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SECTION 1
REVIEW OF FUNDAMENTALS
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TABLE OF CONTENTS
Page
SINGLE DOF SYSTEM 1-3
UNDAMPED FREE VIBRATIONS SDOF SYSTEM 1-6
SINGLE DOF SYSTEM - UNDAMPED FREE VIBRATIONS 1-8
DAMPED FREE VIBRATION SDOF 1-9
DAMPING WITH FORCED VIBRATION 1-13
MSC.NASTRAN DOCUMENTATION 1-23
TEXT REFERENCES ON DYNAMIC ANALYSIS 1-25
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This section will introduce the basics of DynamicAnalysis by considering a Single Degree of Freedom(SDOF) problem
Initially a free vibration model is used to describe the
natural frequency Damping is then introduced and the concept of
critical damping and the undamped solution is shown
Finally a Forcing function is applied and the response
of the SDOF is explored in terms of time dependencyand frequency dependency and compared to theterms found in the equations of motion
SINGLE DOF SYSTEM
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SINGLE DOF SYSTEM (Cont.)
Consider the System Shown
m = mass (inertia)
b = damping (energy dissipation)
k = stiffness (restoring force)
p = applied force
u = displacement of mass
= velocity of mass
= acceleration of mass
u, , and p are time varying in general.
m, b, and k are constants.
m
k b
p(t)
u(t)
u
u
u
u
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Some Theory: The equation of motion is:
In undamped,free vibration analysis, the SDOF equation ofmotion reduces to:
Has a solution of the form:
This form defines the response as being HARMONIC, with a resonant
frequency of:
)()()()( tptkutubtum
0)()( tkutum
tBtAtunn
cossin)(
n
SINGLE DOF SYSTEM (Cont.)
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UNDAMPED FREE VIBRATION SDOFSYSTEM
For an SDOF system the resonant, or natural frequency, isgiven by:
Solve for the constants:
m
kn
n
nn
nnnn
n
tuA
tBt
tBtAtu
u(tBtt
)0(
thus0)sin(,0When
sincos)(
:solutionatingDifferenti
)0thus0)sin(,0When
tutu
tunn
n
cos)0(sin)0(
)(
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UNDAMPED FREE VIBRATION SDOFSYSTEM (Cont.)
The response of the Spring will be harmonic, but the actual form of theresponse through time will be affected by the initial conditions:
If there is no response
If response is a sine function magnitude
If response is a cosine function (180
phase change), magnitude If response is phase and magnitude
dependent on the initial values
0)0(and0)0( uu
0)0(and0)0( uu n
u
0
0)0(and0)0( uu 0u
0)0(and0)0( uu
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SINGLE DOF SYSTEM UNDAMPEDFREE VIBRATIONS
The graph is from a transient analysis of a spring mass system with Initial
velocity conditions only
Time
Disp.
k= 100
m = 1
T = 1/f = 0.63 secs
Hz59.12/f
rad/s10
n
nm
k
10 u
T
Amp
1.0/Amp 0 nu
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DAMPED FREE VIBRATION SDOF
If viscous damping is assumed, the equation of motionbecomes:
There are 3 types of solution to this, defined as: Critically Damped Overdamped
Underdamped
A swing door with a dashpot closing mechanism is a
good analogy If the door oscillates through the closed position it is underdamped
If it creeps slowly to the closed position it is overdamped.
If it closes in the minimum possible time, with no overswing, it iscritically damped.
0)()()( tkutubtum
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DAMPED FREE VIBRATION SDOF (Cont.)
For the critically damped case, there is no oscillation, just adecay from the initial conditions:
The damping in this case is defined as:
A system is overdamped when b > bcr
Generally only the final case is of interest - underdamped
ncr mkmbb 22
mbteBtAtu
2/)()(
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DAMPED FREE VIBRATION SDOF (Cont.)
For the underdamped case b < bcr and the solution is the form:
represents the Damped natural frequency of the system
is called the Critical damping ratio and is defined by:
In most analyses is less than .1 (10%) so
)cossin()( 2/ tBtAetudd
mbt
d
21 nd
crb
b
nd
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The graph is from a transient analysis of the previous spring mass system with
damping applied
Frequency and
period as before
Amplitude is a
function of damping
2% Damping
5% Damping
DAMPED FREE VIBRATION SDOF (Cont.)
Time
Disp.
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DAMPING WITH FORCED VIBRATION
Apply a harmonic forcing function: note that is the DRIVING or INPUT frequency
The equation of motion becomes
The solution consists of two terms:
The initial response, due to initial conditions which decays rapidly in the presence of
damping
The steady-state response as shown:
This equation is described on the next page
tp sin
tptkutubtum sin)()()(
22
2
2
)/2()1(
)sin(/)(
n
n
tkptu
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DAMPING WITH FORCED VIBRATION(Cont.)
This equation deserves inspection as it shows several importantdynamic characteristics:
At = n this term = (2 )^2 and controlsthe scaling of the response
From this is derived the Dynamic
Magnification Factor 1/2
22
2
2 )/2()1(
)sin(/)(
n
n
tkptu
This is the static loading
and dominates as tendsto 0.0
At = n this term = 0.0With no damping present this
results in an infinite response
Phase lead of the response relative to the input
(see next page)
At >> n both terms drive theresponse to 0.0
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is defined as a phase lead in Nastran :
2
21
1/2tan
n
n
DAMPING WITH FORCED VIBRATION(Cont.)
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Summary:
For
Magnification factor 1 (static solution)
Phase angle 360 (response is in phase with the force)
For
Magnification factor 0 (no response)
Phase angle 180 (response has opposite sign of force)
For
Magnification factor 1/2
Phase angle 270
1
n
1
n
1
n
DAMPING WITH FORCED VIBRATION(Cont.)
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A Frequency Response Analysis can be used to explore the
response of our spring mass system to the forcing function.
This method allows us to compare the response of the
spring with the input force applied to the spring over a wide
range of input frequencies
It is more convenient in this case than running multiple
Transient Analyses, each with different input frequencies
Apply the input load as 1 unit of force over a frequency
range from .1 Hz to 5 Hz
Damping is 1% of Critical
DAMPING WITH FORCED VIBRATION(Cont.)
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Magnification Factor = 1/2 = 1/G = 50Static Response = p/k = .01
Peak Response = .5 at 1.59 Hz
Note:
Use of a Log scale helps identify loworder response
Displacement
Frequency (Hz)
DAMPING WITH FORCED VIBRATION(Cont.)
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There are many important factors in setting up a Frequency Response Analysis
that will be covered in a later section
For now, note the response is as predicted by the equation of motion
At 0 Hz result is p/k
At 1.59 Hz result is p/k factored by Dynamic Magnification
At 5 Hz result is low and becoming insignificant
The Phase change is shown here:
In phase up to 1.59 Hz
Out of phase180Degrees after 1 .59 Hz
DAMPING WITH FORCED VIBRATION(Cont.)
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Try a Transient analysis with a unit force applied to the spring at 1.59 Hz Again damping of 1% Critical is applied
The result is shown on the next page:
The response takes around 32 seconds to reach a steady-state solution
After this time the displacement response magnitude stays constant at .45
units The theoretical value of .5 is not reached due to numerical inaccuracy (see
later) and the difficulty of hitting the sharp peak
DAMPING WITH FORCED VIBRATION(Cont.)
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Transient analysis with a unit force applied to the spring at 1.59 Hz
Displacement
Time
DAMPING WITH FORCED VIBRATION(Cont.)
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MSC.NASTRAN DOCUMENTATION
Manuals MSC.NASTRAN Quick Reference Guide
MSC.NASTRAN Reference Manuals
Users Guides
Getting Started with MSC.NASTRAN MSC.NASTRAN Linear Static Analysis
MSC.NASTRAN Basic Dynamic Analysis
MSC.NASTRAN Advanced Dynamic Analysis
MSC.NASTRAN Design Sensitivity and Optimization
MSC.NASTRAN DMAP Module Dictionary MSC.NASTRAN Numerical Methods
MSC.NASTRAN Aeroelastic Analysis
MSC.NASTRAN Thermal Analysis
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MSC.NASTRAN DOCUMENTATION (Cont.)
Other Documentation MSC.NASTRAN Common Questions and Answers
MSC.NASTRAN Bibliography
Documentation available in online form for
workstations and PCs
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TEXT REFERENCES ON DYNAMICANALYSIS
1. W. C. Hurty and M. F. Rubinstein, Dynamics of Structures, Prentice-Hall, 1964.
2. R. W. Clough and J. Penzien, Dynamics of Structures, McGraw-Hill, 1975.
3. S. Timoshenko, D. H. Young, and W. Weaver, Jr., Vibration Problems in Engineering,4th Ed., John Wiley & Sons, 1974.
4. K. J. Bathe and E. L. Wilson, Numerical Methods in Finite Element Analysis, Prentice-Hall, 1976.
5. J. S. Przemieniecki, Theory of Matrix Structural Analysis, McGraw-Hill, 1968.
6. C. M. Harris and C. E. Crede, Shock and Vibration Handbook, 2nd Ed., McGraw-Hill,1976.
7. L. Meirovitch,Analytical Methods in Vibrations, MacMillan, 1967.
8. L. Meirovitch, Elements of Vibration Analysis, McGraw-Hill, 1975.
9. M. Paz, Structural Dynamics Theory and Computation, Prentice-Hall, 1981.
10. W. T. Thomson, Theory of Vibrations with Applications, Prentice-Hall, 1981.
11. R. R. Craig, Structural Dynamics: An Introduction to Computer Methods, John Wiley& Sons, 1981.
12. S. H. Crandall and W. D. Mark, Random Vibration in Mechanical Systems, AcademicPress, 1963.
13. J. S. Bendat and A. G. Piersel, Random Data Analysis and Measurement Techniques,2nd Ed., John Wiley & Sons, 1986.
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