Understanding Damping and Vibration Isolation in Mechanical Systems
Plot Variation Between Frequency Ratio and Phase Angle
The critical damping coefficient is the specific value of damping in a system that brings it back to equilibrium as quickly as possible without oscillation. It is given by:
The damping factor, often denoted by ΞΆ, is a dimensionless measure of damping in an oscillating system. It is the ratio of the actual damping coefficient to the critical damping coefficient:
ΞΆ = C/Cc
- ΞΆ < 1: underdamped
- ΞΆ = 1: critically damped
- ΞΆ > 1: overdamped
The logarithmic decrement is a measure of the rate of decay of oscillations in a damped system. It is defined as the natural logarithm of the ratio of two successive peak amplitudes.
Significance
- Quantifies damping in oscillatory systems.
- Used to calculate the damping factor (ΞΆ) for systems where only displacement data is available.
- Helps in diagnosing the energy dissipation in mechanical systems.
Viscous damping occurs when a system dissipates energy due to resistance proportional to velocity. It is characterized by a damping force given by:
Fd = -Cv
Dry Friction Damping
Mechanism: Energy dissipated due to constant friction between solid surfaces.
Examples: Brake pads, machinery components.
Eddy Current Damping
Mechanism: Energy dissipated as heat due to induced currents in a conductor moving through a magnetic field.
Examples: Magnetic brakes, galvanometers.
Slip or Interfacial Damping
Mechanism: Energy lost due to micro-slippage at contacting surfaces under cyclic loads.
Examples: Bolted or riveted joints in structures.
Why Does a Gyroscopic Couple Occur?
A gyroscopic couple is a turning moment in which the gyroscope’s axis of rotation is inclined so that the changes are opposite. The expression for the gyroscopic couple is given as:
C = I β Ο β Οp
where:
- C is the gyroscopic couple
- I is the moment of inertia
- Ο is the angular velocity
- Οp is the angular velocity of precession
Frequency Ratio vs. Magnification Factor
Vibration Isolation and Transmissibility
The purpose of vibration isolation is to control unwanted vibration so that its adverse effects are kept within acceptable limits. Vibrations originating from machines or other sources are transmitted to a support structure, such as a facility floor, causing a detrimental environment and unwanted levels of vibration. If the equipment requiring isolation is the source of unwanted vibration, the purpose of isolation is to reduce the vibration transmitted from the source to the support structure. Conversely, if the equipment requiring isolation is a recipient of unwanted vibration, the purpose of isolation is to reduce the vibration transmitted from the support structure to the recipient. An isolator is a resilient support that decouples an object from steady-state or forced vibration. To reduce the transmitted vibration, isolators in the form of springs are used. Common springs used are pneumatic, steel coil, rubber (elastomeric), and other pad materials. Natural frequency and damping are the basic properties of an isolator that determine the transmissibility of a system designed to provide vibration and/or shock isolation. Additionally, other important factors must be considered in the selection of an isolator/isolation material.
