The Learning Center
http:/www.rose-hulman.edu/lc
Physics I
Helpful Physics I Information
Constants
0 = 8.85 10 12 C
2
Nm
2
k=
2
1
= 8.99 10 9 Nm 2
C
4 0
e = 1eV = q proton = 16
. 10 19 C
h = 6.63 10 34 Js
c = 3 108 m s
melectron = 9.1 10 31 kg
m proton =
EquationSheet
StaticsandMechanicsofMaterialsI
ForceEquilibriuminTwoandThreeD. Fx = 0 Fy = 0 Fz = 0
N
V
=
An
An
P
P
= cos 2 = cos sin
Ao
Ao
R R i + R y j + Rz k
e= = x
R
Rx2 + R y2 + Rz2
=
Normal/ShearStress
StressonanObliquePlane
Unitvector
A B = A B
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R-2.102
[Rao 5/e, Problem 2.102] The free-vibration response of an electric motor, weighing 500 N,
mounted on a foundation is provided. Identify the foundations spring constant and
damping coefficient.
In Lab 1, you took some acceleration response
R-5.62
[Rao 5/e, Problem 5.62] In the system shown below, the mass ! is excited by a harmonic
force with an amplitude of 15 N and a frequency of 2 Hz. Find the amplitude of steady-
state vibration for each mass if ! = 10 kg, ! = 5 kg, ! = 8000 N/m, and
R-3.64
[Rao 5/e, Problem 3.64] One of the tail rotor blades of a helicopter has an unbalanced mass
of = 0.5 kg at a distance of = 0.15 m from the axis of rotation. The tail section has a
length of 4 m, a mass of 240 kg, a flexural stiffness () of 2.5 M
R-9.73
[Rao 5/e, Problem 9.73] An electric motor is placed on a fixed-fixed beam. The motor
operates at 1350 rpm and has a rotating unbalance of 0.1 kg-m. The beams amplitude of
vibration under steady-state operation of the motor is suppressed by attac
S-1
To enhance the power harvesting capabilities of a piezoelectric
(PZT) patch, you have been asked to design an auxiliary
structure tuned to the first natural frequency of a frame, ! (in
Hz). The PZT patch will be attached to the auxiliary structure
An overview of modal analysis using SolidWorks
1. Create Model
2. Specify materials
3. Simulation/New Study/Frequency
Specify the number of modes
Apply boundary conditions (if any)
4. Run model
Look at results
Look at mesh and refine if necessary
Bracket
Day 28 - Analytical modal analysis for clamped systems
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Today’s objectives:
After this class, you’ll be able to
Day 2 7 - Analytical modal analysis for forced systems
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Today’s obiectives:
After this class, you’ll be able to:
0 Obtain the steady-state re5ponse of a multiudegree-of-
freedom (MDOF) system acted on by a harmonic fo
Day 25 - Orthogonality of the mode shapes
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Today’s obiectives:
After this class, you’lf be able to:
0 Mass-normalize the mode shapes of a multi~degree-of~
freedom (MDOF) system and u
Day 22 - Modal analysis via the ﬁnite element method
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Today’s objectives:
After this class, you’ll be able to:
0 Describe how the ﬁnite element method is used to perform
modal analysis of a continuou
Day l9 ~— Approximate modal analysis using the FRF matrix
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Todax’s objectives:
0 Work through a few activities to get practice with estimating
the natural frequencies and mode shapes of a lightly clamped,
Day 2] - MDOF representation of a continuous system
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Todax’s obiectives:
After this class, you’ll be able to:
0 Approximate the natural frequencies and mode shapes of
a beam by the ﬁni
Day 20 - Analyzing a continuous system as a MDOF system
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Today’s objectives:
After this ciass, you’ll be abie to:
0 Discretize a continuous system using the ﬁnite difference
method.
0 Approx
Day I 7 - Analysis of multi—degree—ofifreedom systems
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Today's obiectives:
0 Work through an activity to get practice with analyzing a
multi-degree-of—freedom (MDOF) system.
EM 406 - Mechanica
Day I6 W Multi—degree—oﬁfreedom systems
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Today's obiectives:
After this class, you’ll be able to:
0 Obtain the natural frequencies, mode shapes, free response,
and harmonic steady~state resp
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Today’s obiectives:
After this class, you’ll be able to:
0 Determine the steady-state response of a viscoust
damped system with two degrees of freedom that is
acted on by a harmonic forcing.
EM 406 - Mechanical Vibrations
Day I I - Undamped, 2-DOF systems with harmonic forcing
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Today’s obiective:
0 Analyze the free response characteristics and steady-
state vibration due to a harmonic ex
Day 26 - Introduction to analytical modal analysis
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Today’s objectives:
After this class, you’ll be able to:
- Determine the free response of a m
Day I 8 - Properties of the FRF matrix
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Today’s obiectives:
After this class, you’ll be able to:
- Estimate the natural frequencies and mode shapes of a
lightly damped, multi-degree-of—fre
Day [4 -Vibration absorbers
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Today’s obiectives:
After this class, you’ll be able to:
0 Explain how a vibration absorber works.
0 Design a vibration absorbe
Day 8 - Introduction to twoudegree-ofifreedom systems
Today's objectives:
After this class, you’ll be able to:
0 Determine the governing equation of motion for a two-
degree-of—freedom (2-DOF) system.
0 Solve for the natural frequencies, mode shapes, an
Day I0 - Harmonic excitation of undamped, 2-DOF systems
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Today’s obiectives:
After this class, you’ll be able to:
0 Determine the steady-state response of an
Day 9 "-— Systems with two degrees of freedom
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Today’s objectives:
After this class, you’ll be able to:
0 Analytically and numerically
Day 2 - Review of mechanical systems modeling
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Today’s objectives:
After this class, you’ll be able to:
0 Describe how simple models of mechanical systems
may be formulated us
Day 7 - Rotating unbalance in SDOF systems
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Todax’s objectives:
After this class, you’ll be able to:
0 Obtain the steady-state response of a viscously damped,
Day 6 - SDOF systems with harmonic base motion
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Todax’s obiectives:
After this class, you’ll be able to:
0 Solve for the steady-state response of a viscoust
dam
Day 4 - Free vibration of SDOF systems with viscous damping
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Todays objectives:
After this class, youll be able to:
0 Explain how the behavior of a viscously clamped system
with a single degree of freedom (SDOF) in free vibration