Structural dynamics

3. Explain Mass, Stiffness & Damping components in the equation of motion 

4. Provide relationship between Natural period (Tn) & Natural frequency (f) and provide their definitions 

5. Explain in detail about Response spectrum & its Graph 

Q1. ) Differentiate Static analysis and Dynamic analysis with proper representing pictures ?

Ans :-  

1. Elastic behavior
2. Inelastic behavior
3. Plastic behavior
4. Non-linear Inelastic behavior

Ans :- 

Elastic behavior :

• The capacity of a material body to resume its initial dimensions and shape after the forces that caused the deformation have been removed.
• This capability is referred to as elastic behavior (or response) in a body.
• The force-displacement graph from point O to a particular point exhibits a linear path; consequently, the body exhibits elastic properties with the applied forces up to a certain point.

Inelastic behavior :

• The deformation of a substance that persists after the force that caused it is removed. Inelastic behavior corresponds to the nonlinear path in the force-displacement graph.
• The body experiences inelastic behavior as soon as the path deviates from a linear curve.

Plastic behavior :

• The changes in a structure's geometry that occur during plastic deformation have an impact on how plastic structures behave after they have been subjected to bending collapse loads.
• The load-deflection relationship for simply supported rectangular reinforced concrete plates is examined in the present paper.

Non-linear Inelastic behavior :

• Materials typically only exhibit elastic behavior up to a certain load, known as the "yield point."
• The deformations that follow are plastic.According to the force-displacement graph, this behavior happens when the line from point "O" to point "a" has a propensity to bend.

Q3.) Explain Mass, Stiffness & Damping components in the equation of motion ?

Ans :-

• The stiffness, damping, and mass component are just three completely independent systems that make up the single degree of freedom system.
• The lack of mass or damping in the frame is the stiffness component.
• If the system is linearly elastic, the external force fs on the stiffness component is related to the displacement u by fs = Ku.
• The frame's damping property, which it possesses without any stiffness or mass, is the damping component.
• FD = c, where fd is the damping force, c is the viscous damping coefficient, and c is the velocity, relates the external force on the damping component to velocity.
• The mass component is the weight of the roof alone, unaffected by the frame's stiffness or damping.
• The acceleration is related to the external force fi on the mass component by the formula fi = m.

                       Then, mű + fd + fs = p(t) --à mű + ců+ ku = p(t)

Q4.) Provide relationship between Natural period (Tn) & Natural frequency (f) and provide their definitions ?

Ans :-

• The system's natural period (Tn), which is measured in seconds, is the amount of time needed for an undamped system to complete one cycle of free vibration.
• It relates to the radians per second (n) natural circular frequency of vibration. By using, Tn = 2/n or n = 2/Tn
• (f) Natural frequency:-
• The number of cycles (or full revolutions) that the system completes in one second with hertz units. If f=1/Tn,
• The mass and stiffness of the structure are the only factors that affect the natural vibration properties n, Tn, and f.

Q5.) Explain in detail about Response spectrum & its Graph ?

Ans :- 

• A response spectrum, which displays the peak response of a straightforward harmonic oscillator when it is subjected to a transient event, is a function of frequency or period. The oscillator's natural frequency and damping have an impact on the response spectrum. As a result, it does not directly depict the frequency content of the excitation (as in a Fourier transform), but rather the impact the signal has on the fictitious system with one degree of freedom (SDOF).
• A technique to gauge the structural reaction to brief, nondeterministic, transient dynamic events is response spectrum analysis. Earthquakes and shocks are two examples of these occurrences. It is challenging to carry out a time-dependent analysis because the load's precise time history is unknown. The short duration of the event prevents it from being categorized as an ergodic ("stationary") process, so a random response approach is also not appropriate.
• A unique kind of mode superposition serves as the foundation for the response spectrum approach. The goal is to create an input that places a cap on how much an event of this kind can excite an eigenmode with a specific natural frequency and damping.

The equation of motion for the mass can, if there are no external forces, be written as

• Dividing by the mass, and using customary notation,

• Here, the undamped natural (angular) frequency is 

• and,the damping ratio is

• For given values of , , and , this equation can be solved for a sufficiently long time. The displacement, velocity, and acceleration response spectra are defined as the maximum values caused by the acceleration history         
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