θ 2 , and θ 3 , and the body-fixed angular velocity and
angular acceleration components. Express your results as a matrix-vector equation. Use
the “two-point” formula for acceleration to help you formulate the equation.
d) Expand the matrix-vector expressions found in parts (a), (b), and (c).
2. Problem 7.10
3. Problem 7.11