TAM 412 Spring 2017: Homework 3
Assigned: Thursday February 9, 2017
Due: Beginning of class on Monday February 20, 2017
Problems from textbook:
1. Problem 1.29
2. Problem 1.27: Hint: Recall the model for the car suspension system we derived in
class.
a. D

TAM 412, Homework 6, Selected answers
Harry Dankowicz
Mechanical Science and Engineering
University of Illinois at Urbana-Champaign
[email protected]
3. By Newtons second law it follows that
m11
r
m22
r
= r1 U (r2 r1 )
= r2 U (r2 r1 )
where r1 U (r2 r1 )

TAM 412, Homework 6, due March 3, 2010
Harry Dankowicz
Mechanical Science and Engineering
University of Illinois at Urbana-Champaign
[email protected]
1. Do exercise 3.1 of the textbook.
2. Do exercise 3.4 of the textbook.
3. Do exercise 3.7 of the textb

TAM 412, Homework 5, Selected answers
Harry Dankowicz
Mechanical Science and Engineering
University of Illinois at Urbana-Champaign
[email protected]
e
3. Let R denote a reference frame xed relative to the wire. Let R denote an inertial reference frame a

TAM 412, Homework 5, due February 24, 2010
Harry Dankowicz
Mechanical Science and Engineering
University of Illinois at Urbana-Champaign
[email protected]
1. Do exercise 1.9 of the textbook.
2. Do exercise 1.13 of the textbook.
3. Do exercise 1.14 of the

TAM 412, Homework 4, Selected answers
Harry Dankowicz
Mechanical Science and Engineering
University of Illinois at Urbana-Champaign
[email protected]
2. Here,
T=
12
q
2
and U =
12
q
2
which implies that
Fq =
and
T
q
T
= Fq q = q
q
(T U )
q
(T U )
=0q+

TAM 412, Homework 4, due February 17, 2010
Harry Dankowicz
Mechanical Science and Engineering
University of Illinois at Urbana-Champaign
[email protected]
1. Do exercise 2.1 of the textbook. Feel free to use the Mathematica VectorPlot and ContourPlot
com

TAM 412, Homework 3, Selected answers
Harry Dankowicz
Mechanical Science and Engineering
University of Illinois at Urbana-Champaign
[email protected]
1.
c. Let
f ( , ) =
2
1
2
2
The conditions of the inverse function theorem are satised at = = 1 and, in

TAM 412, Homework 3, due February 10, 2010
Harry Dankowicz
Mechanical Science and Engineering
University of Illinois at Urbana-Champaign
[email protected]
Please refer to the revised lecture notes on the website for completing this homework assignment.
1

TAM 412, Homework 2, Selected answers
Harry Dankowicz
Mechanical Science and Engineering
University of Illinois at Urbana-Champaign
[email protected]
3. Recall that det (AB ) = det A det B , det (A) = (1)n det A, and det AT = det A. Also, note that
Av =

TAM 412, Homework 7, due March 10, 2010
Harry Dankowicz
Mechanical Science and Engineering
University of Illinois at Urbana-Champaign
[email protected]
1. Do exercise 4.1 of the textbook.
2. Do exercise 4.2 of the textbook.
3. Do exercise 4.3 of the text

TAM 412, Homework 7, Selected answers
Harry Dankowicz
Mechanical Science and Engineering
University of Illinois at Urbana-Champaign
[email protected]
1. Here
T=
where
m 2
x + y2 + z2
_
_
_
2
and U = mgz
x = r cos ; y = r sin ; z = f (r)
It follows that t

TAM 412 Spring 2017: Homework 4
Assigned: Friday February 17, 2017
Due: Beginning of class on Monday February 27, 2017
Problems from text book: 6.8, 6.11
Additional Problems:
TAM 412 Spring 2015
For the following systems, use Hamiltons extended principle

TAM 412 Spring 2017: Homework 1
Assigned: Monday January 23, 2017
Due: Wednesday February 1, 2017
Problems from textbook: 1.1, 1.8, 1.9
Additional Problems:
1. In class we derived the governing nonlinear differential equation for the in-plane
oscillations

TAM 412, Homework 11, due May 5, 2010
Harry Dankowicz
Mechanical Science and Engineering
University of Illinois at Urbana-Champaign
[email protected]
1. Consider the dierential equation
mx + cx + kx = F cos t,
where the dot denotes dierentiation with res

TAM 412, Homework 10, Selected answers
Harry Dankowicz
Mechanical Science and Engineering
University of Illinois at Urbana-Champaign
[email protected]
1. Here,
0
q1
1
1
q2 = C1 eit + C2 eit 0 + C3 eit + C4 eit 1 + C5 ei 2t + C6 ei 2t 1
0
1
q3
0
from wh

TAM 412, Homework 10, due April 21, 2010
Harry Dankowicz
Mechanical Science and Engineering
University of Illinois at Urbana-Champaign
[email protected]
1. Do exercise 6.1 of the textbook.
2. Do exercise 6.2 of the textbook.
3. Do exercise 6.3 of the tex

TAM 412, Homework 9, Selected answers
Harry Dankowicz
Mechanical Science and Engineering
University of Illinois at Urbana-Champaign
[email protected]
1. Introduce ve bases, e, t(1) , t(2) , t(3) , and e, such that e is xed relative to the inertial frame,

TAM 412, Homework 9, due April 7, 2010
Harry Dankowicz
Mechanical Science and Engineering
University of Illinois at Urbana-Champaign
[email protected]
1. Do exercise 5.15 of the textbook.
2. Do exercise 5.16 of the textbook.
3. Do exercise 5.17 of the te

TAM 412, Homework 8, Selected answers
Harry Dankowicz
Mechanical Science and Engineering
University of Illinois at Urbana-Champaign
[email protected]
1. By symmetry, the moment of inertia matrix J about the center of the cube with respect to axes paralle

TAM 412, Homework 8, due March 31, 2010
Harry Dankowicz
Mechanical Science and Engineering
University of Illinois at Urbana-Champaign
[email protected]
1. Do exercise 5.2 of the textbook.
2. Do exercise 5.3 of the textbook.
3. Do exercise 5.4 of the text

TAM 412, Homework 2, due February 3, 2010
Harry Dankowicz
Mechanical Science and Engineering
University of Illinois at Urbana-Champaign
[email protected]
1. Do exercise 1.7 of the textbook.
2. Do exercise 1.8 of the textbook.
3. Do exercise 1.5 of the te

TAM 412, Homework 1, Selected answers
Harry Dankowicz
Mechanical Science and Engineering
University of Illinois at Urbana-Champaign
[email protected]
1. The map : P 7 (x (P ) , y (P ) is onto and one-to one. Also,
s1
0
q2
= y cos x cos
= x cos + y sin

TAM 412, Homework 1, due January 27, 2010
Harry Dankowicz
Mechanical Science and Engineering
University of Illinois at Urbana-Champaign
[email protected]
1. Consider the gure below where and are given angles.
(a) Explain the sense in which the map C (x (

1. A mechanism consisting of particles and rigid bodies has four degrees of freedom. What does this
mean?
2. The generalized forces corresponding to a given applied force on a mechanism and relative to some
chart are all zero. What does this mean?
3. What

Study sheet Stationary Points
Harry Dankowicz
Mechanical Science and Engineering
University of Illinois at Urbana-Champaign
[email protected]
Objective: To investigate the conditions for a local maximum or minimum of a function.
Observation 1 Consider the fu

Sample Mid-Term
TAM 412
Instructions: This test is closed book, closed notes. You may use a calculator, but should not require such
a tool. You may use a ruler for drawing purposes. Unless otherwise noted, all computations should be explicit
and actual al

Programming project
Due date: Monday, April 26, 2010
Instructions: Each student is required to complete a programming project, including a written report and
a computer simulation of the dynamics of a particle-system and/or rigid-body mechanism. The progr

3
Generalized forces
Denition 16 Let C (s) be a curve that passes through C0 at s = 0. The virtual work along C (s) and
evaluated at s = 0 equals the work performed by all physical forces acting on the mechanism under a virtual
displacement corresponding

2
Tangent Vectors
Denition 11 A curve is a subset C (s) of allowable congurations parametrized by a single parameter s.
Denition 12 Two curves C1 (s) and C2 (s) that pass through C0 at s = 0 are equivalent if
d
d
=
.
(C1 (s)
(C2 (s)
ds
ds
s=0
s=0
A tang