# Square root symbol mathcad torrent

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When there are multiple roots, Bisection is not a desirable technique to use, since the function may not change signs at points on either side of the roots. Therefore, a graph of the function must first be drawn before proceeding to do the calculations. Example 3. In an attempt to minimize the number of iterations needed, we will obtain a root value that is correct only to two decimal places, in this case.

Thus the new xstart is 17 and the new xend is Thus the new xstart is The main difference is that while the interval size in the Bisection method is reduced by bisecting it in each step of the iteration , this reduction of interval size is achieved by a linear interpolation fitting the two end points. While the Bisection method is reliable, it is slower than the False Position method in achieving convergence.

Regula Falsi method The procedure for finding roots then is as follows. See Figure 3. Choose the starting and ending points xstart and xend as in the Bisection method. Compute f xstart and f xend. Make sure that f xstart times f xend is still a negative product. If it is not, then there is no root between xstart and xend. Repeat the above procedure until convergence takes place. This is called stagnation of an end point.

This is not desirable since it slows down the convergence process especially when the initial interval is very large or when the function is highly nonlinear. Examples 3. The starting and ending points were 16 and 20 respectively. Thus the new xstart is 16 and the new xend is Thus the new xstart is 1. Thus the new xstart is 3. Thus the new xstart is 4. Thus the new xstart is 5.

In this method, an initial approximation of the root must be assumed, and calculations are started with a "good initial guess". If this initial guess is not a good one, then divergence may occur. From Figure 3. Newton-Raphson method Procedure for Finding Roots 1. Make a good initial guess. Call it X Old. Keep improving the guess using Equation 3. Solution is done when a new improved value X New is almost equal to the previous value X Old Advantages and Disadvantages of the Method While the Newton-Raphson method is faster than the Bisection method, is applicable to the complex domain as well and can be extended to simultaneous nonlinear equations, it may not converge in some situations.

The solution may oscillate about a local maximum or minimum, and if an initial estimate is chosen such that the derivative becomes zero at some point in the iteration process, then a division by zero takes place and convergence will never occur. Although convergence will occur quite rapidly if the initial estimate is sufficiently close to the root, it is possible for it to be be slow when it is far from the root.

Also, if the roots are complex, they will never be generated with real initial guesses. A worthwhile feature of the Newton-Raphson method is that the numerical process will correct itself automatically for minor errors.

Thus, any errors that are made in computing the next guess will simply generate a different point for drawing the tangent line and will not have any effect on the final answer. Convergence Criterion for the Newton-Raphson Method It can be mathematically shown that in order for the Newton-Raphson method to converge to a real root, the absolute value of the derivative of the g xi of Equation 3.

However, in some cases, it may not hold for the initial guess. Start with an initial estimate of 2. Let us start with an initial estimate of However, there is no need to do this every time, since a complex root will always have a complex conjugate associated with it. Here, both f z and z are to be scalar quantities. The procedure to be used will be clear from the following example.

Let us find the root of the function in Example 3. It does not require a guess value. By choosing "matrix" from the "insert" menu, type in a vector "v " as shown , beginning with the constant term, making sure that you insert all coefficients even if they are zero. Then, polyroots v returns all roots at once.

The following steps will find the root of the polynomial of Example 3. Include all coefficients, even zeros. The one difference is that while the derivative fprime x is evaluated analytically in the Newton-Raphson method, it is determined numerically in the Secant method.

Secant method By the Newton-Raphson method, a new improved guess of the root is generated using Equation 3. However, two initial estimates x i-1 and x i of the solution will be required to start the iteration process as can be seen from Equation 3. Make initial estimates x 0 and x 1. Continue obtaining improved estimates x 3 , x Among the disadvantages are convergence to an unintended root at times, and divergence from the root if the initial guesses are bad.

Also, if f x is far from linear near the root, the iterations may start to yield points far away from the actual root. We first draw a graph of the given function to see approximately where a real root lies in the range given. To type a subscript, use the left bracket " [ "and put an integer in the placeholder. Vector elements in Mathcad are ordinarily numbered starting with the first element numbered as the zeroth element.

A look at the Secant Method suggests that this feature relates well with its recursive formula. Put in "N" as the expected number of iterations. Then i goes from 1 to N. To find the required root, begin with the initial estimates x 0 and x 1 and use the recursive formula of the Secant Method to obtain improved estimates.

Continue until convergence to a solution takes place. Clearly, the other complex root will be the complex conjugate -3 - 2i. Resorting to the polyroots function will confirm the roots generated above. Applications of this technique can be found in the use of the Newton-Raphson method and the Secant method. Another name for this method is Fixed Point Iteration. To insure convergence of the iteration, the function selected must be such that the absolute value of its slope is always less than unity.

Otherwise, divergence will occur. Determine the negative root by the method of successive substitution. Using the above recursive relationship, we find revised values of x until convergence occurs as shown below. These roots are difficult to compute by the methods discussed in this chapter for the following reasons. In the case of the Bisection method, the function does not change sign at the root. In the case of Newton-Raphson and Secant methods, the derivative at the multiple root is zero.

Case of multiple roots Because the Newton-Raphson and Secant methods in their orginal forms are methods that resort to linear convergence, they cannot be employed to generate multiple roots. The following modification to the original Newton-Raphson equation has been suggested [ 4 ]. However, this may not be a very practical route since it assumes prior knowledge about the multiplicity of a root. Substitution of Equation 3.

To determine the single root, the above approach can still be employed, but with a different starting estimate, as shown below. The following steps will now demonstrate the application of the modified Secant Method formula, which is Equation 3. In certain situations, it may be necessary to solve nonlinear equations in two or more variables.

In these cases, iteration can be resorted to as presented in the following example. Thus, the computations must be repeated with equations set up in a different format as shown below. Iteration This example illustrates a very serious disadvantage associated with the successive substitution method which is the dependence of convergence on the format in which the equations are put and utilized in the iteration process. Also , even in situations where a converged solution can be attained, initial estimates that are fairly close to the true solution must be resorted to, because, otherwise, divergence may occur and a solution may never be obtained.

The results of the iterative process are summarized in the following table. The procedure is as follows: 1. Provide initial guesses for all unknowns. These initial guesses give Mathcad a place to start searching for the solution. Type the word Given.

This tells Mathcad that what follows is a system of equations. Given can be typed in any combination of upper and lower case letters, and in any font. However, it should not be typed in a text region. Then type the equations and inequalities in any order below the word Given. You can of course separate the left and right sides of an inequality with the appropriate symbol.

Chapter 3: Roots of Equations 87 Example 3. Using the Given and Find functions, solve 2 2 1. The maximum load that the column can carry with a factor of safety of 2. Thus, it must be solved iteratively using trial and error. How this can be accomplished with Mathcad is demonstrated below. Natural Frequencies of Vibration of a Uniform Beam For a uniform beam that is clamped at one end and free at the other, a shown in Figure 3.

For a beam with a length of 20 feet, an EI value of 16 x 10 6 lb-in 2 , and a weight per unit length of 0. Some of these are listed below. Third natural frequency: Solving the Characteristic Equation in Control Systems Engineering A characteristic equation is an algebraic equation that is formulated from the differential equation or equations of a control system [14].

Its solution, which often requires evaluation of the roots of a polynomial of degree higher than two, is crucial in determining system stability and assessing system transient response in terms of its time constant, natural frequencies , damping qualities etc. An application involving the use of Mathcad' s polyroots function in determining the roots of a characteristic polynomial of a control system is presented below.

For varying values of this parameter, the roots of the characteristic polynomial can be evaluated as shown below, and the system transient response and stability studied , following which, an appropriate range for K may be recommended. Horizontal Tension in a Uniform Cable Flexible cables are often used in suspension bridges, transmission lines, telephone lines , mooring lines and many other applications. Plot of g H The use of the Secant method in obtaining the required root is shown below.

This is done below. Label it and give it a title. Verify your answer by using the root function. Also obtain all the roots of the polynomial with the polyroots function. Generate an answer that is good to three decimal places. Obtain all roots of the following polynomial lying in the range Verify your answers by using the root function. Obtain an answer that is correct to at least three decimal places. Obtain an answer that is correct to three decimal places.

Obtain an answer that is correct to four decimal places. Obtain all roots of the following polynomial lying in the range 0 to 1. Obtain all roots of the following polynomial lying in the range 0. Start with initial estimates of 1. Start with initial estimates of 4. Generate an answer that is correct to four decimal places. Generate an answer that is correct to three decimal places. Generate an answer that is correct to threer decimal places. Notice what this plot suggests in terms of the interval where the roots may lie.

Then, draw a second graph for a narrower range Start with an initial estimate of 0. Start with a reasonable initial estimate. Obtain the solution of the following system of nonlinear equations by iteration x 1. Verify your answers with the Given and Find functions.

Using the Given and Find functions of Mathcad, obtain the solution to the following system of equations. Using a range of damping ratios from 0. Determine the first three natural frequencies of a 36 in. E for steel is 30 x 10 6 psi. Determination of the inverse of a matrix is linked with the concepts of transpose, minor, cofactor and adjoint.

Addition and Subtraction: Two matrices with the same number of rows and columns can be added by adding the corresponding terms. Example 4. How to perform matrix operations will be clear from the following examples. Figure 4. Note that, in this case, subscripts will go from 0 to 2. For Example 4. Formulate a set of linear equations that represent the relationship between voltage, current and resistances for the circuit shown in Figure 4.

Some examples of eigenvalue problems are given below. Determination of natural frequencies and mode shapes of oscillating systems. Computation of principal stresses and principal directions 3. Computation of principal moments of inertia and principal axes.

Buckling of structures. Oscillations of electrical networks. Equation 4. For instance, if [A] is a 3x 3 matrix, then Equation 4. This polynomial is called the characteristic polynomial and its roots are called eigenvalues. The nth column of this matrix is an eigenvector corresponding to the nth eigenvalue returned by eigenvals A. For example, the eigenvector associated with the eigenvalue 0.

Mathematical model of two-story building Consider the vibration problem of a two -story building which is mathematically modeled as the two degree of freedom system shown in Figure 4. The square roots of the eigenvalues will then give the natural frequencies and the corresponding eigenvectors will be the mode shapes. Because the system considered here is a two-degree-of-freedom system, there will be two natural frequencies and correspondingly two modes.

Therefore, it is traditional to select a value of unity for one of the components of an eigenvector and compute the other one accordingly. This is called normalization and should be clear from the calculation shown below. Modes shapes of the two-story building Example 4. Investigate the free vibration of a three-story building mathematically modeled as the three-degree-of-freedom system shown in Figure 4.

The natural frequencies in Hz and the corresponding eigenvectors are computed below. Determine the principal stresses and their associated directions. These are. Its x and y components , prindir1, are computed below. Principal stresses and principal directions for two-dimensional stress example Example 4. Chapter 4: Matrices and Linear Algebra Figure 4. Note the orthogonality of the principal stress directions with respect to one another. Principal stresses and principal directions for three-dimensional state of stress.

The following examples will illustrate this situation. Thus, there is no unique eigenvector correponding to the repeated root. In general, iterative procedures must be resorted to, requiring initial guesses as a starting point.

The following procedure utilizes an adaptation of the Taylor series approach in obtaining a solution to a set of nonlinear equations. Defining a, b, c and d as the partial derivatives in Equations 4. These new improved values replace the old values x i and y i in the next step in an effort to obtain still better solutions.

This procedure is repeated until convergence occurs. The following example will clarify the procedure. These answers may now be verified by using the given and find commands. Do calculations on a calculator. Also check your answers with Mathcad 4. Also check your answers with Mathcad. Check your answers with 1 Mathcad, using the inverse and 2 Mathcad's lsolve function. Chapter 4: Matrices and Linear Algebra 4. Check your answers with 1 Mathcad, using Cramer's rule and 2 Mathcad's lsolve function.

Do this problem on a calculator. Check your answers with Mathcad, using the eigenvals, eigenvecs and eigenvec functions 4. Find the eigenvalues of the following matrix. Find an eigenvector corresponding to each eigenvalue. Check your answers with Mathcad, using the eigenvals, eigenvecs and eigenvec functions. Determine the eigenvalues and eigenvectors of the following matrices. Check your answer using the Given and Find functions of Mathcad 4. Check your answer using the Given and Find functions of Mathcad.

A two-degree-of-freedom mathematical model of an automobile suspension system is shown. K1 K2 Figure P 4. An automobile is math-modeled as the two-degree-of-freedom system shown above in Figure P 4. A rigid rod of negligible mass and length 2L is pivoted at the middle point and is constrained to move in the vertical plane by springs and masses shown. The governing differential equations of motion of the double pendulum shown in Figure P 4. Obtain answers that are correct to two decimal places.

Because measured or available data is , typically, not provided in the form of an analytically determined function , the process known as interpolation must be resorted to in order to obtain function values at points other thn the given data points. This process involves the generation of a curve that must pass through the given data points and its use in determining the function value at any intermediate point on this curve. As will be seen in the sections following, a polynomial fitting n data points will be of order n-1 , that is one less than the number of data points given.

Thus, four data points will generate a cubic while three data points will give a quadratic and so on. In general, the higher the order of the interpolating polynomial, the more accurate would be the results of the interpolation process. The given data is put in as follows. Go to the Insert menu, select Matrix, put in the appropriate number of rows and columns and then fill in the placeholders.

Interpolated function using undetermined coefficient method 5. For obtaining a third order polynomial, a fourth data point will be needed , and a fourth order polynomial will require the use of a fourth as well as a fifth data point.

Interpolated function obtained using Gregory-Newton polynomial method Example 5. Fifth order Gregory-Newton interpolating polynomial 5. TABLE 5. L 0 20 40 60 80 R Generate the finite difference table and obtain an interpolating polynomial using Newton's method. Estimate the flux R at a latitude of 45 degrees.

The following finite difference table can be generated using the given data. In such situations, the Lagrangian interpolation method offers a viable means of deducing an interpolating polynomial connecting the dependent variable with the independent variable at intervals that are not necessarily constant.

By this method, given a data sample x1 , y1 ; x2 , y2 ; The Lagrange interpolating polynomial 5. For n data points that are provided, the highest- order interpolating polynomial that can be generated will be of order n Although, in general, the accuracy of the interpolation process increases with the order of the polynomial , there are situations when the accuracy can, in fact, decrease with the polynomial order.

This can happen when the measured data reflects abrupt changes in the dependent variable values for steady changes in the independent variable. In these cases, accuracy can be improved by resorting to lower order polynomials, commonly referred to as splines. Splines normally used are linear, quadratic and cubic. Linear Splines: A linear spline results from connecting adjacent data points with straight lines. These interpolation functions return a curve passing through the points you specify.

Linear interpolation: This is done using the linterp function as shown below. The quantities vx and vy must be specified as vectors and vx must contain real values in ascending order. The linterp function is intended for interpolation and not for extrapolation. Mathcad does this by taking three adjacent points and constructing a cubic polynomial that passes through these points. These cubics are then connected together to make up the interpolated curve. In order to fit a cubic spline curve through a set of points, the procedure shown below must be followed: 1.

Create the vectors vx and vy containing the x and y coordinates of the points of interest through which you want to go through. The elements of vx must be put in in ascending order. This vector vs is a vector of intermediate results to be used with interp. To evaluate the spline at a point x1, do the following: interp vs,vx,vy,x1. Note: Steps 2 and 3 can be combined by doing : interp cspline vx,vy ,vx,vy,x1 Mathcad has two other cubic spline functions as given below: lspline vx,vy : This generates a spline curve that approaches a straight line at the endpoints.

The function interp vs,vx,vy,x returns the interpolated y value corresponding to x. The vector vs is a vector of intermediate results obtained by utilizing the option of using lspline, pspline or cspline on the vectors of given data , namely, vx and vy. For Example 5. The resulting spline function is drawn in Figure 5. Interpolated function obtained with linterp linterp vx , vy , 0.

Comparison of spline functions obtained with Mathcad For the problem of Example 5. The results are sketched in Figure 5. Interpolation with Mathcad For the problem of Example 5. Linear interpolation for Example 5. Stress-Strain Data for Titanium The stress versus strain data given in the table below is obtained from a tensile test of annealed A titanium [ 18 ]. A cubic spline can be developed using Mathcad as follows and the stresses interpolated for any values of strain in the range provided.

The cubic spline interpolation generated together with the given data points is presented in Figure 5. Notice that the spline generated passes through all the given data points as is expected of an interpolating polynomial. Mathcad cubic spline for given stress-strain data 5. These quantities are connected by the following relationship. The data given below provides notch sensitivity q versus notch radius r information for an aluminum alloy and can be used to generate an interpolation from which notch sensitivity values at points other than the given data points can be readily obt ained.

This is done in in Figure 5. Interpolation of notch sensitivity data 5. The preferred speech interference level PSIL was established in an effort to study this effect under the constraint that speech sounds would not be allowed to be reflected back to the listener [ 2, 10 ]. For effective communication at a given voice level , the maximum distance that there can be between the speaker and the listener is a function of the preferred speech interference level existing at the location.

The following data which is provided for communication at the level of a normal male voice can be utilized in an interpolation scheme to determine the maximum PSIL permitted for a given distance between the speaker and the listener.

Interpolation of speech communication data 5. Load-Deflection Data for Elastomeric Mounts Elastomeric mounts are employed when small electrical and mechanical devices have to be isolated from high forcing frequencies [ 9, 10 ]. They are especially useful in the protection of delicate electronic instruments. Mount performance characteristics in the form of load versus static deflection data are provided by manufacturers of vibration control products to enable a designer to select an appropriate isolator for a given application.

The following data is available for an elastomeric mount. Deflection cm : 0 0 0. Check your results against functions obtained with Mathcad splines. Present these comparisons as Mathcad plots with proper labels and titles. Determine the interpolated value of f 0. Calculate f 0. Using the method of undetermined coefficients, derive an interpolating polynomial f x for the data given below.

Compare these results with the true values. Given the following data set : x 0. Estimate f 0. Compare the accuracy of the estimated values if the true values are 0. The measued data is given below. Chapter 5: Numerical Interpolation 5. The amplitude of vibration of an automobile in the vertical direction, after passing over a road bump is found to be as follows.

Time, t seconds 0 0. The Charpy test [ 18 ] provides material toughness data under dynamic conditions. In this test, which is helpful in comparing several materials and in determining low-temperature brittleness and impact strength, the specimen is struck by a pendulum released from a fixed height and the energy absorbed by the specimen, termed the impact value , computed from the height of the swing after fracture.

The following table gives the impact value, V, as a function of temperature, T, for a certain material. F and - 75 deg. F T deg F 0 V ft-lbs 0 1. For the given data, generate both a finite-difference table and an interpolating polynomial. The true value is given to be 0. The following data gives the notch sensitivity , q , of a steel as a functionof the notch radius , r [ 18 ]. Compute a finite-difference table and derive an interpolating polynomial by Newton's method.

The measured data is as follows. When a vibration problem is solved with nonlinearities included, the natural frequencies of vibration become dependent on the amplitudes of vibration [ 9,16,20,21 ]. The behavior of a mooring line employed to control the excursions of a floating ocean structure resembles that of a nonlinear spring with tension-displacement characteristics which depend upon its length, weight, elastic properties, anchor holding capacities and water depth [ 22 ].

In the table below, the horizontal component of mooring line tension , H, is given as a function of the horizontal distance, X, between the ends of the line. X ft The following table gives the pressure P versus temperature T relationship in the liquid-vapor region for water [ 17 ] T deg Kelvin While the value of the endurance limit, S end , of a material is based on its tensile strength, S ult , it is also dependent on the condition of its surface[19 ].

The following data relates to a machined , unnotched specimen subjected to reversed bending. S ult kpsi : 60 80 S end kpsi : 22 30 38 50 60 64 Develop linear and cubic splines using linterp and interp for the above data. Also , graph the functions generated with proper labels and a title. It involves the determination of a function f x that would " best fit " a bunch of experimentally measured values. This function can be a linear function, a polynomial, a nonlinear function, an exponential function , or a linear combination of known functions.

Some examples are shown in Figure 6. In curve- fitting a bunch of data points, typically the number of data points would be much larger than the number of undetermined coefficients in a given problem. Thus, there will be discrepancies between the function f x determined and the data points given and it is very rare for a curve-fit to go through all the given data points exactly. However, these differences are minimized by an adaptation of the Method of Least Squares.

Figure 6. One way to include this weighting is to make multiple inclusions of the associated data point in the regression analysis. For example, if the following data is given and the point 2, 20 is to be assigned a "weighting factor" of 3, this data point must simply be considered thrice in coming up with a curve-fit as shown below. Suppose you want to measure the distance between two points in a field, and let us say that you do this " n " times.

You will come up with " n " measurements which are likely to be somewhat different from one another. Let these be d1 , d2 , d3 , If the given data is x1 ,y1 , x2 , y Example 6. Curve-fit with a linear function 6. Curve-fit with quadratic function 6. Curve-fit with a power function 6.

The function, in this case, has the form. Curve -fit with exponential function A x e Bx 6. Given a set of data points: x1 , y1 , x2 ,y The total number of data points given is "n" while the total number of prescribed functions to be utilized is "m". C m must be zero and the second partials must be positive. These requirements lead to the following matrix equation in the unknowns, C 1 , C Fit the data points: x: 1.

Equation 6. These are done by including a period in the variable name. Whatever follows the period then becomes the subscript. Curve-fit with a linear combination of known functions 6. The function slope vx,vy returns a scalar, which is the slope of the least-squares fitted straight line. The function intercept vx,vy returns a scalar which is the y- axis intercept of this line.

Alternatively, the function line vx,vy can be used, which returns a vector containing the y-intercept as well as the slope of the regression line. Use of these functions is demonstrated below. Determine a linear function that would fit the data of Example 6.

The " regress " function fits a single polynomial of any desired order to fit all given data points. This function will not work very well if your data does not fit into a single polynomial. Unlike "regress", the " loess " function generates different second-order polynomials for different regions of the curve.

It does this by examining data in small regions. The argument "span" controls the size of this region. While smaller values of "span " will make the fitted curve track the data fluctuations in a more precise manner, a larger value of " span " , in general , will generate a smoother fit. The " interp" function is the same as the one we resorted to when doing interpolation.

It returns the interpolated y- values corresponding to the x values. In the examples that follow, the vector " vs1" relates to "regress" while " vs2" is related to the use of the " loess " function. The formats of these functions are given below. Here, vs is the vector that is generated through the use of loess or regress and vx and vy are as defined above Example 6. The given data is. The function linfit vx,vy,F returns a vector that contains the coefficients C 1 , C Here, "F" is a vector representing the functions that are linearly combined to generate the best fit to the given data and vx and vy are vectors representing the given data.

Use of linfit requires that there be at least as many data points as there are terms in F. The use of " linfit" is illustrated in the following steps, as it is applied to the solution of Example 6. Notice that here both F x and S are vectors. Comparison of linfit results with given data 6. Use of this function requires that there be at least as many data points as parameters. Here, vg is an n -element vector of guess values for the parameters u1 , u While it is often faster and less sensitive to poor guess values, this process may fail to converge in situations where the derivative vectors are done incorrectly.

The method also permits a solution that employs numerical approximations for the parameter derivatives. To change methods, right-click on the genfit function and select the desired method from the menu. The use of this Mathcad function is best illustrated by an example as shown below. Curve-fitting with genfit 6.

The use of this function requires that there be at least three data points The vector vg is a three-element vector of real guess values for the parameters a, b and c. If a logarithmic fit is desired that is different from the above form, then use genfit or linfit. The use of this function requires that there be at least two data points. The use of this function requires that there be at least three data points and the vx values must all be greater than or equal to zero.

Negative x-values are not appropriate in this setting because raising them to an arbitrary power can produce complex results that will not correspond to the real y-values. If you need to fit a power function to data in the left-half plane, shift it so that all x-values are positive, then adjust the fitted function accordingly to obtain correct results.

The vector vg is a three-element vector of real guess values for the parameters a, b and c. If a logarithmic fit is desired that is different from the above form, then use genfit. For the data points represented by the vectors vx and vy as given below, and with the use of the vector vg of guess values shown for the coefficients a, b and c..

If a fit has be done on data that has negative x-values, then the data must be shifted to the positive axis. Otherwise, a curve-fit with erroneous values may be generated. The vector vg is a three-element vector of real guess values for the parameters A, b and C, but it is optional, and does not have to be used. However, if it is not used, then expfit generates a guess that fits a line to the logs of vy.

If an exponential fit is desired that is different from the above form, then use genfit. For the data points represented by the vectors vx and vy as given below, and with the use of the vector vg of guess values shown for the coefficients a, b and c ,. Chapter 6: Curve-Fitting The function generated by analysis in Example 6. As was done in Section 6. Compare this with the given data and with the analytical results of Example 6. Compare this with the results of Example 6.

Compare these results with the fit generated in Example 6. Notice that both F x and S are vectors here. The methods discussed in the earlier sections of this chapter will now be used to generate appropriate curve-fits to actual data from several practical applications. It is essentially a relationship between the completely reversed applied stress, S, and fatigue life, L , which is measured in terms of the number of stress reversals to failure.

The following data pertains to a steel with an endurance limit of 40 kpsi and an ultimate strength of 90 kpsi. It is required to curve-fit the data with a suitable function. Temperature Response of an Object Placed in a Hot Stream of Air The following data pertains to the temperatures of a solid steel sphere suspended in a hot stream of air measured at several instants of time [ 17 ].

Given temperature versus time data A glance at the plot of Figure 6. This can be done as shown in the following steps. C vT i 40 f t Given data points g t 20 Analysis fit linfit results 0 0 vt i , t time , seconds Figure 6. The Effect of Operating Temperature on the Strength of a Mechanical Element When a mechanical element is subjected to reversed stresses at temperatures below room temperature, there is a strong possibility of the occurrence of brittle fracture.

On the other hand, when the operating temperatures are higher than room temperature, the yield strength of the material drops off very rapidly with increase in temperature, and yielding can take place.

The temperature-corrected value of tensile strength, then, to be used in design calculations is obtained by multiplying the tensile strength at room temperature by a factor K d which is a function of the temperature of the operating environment. The following data was collected from tests done on carbon and alloy steels and shows the effect of operating temperature on tensile strength [ 18 ].

An appropriate curve-fit to the given data is sought. Kd Temp. F Figure 6. Given Kd versus temperature data We will fit the given data with a second-order polynomial of the following form. Effect of operating temperature on the strength of a mechanical element 6. Drop-Testing of Packaged Articles An interesting parameter that often warrants investigation is the height from which a package can be dropped before it suffers any accountable damage, which can be assessed from the maximum acceleration imparted to the package at the end of the drop [ 20 ].

This is a question that arises when packages must be properly cushioned before shipment to another location. The maximum acceleration to gravity ratio, a , has been found to be a function of the ratio, h , of twice the distance dropped to the static deflection of the package. The following a versus h data is provided , using which a reasonable curve-fit is to be generated. Acceleration ratio versus height ratio curve-fit PROBLEMS In all problems, also obtain results with the Mathcad functions, slope, intercept, regress, loess, interp, expfit, and pwrfit as appropriate and compare the results of your regression analysis with the given data and results generated with Mathcad.

Show this comparison by means of Mathcad plots with proper labels, titles and traces. Determine y 5. Determine y 3. Obtain an answer that is good to two decimal places. For the data set given below, determine a power function that will serve as a least squares fit. Determine f 45 and f 60 6.

The temperature of a a hot surface varies sinusoidally with time. Chapter 6: Curve-Fitting 6. Compute y 3. The following data pertains to the temperatures of an object suspended in a hot stream of air measured at several instants of time t, Time seconds 0.

Given the 4 data points x: 0 2. Apply a weighting factor of 2 to the last two data points. Data defining the stress S versus strain e curve for an aluminum alloy is given below. The following data relating the distance traveled by an object to time was obtained from a test in an experimental test track. The distance traveled, S, is in meters and the time , t, is in seconds. Time, t secs 0. The pressure drag , D P, on an object can be reduced in comparison with the total drag , DT, by streamlining it , that is, making its length, L, in the direction of flow larger with respect to its maximum thickness or diameter, D.

Obtain an appropriate curve-fit to the following data which is provided to demonstrate the effect of streamlining on the pressure drag of a body of symmetrical airfoil cross section. The Brinell hardness number, Bhn , for steel [ 18 ] is given as a function of the tempering temperature, T, in the following table T, deg F.

However, when the strain rate, SR, is increased, as it happens, under conditions of impact loading, the strength of the material also increases. The endurance limit of a steel specimen is, typically, a function of the condition of its surface [ 19 ]. For an unnotched, ground specimen in reversed bending, the endurance limit, S end , is related to its tensile strength, S ult , as given in the following table of data.

Expected voice levels required for speech communication with various back- ground noise levels and separations between the speaker and the listener [ 10 ] are given in the following table of data. Obtain appropriate curve-fits to the given data in the form of separate curves for a Peak shouting b Shouting c Very loud voice d Raised voice and e Normal voice. Determine the voice levels required in the following situations.

We will discuss two methods of computing derivatives as follows 1. Method of finite differences 2. Interpolating polynomial method. These methods will be discussed in detail in the paragraphs following. To compute the derivative of a function using the Mathcad derivative operator, first define the point x at which the derivative is to be computed. Then click on the derivative operator in the calculus palette of Figure 7. Press the equals sign to see the derivative.

Finite differences If a two-step difference is employed, the method of computing derivatives is called the two-step method. Then, the polynomial generated can be differentiated as is done in traditional calculus.

Since 5 data points are given, we can generate a 4 th order polynomial. These are presented in Figures 7. Determination of Velocities and Accelerations from Given Displacement Data The following position versus time data for a train has been experimentally obtained.

Since six data points are given, a fifth order polynomial of the following form will be generated. They are presented in Figure 7. Determination of Shock Absorber Parameters , and Damper and Spring Restoring Forces from Given Vehicle Displacement Data The following displacement x versus time t data is provided for a kg motorcycle with a shock absorber.

Determine a the stiffness and the damping constant of the shock absorber and b the damper and spring restoring forces of the shock absorber as functions of time. The given data is plotted in Figure 7. Since six data points are prescribed , a fifth order polynomial of the following form can be generated. As follows, Mathcad's cspline function can also be used with interp to generate a cubic spline interpolation which is compared with the undetermined coefficients method and with the given data.

Displacement, velocity and acceleration profiles The system settling time is the time taken for the free vibration of the motorcycle to completely die down. This , in this case, is 2. Compare the results with the true value. Obtain the Mathcad interpolated function and its derivative in the given range and graph them with proper labels and titles. Given the following x 1.

Obtain the Mathcad interpolated function and its first and second derivatives in the given range and graph them with proper labels and titles. Graph the polynomials obtained in the given range. Show a comparison of the first derivative obtained by the undetermined coefficient method with that generated using Mathcad. Fit an interpolating polynomial to the data of Problem 7.

Show a comparison of the first derivative and second derivative obtained by the undetermined coefficient method with those generated using Mathcad. The horizontal and vertical positions of a weather balloon with respect to a station are furnished below at various instants of time.

Show a comparison of all results generated with the help of appropriate plots. The deflection curve of a cantilevered beam is given by the following data, in which x is the distance in feet from the fixed end and y is the deflection in feet at x. Also plot the deflection versus x and slope versus x curves 7. Compare these results with those of Part a. The deflection curve of a cantilevered beam is given by the following data, in which x is the distance in feet from the fixed end and y is the deflection at x.

The displacements of an instrument subjected to a random vibration test are given below. Time, t seconds : 0. Integral of a function This analytical solution is possible only when f x is explicitly known. When the evaluation of this integral becomes either impossible or extremely difficult , it is advantageous to resort to numerical techniques. In many science and engineering problems, data is often presented as a given bunch of numbers for given values of the independent variable.

In this case too, the evaluation of the integral, is done by a numerical process. There are several methods available for evaluating an integral numerically. However, only the following will be discussed here. The interpolating polynomial method. Trapezoidal rule. Simpson's rules. Romberg integration. To compute the definite integral of a function usiong the Mathcad integral operator, click on the integral operator in the calculus palette shown in Figure 8.

Then, start filling in the placeholders as required and press the equals sign to see the result. Mathcad resorts to Romberg integration to approximate the integral of an expression over an interval of real numbers. Figure 8. Example 8. Given the data of Example 7. The interpolating polynomial generated in Example 7. The trapezoidal rule The trapezoidal rule estimates the integral of a function on the basis of linear interpolation between the given data points.

Trapezoidal rule with five data points If the intervals are not of equal width, the following equation should be used in place of Equation 8. Compare the result with the true value of the integral. They are the Simpson's one-third rule and the Simpson's three-eighth rule. The first is based on the area under a parabola passing through three equally spaced points, while the second is based on the area under a third-order interpolating polynomial passing through 4 points. If these points are x1 , y1 , x2 , y2 , x3 , y3 and x4 , y4 as shown in Figure 8.

For the data of Example 8. It uses a combination of the trapezoidal rule with an extrapolation technique known as Richardson extrapolation, and provides much better accuracy than the trapezoidal rule. Divide the given range into two segments as shown in Figure 8. Java code for multiplying 2 by 2 matrix, short trivia in linear equation, Integer Worksheet, math solver online, math woksheets. Free online calculator TI84, writing decimals as fractions in simplest form worksheets, laplace in ti Eighth grade calculator online, algebrator demo, 3 equations 3 unknowns complex, "math worksheets".

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