ME – 341 (Spring 2010)

Midterm # 2 (Review of Notes and Top most argued topics)

What is uncertainty? Notes: Error estimation. Uncertainty: An estimate of the probe error in a reported value.

Uncertainty Analysis: A process of identifying the errors in a measurement and quantifying their effects.

Standard Deviation sample:

a = ā ± t Sx (P %) t = Multiplication Factor

For normal distribution: t = 1 for P = 64.3%

t = 2 for P = 95%

Explain types of error? 1. Bias error

2. Precision error : (Usually represented by standard deviation of measured data)

Zero-order Uncertainty Uo : Interpolation error caused by the finest marked resolution of instrument (difficult to

read value below this smallest marking) i.e., uo = ± 2 resolution (95%)

What is Root-Sum-Square (RSS) ? Book: Correct method used to estimate the combined error from different sources.

What is a Wheatstone Bridge Circuit (WBC) ? What does Balance WBC mean? Electronic circuit for measuring minute (small) resistance changes (with high precision).

Example for Standard Deviation (Exp1), RSS method (Exp 2) and Strain Gage (Exp 3&4) on handout. Also

Book 11.3 and 11.4

Gauge Factor The change in resistance of a strain gauge is normally expressed in terms of an empirically determined parameter

called the gauge factor, usually supplied by the manufacturer.

Gauge Factor = relative strain / relative resistance change.

How does a strain gauge work? (Working principle) The strain gage is tightly bonded to a measuring object so that the sensing element (metallic resistive foil) may

elongate or contract according to the strain borne by the measuring object. When bearing mechanical elongation or

contraction, most metals undergo a change in electric resistance. The strain gage applies this principle to strain

measurement through the resistance change. Generally, the sensing element of the strain gage is made of a copper-

nickel alloy foil. The alloy foil has a rate of resist-ance change proportional to strain with a certain constant.

Name the different types of strain gages? Wire gauge, foil gage, semiconductor gage.

Define over-damping, critical damping and under-damping?

1. Over-Damping → ξ2 > 1 ���� no oscillation

2. Critical Damping → ξ2 = 1 ���� no oscillation

3. Under-Damping → ξ2 < 1 ���� oscillation

What is Decay Amplitude Method? (Or Logarithmic Decrement Calculation of Damping Ratio ξ) Logarithmic decrement, δ, is used to find the damping ratio of an under-damped system in the time domain. The

logarithmic decrement is the natural log of the amplitudes of any two successive peaks. By measuring the peak ratio,

ξ can be calculated.

( ) sin( )nw t

dy t De w t

ξ ϕ−= +

The deflection y(t) of the cantilever beam oscillates with amplitude decay with time, which is used

experimentally to determine the damping ratio of the system observed. This is called the decay-amplitude method.

Given that the first, second, and subsequent nth

peaks (amplitude at first, second, and nth

cycles) are defined by the

following equations:

1 1

1 1 1( ) sin( )n nw t w t

dy y t De w t De

ξ ξϕ− −= = + = (4a)

1

2

2 2( )n

d

w tw

y y t De

πξ

− +

= = (4b)

( )1

1 2

( )n

d

nw t

w

n ny y t De

πξ

− − +

= = (4c)

these equations are rearranged to obtain an amplitude ratio between any two peaks of an oscillating damped system,

as defined below:

ln i n n

i d

y wn

y w

ξ+

= −

(5)

Due to the difficulty of obtaining actual deflection measurements of a cantilever beam in vibration, a strain

gage is used to find the analogous values of beam vibration amplitude. The logarithmic decrease in amplitude as

shown below can also defined as a measure of energy loss of the oscillating beam, as described by the following

equation:

( )ln

i i ny y

nδ += (6)

In an extremely underdamped system where ξ << 1, Equations (6) and (1) produce another definition of the

damping ratio based on measured amplitudes of two peaks. This equation is as follows:

( )ln

2

i i ny y

nξ

π+= (7)

Resonance – the resonance frequency is equal to the natural frequency without the damping effect, the system

vibrates with max amplitude. Vibration of large amplitude produced by a relatively small vibration near the same

frequency of vibration as the natural frequency of the resonating system.

Signal Conditioning – modification of instruments signal for proper output. i.e. filtering, amplification, digital

conversion, etc.

What is decibel (dB) ? The decibel (dB) is a logarithmic unit of measurement that expresses the magnitude of a physical quantity.

dB = 20 log M where M = Magnitude ratio or Gain

What is Gain? The ratio of output amplitude to the input amplitude of a dynamic signal. Also known as magnitude ratio “M”.

M (ω) = Output Amplitude [dB]

Input Amplitude

What is the Following? Define frequency filter types?

Filter Types Purpose

Low-Pass Filter Block High Frequency

High-Pass Filter Block Low Frequency

Band-Pass Filter Pass Band Frequency

Notch-Pass Filter Block Band Frequency

What is time constant “τ”? τ=2π/ωd

Time constant measures how fast the initial disturbance decays.

Notes: Measure of “ω” of how fast initial disturbance decay.

Book: System property defining the time required for a first-order system to respond to 63.2% of a step input.

What are the two types of Data Acquisition System (DAS) ? 1. Stand alone DAS

2. PC-Based DAS (with plug-in data acquisition board).

In physics and engineering, the time constant, usually denoted by the Greek letter τ (tau), is the rise-time

characterizing the response to a time-varying input of a first-order, linear time-invariant (LTI) system.

Name the Data Acquisition System (DAS) Components?

1. Signal Conditioning:

Filters (i) - Analog i.e. low-pass filter etc

(ii) - Digital i.e. moving average amplifiers

2. Multiplexer Alternate between input ports for measuring all of them.

3. A/D Converter Converts analog signal to digital signal

4. Central Processing Unit(CPU) Conducts operations as brain

5. Memory

6. Central Bus Connecting various board or peripheral devices

What is a Fourier series? Notes: All instruments and processes generate output signal for a given input signal. From principle of math, a

periodic function (input or output) can be determined into numerous harmonic functions called Fourier series.

Wiki: In mathematics, a Fourier series decomposes a periodic function or periodic signal into a sum of simple

oscillating functions, namely sines and cosines

What if fundamental frequency?

The lowest frequency of a vibrating system

f1 = 1 / T

What is cut-off frequency? The point where a lowpass or highpass filter makes a transition from the passband to the stopband. Normally given as

a -3 dB point. Also called FC The cutoff frequency Fc or Wc is where the gain reduces to 1/√(2)

What is harmonics? Book: Our ability to recognize differences in musical instruments is primarily a result of the higher frequencies

present in the sound, which are usually integer multiples of the fundamental frequency. These higher frequencies are

called harmonics.

What is sampling theorem? Book: The sampling theorem states that to reconstruct the frequency content of a measured signal accurately, the

sample rate must be more than twice the highest frequency contained in the measured signal.

Notes: Sampling theorem provides how frequently we have to sample to capture the signal.

Moving average – in this operation the average window is moving and the avg value of the window is assigned to a

specific point. There is also forward, center and backward moving averages.

Principle of thermocouple – temperature difference between two distinct metals causes the EMF, electro motor force,

to rise. Thus, by measuring the voltage we can find the temperature difference.

Parameters for Under-damped Systems

Peak Overshoot (P) – max overshoot above the steady response

Max Overshoot (% Over Shoot) – P in terms of percentage with respect to steady response

Rise Time (tr) – time required to rise from 10% to 90% of the steady state response

Peak time (tp) – time required to 1st peak

Settling time (ts,a%) – time require for the oscilation to settle withing a% of the steady response.