Mechanical Vibrations and Governors: Theory and Analysis

Porter Governor vs. Hartnell Governor

Frequency Ratio vs. Phase Angle

Plot of variation between frequency ratio and phase angle:

Damping Coefficients and Factors

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 ζ, is a dimensionless measure of damping in an oscillating system. It is the ratio of the actual damping coefficient to the critical damping coefficient (ζ = C/C_c):

• ζ < 1: Underdamped
• ζ = 1: Critically damped
• ζ > 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 (ζ) for systems where only displacement data is available.
• Helps in diagnosing the energy dissipation in mechanical systems.

Types of Damping Mechanisms

Viscous Damping: Occurs when a system dissipates energy due to resistance proportional to velocity. It is characterized by a damping force given by: F_d = -Cv

Dry Friction Damping (Coulomb Damping): Energy is dissipated due to constant friction between solid surfaces. Examples: Brake pads, machinery components.

Eddy Current Damping: Energy is dissipated as heat due to induced currents in a conductor moving through a magnetic field. Examples: Magnetic brakes, galvanometers.

Slip or Interfacial Damping: Energy is lost due to micro-slippage at contacting surfaces under cyclic loads. Examples: Bolted or riveted joints in structures.

Understanding the Gyroscopic Couple

Why Gyroscopic Couple Occurs and Its Derivation

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 · ω · ω_p

Where:

• C is the gyroscopic couple
• I is the moment of inertia
• ω is the angular velocity
• ω_p is the angular velocity of precession

Types of Governors and Centrifugal Governor

Construction:

• Spindle (Vertical shaft): Connected to the engine and rotates with it.
• Flyballs (masses/balls): Two or more masses attached to arms and mounted on the spindle.
• Arms/Links: Connect the flyballs to a central sleeve.
• Sleeve: Mounted on the spindle; moves up and down as the flyballs move outward or inward.
• Throttle Valve Mechanism: The sleeve is connected to a throttle valve.

Working Principle:

• As engine speed increases, the spindle rotates faster.
• This increases centrifugal force on the flyballs, causing them to move outward.
• The outward movement of the balls lifts the sleeve upward via the arms.
• The upward movement of the sleeve partially closes the throttle valve, reducing fuel supply.
• As fuel is reduced, the engine speed drops.
• If the speed decreases too much, the balls move inward due to reduced centrifugal force, lowering the sleeve and increasing fuel supply again.
• This feedback maintains a nearly constant engine speed.

Viscous Damping vs. Coulomb Damping

Vibrometer vs. Accelerometer

Dynamically Equivalent Systems

In order to determine the motion of a rigid body under the action of external forces, it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart, in such a way that:

1. The sum of their masses is equal to the total mass of the body.
2. The center of gravity of the two masses coincides with that of the body.
3. The sum of the mass moment of inertia of the masses about their center of gravity is equal to the mass moment of inertia of the body.

When these three conditions are satisfied, the system is said to be an equivalent dynamical system.

Vibration-Based Condition Monitoring

Fault diagnosis is the process of tracing a fault by means of its symptoms, applying knowledge, and analyzing test results. Accurate diagnosis of faults in complex engineering systems requires acquiring the information through sensors, processing the information using advanced signal processing algorithms, and extracting required features for efficient classification or identification of faults. Identification of faults and subsequent remedial action can increase productivity and reduce maintenance costs in various industrial applications. Machine learning methods involving feature extraction, feature selection, and classification of faults offer a systematic approach to fault diagnosis and can be used in automated or unmanned environments. These are increasingly used in industrial sectors, such as manufacturing, automotive, marine, and aerospace, to maximize equipment uptime and minimize maintenance and operating costs.

Balancing in Rotating Machinery

Balancing is essential in rotating machinery to ensure smooth operation, reduce vibrations, and increase the life of components. It involves correcting the mass distribution in a system so that it does not produce unwanted dynamic forces or moments when in motion.

Static Balancing

Definition: A system is said to be statically balanced when the center of mass of the rotating parts lies on the axis of rotation. In this condition, there is no unbalanced force acting when the object is stationary or rotating at a constant speed without wobbling.

Key Features:

• Achieved by ensuring that the net centrifugal force acting on the rotating system is zero.
• Applicable to single-plane rotating systems (e.g., flywheels, disc-shaped rotors).

Example: A ceiling fan blade is statically balanced if it stays in any position when you stop it manually.

Dynamic Balancing

Definition: A system is said to be dynamically balanced when it is both statically balanced and the couple (torque) caused by unbalanced masses is also zero. This ensures that the system has no unbalanced forces or moments during rotation.

Key Features:

• Required for multi-plane or complex rotating systems like crankshafts, turbines, or car wheels.
• Considers both force balance and couple balance.

Example: A car wheel needs dynamic balancing so it does not wobble or vibrate at high speeds, even if it appears statically balanced.

Frequency Ratio vs. Magnification Factor

Damping States Comparison

Vibration Isolation and Transmissibility

Vibration isolation is a technique used to reduce or prevent the transmission of vibrations from a vibrating machine to its surroundings or from the surroundings to the machine. It is done by inserting an elastic or damping element (like rubber, springs, pads) between the machine and its support.

Purpose:

• To protect the machine from external vibrations
• To protect the supporting structure from machine vibrations
• To reduce noise, wear, and fatigue failures

Transmissibility:

Transmissibility (T) is the ratio of the transmitted force (or motion) to the applied force (or motion) in a vibration isolation system.

• T < 1 → Good isolation (vibration reduced)
• T > 1 → Amplification (vibration increases)

Transmissibility depends mainly on the frequency ratio (ω/ω_n) and damping.

Vibration Isolation Materials:

1. Rubber pads
2. Cork sheets or cork-rubber composites
3. Metal springs (helical springs)
4. Neoprene pads
5. Foam or polyurethane pads
6. Felt pads
7. Pneumatic/air mounts

Logarithmic Decrement and Magnification

Logarithmic Decrement

Definition: Logarithmic decrement is defined as the natural logarithm of the ratio of any two successive amplitudes of a freely damped vibration measured on the same side of the mean position.

δ = ln(X_a / X_b)

Where:

• X_a = first amplitude
• X_b = next successive amplitude

Significance of Logarithmic Decrement:

• It indicates the rate of decay of amplitude in a free damped vibration.
• It measures how quickly the vibration dies out due to damping.
• A higher value of δ means higher damping, therefore the vibration decays faster.
• A lower δ indicates light damping and slow decay of amplitude.
• It is widely used to determine the damping factor experimentally.

Magnification Factor (M.F.)

Definition: The magnification factor is the ratio of the steady-state amplitude of forced vibration to the static deflection produced by a constant force.

M.F. = X / X_st

Where:

• X = amplitude under dynamic (forced) condition
• X_st = static deflection under constant force

Meaning: It shows how much the vibration amplitude is increased (magnified) under forced vibration compared to static loading. At resonance, the magnification factor becomes very high.

Critical Speed of a Light Shaft

When the rotor is mounted on a shaft, its center of gravity (CG) does not coincide with the axis of rotation due to unbalanced force. The centrifugal force causes the shaft to vibrate violently in a direction perpendicular to the axis of the shaft. This phenomenon is called whirling of the shaft, and the corresponding speed is called the critical speed of the shaft.

and

Natural Frequency of a Simple Pendulum via Energy Method