5th ed. Hoboken, NJ: John Wiley, pages, , English, Book; Illustrated, 6. Theory and design for mechanical measurements / Richard S. Figliola, Donald. Theory and Design for Mechanical Measurements solutions manual Figliola 4th ed. 1. SOLUTION MANUAL; 2. PROBLEM FIND: Explain. Theory and design for mechanical measurements fourth edition. Here http Click Here to Download Full PDF echecs16.info Powered.
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Theory and Design for Mechanical Measurements 4th Solutions Microelectronics: Circuit Analysis and Design, 4th edition Chapter 1 By D. A. Neamen. th edition of Theory and Design for Mechanical Measurements. .. In those measurement systems involved in process control, a fourth stage. Theory and Design for Mechanical Measurements, 4th Edition - Free ebook download as PDF File .pdf), Text File .txt) or read book online for free. Solution.
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The period. From Equations 2. From Equation 2. T is a period of y x FIND: Since the function is neither even nor odd. Problem 2. Fourier series for the function y t. The function is approximated as shown below 5 3 1 y t -5 -1 5 15 -3 -5 t Since the function is odd.
An odd periodic extension is assumed. Fourier series for the function y t.. Since the function y t is an even function. Odd Periodic Extension 1 y 0.
The function is extended as shown below with a period of 4. For the odd periodic extension of the function y t shown above.. T b amplitude. A c displacement as a function of time. Construct an amplitude spectrum plot for this series. Amplitude Spectrum 3. The relative importance of the various terms in the Fourier series as discerned from the amplitude of each term would aid in specifying the required frequency response for a measurement system. The corresponding frequency spectrum is shown below Amplitude Spectrum 4.
Signal sources: Sketch representative signal waveforms. FM Radio Wave The time series and the amplitude spectrum are plotted below.
This is important because if we represent the signal by a discrete-time series that has an exact integer number of the periods of the fundamental frequencies. Using the companion software disk. Find the amplitude-frequency spectrum. The signal to be represented contains two fundamental frequencies. Frequency Hz 63 61 59 57 55 53 51 49 47 45 43 41 39 37 35 33 31 29 27 25 23 21 19 17 15 13 11 9 7 5 3 1 Signal Amplitude Amplitude Spectrum 6 5 4 3 2 1 0. Express the signal.
Signal average value. Express the signal as a Fourier series and plot the signal in the time domain. Wall pressure is measured in the upward flow of water and air.
Construct an amplitude spectrum for the signal. The flow is in the slug flow regime. There is clearly a dominant frequency at 0. The figure below shows the amplitude spectrum for the measured data. Pressure measurements were acquired at a sample frequency of Hz. The signal can be reconstructed from the above information. By inspection of Figure 2. The values of An are zero for n even. Assume that Figure 2. Figure 2. Discuss the effects of low amplitude high frequency noise on signals.
The waveform can be altered significantly by the presence of noise. Several aspects of the effects of noise are apparent. So in the transient sense. To do this. Notice how K.
The calibration curve must be a linear one. The forcing function i. Here K is a constant. For a first order system.
Figure 3. Unless noted otherwise. System model FIND: For a second order system. Problem 3. Be certain to always provide units for all answers. Temperature C b The input signal and output signal are shown below. Thermometer similar to Example 3. We saw in Example 3. If a student were to remove a sensor from hot water and transfer it to cold water by hand. Their answers results are not incorrect. The results simply show the effects of a random error. Movement will change the heat transfer coefficient on the sensor by a factor of from 2 to 5 or more.
This uncertainty can be quantified by methods developed in Chapters 4 and 5. By proper test plan design. Solving for time. At time 1. The solution is given by 3. That is. In effect. This system is a more effective as a filter than it is as a measuring system. Associated with this large dynamic error is a large phase shift and associated time lag.
This creates a filtering effect whereby a significant portion of the signal amplitude is attenuated the term "attenuation" refers to a reduction in value and is indicated by a negative dynamic error.
So based on the constraint for dynamic error. For this particular system. Input of the form. For this to be true. To solve for E t. Output signal is linearly proportional to input signal that is.
K is constant FIND: Output response. The full solution then has the form. This places a more restrictive design constraint on the sensor and installation selected. We must set this threshold value. By inspection. Either from the plots or from 3. First-order system i. The test plan should impose the step input and system output recorded at time intervals much less than Td e. The peak values maximum amplitudes. Because the decay is exponential.
Step test imposed on a second order system Damped oscillating response. The transient response is found from the homogeneous solution to the equation model. From the information given.
Also from the Figure. Use Figure 3. Then we want. The system model will have the form. Of course. L remain constant. Values for R. RCL circuit: So the transducer does not meet the constraint over the entire frequency range of interest. The dynamic error. The instrument effectively filters out the information at Hz. So the input signal amplitude is attenuated Inspection of Figure 3. First order system: The steady output will have the form.
Second order system accelerometer: For this system. Seismic Accelerometer of Example 3. The input function has the form. For coupled systems. Two coupled second order systems Input signal: The following is one approach to choosing a system.
But the problem demonstrates one approach to dealing with such an open ended problem. Note also that the phase shift for the system selected is about The 1 phase shift then becomes. The 2 phase shift then becomes.
If we accept. With these values selected. First stage: A second stage with a higher natural frequency would bring M2 closer to unity and decrease the phase shift.
Translate this specification into words SOLUTION A typical audio amplifier increases the output amplitude relative to the input amplitude by some amount and that amount is called its gain. Under normal circumstances.
Music signals are a series of sinusoidal frequency terms. In literature pertaining to amplifiers. So between 0 and This system specification states that for an input frequency within 0 to The harmonics give distinction to the source of the sound so that different instruments are recognizable.
Even single notes.
Another way to write this is: But the specification is explicit that the amplitude never varies by more than the 1dB. A typical audio amplifier will have some spikes and troughs across its frequency response. So the product KM f is frequency dependent and therefore the amplifier gain is frequency dependent. So the amplifier gain is stated at some reference frequency. This power is simply another way to state the gain. Now in our equations. Determine if measurement system specifications are adequate for the input signal.
That is not good. We offer one possible solution to illustrate the design selection approach. The spectrum measurement device will not be a factor owing to its wide frequency response relative to these input frequencies.
A quick inspection of Figure 3. Now we should meet our criterion without resonance problems. The second order displacement transducer will be most heavily tested at the highest input frequency. The slope is —0.
Each student understands material in their own individual way. So this is an opportunity for written technical communication between instructor and student. This is a step function input as the circuit sees it. RL circuit only one reactive element here. Around the loop: For the capacitor to reach 1 this energy level.
So we seek the time 2 required to charge the capacitor to this voltage level. Lets start by estimating the total energy that can be stored in the capacitor. The input magnitude is controlled by the user and may be varied with time. The system response slows down relative to time as the time constant is increased independent of the magnitude of the step change imposed. The user should select a time constant and an amplitude.
The instrument time constant is set by the user and may be varied with time. Interpolation gives P 2. Toss of four coins FIND: Develop the histogram for the outcome of any toss.
State the probability of obtaining three heads on one toss. The possible outcomes are: Number of Heads. The histogram is shown below. Because of the few number of tosses, the development of the histogram is primitive. But the symmetry is obvious. This type of distribution is best. As the number of possible outcomes number of coins tossed becomes large say 30 or more , the two distributions become nearly identical over a wide interval about the mean and the.
This is because each shot is independent of the other and each shot differs from the other by random variation. A 'better' player will have a mean distance in the outcome that is close to the target point and have a low variance.
That is, the player will have a low systematic error average distance from target point and low random error variations from the average point. This game and its interpretation are similar to the dart game discussed in Chapter 1 in the discussion of random and systematic error.
In the US, a variation of this game is called 'matchbook football' where the objective is to slide the matchbook across a table so that it just overhangs the table edge. Histogram for Table 4. Frequency distribution for Table 4. Compare and discuss histograms for Table 4. The variations seen are likely a consequence of 1 normal variation due to finite data sets, 2 random errors in each measurement.
Each histogram clearly shows a central tendency and in each case it is in bin 4. Three datasets from Table 4. Data from Table 4. This is what is meant by a central tendency in a population — a tendency for a data point to have or be close to one value over all others.
Column 1. Then from Table 4. Column 2: Data of Table 4. While the sample mean value defines the mean value of the. They are very different! Compare the meaning of this statement to that found in Problem 4. Different data sets of the same variable will give somewhat different mean values.
Remember this assumes that there is no systematic error acting on the measurement. As N becomes large. From Table 4. Columns 1. We learn that the more information measurements we have about a dataset. Data sets in Columns 1. The number of measurements. While column 3 predicts the ensemble mean value the closest.
Table 4. The expected number of outcomes for a normal distribution is estimated to be 2. For 7 intervals and using calculated values of the 2 statistical quantities. This is the deviation between expected and observed.
The data lends itself to 7 intervals: Number of A. Large sample of grades i. The average expected life of the bulb is hours Doubling N from 10 to 20 is more efficient than increasing N from to The value of S x x represents the standard random error in the mean value. This is the "diminishing returns" in using N to improve random error. There is some gain as the value of t drops as N becomes larger.
As the number of measurements. But as N increases. So that P So in the absence of other errors. Please review "replication" discussed in Chapter 1. The variations in the statistics between each data set reflect 1 the ability to duplicate the operating conditions for each test exactly. The three data sets are replicate measurements of a variable under similar conditions.
Time to break up and repour! In particular. Fixed operating conditions. Check for outliers. No outliers are detected in this data set. Using t9. Measured variable has a normal distribution. Is claim supported by the data set?
Sample is representative of the batch. But a third order fit reduces tSyx to a minimum and appears to fit the data well. The data are plotted below. Data set provided. The data can be fit to: The differences between the two results are nearly indistinguishable hence. Not knowing that. For t4. The t value is found to be t2. Note how the r value is not very sensitive here and is generally not a good indicator of how 'well' data fits a curve.
A first order fit does not appear reasonable. So there is about a 1. Rejecting this batch is prudent or at least prudent with Is rejection of the batch based on S x2 prudent? The problem is not in the random error random variations in diameter between bearings.
The whole setup is shifted to making too large a bearing. Stop the machine and reset the tooling set-up. Does the sample mean support the intention so that the constraint is met? This data set suggests 5.
COMMENT Clearly there is a systematic error in the bearing grinding setup causing the bearing diameter mean value to be greater than the intended bearing diameter. So the criterion is met. This would be tested again following these measurements. To reduce the interval of Based on the data set provided: Interpolation of Table 4.
This result is equivocal. For all the intervals: The expected occurrences n'j are listed above. Sx Test the hypothesis of a normal distribution. The statistics should then be recomputed to verify that the constraint is met. Because N1 is quite small relative to NT. NT required to achieve CI within 0. This CI is a tight constraint on random error given the large variations seen in the dataset Sx1 value.
Expect 0. This is because the former is the limiting case of the latter. So pb will take the form: For x defective parts in any samples x pb pp 0 1 2 4 10 0. We see that the results above come close to this condition. Particle passage through a small volume Average particles passing in time period t1 is 4: The user can vary the number of intervals K and the number of data points N used in each histogram.
The user can vary the number of intervals K and the number of data points N used in each plot. The signal has its own statistics and pdf. In most cases. The program creates a new pdf each time the program is rerun. The program updates the histogram for each block of N data points. The pdf will change with each new data set because the finite statistics of the data set finite population do not exactly estimate the statistics of the infinite data set infinite population.
With the appropriate number of intervals for the N samples. As interval number is increased. The signal is continually measured and the 0. But as the number of samples N is increased the signal increases with time. The values of the finite statistics change as new values are added to the data set. Unlike systematic errors. Randomization methods break up some of the trends brought on by interference.
Random errors are manifested by measured data scatter and their effects on the estimate of the true value of the measured variable can be estimated statistically using the methods discussed previously in Chapter 4. Random error leads to scatter in the measured values obtained during the measurement of a variable under otherwise fixed operating conditions.
Randomization makes systematic errors behave as random errors. Calibration is an excellent way to isolate. An error refers to a difference between the measured value and the true value. Both systematic and random errors are present in any measurement to some degree. Incorporating replication strengthens the random error estimates and allows estimates of control to be made. Because this exact amount is. For the engineer. These methods include: By incorporating repetition into a test plan.
Questions 5. This numerical estimate is the uncertainty. Estimate of the precision and bias errors involved in estimating the value of a variable.
It is based on the data set and the precision and the bias errors involved in the measurement. The range or interval of values within which the true value is expected to lie with some probability. Exact statement of the mean or central tendency of a measured data set. The confidence interval is in part based on the precision-based interval tSx discussed in Chapter 4 due solely on the variations in the measured data set. The nearest approximation of the true value that can be made with the data set available.
The mean value of a finite data set is given by its sample mean value. Most often refers to the true mean value of the variable which would result from an infinite sampling under perfect test control.
The actual value of the measured variable. It is usually offered by the sample mean value and qualified by its precision interval. The value sought by measurement. It is the range of probable errors which affect the outcome of a measurement. So u1 might be estimated by a simple surrogate experiment whereby a fixed pressure is repeatedly measured say 20 times and the outcome stated as tSp. This gives us an estimate for how the pressure might be affected during our normal single measurement.
It can also include other sources of error known at the time of analysis. Nth order uncertainty is applied to those measurement situations where a number of repeated measurements cannot or will not be taken. The Nth order uncertainty includes all conceivable errors. Assuming that the pressure did not change. Suppose we want to measure the pressure of a tire and make only one reading per tire during a normal tire installation or during a tire pressure check.
This is the way we tend to do it — isn't it? How good might we expect that one value of pressure to be? To quantify our methodology. The values for ud and uN differ by the errors which enter during the conduct and control of the test. The analysis focuses on the control of the test and how that would affect the outcome.
This value provides a good guess of the uncertainty to be expected when minimal information is available. Tire pressure gauge. We bank this information for subsequent use. Instrument error is negligible compared with error due to resolution. Micrometer Resolution: At higher speeds the instrument errors dominate.
Insist on a detailed description. Analog Tachometer Resolution: At low speeds it is dominated by the ability to read the tachometer resolution. This misnomer causes confusion. We suspect the term to describe the combined effects of all known elemental errors.
Speedometer Resolution: This made the vehicle slightly more attractive to a fuel cost conscious consumer. Unless the consumer actually made a careful test of the speedometer and odometer performance. Caveat emperor! During the tight fuel availability times in the western hemisphere of the 's. In either case. Temperature sensor Error limit: Consider as Case 1. Although each individual resistor in Case 2 has a larger absolute uncertainty than those in Case 1.
We should proceed using this design. The combination in Case 2 is just less sensitive to the individual uncertainties. Our design results from a close analysis of the sensitivity of the resultant to each contributing uncertainty. This is not obvious by inspection alone. At full-scale. A true problem found several times in catalog of a major supplier of engineering sensors and readouts in US. Meter resolution does not affect the uncertainty to any practical extent.
Note that temperatures must be in absolute and that a 1 oC change equals a 1 K change. In most engineering processing plant applications. This problem solution provides the propagation of uncertainty from the variables to the result. Heat transfer from a rod is to be determined.
The equations would not be logical otherwise. Focusing on reducing the uncertainty in the resistance measurement would be a good starting place to reduce u d E. For the nominal values of power and resistance given. Note that the units in each of the working equations are consistent.
Such an analysis builds on a design-stage analysis estimate to include estimates of uncertainty. The analysis assumes perfect control of the measurement process and its procedure. Used when only a single or very few measurement s is planned. In an advanced-stage analysis. Repetition provides a measure of repeatability during the same test.
It does this by permitting the test engineer to quantify the differences in test results obtained from distinctly duplicate tests conducted under nominally identical conditions. In a multiple-measurement analysis. Advanced-Stage Analysis: Provides an accurate estimate of the uncertainty in a result based on a detailed knowledge of the measurement process.
Provides a quick but not accurate estimate in uncertainty based on a planned approach. In general. Displacement Transducer Instrument specifications Linearity: Keep in mind that at this level of uncertainty only the intrinsic instrument errors and no procedural errors are considered.
Measurement system of Problem 5. Measurement is sufficiently controlled such that all errors have been randomized and considered in the measured data. SOLUTION The measurements have provided additional information about the uncertainty involved in the using this measurement system and involved in measuring this particular variable.
We will approach the problem as a multiple measurement uncertainty analysis problem. Using the procedure of problem 5. Then, we can identify two elements of systematic error at the data acquisition source see Table 5. We can set the random error in both of these elements to zero no data provided. The measurements provide a set of finite statistics from which a precision interval in estimating the true mean value can be made. Begin by estimating the random error due to temporal variation in the measurand:.
The measurement systematic error can be expressed as:. COMMENT Note that we could use this information as the basis for a single sample uncertainty estimate by using the test performance obtained at This information could be used to update the design stage analysis in the previous Problem. Nominal pressure value to be measured is psi at 70oF Pressure transducer Accuracy: This value would be adjusted accordingly in an actual situation.
Systematic and random source errors in a measurement of force. If we assume that the measurand does not change. The contribution from the instrument itself is barely 0. P listed as P due to temporal variation in Table 5. For t5. Because P4 is the only random error involved. We use this sensitivity to convert between psi and mV.
Sample ingot is cylindrical in shape. Density of metal composite is determined by mass estimation. The variation in readings taken at each cross-section location is estimated by the pooled standard deviation relative to the pooled mean.
This yields a measure of the standard random error due to data scatter P2. Sample mean and sample standard deviation values.
Elemental errors from data acquisition sources will consist at least as a systematic error due to instrument error B1. We often think of such non-temporal variables as being fixed and absolute.
Because they are statistical statements. Even though all the readings appear to be well behaved with some variations in position and between positions. This problem also provides a nice example of how the statement of a value of a variable such as the diameter of an ingot or of a shaft is actually a statistical statement. From the previous problem. These give: We will also use the systematic and random error estimates from the diameter measurement information in Problem 5.
Measurements of mass and length and results of Problem 5. The systematic errors are the same as the uc values used previously.
L and d. In this updated analysis. Calibration curve fit. From the problem statement. Calibration against a standard. Uncertainty in any estimated T. Lacking other information. Random errors in the measurement system are presumed included in the calibration curve.
We see that a very small change in voltage corresponds to a large change in temperature! This is typical of thermocouples.
Voltage measurement system systematic errors are not indicated but will be present in the measured data. Calibration data are provided.
To compute a calibration curve fit. It corresponds to the curve: In practice. This determines R and I for analysis. With the information available. A broader look of this problem would vary E and optimize to determine a basis for preferred operating conditions in E.
Method 2: If uncertainty in the oven test temperature is to be estimated. This is an excellent problem for an instructor-directed group discussion. At any set temperature: In both cases the effects will be the same for either method. Composite material is function of cure temperature.
By setting the oven to one representative condition. We also neglect data reduction errors. In order to simplify the solution we assume that sensor installation effects and measurement system operating conditions are properly controlled. Possible cure temperature range: Oven divided into quadrants: This test is really one of oven performance.
This will be an important omission if we intend to use the oven for batch production. One alternative performs a multiple measurement analysis to estimate our ability to control the oven at 30oC. Evaluate an NT based on the available S1.
This value reflects the manufacturer accuracy statement. The spatial error arises because the diameter is not uniform along its length such that the mean values vary. Instrument errors will be treated as a systematic error. We will restrict the solution to errors due to data acquisition sources. Collecting random errors. We do this by running a trial set of data for pressure set point versus actual process pressure: Single measurement analysis: From the manufacturer statement.
We are asked to estimate the uncertainty in the set pressure at any measurement. Pressure is measured using a dial gauge. Pressure will be set at p and readings are to be taken. Notice how this is quite different then trying to determine the average set point over 30 trials which works out to be only 0.
Estimate the uncertainty in vessel pressure. The uncertainty in the actual mean set point is: This is an example of the advanced stage analysis applied to access uncertainty in a single trial sample.
For the transducer. For the voltmeter: First-order system: As a class exercise. As a guide: Neglect uncertainty in gas constant R. Subbing terms. Neglect uncertainty in gas constant. As a concomitant check. Then with. But friction effects are built into the first estimate and the differences will reflect the magnitude of this systematic error.
Vo and launch angle. The units of angle used are in the equations is radians. Note that the only significant contribution to uncertainty results from uncertainty in the initial velocity. Measured data relating golf ball carry distance to launch angle and initial velocity FIND: Uncertainty in carry distance.
Measured data relating pressure drop and volumetric flow rate FIND: Required uncertainty in pressure drop to yield a 0. The accuracy of pressure transducers is often reported as a percentage of the full scale reading.
For this problem. It is not likely that a single transducer could yield the desired accuracy over the range of pressures required in the present problem. Plugging in the known values. A portion of the uncorrelated error could come the use of different measuring devices. If so. On the other hand. The functional relationship can have an impact on whether correlated errors have an effect on the uncertainty.
Torque on the current loop. From 6. The magnetic field is oriented at an angle of 90o to the current flow direction. Since Eo is one-half of Ei. As a percentage of full-scale output. FL Eo 0. Bridge circuit of Figure 6. A bridge circuit is to be used to calibrate a Hz frequency source. Bridge components L and C to yield a resonance frequency of Hz. The meter resistance Rg is infinite. Clearly the bridge output is non-linear with R1 over this range of values for R1.
Plot the output voltage for: Potentiometer as shown in Figures 6. Loading errors should be included in the analysis.
Design stage uncertainty in a measured value of voltage at nominal values of 2 and 8 V. Sinusoidal inputs having specific phase relations FIND: See Figures below.
Show that the phase relationships can be determined from a Lissajous diagram using the equation above. Eliminating the amplitude A between these equations. Referring to Fig.
See Problem 6. Phase lag occurs in electronic circuits FIND: The arrangement shown below will allow measurement of phase lag in an electronic circuit. Two sinusoidal signals are to be compared using a dual trace oscilloscope. Add to Basket. Compare all 6 new copies. John Wiley and Sons, Hardcover. Book Description Condition: AS NEW may have shelfwear. Choose expedited shipping for superfast delivery business days. We also ship to PO Box addresses.
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Wiley , This specific ISBN edition is currently not available. View all copies of this ISBN edition: Synopsis About this title This textbook provides an in-depth introduction to the theory of engineering measurements, measurement system performance, and instrumentation. From the Publisher: From the Back Cover: Features of the Fourth Edition: Expanded treatment of mechatronics concepts Chapter 12, Appendix C. More practical use of operational amplifiers in discussion of signal conditioning circuits.
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