Introduction of Vibrations

Vibrations: Vibration refers to mechanical oscillations about an equilibrium point. In its simplest form, vibration can be considered to be the oscillation or repetitive motion of an object around an equilibrium position.

Vibrations or mechanical oscillations are of many types as given below

• Free Vibration (Natural vibration) Vibration over an interval of time during which the system is free from excitation is known as free vibration.
• Damped and Undamped Vibration: Energy of a vibrating system is gradually dissipated by friction and other resistance. If damping is present, then the resulting vibration is damped vibration and when damping is absent it is undamped vibration. The damped vibration can again be classified as under-damped, critically-damped and over-damped system depending on the damping ratio of the system
• Forced Vibration When a repeated force continuously acts on a system, the vibrations are said to be forced.
• Linear Vibration: If all the basic components of a vibratory system‐the spring, the mass, and the damper, behave linearly, the resulting vibration is known as linear vibration. The differential equations that govern the behaviour of vibratory linear systems are linear. Therefore, the principle of superposition holds.
• Nonlinear vibration: vibration: If however, however, any of the basic components behave nonlinearly, the
• vibration is called ‘nonlinear vibration’. The differential equations that govern the behaviour of vibratory non‐linear systems are non-linear. Therefore, the principle of superposition does not hold.
• Deterministic Vibration: If the value or magnitude of the excitation (force or motion) acting on a vibratory system is known at any given time, the excitation is called ‘deterministic’. The resulting vibration is known as ‘deterministic vibration’.
• Nondeterministic vibration: In some cases, the excitation is non‐deterministic or random; the value of excitation at a given time cannot be predicted. In these cases, a large collection of records of the excitation may exhibit some statistical regularity. It is possible to estimate averages such as the mean and mean square values of the excitation.
• Harmonic Vibration: Vibration in which the motion is a sinusoidal function of time.
• Fundamental Vibration: Harmonic component of a vibration with the lowest frequency.
• Steady State Vibration: When the particles of the body move in steady state condition or continuing period vibration is called steady state vibration.
• Transient Vibration: Vibratory motion of a system other than steady state.
• Longitudinal Vibration: Vibration parallel to the longitudinal axis of a member
• Transverse Vibration: Vibration in a direction perpendicular to the longitudinal axis or central plane of a member.
• Torsional Vibration: Vibration that involves torsion of a member.

Mode of Vibration: Configuration of points of a SHM is called the mode of vibration.

Natural Frequency: Frequency of free simple harmonic vibration of an undamped linear system.

Time Period: Time taken for one oscillation is called time period.

Simple Pendulum: If time period of the pendulum is 1s, then pendulum is called simple pendulum.

Time period is given by

  (for small amplitude sin θ ≈ θ)

 [large amplitude (θ₀)]

• If the time period of a simple pendulum is 2s, it is called second pendulum.
• If a simple pendulum is in a carriage which is accelerating with acceleration a_r then g_eff = g-a
• If the acceleration a is upward

g_eff = g+a

• If the acceleration a is downward g_eff = g-a

• If the pendulum moves on edge which makes angle θ from horizontal.

• Simple pendulum consists of small sphere of mass m with charge q suspended by a thread of length l.