Ch 01 introduction

8/25/2013 System Dynamics 1.01 Introduction System Dynamics 1.02 Introduction §1.Introduction to System Dynamics - System: a combination of elements intended to act together to accomplish an objective - Input and Output: an input is a cause; an output is an effect due to the input Introduction - Input-output relation: a description of how the output is effected by the input HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics 1.03 Nguyen Tan Tien Introduction HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics Nguyen Tan Tien 1.04 Introduction §1.Introduction to System Dynamics - Static element: element’s output value depends only on its input value Ex: the current flowing through a resistor depends only on the present value of the applied voltage - Dynamic element: element’s output value depends on past its input value Ex: the present position of a bike depends on what its velocity has been from the start - A static system: one whose output at any given time depends only on the input at that time - A dynamic system: one whose present output depends on past inputs - System dynamics: the study of systems that contain dynamic elements §1.Introduction to System Dynamics - Modeling: simplifying the problem sufficiently and applying the appropriate fundamental principles The resulting mathematical description is called a mathematical model, or just a model - Steps in engineering problem solving HCM City Univ of Technology, Faculty of Mechanical Engineering HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics 1.05 Nguyen Tan Tien Introduction §1.Introduction to System Dynamics If you use a program to solve the problem, hand check the results using a simple version of the problem Checking the dimensions and units, and printing the results of intermediate steps in the calculation sequence can uncover mistakes 10 Perform a “reality check” on your answer Does it make sense? Understand the purpose of the problem Collect the known information Realize that some of it might turn out to be not needed Determine what information you must find Simplify the problem only enough to obtain the required information State any assumptions you make Draw a sketch and label any necessary variables Determine what fundamental principles are applicable Think generally about your proposed solution approach and consider other approaches before proceeding with the details Label each step in the solution process System Dynamics 1.06 Nguyen Tan Tien Introduction §1.Introduction to System Dynamics - Control system: dynamic systems require a control system to perform properly - Theme applications Estimate the range of the expected result and compare it with your answer Do not state the answer with greater precision than is justified by any of the following a.The precision of the given information b.The simplifying assumptions c.The requirements of the problem Interpret the mathematics If the mathematics produces multiple answers, not discard some of them without considering what they mean The mathematics might be trying to tell something, and you might miss an opportunity to discover more about the problem HCM City Univ of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien A robot arm HCM City Univ of Technology, Faculty of Mechanical Engineering Mechanical drive for a robot Nguyen Tan Tien 8/25/2013 System Dynamics 1.07 Introduction §1.Introduction to System Dynamics System Dynamics 1.08 Introduction §1.Introduction to System Dynamics - Computer methods: steps for developing a computer solution State the problem concisely Specify the data to be used by the program This is the “input” Specify the information to be generated by the program This is the “output” Work through the solution steps by hand or with a calculator; use a simpler set of data if necessary Write and run the program Check the output of the program with your hand solution Run the program with your input data and perform a reality check on the output Mechanical drive for a conveyor system HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics 1.09 A vehicle suspension system Nguyen Tan Tien Introduction §2.Units - Every quantity is measured in terms of some arbitrary, but internationally accepted units, called fundamental units - Some units are expressed in terms of other units, which are derived from fundamental units, are known as derived units e.g the unit of area, velocity, acceleration, pressure, etc - There are only four systems of units + Centimeter-Gram-Second system of units: CGS + Foot-Pound-Second system of units: FPS + Met-Kilogram-Second system of units: MKS + International Systems of units: SI If you will use the program as a general tool in the future, test it by running it for a range of reasonable data values, and perform a reality check on the results Document the program with comment statements, flow charts, pseudo-code, or whatever else is appropriate HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics System Dynamics 1.11 Nguyen Tan Tien Introduction Introduction §2.Units - SI and FPS units Quantity SI unit Time second (𝑠) Length meter (𝑚) Force newton (𝑁) Mass kilogram (𝑘𝑔) Energy joule (𝐽) Power Temperature HCM City Univ of Technology, Faculty of Mechanical Engineering 1.10 Nguyen Tan Tien FPS unit second 𝑠𝑒𝑐 foot (𝑓𝑡) pound (𝑙𝑏) slug (𝑠𝑙𝑢𝑔) foot-pound (𝑓𝑡˗𝑙𝑏) Btu (= 778𝑓𝑡˗𝑙𝑏) watt (𝑊) 𝑓𝑡˗𝑙𝑏/𝑠𝑒𝑐 horsepower (ℎ𝑝) degree Celsius ( 0𝐶) degrees Fahrenheit (0𝐹) degrees Kelvin (𝐾) degrees Rankine ( 0𝑅) HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics 1.12 Nguyen Tan Tien Introduction §2.Units - Unit conversion factors Length 1𝑚 = 3.281𝑓𝑡 1𝑓𝑡 = 0.3048𝑚 1𝑚𝑖𝑙𝑒 = 5280𝑓𝑡 1𝑘𝑚 = 1000𝑚 Speed 1𝑓𝑡/𝑠𝑒𝑐 = 0.6818𝑚𝑖/ℎ𝑟 1𝑚𝑖/ℎ𝑟= 1.467𝑓𝑡/𝑠𝑒𝑐 1𝑚/𝑠 = 3.6𝑘𝑚/ℎ 1𝑘𝑚/ℎ = 0.2778𝑚/𝑠 1𝑘𝑚/ℎ𝑟= 0.6214𝑚𝑖/ℎ𝑟 1𝑚𝑖/ℎ𝑟= 1.609𝑘𝑚/ℎ Force 1𝑁 = 0.2248𝑙𝑏 1𝑙𝑏 = 4.4484𝑁 Mass 1𝑘𝑔 = 0.06852𝑠𝑙𝑢𝑔 1𝑠𝑙𝑢𝑔 = 14.594𝑘𝑔 Energy 1𝐽 = 0.7376𝑓𝑡˗𝑙𝑏 1𝑓𝑡˗𝑙𝑏 = 1.3557𝐽 Power 1ℎ𝑝 = 550𝑓𝑡˗𝑙𝑏/𝑠𝑒𝑐 1ℎ𝑝 = 745.7𝑊 1𝑊 = 1.341 × 10−3ℎ𝑝 Temperature 𝑇 0𝐶 = 5(𝑇 0𝐹 − 32)/9 𝑇 0𝐹 = 9𝑇 0𝐶/5 + 32 §3.Developing Linear Model - A linear model of a static system element has the form 𝑦 = 𝑚𝑥 + 𝑏 𝑥: the input 𝑦: the output of the element - Developing linear model from data If we are given data on the input-output characteristics of a system element, we can first plot the data to see whether a linear model is appropriate, and if so, we can extract a suitable model HCM City Univ of Technology, Faculty of Mechanical Engineering HCM City Univ of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien Nguyen Tan Tien 8/25/2013 System Dynamics 1.13 Introduction §3.Developing Linear Model - Ex.1.3.1 A Cantilever Beam Deflection Model The deflection of a cantilever beam 𝑥 is the distance its end moves in response to a force 𝑓 applied at the end A linear relation exists between 𝑓 and 𝑥 ? HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics Nguyen Tan Tien 1.15 Introduction System Dynamics 1.14 Introduction §3.Developing Linear Model Solution The data lies close to a straight line ⟹ we can use the linear function 𝑥 = 𝑎𝑓 to describe the relation 0.78 − 𝑎= = 9.75 × 10−4 𝑖𝑛/𝑙𝑏 800 − + Interpolation (nội suy) / Extrapolation (ngoại suy) HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics 1.16 Nguyen Tan Tien Introduction §3.Developing Linear Model - Linearization: obtain a linear model (an accurate approximation) over a limited range of the independent variable - Ex.1.3.2 Linearization of the Sine Function Obtain linear approximation of 𝑓 = 𝑠𝑖𝑛𝜃 at 0; 𝜋/3 and 2𝜋/3 Solution Replace the plot of the nonlinear function with a straight line that passes through the reference point and has the same slope as the nonlinear function at that point Note that the slope of the sine function is its derivative, 𝑓 ′ = 𝑑𝑠𝑖𝑛𝜃/𝑑𝜃 = cos𝜃 §3.Developing Linear Model 𝜃 = 00 𝑓 = 𝑠𝑖𝑛0~0; 𝑓 ′ = 𝑐𝑜𝑠0 = ⟹ 𝑓 = 𝜃 − + = 𝜃 𝜋 𝜃= 𝜋 𝜋 𝜋 𝑓 = 𝑠𝑖𝑛 ~0.866; 𝑓 ′ = 𝑐𝑜𝑠 = 0.5 ⟹ 𝑓 = 0.5 𝜃 − + 0.866 3 2𝜋 𝜃= 2𝜋 𝑓 = 𝑠𝑖𝑛 ~0.866 2𝜋 𝑓 ′ = 𝑐𝑜𝑠 = −0.5 2𝜋 ⟹ 𝑓 = −0.5 𝜃 − + 0.866 HCM City Univ of Technology, Faculty of Mechanical Engineering HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics Nguyen Tan Tien 1.17 Introduction §3.Developing Linear Model - The linear approximation can also be developed with an analytical approach based on the Taylor series - A function 𝑓(𝜃) in the vicinity of 𝜃 = 𝜃𝑟 𝑑𝑓 𝑑2𝑓 𝑓 𝜃 = 𝑓 𝜃𝑟 + 𝜃 − 𝜃𝑟 + 𝜃 − 𝜃𝑟 𝑑𝜃 𝜃=𝜃 𝑑𝜃 𝜃=𝜃 𝑟 𝑟 +⋯+ 𝑑𝑘 𝑓 𝑘! 𝑑𝜃 𝑘 𝜃 − 𝜃𝑟 𝑘 𝜃=𝜃𝑟 - If 𝜃 is close enough to 𝜃𝑟 𝑑𝑓 𝑓 𝜃 ≈ 𝑓 𝜃𝑟 + 𝜃 − 𝜃𝑟 𝑑𝜃 𝜃 System Dynamics 1.18 Nguyen Tan Tien Introduction §3.Developing Linear Model - If 𝜃 is close enough to 𝜃𝑟 𝑑𝑓 𝑓 𝜃 ≈ 𝑓 𝜃𝑟 + 𝜃 − 𝜃𝑟 𝑑𝜃 𝜃=𝜃 𝑟 Let 𝑚≡ 𝑑𝑓 𝑑𝜃 𝜃=𝜃𝑟 𝑦 ≡ 𝑓 𝜃 − 𝑓 𝜃𝑟 𝑥 ≡ 𝜃 − 𝜃𝑟 ⟹ 𝑦 = 𝑚𝑥 (linear form) 𝑟 HCM City Univ of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien HCM City Univ of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien 8/25/2013 System Dynamics 1.19 Introduction §3.Developing Linear Model - Ex.1.3.3 Linearization of a Square-Root Model The models of many fluid systems involve the square-root function ℎ, which is nonlinear Obtain a linear approximation of 𝑓(ℎ) = ℎ valid near ℎ = Solution The truncated Taylor series for the function 𝑓(ℎ) = ℎ 𝑑 ℎ 𝑓 ℎ ≈ 𝑓 ℎ𝑟 + ℎ − ℎ𝑟 𝑑𝜃 ℎ 𝑟 with ℎ𝑟 = 𝑓 ℎ ≈ + ℎ−0.5 ℎ − 9 = + (ℎ − 9) HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics 1.21 Nguyen Tan Tien Introduction §3.Developing Linear Model Solution a Taylor series approximation near 𝑣 = 600𝑓𝑡/𝑠𝑒𝑐 (line 𝐵) 𝑑𝐷 𝐷≈𝐷 + 𝑣 − 600 = 201.6 + 0.672(𝑣 − 600) 𝑑𝑣 𝑣=600 𝑣=600 b The linear model that gives a conservative estimate of the drag is the straight-line model that passes through the origin and the point at 𝑣 = 1000 This is the equation 𝐷 = 0.56𝑣 shown by the straight line 𝐶 HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics 1.23 Nguyen Tan Tien Introduction §4.Function Identification and Parameter Estimation - Each function gives a straight line when plotted using a specific set of axes + Linear function 𝑦(𝑥) = 𝑚𝑥 + 𝑏 gives a straight line when plotted on rectilinear axes + Power function 𝑦(𝑥) = 𝑏𝑥 𝑚 gives a straight line when plotted on log-log axes + Exponential function 𝑦(𝑥) = 𝑏10𝑚𝑥 or𝑦 = 𝑏𝑒 𝑚𝑥 , give a straight line when plotted on semilog axes with a logarithmic 𝑦 axis System Dynamics 1.20 Introduction §3.Developing Linear Model - Ex.1.3.4 Modeling Fluid Drag The drag force 𝐷 on an object moving through a liquid or a gas is a function of the velocity 𝐷 = 12𝜌𝐴𝐶𝐷 𝑣 𝜌: mass density of the fluid, 𝑠𝑙𝑢𝑔/𝑓𝑡 𝐴: object’s cross-sectional area normal to the relative flow, 𝑓𝑡 𝑣: object’s velocity relative to the fluid, 𝑓𝑡/𝑠𝑒𝑐 𝐶𝐷 : drag coefficient For Aerobee rocket with 1.25𝑓𝑡 diameter, 𝐶𝐷 = 0.4, 𝜌 = 0.0023 , and 𝐷 = 0.00056𝑣 2, obtain linear approximation near a 𝑣 = 600𝑓𝑡/𝑠𝑒𝑐 b ≤ 𝑣 ≤ 1000𝑓𝑡/𝑠𝑒𝑐 HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics 1.22 Nguyen Tan Tien Introduction §4.Function Identification and Parameter Estimation - Function identification: the process of identifying or discovering a function that can describe a particular set of data The term curve fitting is also used to describe the process of finding a curve, and the function generating the curve, to describe a given set of data - Parameter estimation: the process of obtaining values for the parameters/coefficients, in the function that describes the data - Three function types can often describe physical phenomena Linear function: 𝑦(𝑥) = 𝑚𝑥 + 𝑏 Power function: 𝑦(𝑥) = 𝑏𝑥 𝑚 Exponential function: 𝑦(𝑥) = 𝑏10𝑚𝑥 or 𝑦 = 𝑏𝑒 𝑚𝑥 HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics 1.24 Nguyen Tan Tien Introduction §4.Function Identification and Parameter Estimation - Step for function identification 1.Examine the data near the origin 𝑦(𝑥) = 𝑏10𝑚𝑥 or 𝑦 = 𝑏𝑒 𝑚𝑥 can never pass through the origin 𝑦(𝑥) = 𝑚𝑥 + 𝑏 can pass through the origin only if 𝑏 = 𝑦(𝑥) = 𝑏𝑥 𝑚 can pass through the origin but only if 𝑚 > Power function 𝑦 = 2𝑥 −0.5 , Exponential function 𝑦 = 10 × 10−𝑥 HCM City Univ of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien HCM City Univ of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien 8/25/2013 System Dynamics 1.25 Introduction §4.Function Identification and Parameter Estimation 2.Plot the data using rectilinear scales + If data forms a straight line, then it can be represented by the linear function + If we have data at 𝑥 = 0, then a If 𝑦(0) = 0, try the power function b If 𝑦(0) ≠ 0, try the exponential function + If data is not given for 𝑥 = 0, proceed to step 3.If you suspect a power function, plot the data using log-log scales Only a power function will form a straight line If you suspect an exponential function, plot it using semilog scales Only an exponential function will form a straight line HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics 1.27 Nguyen Tan Tien Introduction §4.Function Identification and Parameter Estimation - Ex.1.4.1 Temperature Dynamics of Water Water in a glass measuring cup was allowed to cool after being heated to 2040 𝐹 The ambient air temperature was 700 𝐹 The measured water temperature at various times is given in the following table Obtain a functional description of the water temperature 𝑇 versus time 𝑡 Solution Examine data near the origin Consider the relative temperature ∆𝑇 = 𝑇 − 700 The data does not pass through the origin ⟹ using the linear function and the power function as candidates HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics 1.29 Nguyen Tan Tien Introduction §4.Function Identification and Parameter Estimation Obtain the coefficient 1.26 Introduction §4.Function Identification and Parameter Estimation - Obtain the coefficient: If the data lie very close to a straight line, we can draw the line through the data using a straightedge and then read two points (𝑥1 , 𝑦1 ) and (𝑥2 , 𝑦2 ) Linear function 𝑦(𝑥) = 𝑚𝑥 + 𝑏 𝑚= 𝑦2 −𝑦1 𝑥2 −𝑥1 , 𝑏 = 𝑦1 − 𝑚 𝑥1 Power function 𝑦(𝑥) = 𝑏𝑥 𝑚 𝑚= log(𝑦2 /𝑦1 ) log(𝑥2 /𝑥1 ) , 𝑏 = 𝑦1 𝑥1−𝑚 Exponential function 𝑦(𝑥) = 𝑏10𝑚𝑥 𝑚= 𝑥2 −𝑥1 log 𝑦2 𝑦1 , 𝑏 = 𝑦1 10−𝑚𝑥1 Exponential function 𝑦(𝑥) = 𝑏𝑒 𝑚𝑥 𝑚= 𝑥2 −𝑥1 log 𝑦2 𝑦1 , 𝑏 = 𝑦1 𝑒 −𝑚𝑥1 HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics 1.28 Nguyen Tan Tien Introduction §4.Function Identification and Parameter Estimation Plot the data Plot of temp vs time Semilog plot of temp vs time Plot the data ∆𝑇(𝑡) on a rectilinear and semilog axes ⟹ The data lie close to a straight line, so we can use the exponential function to describe the relative temperature ∆𝑇(𝑡) = 𝑏𝑒 𝑚𝑡 HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics 1.30 Nguyen Tan Tien Introduction §4.Function Identification and Parameter Estimation Obtain the coefficient Plot of temp vs time Semilog plot of temp vs time Choose two point 515,90 and (1090,60) 60 𝑚= log = −0.0007, 𝑏 = 90𝑒 −0.0007×515 = 129 1090 − 515 90 HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics Nguyen Tan Tien Plot of temp vs time Comparison of the fitted function with the data The estimated function ∆𝑇 = 129𝑒 −0.0007𝑡 or T = ∆𝑇 + 70 = 129𝑒 −0.0007𝑡 + 70 HCM City Univ of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien 8/25/2013 System Dynamics 1.31 Introduction §4.Function Identification and Parameter Estimation - Ex.1.4.2 Orifice Flow A hole ∅6𝑚𝑚 was made in a translucent milk container A series of marks 1𝑐𝑚 apart was made above the hole While adjusting the tap flow to keep the water height ℎ constant, the time 𝑡 for the outflow to fill a 250𝑚𝑙 cup was measured (1𝑚𝑙 = 10−6 𝑚3 ) This was repeated for several heights The data are given in the following table Obtain a functional description of the volume outflow rate 𝑓 as a function of water height ℎ above the hole Solution Obtain the flow rate data 𝑓 = 250/𝑡 (𝑚𝑙/𝑠) HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics 1.33 Nguyen Tan Tien Introduction §4.Function Identification and Parameter Estimation Obtain the coefficient Plot of flow rate data System Dynamics 1.32 Plot of flow rate data Log-log plot of flow rate data The log-log plot shows that the data lie close to a straight line ⟹ using the power function to describe the flow rate as a function of height 𝑓 = 𝑏ℎ𝑚 HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics 1.34 1.35 Introduction - According to the least-squares criterion, the line that gives the best fit is the one that minimizes 𝐽 (𝑚𝑥𝑖 + 𝑏 − 𝑦𝑖 )2 𝐽= log(30/9.4) 𝑚= = 0.558, 𝑏 = 9.4 1−0.558 = 9.4 log(8/1) ⟹ 𝑓 = 9.4ℎ0.558 System Dynamics Nguyen Tan Tien §5.Fitting Models to Scattered Data - In practice the data often will not lie very close to a straight line ⟹ A systematic and objective way of obtaining a straight line describing the data is the least-squares method - Suppose we want to find the coefficients of the straight line 𝑦 = 𝑚𝑥 + 𝑏 that best fits the following data Comparison of the flow rate function and the data HCM City Univ of Technology, Faculty of Mechanical Engineering Introduction §4.Function Identification and Parameter Estimation Plot of the resulting flow rate data 𝑖=1 Nguyen Tan Tien Introduction HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics 1.36 Nguyen Tan Tien Introduction §5.Fitting Models to Scattered Data - The function 𝐽 𝐽 = (0𝑚 + 𝑏 − 2)2 +(5𝑚 + 𝑏 − 6)2 +(10𝑚 + 𝑏 − 11)2 The value of 𝑚 and 𝑏 that minimizes 𝐽 can be found from 𝜕𝐽 𝜕𝐽 = 0, =0 𝜕𝑚 𝜕𝑏 or 𝜕𝐽 = 5𝑚 + 𝑏 − + 10𝑚 + 𝑏 − 11 10 = 𝜕𝑚 𝜕𝐽 = 𝑏 − + 5𝑚 + 𝑏 − + 2(10𝑚 + 𝑏 − 11) = 𝜕𝑏 250𝑚 + 30𝑏 = 280 ⟹ 30𝑚 + 6𝑏 = 38 11 11 ⟹𝑚= ,𝑏 = : 𝑦= 𝑥+ 10 10 §5.Fitting Models to Scattered Data - If we evaluate this equation at the data values 𝑥 = 0,5, and 10, we obtain the values 𝑦 = 1.8333,6.3333, and 10.8333 These values are different than the given data values 𝑦 = 2,6, and 11 because the line is not a perfect fit to the data - The value of 𝐽 𝐽 = (1.8333 − 2)2 +(6.3333 − 6)2 +(10.8333 − 11)2 = 0.1666 - No other straight line will give a lower value of 𝐽 for these data HCM City Univ of Technology, Faculty of Mechanical Engineering HCM City Univ of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien Nguyen Tan Tien 8/25/2013 System Dynamics 1.37 Introduction §5.Fitting Models to Scattered Data - General linear case: formulas for the coefficients 𝑚 and 𝑏 in the linear equation 𝑦 = 𝑚𝑥 + 𝑏 with 𝑛 data points 𝑛 System Dynamics 1.38 Introduction §5.Fitting Models to Scattered Data - Ex.1.5.1 Fitting Data with the Power Function Find a functional description of the following data (𝑚𝑥𝑖 + 𝑏 − 𝑦𝑖 )2 𝐽= 𝑖=1 Solution Plot the data using rectilinear scales - The value of 𝑚 and 𝑏 that minimizes 𝐽 can be found from 𝑥𝑖2 + 𝑏 (𝑚𝑥𝑖 + 𝑏 − 𝑦𝑖 )𝑥𝑖 = ⟹ 𝑚 𝑖=1 𝑛 𝑛 𝑖=1 𝑛 (𝑚𝑥𝑖 + 𝑏 − 𝑦𝑖 ) = ⟹𝑚 𝑖=1 𝑛 𝑥𝑖 = 𝑖=1 𝑛 80 0.6 10 70 0.6 60 x-y plot 50 𝑥𝑖 + 𝑏𝑛 = 𝑖=1 𝑥𝑖 𝑦𝑖 𝑖=1 𝑦𝑖 40 10 x-logy 0.5 10 logx-logy logy 𝑛 logy 𝜕𝐽 =2 𝜕𝑏 𝑛 y 𝜕𝐽 =2 𝜕𝑚 0.4 10 30 𝑖=1 20 0.3 10 0.5 10 10 1.5 2.5 x 3.5 1.5 2.5 x 3.5 -0.1 10 0.1 10 logx 10 These data lie close to a straight line when plotted on log-log axes ⟹ Power function 𝑦(𝑥) = 𝑏𝑥 𝑚 can describe the data HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics Nguyen Tan Tien 1.39 Introduction §5.Fitting Models to Scattered Data Using the transformations 𝑋 = 𝑙𝑜𝑔𝑥 and 𝑌 = 𝑙𝑜𝑔𝑦 From this we obtain 4 𝑋𝑖 = 1.3803 , 𝑖=1 𝑌𝑖 = 5.5525, 𝑖=1 𝑋𝑖2 = 0.6807 𝑋𝑖 𝑌𝑖 = 2.3208, 𝑖=1 𝑛 𝑛 𝑋𝑖2 + 𝑏 𝑖=1 𝑛 𝑚 𝑖=1 𝑋𝑖 + 𝑏𝑛 = 𝑖=1 ⟹ 𝑌𝑖 ⟹ 1.3803𝑚 + 4𝐵 = 5.5525 𝑚 = 1.9802 ⟹ The desired function 𝑦 = 5.068𝑒1.9802 𝐵 = 0.7048 System Dynamics Nguyen Tan Tien 1.41 §5.Fitting Models to Scattered Data - Ex.1.5.2 Consider the given data Introduction Point Constraint The best-fit line is 𝑦 = (9/10)𝑥 + 11/6 Find the best-fit line that passes through the point 𝑥 = 10, 𝑦 = 11 Solution Apply new variables 𝑋 = 𝑥 − 10, 𝑌 = 𝑦 − 11: 𝑚 3𝑖=1 𝑋𝑖2 = 3𝑖=1 𝑋𝑖 𝑌𝑖 𝑖=1 𝑋𝑖 = (−10)2 +52 + = 125 ⟹𝑚= 115 125 = 23 25 𝑖=1 𝑋𝑖 𝑌𝑖 = −10 −9 + −5 −5 + = 115 23 23 ⟹𝑌= 𝑋 ⟹ 𝑥 − 10 = 𝑦 − 11 25 25 23 ⟹𝑦= 𝑥+ 25 HCM City Univ of Technology, Faculty of Mechanical Engineering Introduction 𝜕𝐽 =0⟹𝑚 𝜕𝑚 𝑛 𝑛 𝑥𝑖2 = 𝑖=1 𝑥𝑖 𝑦𝑖 ⟹ 𝑚 𝑖=1 If the model is required to pass through a point not at the origin, say the point (𝑥0 , 𝑦0 ), subtract 𝑥0 from all the 𝑥 values, subtract 𝑦0 from all the 𝑦 values, and then use the above equation to find the coefficient 𝑚 The resulting equation will be of the form 𝑦 = 𝑚 𝑥 − 𝑥0 + 𝑦0 𝑖=1 HCM City Univ of Technology, Faculty of Mechanical Engineering (𝑚𝑥𝑖 − 𝑦𝑖 )2 ⟹ 𝑖=1 𝑖=1 𝑛 1.40 𝑛 𝑋𝑖 𝑌𝑖 ⟹ 0.6807𝑚 + 1.3803𝐵 = 2.3208 Nguyen Tan Tien §5.Fitting Models to Scattered Data - Constraining models to pass through a given point In general the least-squares method will give a nonzero value for 𝑏 in 𝑦 = 𝑚𝑥 + 𝑏 because of the scatter or measurement error that is usually present in the data To obtain a zero-intercept model of the form 𝑦 = 𝑚𝑥, we must derive the equation for 𝑚 from basic principles 𝐽= 𝑛 𝑋𝑖 = System Dynamics 𝑖=1 Using 𝑋, 𝑌, and 𝐵 = 𝑙𝑜𝑔𝑏 instead of 𝑥, 𝑦, and 𝑏, we obtain 𝑚 HCM City Univ of Technology, Faculty of Mechanical Engineering HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics Nguyen Tan Tien 1.42 Introduction §5.Fitting Models to Scattered Data - Constraining a coefficient The given data can be described by a function with a specified form and specified values of one of more of its coefficients ⟹ modify the least-squares method to find the best-fit function of a specified form - Ex.1.5.3 Fitting a Power Function with a Known Exponent Fit the power function 𝑦 = 𝑏𝑥 𝑚 to the data 𝑦𝑖 with known 𝑚 Solution The least squares criterion 𝐽 = 𝑛𝑖=1(𝑏𝑥 𝑚 − 𝑦𝑖 )2 To obtain the value of 𝑏 that minimizes 𝐽, we solve 𝜕𝐽 𝜕𝑏 =0 This gives 𝑏= Nguyen Tan Tien 𝑛 𝑚 𝑖=1 𝑥𝑖 𝑦𝑖 𝑛 2𝑚 𝑖=1 𝑥𝑖 HCM City Univ of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien 8/25/2013 System Dynamics 1.43 Introduction §5.Fitting Models to Scattered Data - The quality of curve fit • Experiment data: (𝑥𝑖 ,𝑦𝑖 ) • Modeling function: 𝑦=𝑓 𝑥 • Modeling error: 𝑒𝑖 = 𝑓 𝑥𝑖 − 𝑦𝑖 • Residuals function 𝑛 [𝑓 𝑥𝑖 − 𝑦𝑖 ]2 𝐽= 𝑖=1 • Sum of the squares of the deviation (𝑦 : the mean of 𝑦 values) 𝑛 System Dynamics • The coefficient of determination 𝐽 𝑟2 = − 𝑆 For a perfect fix, 𝐽 = and 𝑟 = HCM City Univ of Technology, Faculty of Mechanical Engineering 1.45 (𝑚𝑥 − 𝑎𝑥 𝑛 )2 𝑑𝑥 𝐽= 𝑖=1 System Dynamics Introduction 𝐿 [𝑦𝑖 − 𝑦]2 𝑆= 1.44 §5.Fitting Models to Scattered Data - Integral form of the least squares criterion - Ex.1.5.4: Fitting a Linear Function to a Power Function Fit the linear function 𝑦 = 𝑚𝑥 to the power function 𝑦 = 𝑎𝑥 𝑛 over the range ≤ 𝑥 ≤ 𝐿 The values of a and 𝑛 are given Solution The appropriate least-squares criterion is the integral of the square of the difference between the linear model and the power function over the stated range To obtain the value of 𝑚 that minimizes 𝐽 , solve 𝜕𝐽 =2 𝜕𝑚 Nguyen Tan Tien Introduction §6.Matlab and Least Squares Method - Integral form of the least squares criterion Matlab’s polyfit function is based on the least-squares method 𝑝 = 𝑝𝑜𝑙𝑦𝑓𝑖𝑡(𝑥, 𝑦, 𝑛) 𝑝: the row vector of length 𝑛 + that contains the polynomial coefficients in order of descending powers 𝑛: degree of polynomial 𝑥,𝑦: data vectors, 𝑥 is the independent vector 𝐿 𝜕𝐽 𝜕𝑚 =0 3𝑎 𝑛−1 𝑥(𝑚𝑥 − 𝑎𝑥 𝑛 )2 𝑑𝑥 = ⟹ 𝑚 = 𝐿 𝑛+2 HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics Nguyen Tan Tien 1.46 Introduction §6.Matlab and Least Squares Method - Ex.1.6.1 Fitting First and Second Degree Polynomials Find the first and second degree polynomials that fit the following data in the least-squares sense Evaluate the quality of fit for each polynomial Solution >>x = (0:10); y = [48, 49, 52, 63, 76, 98, 136, 150, 195, 236, 260]; p_first = polyfit(x,y,1) p_second = polyfit(x,y,2) Result p_first = 22.4636 11.5909 p_second = 2.2343 0.1210 45.1049 1.47 Introduction §6.Matlab and Least Squares Method Plot polinomials and evaluate the “quality of fit” quantities 𝐽, 𝑆, 𝑟 >>x = (0:10); y = [48, 49, 52, 63, 76, 98, 136, 150, 195, 236, 260]; mu = mean(y); xp = (0:0.01:10); for k = 1:2 yp(k,:) = polyval(polyfit(x,y,k),xp); J(k) = sum((polyval(polyfit(x,y,k),x)-y).^2); S(k) = sum((polyval(polyfit(x,y,k),x)- mu).^2); r2(k) = 1-J(k)/S(k); end subplot(2,1,1) plot(xp,yp(1,:),x,y,'o'),axis([0 10 300]),xlabel('x'), ylabel('y'),title('First-degree fit') subplot(2,1,2) plot(xp,yp(2,:),x,y,'o'),axis([0 10 300]),xlabel('x'), ylabel('y'),title('Second-degree fit') disp('The J values are:'),J disp('The S values are:'),S disp('The r^2 values are:'),r2 HCM City Univ of Technology, Faculty of Mechanical Engineering HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics Nguyen Tan Tien 1.48 Introduction §6.Matlab and Least Squares Method First-degree fit 300 Result The J values are: J= 1.0e+03 * 4.6793 0.3962 The S values are: S= 1.0e+04 * 5.5508 5.9791 The r^2 values are: r2 = 0.9157 0.9934 200 y System Dynamics Nguyen Tan Tien ⟹ 𝑦 = 2.2343𝑥 + 0.1210𝑥 + 45.1049 100 2 x Second-degree fit 10 10 200 100 𝑟2 300 y HCM City Univ of Technology, Faculty of Mechanical Engineering ⟹ 𝑦 = 22.4636𝑥 + 11.5909 x > 𝑟2 second−degree polynomial first−degree polynomial ⟹ According to the 𝑟 criterion, the second-degree polynomial represents the data better than the first-degree polynomial Nguyen Tan Tien HCM City Univ of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien 8/25/2013 System Dynamics 1.50 Introduction §6.Matlab and Least Squares Method - Ex.1.6.2 A Cantilever Beam Deflection Model The force-deflection data for the cantilever beam is given in the following table Obtain a linear relation between 𝑥 and 𝑓, estimate the stiffness 𝑘 of the beam, and evaluate the quality of the fit Solution HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics Nguyen Tan Tien 1.50 Introduction §6.Matlab and Least Squares Method Result 0.7 0.6 0.5 0.4 Introduction >>x = [0, 0.15, 0.23, 0.35, 0.37, 0.5, 0.57, 0.68, 0.77]; f = (0:100:800); p = polyfit(f,x,1) k = 1/p(1) fp = (0:800); xp = p(1)*fp+p(2); plot(fp,xp,f,x,'o'), xlabel('Applied Force f (lb)'), ylabel('Deflection x (in.)'), axis([0 800 0.8]) J = sum((polyval(p,f)-x).^2) S = sum((polyval(p,f)-mean(x)).^2) r2 = - J/S HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics 1.51 Nguyen Tan Tien Introduction 0.3 0.2 0.1 0 100 200 HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics 1.50 §6.Matlab and Least Squares Method Solution §6.Matlab and Least Squares Method - Ex.1.6.3 Constraining the Curve Fit Use Matlab to fit a straight line to the beam force-deflection data given in Ex.1.6.2, but constrain the line to pass through the origin Solution 0.8 Deflection x (in.) p= 0.0009 0.0356 k= 1.0909e+03 J= 0.0048 S= 0.5042 r2 = 0.9905 System Dynamics 1.52 300 400 500 Applied Force f (lb) 600 700 800 Nguyen Tan Tien Introduction §6.Matlab and Least Squares Method - Ex.1.6.4 Temperature Dynamics of Water Water in a glass measuring cup was allowed to cool after being heated to 2040 𝐹 The ambient air temperature was 700 𝐹 The measured water temperature at various times is given in the following table Obtain a functional description of the water temperature vs time Solution HCM City Univ of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics 1.53 Nguyen Tan Tien Introduction §6.Matlab and Least Squares Method - Ex.1.6.5 Orifice Flow A hole 6𝑚𝑚 in diameter was made in a translucent milk container While adjusting the tap flow to keep the water height constant, the time for the outflow to fill a 250𝑚𝑙 cup was measured This was repeated for several heights The data are given in the following table Obtain a functional description of the volume outflow rate 𝑓 as a function of water height ℎ above the hole Solution HCM City Univ of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien 8/25/2013 System Dynamics 1.54 Introduction §6.Matlab and Least Squares Method - Ex.1.6.6 Orifice Flow with Constrained Exponent Consider the data of Example 1.6.5 Determine the best-fit value of the coefficient 𝑏 in the square-root function 𝑓 = 𝑏ℎ1/2 Solution HCM City Univ of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien 10 ... Univ of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien HCM City Univ of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien 8/25/ 2013 System Dynamics 1.25 Introduction. .. Univ of Technology, Faculty of Mechanical Engineering HCM City Univ of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien Nguyen Tan Tien 8/25/ 2013 System Dynamics 1.37 Introduction. .. 0.866 HCM City Univ of Technology, Faculty of Mechanical Engineering HCM City Univ of Technology, Faculty of Mechanical Engineering System Dynamics Nguyen Tan Tien 1.17 Introduction §3.Developing