ME- 495Mechanical and Thermal Systems Lab

Fall 2011

Chapter 4 - THE ANALOG MEASURAND:

TIME-DEPENDANT CHARACTERISTICS

Professor: Sam Kassegne

THE ANALOG MEASURAND:TIME-DEPENDANT CHARACTERISTICS

Two types of measurement– Static – constant over time– Dynamic – varies over time

Steady state – periodic over timeNon-repetitive or Transient

– Single Pulse – aperiodic– Continuing or random

SIMPLE HARMONIC RELATIONS S = Sosin(t)

– S = instantaneous displacement from equilibrium– So = amplitude (maximum displacement)

= circular frequency (rad/s)

Velocity = ds/dt = So cos(t)– Vo = So

Acceleration = dV/dt = - So 2 sin(t)

– ao = - So 2

Exercise: Give some practical examples of such a motion.

CIRCULAR & CYCLIC FREQUENCY 1Hz (hertz) = 1 cycle/second f = cyclic frequency (Hz) = circular frequency = 2f S = Sosin(2ft)

Scotch-yoke mechanism – Harmonic motion

COMPLEX RELATIONS

– Ao, An, Bn = amplitude determining constants called harmonic constants.

– n = integers from 1 to called “harmonic orders”

The above equation is also called Fourier Series.

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Most complex dynamic mechanical signals, steady-state or transient (such as pressure, displacement, or strain) may be expressed as a combination of simple harmonic components.

Example:

A 2-harmonic term pressure-time function.

P = 100 sin 80t + 50 cos (160t-4)

The circular frequency of the fundamental harmonic = 80 rad/sec or 80/2 = 12.7 Hz.

The period = 1/12.7 = 0.0788 sec

The circular frequency of the second harmonic = 160rad/sec or 160/2p = 25.4 Hz.

The period = 1/25.4 = 0.039 sec

It also lags the fundamental by 1/8th cycle (/4)

FREQUENCY SPECTRUMType of plot where frequency (instead of time) is the independent variable and amplitude is the ordinate.

Special Wave-forms. (infinite series)

Frequency spectrum is useful because it allows identifying – at a glance – the frequencies present in a signal (say natural frequencies).

Interest in FS plots increased because of spectrum analyzers.

Two ways to perform a spectrum analysis Spectrum Analyzers electronic device which displays

frequency spectrum on CRT using circuitry Fast Fourier Transform computer algorithm that computes

spectrum

FREQUENCY SPECTRUM (ctd.)

Harmonic or Fourier Analysis

The process of determining the frequency spectrum of a known waveform is called harmonic analysis, or Fourier analysis.

The case where y(t) is known only at discrete points in time is important is practice because of wide use of computers for recording signals.

DISCREET FOURIER TRANSFORM

N = total points recorded T = time period t = time interval n = harmonic order r = sample count = 1,2,3,…N

DISCREET FOURIER TRANSFORM (ctd)

• The process starts with an analog time dependent signal which is then digitized by selecting discrete values at predetermined time intervals.

• These values are then processed by harmonic analysis through which frequency contribution and relative amplitudes are determined.

• This provides a composite functional relationship that defines the original relationship (Equation in previous slide).

If one assumes that one’s experimental data can be expressed as cosine/sine function

(say y = cos(-x+phi); y = sin (x+phi1), etc),

the harmonic analysis then becomes a problem of determining appropriate values for harmonic orders (n), coefficients and phase angles.

EXAMPLE OF DFT ANALYSIS 3 signals: 500, 1000, 1500 Same amplitude: 50 mV Out of phase with each other Sample rate fs = 9000Hz N = 18 flowest = 500 Hz therefore T = 1/fL=0.002sec

and t= 1/fs t=0.11ms Harmonic orders evaluated: n=N/2=9

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Method for calculating the Harmonic constants using Excel Spreadsheet

EXAMPLE OF DFT ANALYSIS (ctd.)