applications of differential equations in civil engineering problems

applications of differential equations in civil engineering problems2020/09/28

where \(\alpha\) and \(\beta\) are positive constants. W = mg 2 = m(32) m = 1 16. Such a circuit is called an RLC series circuit. Consider a mass suspended from a spring attached to a rigid support. where \(P_0=P(0)>0\). In some situations, we may prefer to write the solution in the form. 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RLC circuit, Force equation idea versus mathematical idea, status page at https://status.libretexts.org, \(v_{i+1} = v_i + (g - \frac{c}{m}(v_i)^2)(t_{i+1}-t_i)\), \(-Ri(t)-L\frac{di(t)}{dt}-\frac{1}{C}\int_{-\infty}^t i(t')dt'+V(t)=0\), \(RC\frac{dv_c(t)}{dt}+LC\frac{d^2v_c(t)}{dt}+v_c(t)=V(t)\). below equilibrium. Let us take an simple first-order differential equation as an example. The relationship between the halflife (denoted T 1/2) and the rate constant k can easily be found. Watch this video for his account. We solve this problem in two parts, the natural response part and then the force response part. \end{align*} \nonumber \]. Problems concerning known physical laws often involve differential equations. This suspension system can be modeled as a damped spring-mass system. After learning to solve linear first order equations, youll be able to show (Exercise 4.2.17) that, \[T = \frac { a T _ { 0 } + a _ { m } T _ { m 0 } } { a + a _ { m } } + \frac { a _ { m } \left( T _ { 0 } - T _ { m 0 } \right) } { a + a _ { m } } e ^ { - k \left( 1 + a / a _ { m } \right) t }\nonumber \], Glucose is absorbed by the body at a rate proportional to the amount of glucose present in the blood stream. Find the equation of motion if it is released from rest at a point 40 cm below equilibrium. Let's rewrite this in order to integrate. Since the motorcycle was in the air prior to contacting the ground, the wheel was hanging freely and the spring was uncompressed. This can be converted to a differential equation as show in the table below. : Harmonic Motion Bonds between atoms or molecules written as y0 = 2y x. Therefore, the capacitor eventually approaches a steady-state charge of 10 C. Find the charge on the capacitor in an RLC series circuit where \(L=1/5\) H, \(R=2/5,\) \(C=1/2\) F, and \(E(t)=50\) V. Assume the initial charge on the capacitor is 0 C and the initial current is 4 A. hZ }y~HI@ p/Z8)wE PY{4u'C#J758SM%M!)P :%ej*uj-) (7Hh\(Uh28~(4 The steady-state solution is \(\dfrac{1}{4} \cos (4t).\). Graphs of this function are similar to those in Figure 1.1.1. The motion of the mass is called simple harmonic motion. A 200-g mass stretches a spring 5 cm. In many applications, there are three kinds of forces that may act on the object: In this case, Newtons second law implies that, \[y'' = q(y,y')y' p(y) + f(t), \nonumber\], \[y'' + q(y,y')y' + p(y) = f(t). Let \(P=P(t)\) and \(Q=Q(t)\) be the populations of two species at time \(t\), and assume that each population would grow exponentially if the other did not exist; that is, in the absence of competition we would have, \[\label{eq:1.1.10} P'=aP \quad \text{and} \quad Q'=bQ,\], where \(a\) and \(b\) are positive constants. Gravity is pulling the mass downward and the restoring force of the spring is pulling the mass upward. 4. The general solution of non-homogeneous ordinary differential equation (ODE) or partial differential equation (PDE) equals to the sum of the fundamental solution of the corresponding homogenous equation (i.e. P,| a0Bx3|)r2DF(^x [.Aa-,J$B:PIpFZ.b38 Thus, the study of differential equations is an integral part of applied math . \nonumber \]. `E,R8OiIb52z fRJQia" ESNNHphgl LBvamL 1CLSgR+X~9I7-<=# \N ldQ!`%[x>* Ko e t) PeYlA,X|]R/X,BXIR To save money, engineers have decided to adapt one of the moon landing vehicles for the new mission. Integral equations and integro-differential equations can be converted into differential equations to be solved or alternatively you can use Laplace equations to solve the equations. The course stresses practical ways of solving partial differential equations (PDEs) that arise in environmental engineering. shows typical graphs of \(T\) versus \(t\) for various values of \(T_0\). Such circuits can be modeled by second-order, constant-coefficient differential equations. Therefore, if \(S\) denotes the total population of susceptible people and \(I = I(t)\) denotes the number of infected people at time \(t\), then \(S I\) is the number of people who are susceptible, but not yet infected. Now suppose \(P(0)=P_0>0\) and \(Q(0)=Q_0>0\). \end{align*}\], Therefore, the differential equation that models the behavior of the motorcycle suspension is, \[x(t)=c_1e^{8t}+c_2e^{12t}. The solution of this separable firstorder equation is where x o denotes the amount of substance present at time t = 0. In the case of the motorcycle suspension system, for example, the bumps in the road act as an external force acting on the system. The arrows indicate direction along the curves with increasing \(t\). Therefore the wheel is 4 in. illustrates this. In this paper, the relevance of differential equations in engineering through their applications in various engineering disciplines and various types of differential equations are motivated by engineering applications; theory and techniques for . You learned in calculus that if \(c\) is any constant then, satisfies Equation \ref{1.1.2}, so Equation \ref{1.1.2} has infinitely many solutions. Thus, a positive displacement indicates the mass is below the equilibrium point, whereas a negative displacement indicates the mass is above equilibrium. When \(b^2=4mk\), we say the system is critically damped. They are the subject of this book. \nonumber \], Applying the initial conditions \(q(0)=0\) and \(i(0)=((dq)/(dt))(0)=9,\) we find \(c_1=10\) and \(c_2=7.\) So the charge on the capacitor is, \[q(t)=10e^{3t} \cos (3t)7e^{3t} \sin (3t)+10. The last case we consider is when an external force acts on the system. Displacement is usually given in feet in the English system or meters in the metric system. We willreturn to these problems at the appropriate times, as we learn how to solve the various types of differential equations that occur in the models. We measure the position of the wheel with respect to the motorcycle frame. After only 10 sec, the mass is barely moving. The curves shown there are given parametrically by \(P=P(t), Q=Q(t),\ t>0\). We first need to find the spring constant. If\(f(t)0\), the solution to the differential equation is the sum of a transient solution and a steady-state solution. Thus, \[ x(t) = 2 \cos (3t)+ \sin (3t) =5 \sin (3t+1.107). Consider the forces acting on the mass. However it should be noted that this is contrary to mathematical definitions (natural means something else in mathematics). A homogeneous differential equation of order n is. DIFFERENTIAL EQUATIONS WITH APPLICATIONS TO CIVIL ENGINEERING: THIS DOCUMENT HAS MANY TOPICS TO HELP US UNDERSTAND THE MATHEMATICS IN CIVIL ENGINEERING Separating the variables, we get 2yy0 = x or 2ydy= xdx. Differential Equations of the type: dy dx = ky Since, by definition, x = x 6 . The idea for these terms comes from the idea of a force equation for a spring-mass-damper system. physics and engineering problems Draw on Mathematica's access to physics, chemistry, and biology data Get . Here is a list of few applications. If an equation instead has integrals then it is an integral equation and if an equation has both derivatives and integrals it is known as an integro-differential equation. \(x(t)=\dfrac{1}{2} \cos (4t)+ \dfrac{9}{4} \sin (4t)+ \dfrac{1}{2} e^{2t} \cos (4t)2e^{2t} \sin (4t)\), \(\text{Transient solution:} \dfrac{1}{2}e^{2t} \cos (4t)2e^{2t} \sin (4t)\), \(\text{Steady-state solution:} \dfrac{1}{2} \cos (4t)+ \dfrac{9}{4} \sin (4t) \). However, the model must inevitably lose validity when the prediction exceeds these limits. Under this terminology the solution to the non-homogeneous equation is. If the lander is traveling too fast when it touches down, it could fully compress the spring and bottom out. Bottoming out could damage the landing craft and must be avoided at all costs. They're word problems that require us to create a separable differential equation based on the concentration of a substance in a tank. We are interested in what happens when the motorcycle lands after taking a jump. What is the natural frequency of the system? Letting \(=\sqrt{k/m}\), we can write the equation as, This differential equation has the general solution, \[x(t)=c_1 \cos t+c_2 \sin t, \label{GeneralSol} \]. Let time \(t=0\) denote the instant the lander touches down. Second-order constant-coefficient differential equations can be used to model spring-mass systems. Consider the differential equation \(x+x=0.\) Find the general solution. If the motorcycle hits the ground with a velocity of 10 ft/sec downward, find the equation of motion of the motorcycle after the jump. When the motorcycle is lifted by its frame, the wheel hangs freely and the spring is uncompressed. This system can be modeled using the same differential equation we used before: A motocross motorcycle weighs 204 lb, and we assume a rider weight of 180 lb. A 16-lb weight stretches a spring 3.2 ft. What is the transient solution? Engineers . Differential equations for example: electronic circuit equations, and In "feedback control" for example, in stability and control of aircraft systems Because time variable t is the most common variable that varies from (0 to ), functions with variable t are commonly transformed by Laplace transform Figure \(\PageIndex{6}\) shows what typical critically damped behavior looks like. Start with the graphical conceptual model presented in class. This website contains more information about the collapse of the Tacoma Narrows Bridge. The amplitude? \(x(t)=0.1 \cos (14t)\) (in meters); frequency is \(\dfrac{14}{2}\) Hz. (See Exercise 2.2.28.) According to Hookes law, the restoring force of the spring is proportional to the displacement and acts in the opposite direction from the displacement, so the restoring force is given by \(k(s+x).\) The spring constant is given in pounds per foot in the English system and in newtons per meter in the metric system. ), One model for the spread of epidemics assumes that the number of people infected changes at a rate proportional to the product of the number of people already infected and the number of people who are susceptible, but not yet infected. Last, let \(E(t)\) denote electric potential in volts (V). The difference between the two situations is that the heat lost by the coffee isnt likely to raise the temperature of the room appreciably, but the heat lost by the cooling metal is. Such equations are differential equations. E. Linear Algebra and Differential Equations Most civil engineering programs require courses in linear algebra and differential equations. Find the equation of motion of the lander on the moon. Then the rate of change of the amount of glucose in the bloodstream per unit time is, where the first term on the right is due to the absorption of the glucose by the body and the second term is due to the injection. That note is created by the wineglass vibrating at its natural frequency. International Journal of Mathematics and Mathematical Sciences. The function \(x(t)=c_1 \cos (t)+c_2 \sin (t)\) can be written in the form \(x(t)=A \sin (t+)\), where \(A=\sqrt{c_1^2+c_2^2}\) and \( \tan = \dfrac{c_1}{c_2}\). This may seem counterintuitive, since, in many cases, it is actually the motorcycle frame that moves, but this frame of reference preserves the development of the differential equation that was done earlier. Natural response is called a homogeneous solution or sometimes a complementary solution, however we believe the natural response name gives a more physical connection to the idea. Different chapters of the book deal with the basic differential equations involved in the physical phenomena as well as a complicated system of differential equations described by the mathematical model. After youve studied Section 2.1, youll be able to show that the solution of Equation \ref{1.1.9} that satisfies \(G(0) = G_0\) is, \[G = \frac { r } { \lambda } + \left( G _ { 0 } - \frac { r } { \lambda } \right) e ^ { - \lambda t }\nonumber \], Graphs of this function are similar to those in Figure 1.1.2 Natural response part and then the force response part separable firstorder equation is where x denotes. Be converted to a differential equation as an example mass is barely moving ( x+x=0.\ ) find equation. To physics, chemistry, and biology data Get definitions ( natural means something else in mathematics.! Definition, x = x 6 partial differential equations motorcycle frame constant-coefficient differential equations can used! X+X=0.\ ) find the general solution x o denotes the amount of substance present time. Volts ( V ) system is critically damped last case we consider is when an external force on... 1 16 damped spring-mass system denote electric potential in volts ( V ) last we... ) =Q_0 > 0\ ) exceeds these limits an example a jump relationship between the halflife denoted... This can be modeled as a damped spring-mass system = 2 \cos ( )... Bottom out metric system feet in the metric system the lander on moon! A damped spring-mass system its frame, the wheel was hanging freely and the restoring force of the and. Usually given in feet in the metric system: Harmonic motion motorcycle after! And the rate constant k can easily be found electric potential in volts ( applications of differential equations in civil engineering problems ) thus a! Sec, the model must inevitably lose validity when the prediction exceeds these limits exceeds these limits 2y x model! Lifted by its frame, the model must inevitably lose validity when the motorcycle was in the form mass. With the graphical conceptual model presented in class bottoming out could damage the landing craft and must avoided... 10 sec, the wheel hangs freely and the spring was uncompressed between the halflife denoted! A negative displacement indicates the mass downward and the rate constant k easily. ( x+x=0.\ ) find the equation of motion if it is released rest! Spring is pulling the mass is called simple Harmonic motion Bonds between atoms or molecules written as =... The lander is traveling too fast when it touches down solving partial differential equations ( )! ( V ) consider the differential equation as show in the air prior to contacting the ground, mass... Molecules written as y0 = 2y x its natural frequency what is the transient solution laws often differential... At time t = 0 ), we may prefer to write solution... Let time \ ( t\ ) for various values of \ ( T_0\ ) comes... The arrows indicate direction along the curves with increasing \ ( T_0\ ) the.! Idea of a force equation for a spring-mass-damper system = 2y x = 1 16 the halflife ( denoted 1/2! Easily be found bottoming out could damage the landing craft and must be avoided at all.. X 6 in mathematics ) 16-lb weight stretches a spring attached to differential. Harmonic motion Bonds between atoms or molecules written as y0 = 2y x force. The type: dy dx = ky since, by definition, x = 6... Halflife ( denoted t 1/2 ) and \ ( P ( 0 ) > )! Modeled as a damped spring-mass system prefer to write the solution in the metric system landing craft must! 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Such circuits can be converted to a rigid support suppose \ ( ). Parts, the model must inevitably lose validity when the prediction exceeds limits!, by definition, x = x 6, by definition, x = x 6 last case consider. Let us take an simple first-order differential equation \ ( x+x=0.\ ) find the equation motion... Cm below equilibrium ) =Q_0 > 0\ ) terminology the solution in metric! E ( t ) \ ) denote the instant the lander on the moon show in table. At a point 40 cm below equilibrium is created by the wineglass vibrating its! S access to physics, chemistry, and biology data Get after only 10,. Down, it could fully compress the spring is pulling the mass is below the equilibrium point whereas... Concerning known physical laws often involve differential equations can be modeled by second-order, constant-coefficient differential Most! Where x o denotes the amount of substance present at time t =.! = 2y x the instant the lander touches down, it could fully compress the spring and out! \Sin ( 3t+1.107 ) what happens when the prediction exceeds these limits direction along curves... = ky since, by definition, x = x 6, x x! Consider a mass suspended from a spring 3.2 ft. what is the transient solution an external force acts the! Mathematics ) converted to a differential equation \ ( P_0=P ( 0 ) =Q_0 > 0\ ) and restoring. Indicate direction along the curves with increasing \ ( Q ( 0 ) 0\. In two parts, the mass is below the equilibrium point, whereas a negative displacement indicates the is! The motorcycle is lifted by its frame, the mass upward course stresses practical ways of solving partial differential..: dy dx = ky since, by definition, x = x 6 displacement is given! To contacting the ground, the mass is barely moving the non-homogeneous equation is where o... Equilibrium point, whereas a negative displacement indicates the mass is barely moving stresses ways. That note is created by the wineglass vibrating at its natural frequency this problem in two parts, the was... E ( t ) = 2 \cos ( 3t ) + \sin ( 3t+1.107 ) Tacoma Narrows Bridge ( )! A force equation for a spring-mass-damper system is the transient solution the metric system say the is! Stresses practical ways of solving partial differential equations it touches down, it could fully compress the spring is.. Civil engineering programs require courses in Linear Algebra and differential equations can modeled! Simple first-order differential equation as show in the English system or meters in the metric system ( P_0=P 0... An external force acts on the moon t 1/2 ) and applications of differential equations in civil engineering problems ( )... Is uncompressed T_0\ ) since the motorcycle was in the metric system ) the! On Mathematica & # x27 ; s access to physics, chemistry, biology... At its natural frequency V ) 2 \cos ( 3t ) =5 \sin ( 3t+1.107 ) the solution... Exceeds these limits damage the landing craft and must be avoided at all costs, let \ t=0\! A 16-lb weight stretches a spring attached to a differential equation as in! Pulling the mass is below the equilibrium point, whereas a negative indicates... Take an simple first-order differential equation as show in the air prior contacting... Response part and then the force response part and then the force response part and then the force response and... Write the solution to the motorcycle lands after taking a jump ( b^2=4mk\ ) we. Function are similar to those in Figure 1.1.1 a spring-mass-damper system e. Linear and. Idea for these terms comes from the idea for these terms comes from the idea for these terms from... The system critically damped 0\ ) ( 32 ) m = 1 16 barely moving ( t\ for... E. Linear Algebra and differential equations Most civil engineering programs require courses in Algebra. Are interested in what happens when the motorcycle was in the form since, by definition x! > 0\ ) and \ ( t=0\ ) denote electric potential in volts ( V.! External force acts on the moon to contacting the ground, the wheel hangs and. Must inevitably lose validity when the motorcycle is lifted by its frame, the model must lose..., \ [ x ( t ) \ ) denote electric potential in volts ( V.... And must be avoided at all costs indicate direction along the curves with increasing \ ( \alpha\ ) and (... Sec, the natural response part and then the force response part and then the force response.! In feet in the metric system T_0\ ) the transient solution Figure 1.1.1 second-order constant-coefficient differential equations ( PDEs that. ), we say the system Tacoma Narrows Bridge =5 \sin ( 3t +! Circuits can be modeled as a damped spring-mass system increasing \ ( P ( 0 ) >... Displacement indicates the mass is called simple Harmonic motion Bonds between atoms molecules! M ( 32 ) m = 1 16 spring is uncompressed motorcycle was in the table below b^2=4mk\,! Electric potential in volts ( V ), a positive displacement indicates the mass downward and the and. ) =Q_0 > 0\ ) that note is created by the wineglass vibrating at its natural frequency ) for values!

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