Mechanical Vibrations Theory

Fundamental concepts and mathematical foundations of mechanical vibration systems.

Fundamentals of Vibration

Mechanical vibration is the study of oscillatory motion in mechanical systems. It occurs when a system is displaced from a position of stable equilibrium and is acted upon by restoring forces that tend to return the system to equilibrium.

Basic Terminology

Degree of Freedom (DOF): The number of independent coordinates required to completely describe the motion of a system.
Natural Frequency: The frequency at which a system tends to oscillate in the absence of damping or driving forces.
Damping: The dissipation of energy in a vibrating system, causing the amplitude of vibration to decrease over time.
Resonance: The phenomenon that occurs when a system is excited at a frequency close to its natural frequency, resulting in large amplitude vibrations.

Elements of Vibrating Systems

Mass/Inertia Element

Stores kinetic energy and provides inertial resistance to acceleration.

Newton's Second Law

\[F = m·a\]

Spring/Elastic Element

Stores potential energy and provides resistance proportional to displacement.

Hooke's Law

\[F = k·x\]

Damper/Dissipative Element

Dissipates energy and provides resistance proportional to velocity.

Viscous Damping

\[F = c·v\]

Single Degree of Freedom (SDOF) Systems

A Single Degree of Freedom (SDOF) system is the simplest vibrating system, consisting of a mass, spring, and damper. The motion of an SDOF system is described by a second-order differential equation.

Equation of Motion

General equation of motion for an SDOF system

\[m·ẍ + c·ẋ + k·x = F(t)\]

For free vibration (F(t) = 0), the solution depends on the damping ratio:

Underdamped (ζ < 1)

The system oscillates with decreasing amplitude.

Where ωₙ = √(k/m) is the natural frequency and ζ = c/(2√(km)) is the damping ratio

\[x(t) = e^(-ζωₙt)·[A·cos(ωₙ√(1-ζ²)·t) + B·sin(ωₙ√(1-ζ²)·t)]\]

SDOF Systems with External Excitation

When an external force is applied to an SDOF system, the response consists of two parts: the transient response (which decays with time) and the steady-state response (which persists as long as the force is applied).

Types of External Forces

Step Force

A constant force suddenly applied to the system.

Where H(t) is the Heaviside step function

\[F(t) = F₀·H(t)\]

Harmonic Force

A sinusoidal force applied to the system.

Where ω is the forcing frequency

\[F(t) = F₀·sin(ωt)\]

Impulse Force

A large force applied for a very short duration.

Where δ(t) is the Dirac delta function

\[F(t) = I·δ(t)\]

Harmonic Response and Resonance

When a harmonic force is applied to an SDOF system, the steady-state response is also harmonic but with a phase difference. The amplitude of the response depends on the frequency ratio (r = ω/ωₙ) and damping ratio (ζ).

Where φ = tan⁻¹(2ζr/(1-r²)) is the phase angle

\[x(t) = (F₀/k)·(1/√((1-r²)² + (2ζr)²))·sin(ωt - φ)\]

The Dynamic Amplification Factor (DAF) is defined as:

Ratio of dynamic to static displacement

\[DAF = 1/√((1-r²)² + (2ζr)²)\]

When r = 1 (ω = ωₙ), resonance occurs, and for lightly damped systems, the response amplitude can be very large.

Two Degrees of Freedom (2DOF) Systems

A Two Degrees of Freedom (2DOF) system requires two coordinates to describe its motion. Examples include two masses connected by springs and dampers, or a mass with both translational and rotational motion.

Equations of Motion

For a system with two masses connected by springs and dampers:

Equation of motion for mass 1

\[m₁·ẍ₁ + c₁·ẋ₁ + k₁·x₁ + c₂·(ẋ₁ - ẋ₂) + k₂·(x₁ - x₂) = F₁(t)\]

Equation of motion for mass 2

\[m₂·ẍ₂ + c₂·(ẋ₂ - ẋ₁) + k₂·(x₂ - x₁) = F₂(t)\]

These equations can be written in matrix form:

Where [M], [C], and [K] are the mass, damping, and stiffness matrices

\[[M]{ẍ} + [C]{ẋ} + [K]{x} = {F(t)}\]

Natural Frequencies and Mode Shapes

A 2DOF system has two natural frequencies and corresponding mode shapes. These are found by solving the eigenvalue problem:

Where ω are the natural frequencies and {φ} are the mode shapes

\[([K] - ω²[M]){φ} = {0}\]

In the first mode (lower frequency), the masses typically move in the same direction. In the second mode (higher frequency), they move in opposite directions.

Transmissibility

Transmissibility is a measure of how much force or motion is transmitted through a vibrating system. It is a key concept in vibration isolation and control.

Force Transmissibility

Force transmissibility is the ratio of the force transmitted to the foundation to the force applied to the mass.

Where r = ω/ωₙ is the frequency ratio

\[TR = |F_T/F₀| = √(1 + (2ζr)²) / √((1-r²)² + (2ζr)²)\]

Key characteristics:

At low frequencies (r < 1), TR ≈ 1
At resonance (r = 1), TR reaches maximum value of 1/(2ζ)
At r = √2, TR = 1 regardless of damping
For r > √2, TR < 1 (isolation region)

Motion Transmissibility

Motion transmissibility is the ratio of the displacement amplitude of the mass to the displacement amplitude of the base.

Where x is the mass displacement and y is the base displacement

\[TR = |x/y| = √((1+2ζ²r²)² + (2ζr(1-r²))²) / √((1-r²)² + (2ζr)²)\]