What is Transfer Function? System Response Characterization • Portable balancer, vibration analyzer "Balanset" for dynamic balancing crushers, fans, mulchers, augers on combines, shafts, centrifuges, turbines, and many others rotors What is Transfer Function? System Response Characterization • Portable balancer, vibration analyzer "Balanset" for dynamic balancing crushers, fans, mulchers, augers on combines, shafts, centrifuges, turbines, and many others rotors

What is Transfer Function? System Response Characterization

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Understanding Transfer Function

Definition: What is a Transfer Function?

Transfer function (also called frequency response function or FRF) is a complex-valued function that describes how a mechanical system responds to input forces or motions as a function of frequency. Mathematically, it is the ratio of the output vibration response to the input excitation at each frequency: H(f) = Output(f) / Input(f). The transfer function contains both magnitude information (how much the system amplifies or attenuates at each frequency) and phase information (time delay or resonance characteristics).

Transfer functions are fundamental to understanding machinery dynamics because they characterize the system’s inherent response characteristics—natural frequencies, damping, mode shapes—independent of the specific forcing that may be present during operation. They are essential for modal analysis, structural modification prediction, and vibration isolation design.

Mathematical Formulation

Basic Definition

  • H(f) = Y(f) / X(f)
  • Where Y(f) = output (response) spectrum
  • X(f) = input (excitation) spectrum
  • Both measured simultaneously

Using Cross-Spectrum

For noisy measurements:

  • H(f) = Gxy(f) / Gxx(f)
  • Gxy = cross-spectrum between input and output
  • Gxx = auto-spectrum of input
  • Reduces bias from output noise
  • Standard method in practice

Components

  • Magnitude |H(f)|: Amplification factor at each frequency
  • Phase ∠H(f): Phase lag between output and input
  • Real Part: In-phase response
  • Imaginary Part: Quadrature response

Physical Meaning

Magnitude Interpretation

  • |H| > 1: System amplifies at this frequency (resonance region)
  • |H| = 1: Output equals input (neutral)
  • |H| < 1: System attenuates (isolation, off-resonance)
  • Peaks: Occur at natural frequencies (resonances)
  • Peak Height: Related to damping (higher peaks = less damping)

Phase Interpretation

  • 0°: Output in-phase with input (stiffness-controlled, below resonance)
  • 90°: Output lags input by quarter cycle (at resonance)
  • 180°: Output opposite to input (mass-controlled, above resonance)
  • Phase Through Resonance: Characteristic 180° shift from below to above

Measurement Methods

Impact Testing (Bump Test)

Most common for machinery:

  • Input: Instrumented hammer strike (measures force)
  • Output: Accelerometer on structure (measures response)
  • Advantages: Quick, simple, no special equipment beyond hammer and accelerometer
  • Limitations: Single impact = limited averaging, force spectrum quality

Shaker Testing

  • Controlled electromagnetic shaker applies force
  • Random, swept sine, or chirp excitation
  • Excellent force control and spectral content
  • Gold standard but requires shaker equipment

Operational Measurement

  • Use operating forces as input (running machine)
  • Less controlled but real operating conditions
  • Requires identifying input (force measurement or reference point)

Applications

1. Modal Analysis

Identifying natural frequencies and mode shapes:

  • Peaks in transfer function magnitude = natural frequencies
  • Phase through peaks confirms resonance
  • Peak width indicates damping
  • Multiple measurement points reveal mode shapes

2. Resonance Diagnosis

  • Determine if operating frequency near natural frequency
  • Assess separation margin
  • Identify problematic resonances
  • Guide modification strategies

3. Vibration Isolation Design

  • Predict isolator effectiveness
  • Transfer function shows transmission vs. frequency
  • Isolator natural frequency visible as peak
  • Above 2× isolator frequency, good isolation (|H| < 1)

4. Structural Modification Prediction

  • Predict effect of mass, stiffness, or damping changes
  • Before/after comparison validates modifications
  • Optimize modifications through modeling

Interpretation in Machinery Context

Rotor-Bearing System

  • Input: Unbalance force on rotor
  • Output: Bearing vibration
  • Transfer function shows how unbalance creates vibration
  • Peaks at critical speeds
  • Used in rotor dynamics analysis

Foundation Transmission

  • Input: Bearing housing vibration
  • Output: Foundation or floor vibration
  • Shows vibration transmission path
  • Identifies problematic transmission frequencies
  • Guides isolation or stiffening

Relationship to Other Functions

Transfer Function vs. Frequency Response

  • Terms often used interchangeably
  • Frequency Response Function (FRF) is same as transfer function in vibration context
  • Both describe system response vs. frequency

Transfer Function and Coherence

  • Coherence validates transfer function quality
  • High coherence (>0.9) = reliable transfer function
  • Low coherence = poor measurement or uncorrelated noise
  • Always check coherence when using transfer functions

Transfer function is a powerful analytical tool that characterizes mechanical system dynamics through the fundamental relationship between input and output. Understanding transfer function measurement, interpretation—particularly recognizing resonances from magnitude peaks and phase transitions—and application enables modal analysis, resonance diagnosis, structural modification prediction, and comprehensive vibration transmission analysis essential for advanced machinery dynamics and vibration control.

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