Structural dynamics

1. Aim: To differenciate about Static analysis and dynamic analysis with proper representing pictures.

Introduction: To differenciate about Static analysis and dynamic analysis with proper representing pictures.

Procedure:

• Dynamic analysis is the process of testing and evaluating a program - while software is running.
• Systems consisting of mass 'm', Stiffness 'k' and a viscous damper.
• An idealization of a one-story structure.
• The number of independent dispalcements required to define the displaced positions of all the massive relative to their original position is called the number of Degrees of Freedom (DOFs) for dynamic analysis.
• A single degree of freedom system subjected to externally applied static force with no dynamic excitation.
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2. Aim: To explain about the force displacement topics as in the given question.

Introduction: To explain about the force displacement topics as in the given question.

Procedure:

A. Elastic behaviour:

• For a linear system the relationship between the lateral force fs and the resulting deformation u is linear, i.e.,

fs = ku

where k is the lateral stiffness of the system, its units are force/length.

Consider the frame belowe with bay width L, height h, elastic modulus E and the second moment of the cross-sectional area about the axis of bending lb and lc for the beam and columns, respectively; the columns are fixed at the base

D. Non-Linear Inelastic Behaviour:-

• Force displacement relationship becomes non-linear and inelastic at large deformations.
• Force fs corresponding to deformation is not single valued and depends on history of deformations and whether deformations is increasing or decreasing.
• Thus the resisting force can be expressed as
• fs = fs(u,v)
• Where the constant c is the viscous damping coefficient, it has units of force x time/length.

3. Aim: To explain about the Mass, Stiffness and damping components in the equation of motion.

Introduction: To explain about the Mass, Stiffness and damping components in the equation of motion.

Procedure:

• The Single degree of freedom system can be viewed as a combination of three pure, independent systems.
• The stiffness component: Frame without damping or mass. The external force fs on the stiness component is related to displacement u by fs = ku, if the system is linearly elastic.
• The damping component: Frame with its damping property but no stiffness or mass.
• The external force on the damping component is related to velocity by fd = cv.
• The mass component: The roof mass without the stiffness or damping of the frame. The external force fi on the mass component is related to the accleration by
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• The displacement of the ground is denoted by ug, the total (or absolute) displacement of the mass by uT, and the relative displacement between the mass and ground by u.
• At each instant of time these displacements are related by:
• uT(t) = u(t) + ug(t)
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• From the free body diagram including the inertia force fi, the euation of dynamic equilibrium is :
• fl + fD + fs = o
• Thus for a linear elastic system, the inertia force fl is related to the acceleration u of the mass by:
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• Thus, the euation of motion governing the relative displacement or deformation u(t) of the linear structure subjected to the ground accleration ug(t):
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• Comparision of two equations shows that the equations of motion for the structure subjected to two separate excitations - ground accleration ug(t) and external force = -mug(t) are one and the same.
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• Thus the relative displacement or deformaation u(t) of the structure due to the ground accleration ug(t) will be identiical to the displacement u(t) of the structure if its base were stationary and if it were subjected to an external force = -mug(t).

Damping Forces:

• Its impact is evident from the steady decline in amplitude of a vibrating body.
• Sources of damping in vibrating structures
• Friction at steel connections
• Opening and closing of microcracks in concrete.
• Friction between the structural and non structural elements such as partition walls.
• Damping in structure is represented in an idealized manner called equivalent viscous damping.
• The damping forces fd is related to the velocity across the linear viscous damper by
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• The differential equation governing the displacement u(t) of an idealized single story frame subjected to an externally applied force p(t) can be derived using Newton's second Law of the Motion.
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4. Aim: To provide the relation between Natural period (Tn) and Natural frequency (f) and their definitions.

Introduction: To provide the relation between Natural period (Tn) and Natural frequency (f) and their definitions.

Procedure:

Natural Period:-

• Natural period of vibration of the system (Tn) is the time required for the undamped system to complete one cycle of free vibration (Unit is seconds).
• It is related to the natural circular frequency of vibration, ωn, in units of radians per second
• Tn = 2π/ωn or ωn = 2π/Tn

Natural Frequency:-

• Natural cyclic frequency of vibrationn of the system is the number of cycles (or one complete revolution) executed by system in 1 second(unit is hertz)
• fn = 1/Tn
• The natural vibration properties ωn, Tn and fn depend only on mass and stiffness of the structure.
• k = mω□2
• The stiffer of two Single Degree of freedom systems having the same mass will have the shorter natural period (less time to complete one cycle) and the higher natural frequency.
• Similarly, the heavier (more mass) of two structures having the same stiffness will have the longer natural period and the lower natural frequency.
• ωn = √ k/ m

5. Aim: To explain about Response spectrum and it's graph.

Introduction: To explain about Response spectrum and it's graph.

Procedure:

Response spectrum:-

• A central concept in earthquake engineering, response spectrum is a practical means of characterizing ground motions and their effects on structures of different Tn.
• It provides a convenient means to summarize the peak response of all possible linear Single degree of freedom systems to a particular component of ground motion.
• It also provides a practical approach to apply the knowledge of structural dynamics to the design of structures and to the development of lateral force requirements in building codes.
• A plot of the peak value of a response quantity (u,v and a) as a function of the natural vibration period Tn of the system, is called Response spectrum for that quantity.
• Each such plot is for Single Degree of Freedom systems having a fixed damping ratio and several such plots for different values of damping ratio are included to cover the range of damping values encountered in actual structures.
• The response spectrum can be of 3 typess.
• The deformation response spectrum is a plot of peak u, against Tn for fixed damping ratio.
• A similar plot for peak `velocity u`is the relative velocity of response spectrum, and for peak uT is the accleration response spectrum.