6Problems and Solutions Section 6.3 (6.8 through 6.29) 6.8 Calculate the natural frequencies and mode shapes for a free-free bar. Calculate the temporal solution of the first mode. Solution: Following
2009-04-22
1
MAE351 Mechanical Vibrations
Lecture 26
(Chap. 6.4-6.5)
2
6.4 Torsional Vibrations
d dx from calculus x ( x, t ) GJ from solid x
mechanics
G: shear modulus
J : polar moment of cross-secti
2009-04-22
1
MAE351 Mechanical Vibrations
Lecture 25
(Chap. 6.1-6.3)
2
Chapter 6 Distributed Parameter Systems
Extending the first 5 chapters to systems with distributed mass and stiffness properties:
2009-05-06
Review Questions 5
1. If two different dynamic systems have the same transfer function for input force and output response, is it true that their dynamic responses in operating conditions a
1
MAE351 Mechanical Vibrations
Lecture 23
(Chap. 5.5-5.6)
2
5.5 Optimization of Vibration Absorber
A certain amount of damping, which is expected to be able to improve the performance of vibration ab
2009-04-18
Review Questions 4
1. How do you determine the number of DOF of a lumped-mass system? 2. How many DOFs does an automobile in driving have, if it is treated as (a) rigid body, and (b) an ela
2009-03-17
1
MAE351 Mechanical Vibrations
Lecture 19
(Chap. 4.7-4.8)
2
4.7 Lagrange's Equations
d T dt qi T U Qi , i 1, 2,., n qi qi
where T : kinetic energy, U : potential energy Qi
W : generalized
2009-03-17
1
MAE351 Mechanical Vibrations
Lecture 17
(Chap. 4.4-4.5)
2
Section 4.4 More than 2 DOF (Multi-DOF Systems)
Extending previous section to any number of degrees of freedom
1
2009-03-17
3
Man
1
MAE351 Mechanical Vibrations
Lecture 16
(Chap. 4.3)
Review: Window 4.2- Normalization
Orthonormal Vectors similar to the unit vectors of statics and dynamics
T T x1 and x 2 are both normal if x1 x1
2009-03-23
Review Questions: Chap. 3
1. What is the basis for expressing the response of a system under periodic excitation as a summation of several harmonic responses? 2. What is the nonperiodic for
1
MAE351 Mechanical Vibrations
Lecture 13
(Chap. 3.6, 3.9-3.10)
2
Sec 3.6 Shock (or Response) Spectrum
Spectrum of a given forcing function is a plot of a response quantity (x) against the ratio of th
1
MAE351 Mechanical Vibrations
Lecture 12
(Chap. 3.3-3.4)
2
3.3 Periodic Forcing Functions
2 2n x n x F (t ) where F (t ) F (t T ) x
We know that periodic
functions can be represented by a series of
2009-03-05
Review Questions: Chap. 2
1. What is the physical meaning of homogeneous and particular solutions to the forced, underdamped, single degree-offreedom mechanical oscillator model? 2. What is
1
MAE351 Mechanical Vibrations
Lecture 9
(Chap. 2.6, 2.8-2.9)
2.6 Measurement Devices (Vibration Pick-Up)
Vibration is measured using a single DOF
2
vibrating system
k
x(t)
m
y(t) c
1
3
Accelerometer
1
MAE351 Mechanical Vibrations
Lecture 8
(Chap. 2.4-2.5)
2
(extension of Sec. 2.4)
Section 5.2 Isolation
A major job of vibration engineers is to isolate systems from vibration disturbances or vice ve
1
MAE351 Mechanical Vibrations
Lecture 7
(Chap. 2.3-2.4)
2
Section 2.3 Alternative Representations
A variety of methods for solving differential equations So far, we used the method of undetermined c
2009-03-05
Review Questions: Chap. 1
1. Consider a pendulum with a concentrated mass m hanging in a massless cord with length l. Is the resultant oscillatory motion linear or nonlinear? 2. What h 2 Wh
MAE351 Mechanical Vibrations
Lecture 4
(Chap. 1.7-1.10)
2
Design Considerations (Section 1.7) 1 7)
Using the analysis so far to guide the l ti th selection of components. f t
1
3
Example 1.7.1:
design
MAE351 Mechanical Vibrations
Lecture 3
(Chap. 1.4-1.6)
1
Modeling and Energy Methods
(Section1.4)
Modeling: An art or process of writing an equation or p g q systems of equation to describe the motion
MAE351 Mechanical Vibrations
Lecture 2
(Chap. 1.3)
1
Basic Mechanical Elements of Vibrations
x m k c
(Courtesy of Dr. D. Russell)
m = mass k = stiffness c = damping
2
1
Viscous Damping
All real system
MAE351 Mechanical Vibrations
Lecture 1
(Chap. 1.1-1.2) (Chap 1 1-1 2)
1
1
Basic Mechanical Elements of Vibrations
x m k c
(Courtesy of Dr. D. Russell)
m = mass k = stiffness c = damping =
2
2
Stiffne
MAE351 Mechanical Vibrations
Lecture 1
(Chap. 1.1-1.2)
1
Basic Mechanical Elements of Vibrations
x m k c
(Courtesy of Dr. D. Russell)
m = mass k = stiffness c = damping
2
1
Stiffness
From strength of
2009-05-10
Review Questions 6
1. How many natural frequencies does a continuous system have? 2. How does a continuous system differ from a discrete system in the nature of its equation of motion? 3. A
Problems and Solutions Section 7.6 (7.25-7.31) 7.25 Referring to Section 5.4 and Window 5.3, calculate the receptance matrix of equation (7.25) for the following two-degree-of-freedom system, without