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 example 6.31 (with different B.C.'s), the spatial resp
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-section
Newton's N t ' 2nd law on the element d : l th l t d
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: strings, rods, beams, membranes, and plates
("continuo
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 at the same frequency are same? 2. In usual machineries,
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 absorber, may l d t unexpectedly worse results. b b lead
1
MAE351 Mechanical Vibrations
Lecture 22
(Chap. 5.3-5.4)
2
Coupled oscillator
Vibration absorber
f=0.67
f=1.0
f=1.3
(Courtesy of Dr. D. Russell)
1
3
5.3 Vibration Absorbers
F(t)=F0sint
Primary mass
x
m ka xa K /2
absorber
K /2
ma
Primary system experie
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 elastic body? t eated igid bod bod ? 3. What are physical
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 force at the i - th coordinate q i
For a conservati ve
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
Many systems have large numbers of DOF
Process stays the s
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 1 and xT x 2 1; orthogonal if x1 x 2 0 2
2
This is abbr
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 force? Give an example. 3. What is the nonstationary force
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 the forcing characteristic (such as rise time) to the nat
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 sines and cosines (Fourier) or a series of ejt and e-jt
1
MAE351 Mechanical Vibrations
Lecture 11
(Chap. 3.1-3.2)
2
Types of Input Signals (Forces)
(adapted from B&K Technical Note BA767412)
1
Chap. 3. General Forced Response: Linear superposition
If x1 , x2 are both solutions, then x a1 x1 a2 x2 is a solution
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 the frequency of the response of a viscously damped sy
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
Accelerometers & Load Cells
Piezo-electric Type Accelerometers
4
2
5
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 versa. versa Uses material heavily from Sections 2.4 on B
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 coefficients Now we look at 3 alternatives: g pp a geome
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 t happens to the natural frequency, if the mass t th t
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 of damper for varying mass system
Mass 2 kg < m < 3kg
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 of a physical device, done mostly by Newton's 2nd law
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 systems dissipate energy when they vibrate. To account for th
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
Stiffness
From strength of materials, recall:
Hooke's Law: H
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 materials recall:
x0
x1
x2
x3
g
f fkk 103 N
(or F)
no
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. Are the boundary conditions important in a discrete syst
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 using the system's mode shapes.
2 0 x1 3 -1 x1 6 -2 x1