Vibrations
What is Vibrations?
Oscillatory motion of a mechanical system about an equilibrium position.
Key formula / rule: Natural Frequency (Undamped SDOF)
Key points
- Define and classify different types of vibrations.
- Derive equations of motion for SDOF systems (spring-mass-damper).
- Calculate natural frequency for undamped and damped SDOF systems.
- Analyze the response of SDOF systems to harmonic excitation (forced vibration).
Common exam trap
Confusing natural frequency with damped natural frequency.
Definitions
- Term
Vibration
- Meaning
Oscillatory motion of a mechanical system about an equilibrium position.
- Term
Natural Frequency (ωn)
- Meaning
The frequency at which a system will oscillate if disturbed and then left to vibrate freely, without any external periodic force or damping.
- Term
Damping
- Meaning
The dissipation of energy from an oscillating system, typically due to resistance forces like friction or air resistance.
- Term
Resonance
- Meaning
A phenomenon where a system responds with maximum amplitude when the frequency of the external excitation force matches or is close to its natural frequency.
- Term
Degree of Freedom (DOF)
- Meaning
The minimum number of independent coordinates required to completely describe the motion of a system.
Learning objectives
Define and classify different types of vibrations.
Derive equations of motion for SDOF systems (spring-mass-damper).
Calculate natural frequency for undamped and damped SDOF systems.
Analyze the response of SDOF systems to harmonic excitation (forced vibration).
Understand and calculate concepts like damping ratio, logarithmic decrement, magnification factor, and transmissibility.
Identify and mitigate resonance conditions in engineering systems.
Formulae
- Name
Natural Frequency (Undamped SDOF)
- Note
For a simple spring-mass system, where k is stiffness and m is mass.
- Expression
ωn = √(k/m)
- Name
Damping Ratio
- Note
Where c is damping coefficient, m is mass, and ωn is undamped natural frequency.
- Expression
ζ = c / (2 * m * ωn)
- Name
Damped Natural Frequency
- Note
Frequency of oscillation for an underdamped system (ζ < 1).
- Expression
ωd = ωn * √(1 - ζ2)
- Name
Logarithmic Decrement
- Note
Used to determine damping ratio from successive amplitudes in free vibration decay.
- Expression
δ = (1/n) * ln(Xi / Xi+n) = 2πζ / √(1 - ζ2)
- Name
Magnification Factor (Forced Vibration)
- Note
Ratio of dynamic amplitude to static deflection, where r = ω/ωn (frequency ratio).
- Expression
MF = 1 / sqrt((1 - (r2))2 + (2ζr)2)
- Name
Transmissibility Ratio
- Note
Ratio of transmitted force to exciting force, or transmitted amplitude to base excitation amplitude.
- Expression
TR = sqrt(1 + (2ζr)2) / sqrt((1 - r2)2 + (2ζr)2)
Prerequisites
Engineering Mechanics (Statics and Dynamics)
Strength of Materials (for stiffness calculations)
Differential Equations (for solving equations of motion)
Basic Calculus and Algebra
Common mistakes
Confusing natural frequency with damped natural frequency.
Incorrectly applying damping ratio (ζ) in formulas.
Ignoring the phase angle in forced vibration problems.
Not considering the effect of gravity on static deflection for natural frequency calculations (for vertical systems).
Assuming undamped conditions when damping is clearly present.
Keywords
Vibration
Oscillation
Natural Frequency
Damping
Resonance
SDOF
Free Vibration
Forced Vibration
Magnification Factor
Transmissibility
Logarithmic Decrement
Practice preview
Which type of vibration occurs when a system oscillates at its natural frequency without any external periodic force?…
easy
For a single-degree-of-freedom system, which of the following statements is TRUE regarding damped free vibration?…
medium
A mass of 10 kg is attached to a spring with a stiffness of 1000 N/m. What is the natural frequency of the undamped system in rad/s?…
medium