Theory of Vibrations:
- The study of vibrations is concerned with the oscillatory motion of bodies and the forces associated with them
- Vibrations are initiated when the energy is imparted to the elastic system by any external source
Free vibration system:
- Free vibration is a type of vibration in which a force is applied once and the structure or part is allowed to vibrate at its natural frequency..
- Free vibration occurs when a system is set off with an initial input and then allowed to vibrate freely.
- System oscillates under the action of forces inherent in the system itself and in absence of any externally applied forces
- The system will vibrate at one or more of its natural frequencies, which are properties of the dynamical system established by its mass and stiffness distribution
- The response of a system is called free vibration when it is disturbed and then left free to vibrate about some mean position.
- Eg: Example: Oscillation of simple pendulum about a vertical equilibrium position
Forced vibration system:
- Forced vibrations occur if a system is continuously driven by an external agency.
- Vibrations that take place under the excavation of external forces.
- When excitation is oscillatory, the system is forced to vibrate at excitation frequency.
- Eg: Child swing, electric bells, machine tools etc
Linear and non-linear vibration:
- The frequency of steady state vibration response resulting from the application of periodic, harmonic input is equal to the frequency of applied force or motion with response magnitude being dependent on the actual mechanical system
- Governed by linear differential equations, follows the law of superposition.
- Linear vibration becomes non linear for very large amplitude of vibration. It does not follow low of super position in this case.
DEGREES OF FREEDOM
Degrees of freedom are a set of independent displacements/ rotations that completely define the displaced position of the mass with respect to its initial position. It is the number of parameters that determine the state of a physical system and is important to the analysis of systems of bodies in Engineering.
Degrees of freedom of a body is defined as the number of independent movements it has. Or in other words it is the number of independent coordinates needed to define position of a body.
CONTINUOUS LOAD PATH:
Continuous load path is like a chain that ties the structure/building together from the roof to the foundation. A continuous load path is critical during earthquake or cyclone because it help to hold the structural elements together when ground forces or high winds try to pull the structure apart.
CONTINUOUS SYSTEM:
In Continuous system, the mass ad elasticity are continuously distributed. Such systems are also known as distributed parameter systems
Eg: Strings, rods, beams, plates and shells.
LUMPED MASS IDEALIZATION:
In lumped system, the components are discrete with the mass assumed to be rigid and concentrated at individual points.
CONTINUOUS LOAD PATH:
Continuous load path is like a chain that ties the structure/building together from the roof to the foundation. A continuous load path is critical during earthquake or cyclone because it help to hold the structural elements together when ground forces or high winds try to pull the structure apart.
CONTINUOUS SYSTEM:
In Continuous system, the mass ad elasticity are continuously distributed. Such systems are also known as distributed parameter systems
Eg: Strings, rods, beams, plates and shells.
LUMPED MASS IDEALIZATION:
In lumped system, the components are discrete with the mass assumed to be rigid and concentrated at individual points.
Taking an example of above cantilever beam, the displacements and accelerations for each point along the axis of the beam will be required because the mass of the beam is continuously distributed along its length. This reinforces the formulation in terms of partial differential equations because the position along the span and time must be taken as independent variables. such models are called continuous models. A continuous system indicates infinite degrees of freedom (DOF) system
To simplify the analysis, it may be assumed that the mass of the beam is concentrated in a series of discrete points and that inertia forces will develop only at these mass points. Such discrete points are called lumps and the concentrated mass in these points is called lumped mass. In this case, the displacements and accelerations need to be defined only at these mass points. The lumped mass models indicate particular DOF system
OSCILLATORY MOTION:
It can be a periodic motion that repeats itself in a regular cycle such as sine wave, the side-to-side wing of pendulum or up and down motion of spring with weight. It is also known as periodic motion.
SIMPLE HARMONIC MOTION:
It is repetitive movement back and forth through an equilibrium or central position, so that the maximum displacement on one side of this position is equal to the maximum displacement on other side.
In other words, SHM is simply that the acceleration causing the motion of a particle or object is proportional and in opposition to the displacement x from its equilibrium position
The main difference between oscillation, vibration and simple harmonic motion is that oscillation refers to any repeated variation about a central value, while the term vibration refers to mechanical oscillations. Simple harmonic motion refers to cases where an object oscillates about an equilibrium position under a restoring force which is directly proportional to the eobject's displacement
For example, If you are swinging on a swing and you friend pushing you every time when swing comes towards her. Then your motion is said to be oscilaltion
If your friend pulled back the swing and let the swing move back and forth on its own that is an example of natural vibration, however if your friend pushed the swing each time it came down is an example of forced vibration
Musical instrument string can be compared to simple harmonic motion
