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Chapter 2 Chapter 2
Analytical Modeling
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Real structure Structural dynamics analysis procedure:
Real structure
Appropriate mathematical Model
Equation of motion
Equation solving
2.1 Mathematical Model
- For carrying out any dynamic analysis, the real structure must be represented by a mathematical model.
- To define a mathematical model, we should take into account full complexity of the physical structure including:
a. geometry (layout) and connection details.
b. non-structural attachments.
c. material properties.
d interaction with the supports and medium
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d. interaction with the supports and medium.
2.2 Simplifications
- Mathematical models should be as simple as possible and include all necessary information.
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- Simplifying assumptions in modeling:
a. Linear structure:
the dynamic parameters (mass, stiffness, damping) do not depend on time or on amount of vibration.
amount of vibration.
b. Single-degree-of-freedom (SDOF):
one time-dependent coordinate describes the vibration of whole structure/element.
c. Simplification of geometry and the boundary conditions.
d Id ifi i f h l
d. Identification of the structural parameters:
mass (e.g. lumped mass concept), stiffness (e.g. linear over limited range), damping(e.g. proportional to velocity), etc.
e. Modeling the dynamic forces:
wind loads, blast loads, periodic and non periodic forces, etc.
2.3 Formulation of the Equation of Motion
2.3.1 Force approach ( Direct equilibration of forces) I.
II.
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2.3.2 Energy approach
The equation of motion can be also derived by formulation of:
- kinetic energy - potential energy
- virtual work expression associated with non conservative forces and applying Hamilton’s principle.
Hamilton’s principle as of a variational approach is of great interest for
i i it i i t t l i d li ith d i
engineer, since it gives very convenient tool in dealing with dynamic problems.
2.4 Damping
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One simplified model of damping mechanism is the dashpot (like an oil damper), where the relative velocity of piston, , and the transmitted force may be related approximately as:
2.5 Discritization of spatial domain
-The simplest SDOF model although is easy to analyze may not give Appropriate mathematical Model:
-The simplest SDOF model, although is easy to analyze, may not give enough information in most engineering problems.
-Thus, we need to use the concept of multi-degree-of-freedom (MDOF) modeling.g
A frame discritization
2.6 Degree of freedom
- The number of coordinate which describe the motion of the structure is called the number of degrees of freedom.
For example,
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- A particle has 3 DOF (translations)
- A rigid body has 6 DOF (translations and rotations)
- A cantilever 2D beam can be modeled with 18 DOF
2.7 Linearity and Superposition principle
As mentioned before, linearity of structure means that the dynamic parameters (e.g. m, c, k) are independent of time (t), and amount of vibration (displacement, velocity, or acceleration)( p y )
Principle of superposition
For linear systems, the responses to a given number of distinct excitations can be obtained separately and then combined to obtain the aggregate response.
2.8 Nonlinearity
- Some examples of vibration phenomenon can be understood only by considering changes in dynamic parameters (e.g. m, c, k) according to time (t), and amount of vibration (displacement, velocity, or acceleration).
-These changes in dynamic parameters can be caused by:
1. nonlinear relation of stress and strain 2. cracks
3. large displacement 4 interaction with fluid 4. interaction with fluid.
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