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In real scenarios, structural foundations have to bear many kind of dynamic loadings, such as the loading due to the earthquakes, blastings, wind loads, pile driving, machine works, quarrying and fast moving moving traffic.
The characteristics of the dynamic load is that, they varies with time. Purely dynamic loads do not occur in nature. Loads are the combination of the static and dynamic load. Static loads are caused by the dead weight of the building itself, while the dynamic loads may occur due to the causes mentioned above.
The pattern of variation of a dynamic load with respect to time may either be periodic or transient. The periodical motions can be resolved into sinusoidally varying components e.g. vibrations in the case of reciprocating machine foundations. Transient vibrations may have very complicated non-periodic time history e.g. vibrations due to earthquake and quarry blasts.
The structures subjected to dynamic loads may vibrate along its extension, shearing, bending and may be along torsional deformation of the structure. The form of vibration mainly depends of the mass, stiffness distribution and end conditions of system.
To study the response of a vibratory system, in many cases it is satisfactory to reduce it to the idealized system of lumped parameters. The simplest model consists of mass, spring and dashpot.
First of all we must know, what a Degree of freedom means, well!, the number of independent coordinates which are required to define the position of a system during vibration, is called the degrees of freedom of the system. In order to understand the motion of the systems with the multi degrees of freedom, we should first analyse the single motion with the single degree of freedom.
A simple harmonic motion can be described by a simple sinusoidal function, having a time period of T. In this article we are going to discuss the vibrations of a single degree of freedom system shown in fig.3 above, consisting of a rigid mass m supported by a spring and dashpot.
Damping in this system is represented by the dashpot, and the resulting damping force is proportional to the velocity. The system is subjected to an external time dependent force F(t).
Figure above, shows the free body diagram of the mass m at any instant during the course of vibrations. The forces acting on the mass m are:
In real scenarios, structural foundations have to bear many kind of dynamic loadings, such as the loading due to the earthquakes, blastings, wind loads, pile driving, machine works, quarrying and fast moving moving traffic.
The characteristics of the dynamic load is that, they varies with time. Purely dynamic loads do not occur in nature. Loads are the combination of the static and dynamic load. Static loads are caused by the dead weight of the building itself, while the dynamic loads may occur due to the causes mentioned above.
The pattern of variation of a dynamic load with respect to time may either be periodic or transient. The periodical motions can be resolved into sinusoidally varying components e.g. vibrations in the case of reciprocating machine foundations. Transient vibrations may have very complicated non-periodic time history e.g. vibrations due to earthquake and quarry blasts.
The structures subjected to dynamic loads may vibrate along its extension, shearing, bending and may be along torsional deformation of the structure. The form of vibration mainly depends of the mass, stiffness distribution and end conditions of system.
To study the response of a vibratory system, in many cases it is satisfactory to reduce it to the idealized system of lumped parameters. The simplest model consists of mass, spring and dashpot.
First of all we must know, what a Degree of freedom means, well!, the number of independent coordinates which are required to define the position of a system during vibration, is called the degrees of freedom of the system. In order to understand the motion of the systems with the multi degrees of freedom, we should first analyse the single motion with the single degree of freedom.
Damping in this system is represented by the dashpot, and the resulting damping force is proportional to the velocity. The system is subjected to an external time dependent force F(t).
- Exciting force, F(t): It is the externally applied force that causes the motion of the system.
- Restoring force, Fr: It is the force exerted by the spring on the mass and tends to restore the mass to its original position. For a linear system, restoring force is equal to K.z, where K is the spring constant and indicates the stiffness. This force always acts towards the equilibrium position of the system.
- Damping force, Fd: The damping force is considered directly proportional to the velocity and is given by C.z' where C is called the co-efficient of viscous damping; this force always opposes the motion.
- Inertia force, Fi: It is due to the acceleration of the mass and is given by m.z''. This force can be justified by the De-Alembert's principle.
The equation of equilibrium for the mass m is given as:
m.z'' + C.z' + K.z = F(t)
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