The document discusses structural dynamics and analysis of multi-degree of freedom systems. It covers key concepts such as dynamic differential equations, natural frequencies, damping, modal analysis, and different analysis methods including direct integration and frequency-based approaches. Linear dynamics allows use of natural modes of vibration while nonlinear dynamics accounts for large motions and changing geometry. Damping dissipates energy through velocity-dependent forces. Analysis can be performed via direct time integration or modal superposition depending on the problem characteristics and dominant frequencies.

1. STRUCTURAL DYNAMICS
MULTIPLE DEGREE OF FREEDOM
SYSTEM
Presented By: Abdul Majid (明晢)
Student ID: S32002006W
School of Aerospace and Civil Engineering
Harbin Engineering University

2. OVERVIEW
 Dynamic Differential Equation Classic Dynamics versus Nonlinear
Dynamics
 Inertial Effects Damping Effects
 Natural Frequency Extraction Free Systems Harmonic Systems
 Base Motion
 Damping Methods
 Direct Linear Dynamics
 Versus Frequency Based Dynamics
 Damping in Direct Linear Dynamics
 Controlling Accuracy of Calculations
 Nonlinear Dynamics

3. Linear Dynamics allows
effective use of the “natural
modes” of vibration of a
structure.
Example of a Modal shape for
a flat circular disc with
centered circular hole
modeled with shell elements
DYNAMICS
 Dynamic analysis differs from static analysis
in three fundamental aspects:
 Inertial effects are included
 Dynamic loads vary as a function of time.
 The time-varying load application induces a
time-varying structural response.
 Mass and Density need to be accounted for
 Must be in proper (consistent) units

4. [M]{ü} + [C]{ú} + [K]{u} – {P} = 0
DYNAMICS DIFFERENTIAL EQUATION
Where:
 [M]{ü} represents the inertial forces
[M] – Mass matrix
{ü} – Acceleration
 [C]{ú} represents the dissipative forces
[C] – Dissipative matrix
{ú} – Velocity
 [K]{u} represents the stiffness forces
[K] – Stiffness matrix
{u} – Displacement
 {P} represents the external forces

5. DYNAMICS CONCEPTS
 Static
- Events in which time parameters and inertia effects do not play a
significant role in the solutions.
 Dynamic
- A significant time dependent behavior exists in the problem because of
inertial forces (d’Alembert forces). Hence, a time integration of the
equations of motion is required.
 Linear Dynamic
- The motion or deformation produced by a dynamic behavior is small
enough so that the frequency content of the system remains relatively
constant.
 Nonlinear Dynamic
- The motion or deformation produced by a dynamic behavior
of the structure is large enough that we must account for changes in
geometry, material or contact changes in the model.

6. DYNAMICS CONCEPTS (CONT.)
 Direct Integration (over time)
- All kinematic variables are integrated through time. It can be used to
solve linear or nonlinear problems.
 Natural Frequency
- The frequencies at which the structure naturally tends to vibrate if it is
subjected to a disturbance.
 Modal Dynamics
- A dynamic solution is obtained by superimposing the natural frequencies
and mode shapes of a structure to characterize its dynamic response in the
linear regime.
 Damping
- The dissipative energy produced by a structure’s motion.

7. NATURAL FREQUENCY
 Natural Frequency Solution
 The natural frequencies of a structure are
the frequencies at which the structure
naturally tends to vibrate if it is
subjected to a disturbance
 When an applied oscillatory load
approaches a natural frequency of a
structure, the structure will resonate.
This is a phenomenon in which the
amplitude of the displacement of an
oscillating structure will dramatically
increase at particular frequencies.

8. The natural frequency solution, or eigenvalue
analysis, is the basis for many types of
dynamic analyses.
The structure may include preload before the
eigenvalues are calculated. This affects the
results.
The natural frequency for a Single Degree
Of Freedom (SDOF) system is given by
The frequency procedure extracts
eigenvalues of an undamped system:
NATURAL FREQUENCY OF
FREE UNDAMPED SYSTEM

9. The structure may include preload before
the eigenvalues are calculated. This affects
the results.
The frequency procedure extracts
eigenvalues of a damped system:
The natural frequency for the Damped
Single Degree Of Freedom (SDOF) system
is given by the same equation of the
undamped system:
NATURAL FREQUENCY OF
FREE DAMPED SYSTEM

10. NATURAL FREQUENCY OF FREE
DAMPED SYSTEM (CONT.)

11. When the Damped system is loaded with
an exponential function of a single
frequency, the resultant oscillations are
called harmonic:
HARMONIC OSCILLATIONS

12. HARMONIC OSCILLATIONS (CONT.)

13. MULTIPLE DEGREE OF FREEDOM SYSTEM

14. MULTIPLE DOFS - FREE VIBRATION ANALYSIS

15. MULTIPLE DOFS:
MODAL SUPERPOSITION METHOD

16. MULTIPLE DOFS – HARMONIC
ANALYSIS

17. Example:
Third Modal Shape of a Cantilevered Plate
NATURAL FREQUENCIES,
PRELOADING AND FEM
 Preloading changes the structural
stiffness and as a result, changes
the results.
 A finite element mesh must be
sufficiently fine enough to
capture the mode shapes that
will be excited in the response.
 Meshes suitable for static
simulation may not be suitable
for calculating dynamic response
to loadings that excite high
frequencies.
 As a general rule of thumb, you
should have a minimum of 7
elements spanning a sine wave.

18. Example:
Impact Test using Explicit Dynamics
Reaction Force at Wall
FREQUENCY BASED DYNAMICS
 When a linear structural
response is dominated by a
relatively small number
modes, modal superposition
can lead to a particularly
different method of
determining the response.
 Modal based solutions
require extraction of the
natural frequency and mode
shapes first (i.e. requires
running a Natural Frequency
solution first)

19. DYNAMIC ANALYSIS METHODS IN MSC.MARC
 Eigenvalue extractions linear with
preloading
 Lanczos method
 Power Sweep
 Harmonic response linear with
preloading
 Real (no Damping)
 Imaginary (Damping)
 Transient analysis linear and
nonlinear
 Explicit
 Implicit
 Contact

20. DYNAMIC ANALYSIS METHODS IN MSC.MARC
(CONT.)
 Modal-based Solutions include:
 Steady State Dynamics (i.e.: rotating
machinery in buildings)
 Harmonic responses for the steady state
response of a sinusoidal excitation
 Modal Linear Transient Dynamics (i.e.:
diving board or guitar spring)
 Modal superposition for loads known as a
function of time
 Response Spectrum Analysis (i.e.:
seismic events)
 Provides an estimate of the peak
response when a structure is subjected
to a dynamic base excitation

21. DYNAMIC ANALYSIS METHODS IN MSC.MARC
(CONT.)
 Frequency based dynamics should have the following characteristics:
 The system should be linear.
(but for nonlinear preloading)
 Linearized material behavior
 No change in contact conditions
 No nonlinear geometric effects other than those resulting from preloading.
 The response should be dominated by relatively few frequencies.
 As the frequency of the response increases, such as shock analysis, modal based
dynamics become less effective
 The dominant loading frequencies should be in the range of the extracted
frequencies to insure that the loads can be described accurately.
 The initial accelerations generated by any sudden applied loads should be
described by eigenmodes.
 The system should not be heavily damped.

22. BASE MOTION
 Base motion specifies the
motion of restrained nodes.
 The base motion is defined by a
single rigid body motion, and
the displacements and rotations
that are constrained to the body
follow this rigid body motion.
 Example: Launch excitation of
mounted electronics packages or
hardware.
 Base motion is always specified
in the global directions.

23. Frequency Value
0.0001 0.0000975
0.0005 0.0004875
0.01 0.00975
0.2 0.195
0.3 0.2925
1 0.975
2.5 2.5
3 2.5
4.5 2.5
6.6 2.5
8 2.25
10 2
100 1.1
1000 1.01
This is a typical
earthquake spectrum for
rocklike material with a
soil depth less than 200
ft, as provided by the
UBC
POWER TRANSMISSION TOWER BASE
MOTION EXAMPLE

24. [M]{ü} + [C]{ú} + [K]{u} - P = 0
Where
[C]{ú} - Dissipative forces
[C] - Damping matrix
{ú} - Velocity of the structure
DAMPING
 Damping is the energy
dissipation due to a
structure’s motion.
 In an undamped structure, if
the structure is allowed to
vibrate freely, the magnitude
of the oscillations is constant.
 In a damped structure, the
magnitude of the oscillations
decreases until the oscillation
stops.
 Damping is assumed to be
viscous, or proportional to
velocity
 Dissipation of energy can be
caused by many factors
including:
 Friction at the joints of a structure
 Localized material hysteresis

25. Damped natural frequencies are related to
undamped frequencies via the following
relation:
where
wd the damped eigenvalue
wn the undamped eigenvalue
x = c/co the fraction of critical damping or
damping ratio
c the damping of that mode
shape
co the critical damping
2
1 

 
 n
d
Damping exhibits three characteristic forms:
DAMPING (CONT.)
 Under damped systems (z < 1.0)
 Critically damped systems (z = 1.0)
 Over damped systems (z > 1.0)

26. MODAL DAMPING
 Damping in Modal Analysis
 Direct Damping
 Allows definition of damping as a fraction of
critical damping.
 Typical value is between 1% and 10% of the
critical damping.
 The same damping values is applied to
different modes.

27. Direct dynamic solutions assemble the
mass, damping and stiffness matrices and
the equation of dynamic equilibrium is
solved at each point in time.
Direct method is favored in wave
propagation and shock loading problems, in
which many modes are excited and a short
time of response is required.
Since these operations are computationally
intensive, direct integration is more
expensive than the equivalent modal
solution.
Direct dynamic solutions can be used to solve
linear transient, steady state and nonlinear
solutions using Rayleigh damping.
Rayleigh damping is assumed to be made up
of a linear combination of mass and stiffness
matrices:
[C] =  [M] + (+gt)[K]
Many direct integration analyses often define
energy dissipative mechanisms as part of the
basic model (dashpots, inelastic material
behavior, etc.)
For these cases, generic damping is usually
not important.
DAMPING IN DIRECT LINEAR
AND NONLINEAR DYNAMICS

28. The damping terms for direct
integration are defined in the
materials form:
DAMPING IN DIRECT LINEAR AND
NONLINEAR DYNAMICS (CONT.)
 Mass Proportional
Damping
 Introduces damping
forces caused by
absolute velocities in
the model
 Stiffness Proportional
Damping
 Introduces damping
which is proportional to
strain rate.

29. Nonlinear dynamic procedure
uses implicit time integration,
such as Central Difference or
Newmark-beta methods.
DAMPING IN DIRECT LINEAR AND
NONLINEAR DYNAMICS (CONT.)
 Solution includes an
automatic impact solution
for velocity and acceleration
jumps due to contact bodies
including rigid structure.
 The high frequency
response, which is important
initially, is damped out
rapidly by the dissipative
mechanisms in the model

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