Mathematics for Machine Technology
7th Edition
ISBN: 9781133281450
Author: John C. Peterson, Robert D. Smith
Publisher: Cengage Learning
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Chapter 47, Problem 2AR
To determine
(a)
To find out the value of the given algebraic expression
To determine
(b)
To find out the value of the algebraic expression
To determine
(c)
To find out the value of the algebraic expression
To determine
(d)
To find out the value of the algebraic expression
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Show that these two matrices generate the algebra M₂(Q) over the field Q.
0
1
S :=
and T =
==
-1
-
-1
(13)
Let G = S3, the symmetric group on 3 letters. Show that
K(X,Y)
KG
-
(X21, YX XY², Y³ —– 1) '
-
(Hint: Write S3
(12) and Y
X
S3 as a group.)
=
{id, (12), (23), (13), (123), (132)}. Consider the map from
→ (123). You may assume that these two elements generate
Consider a simplified two-joint robotic arm where the two joints are connected by rigid links.Each joint has a mass, and the links are modeled as idealized rods. The motion of each jointaffects the other through the coupling of forces. The system can be modeled by the followingsecond-order coupled ODEs:
m1θ1" + k1(θ1 − θrest) + k3(θ1 − θ2) = 0 (equation 1)m2θ2" + k2(θ2 − θrest) + k3(θ2 − θ1) = 0 (equation 2)
where θ1(t) and θ2(t) represent the angular positions of the first and second joints, respec-tively. m1 and m2 are the effective masses at each joint. k1, k2 and k3 are the stiffnessconstants representing the elastic restoring forces in the joints and links. θrest = 0 representsthe equilibrium position for each joint.The parameters are:m1 = 1, m2 = 1, θrest = 0, k1 = 1, k2 = 2, k3 = 3, θrest = 0The initial conditions are:θ1(0) = 1, θ2(0) = 2Please solve by hand to get θ1(t) and θ2(t) using eigenvectors and eigenvalues. Show all steps/calculations, and provide a written description…
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