A pair of parallel-axis helical gears has 14 normal pressure angle, diametral pitch of
6 teeth/in, and 45 helix angle. The pinion has 15 teeth, and the gear has 24 teeth.
Calculate the transverse and normal circular pitch, t
Find the diametral pitch of a pair of gears having 32 and 84 teeth, respectively, whose
center distance is 3.625 in.
N 2 N 3 32 + 84
= 3.625 in
= 16 teeth/in
2 ( 3.625 in )
R2 + R3 =
Find the numbe
Position and Displacement
Describe and sketch the locus of a point A which moves according to the equations
RA = atcos ( 2t ) , RA = atsin ( 2 t ) , RA = 0 .
Find the position difference from point P
to point Q on the curve y = x
The position vector of a point is given by the equation R = 100e j t , where R is in
inches. Find the velocity of the point at t = 0.40 s.
R ( t ) = 100e j t
R ( t ) = j 100e j t
R ( 0.40 ) = j 100e j 0.40
= j 100 ( cos 0.40
Worms and Worm Gears
A worm having 4 teeth and a lead of 1.0 in drives a worm gear at a velocity ratio of 7.5.
Determine the pitch diameters of the worm and worm gear for a center distance of 1.75
px = l N 2 = 1 in 4 teeth = 0.25 in/
A pair of straight-tooth bevel gears is to be manufactured for a shaft angle of 90. If the
driver is to have 18 teeth and the velocity ratio is to be 3:1, what are the pitch angles?
N 2 = 18 teeth ,
2 = tan 1 ( N 2 N 3 ) = 1
DESIGN OF MECHANISMS
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The reciprocating radial roller follower of a plate cam is to rise 2 in with simple
harmonic motion in 180o of cam rotation and return with simple harmonic motion i
The position vector of a point is defined by the equation R = 4t t 3 + 10 where R
is in inches and t is in seconds. Find the acceleration of the point at t = 2 s.
R ( t ) = ( 4 t2 )
R ( t ) = 4t t 3 3 + 10
F ( j )
f (t )e jt dt
f (t )
F ( j )e jt d
C ( j ) G ( j ) R ( j )
G ( j ) R ( ) jX ( )
M ( ) R 2 ( ) X 2 ( )
( ) tan 1
X ( )
R ( )
r (t ) 10 sin(100t 20o )
c(t ) 10 G ( j100) sin(100
Find the speed and direction of gear 8 in the figure. What is the kinematic coefficient of
N 2 N 4 N 5 N 7 18 15 33 16
N 3 N 5 N 6 N 8 44 33 36 48
8 = 8/ 2 2 = ( +5 88 ) ( 1 200 rev/min ccw ) = 68.
Synthesis of Linkages
A function varies from 0 to 10. Find the Chebychev spacing for two, three, four, five,
and six precision positions.
With x0 = 0.0 and xn +1 = 10.0 , Eq. (11.5) becomes:
x j = 5.0 5.0 cos
Mathematical Models, Analysis and Simulation, Part I, Fall 2005
Homework 5, Phase Portraits.
Three problems are to be solved in this homework assignment. The rst consists in scaling
of a homogeneous and a nonhonogeneous differential equation.
Engineering vibration (PME302) HW # 1. Due: Sep. 28
1. Derive the equation of motion for the above systems.
(: PME302) # 2. : 10 9
1. m kx 0
rad/s, x0 = 1 mm, v0 =
. (2 )
2. 1 mm , (phase shift at t=0) 2 rad 5 rad/s .
. (2 )
3. An undamped system vibrates with a frequency of 10 Hz and amplitude 1 mm. Calculate the maximum amplitude of the
Plot the response x(t) of an underdamped system with
rad/s, = 0.1,
and v0=0 for the following initial displacements: x0= 1 mm, x0= 5 mm, x0= 10
for the following initial displacements:
mm, and x0= 100 mm.
Engineering vibration (PME302) HW # 3. Due: Oct. 17
1. A wave consisting of the wave from a passing boat impacts a seawall. It is desired to calculate the resulting vibration.
Figure P3.15 illustrates the situation and suggests a model. This force in Figu
The Z-Plane and the Unit Circle
Properties of the z-transform
Transfer Function, Poles and Zeroes
Physical Interpretation of Poles and Zeroes
Nonlinear System Theory
The Volterra/Wiener Approach
Wilson J. Rugh
Originally published by The Johns Hopkins
University Press, 1981 (ISBN O-8018-2549-0). Web
version prepared in 2002.
CHAPTER 1 Input/Output Representations in the Time
Minimum time control of the Kepler equation
Jean-Baptiste Caillau, Joseph Gergaud,
and Joseph Noailles
ENSEEIHT-IRIT, UMR CNRS 5505
2 rue Camichel
cfw_caillau, gergaud, [email protected]
Van der Pols Oscillator under Delayed Feedback
Fatihcan M. Atay
Department of Mathematics
Istinye 80860, Istanbul, Turkey
E-mail: [email protected]
Preprint. Final version in
J. Sound and Vibration 218(2):333-339, 1998
o has memory (integrator),
o responds based on not only current but also past input,
o is represented by a differential equation, or
o has energy-storing elements (spring or mass)
Transfer function is the ratio of the output to the input in s domain.
G (s ) =
Y (s )
U (s )
Transfer function is a complex rational function
p(s ) bm s m + bm -1s m -1 + + b1s + b0
2. State-space representation
3. Relationship with transfer function
o Non-uniqueness of state variables
Copyright Dongik Shin, 2010
An n-th order differential equation has n i
Excerpted from "A Mathematical Introduction to Robotic Manipulation"
by R. M. Murray, Z. Li and S. S. Sastry
Lyapunov Stability Theory
In this section we review the tools of Lyapunov stability theory. These
tools will be used in the next section to anal