**MY-ASSIGNMENTEXPERT™**可以为您提供engineering MATH4604 Finite Element MethoD有限元方法的代写代考和辅导服务！

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#### 这是普渡大學有限元方法的代写成功案例。

**CE595**课程简介

• Academic integrity is expected of all students at all times. Further information on academic integrity policies may be found in the handbook University Regulations and on the Web at http://www.purdue.edu/ODOS/osrr/integrity.htm.

• Homework is due in class on the date indicated. In general, no late homework will be accepted. If you feel that you have serious extenuating circumstances (eg., illnesses or accidents requiring medical attention, personal or family crises), you must discuss your situation with Dr. Varma as soon as possible. In particular, foreseeable conflicts with due dates (eg., interviews, participation in sports activities, religious observances, etc.) must be brought to our attention before the due date.

• It is anticipated, even encouraged, that students will consult with each other on homework assignments. It is expected, however, that all work submitted by the student represent his/her own effort. Instances of plagiarism on an assignment will result in full loss of credit for that assignment.

• Instances of cheating in any form during an exam will result in full loss of credit for that exam. Additional measures, including immediate failure of the course, may be applied at the discretion of the instructor and/or University staff.

• Students who have documented disabilities and require accommodations must make an appointment with Dr. Varma to discuss their needs by the end of the second week of class. Students with disabilities must be registered with Adaptive Programs in the Office of the Dean of Students before classroom accommodations can be provided.

## Prerequisites

Grades will be based upon the following elements:

• Homework (25% of total grade) Due in class at dates to be announced.

• Hourly Exam #1 (25% of total grade) Tentatively scheduled for February last week.

• Hourly Exam #2 (25% of total grade) Tentatively scheduled for April first week.

• Course Project or Final Exam (25% of total grade) Schedule will be announced later in the semester.

**CE595 Finite Element Method** HELP（EXAM HELP， ONLINE TUTOR）

Problem statement (Planar motion of a pendulum): A simple pendulum (such as the one used in a clock) consists of a bob of mass $m(\mathrm{~kg})$ attached to one end of a rod of length $\ell(\mathrm{m})$ with the other end pivoted to a fixed point $\mathrm{O}$, as shown in Fig. 1.2.1(a). Make necessary simplifying assumptions to derive the governing equation for the simplest linear motion of the pendulum. Also, determine the analytical solution of the resulting equation for the case in which the initial conditions on the angular displacement $\theta$ and its derivative are specified.

Solution: In order to derive the governing equation of the problem, we must make certain assumptions concerning the system (the bob and rod), consistent with the goal of the analysis. If the goal is to study the simplest linear motion of the pendulum, we assume that the bob as well as the rod are rigid (i.e., not deformable) and the rod is massless (i.e., compared to the mass of the bob). In addition, we assume that there is no friction at the pivot point $\mathrm{O}$ and the resistance offered by the surrounding medium to the pendulum is also negligible.

Under these assumptions, the equation governing the motion of the system can be formulated using the principle of conservation of linear momentum (or simply Newton’s second law), which states, in the present case, that the vector sum of externally applied forces on a system is equal to the time rate of change of the linear momentum (mass times velocity) of the system:

$$

\mathbf{F}=\frac{d}{d t}(m \mathbf{v})=m \mathbf{a}

$$

where $\mathbf{F}$ is the vector sum of all forces acting on the system, $m$ is the mass of the system, $\mathbf{v}$ is the velocity vector, and $\mathbf{a}$ is the acceleration vector of the system. To write the equation governing the angular motion, we set up a coordinate system, as shown in Fig. 1.2.1(b). Applying Newton’s second law to the $x$-direction [note that the dynamic equilibrium of forces in the $y$ direction gives the reaction force $R(\mathrm{~N})$ in terms of the weight $\mathrm{mg}$ of the bob], we obtain

$$

F_x=m \frac{d v_x}{d t}, \quad \text { where } F_x=-m g \sin \theta, v_x=\ell \frac{d \theta}{d t}

$$

and $\theta$ is the angular displacement (radians), $v_x$ is the component of velocity $(\mathrm{m} / \mathrm{s})$ along the $x$ coordinate, and $t$ denotes time (sec). Thus, the equation for angular motion becomes

$$

-m g \sin \theta=m \ell \frac{d^2 \theta}{d t^2} \quad \text { or } \quad \frac{d^2 \theta}{d t^2}+\frac{g}{\ell} \sin \theta=0

$$

Equation (1.2.3) is nonlinear on account of the term $\sin \theta$. For small angular motions (consistent with the goal of the study), $\sin \theta$ is approximated as $\sin \theta \approx \theta$ (i.e., linearized). Thus, the angular motion is described by the linear differential equation

$$

\frac{d^2 \theta}{d t^2}+\frac{g}{\ell} \theta=0

$$

Equations (1.2.3) and (1.2.4) represent mathematical models of nonlinear and linear motions, respectively, of a rigid pendulum. Their solution requires knowledge of conditions at time $t=0$ on $\theta$ and its time derivative $\dot{\theta}$ (angular velocity). These conditions are known as the initial conditions. Thus, the linear problem involves solving the differential equation in Eq. (1.2.4) subjected to the initial conditions

$$

\theta(0)=\theta_0, \quad \frac{d \theta}{d t}(0)=v_0

$$

The problem described by Eqs. (1.2.4) and (1.2.5) is called an initial-value problem (IVP).

The linear problem described by Eqs. (1.2.4) and (1.2.5) can be solved analytically. The general analytical solution of the linear equation in Eq. (1.2.4) $\left(\ddot{\theta}+\lambda^2 \theta=0\right)$ is

$$

\theta(t)=A \sin \lambda t+B \cos \lambda t, \quad \lambda=\sqrt{\frac{g}{\ell}}

$$

where $A$ and $B$ are constants of integration, which can be determined using the initial conditions in Eq. (1.2.5). We obtain

$$

A=\frac{v_0}{\lambda}, \quad B=\theta_0

$$

and the solution to the linear problem becomes

$$

\theta(t)=\frac{v_0}{\lambda} \sin \lambda t+\theta_0 \cos \lambda t

$$

For zero initial velocity $\left(v_0=0\right)$ and nonzero initial position $\theta_0$, we have

$$

\theta(t)=\theta_0 \cos \lambda t

$$

which represents a simple harmonic motion.

Problem statement (One-dimensional heat flow): Derive the governing equations (i.e., develop the mathematical model) of steady-state heat transfer through a cylindrical bar of nonuniform cross section, as shown in Fig. 1.2.2(a). Assume that there is a heat source within the rod generating energy at a rate of $f(\mathrm{~W} / \mathrm{m})$. In practice, such energy source can be due to nuclear fission or chemical reactions taking place along the length of the bar, or due to the passage of electric current through the medium (i.e., volume heating). Assume that temperature is uniform at any section of the bar, $T=T(x)$. Due to the difference between the temperatures of the bar and the surrounding medium, there is convective heat transfer across the surface of the body and at the right end. Also, analytically solve the resulting equation for constant material and geometric parameters and uniform internal heat generation $f$. The bar is subject to a known temperature $T_0\left({ }^{\circ} \mathrm{C}\right)$ at the left end and exposed, both on the surface and at the right end, to a medium (such as cooling fluid or air) at temperature $T_{\infty}$.

Solution: The principle of balance of energy, also known as the first law of thermodynamics, can be used to derive the governing equations of the problem. The principle of balance of energy requires that the rate of change (increase) of internal energy is equal to the sum of heat gained by conduction, convection, and internal heat generation (radiation not included). For a steady process the time rate of internal energy is zero.

Consider a volume element of length $\Delta x$ and having an area of cross section $A(x)\left(\mathrm{m}^2\right)$ at a distance $x$ along the length of the bar, as shown in Fig. 1.2.2(b). If $q$ denotes the heat flux (heat flow per unit area, $\mathrm{W} / \mathrm{m}^2$ ), then $[A q]x$ is the net heat flow into the volume element at $x,[A q]{x+\Delta x}$ is the net heat flow out of the volume element at $x+\Delta x$, and $\beta P \Delta x\left(T_{\infty}-T\right)$ is the heat flow through the surface into the bar. Here $\beta$ denotes the film (that is formed the temperature of the surrounding medium, and $P$ is the perimeter (m) of the bar. Then the energy balance gives

$$

[A q]x-[A q]{x+\Delta x}+\beta P \Delta x\left(T_{\infty}-T\right)+f \Delta x=0

$$

Dividing throughout by $\Delta x$,

$$

-\frac{[A q]{x+\Delta x}-[A q]_x}{\Delta x}+\beta P\left(T{\infty}-T\right)+f=0

$$

and taking the limit $\Delta x \rightarrow 0$, we obtain

$$

-\frac{d}{d x}(A q)+\beta P\left(T_{\infty}-T\right)+f=0

$$

We can relate the heat flux $q\left(\mathrm{~W} / \mathrm{m}^2\right)$ to the temperature gradient using a material law. Such a relation is provided by the Fourier law

$$

q(x)=-k \frac{d T}{d x}

$$

where $k$ denotes the thermal conductivity $\left[\mathrm{W} /\left(\mathrm{m} \cdot{ }^{\circ} \mathrm{C}\right)\right]$ of the material. The minus sign on the right side of the equality in Eq. (1.2.12) indicates that heat flows from high temperature to low temperature. Equation (1.2.12) is a constitutive relation between the heat flux and the temperature gradient.

Now using the Fourier law, Eq. (1.2.12) in Eq. (1.2.11), we arrive at the heat conduction equation

$$

\frac{d}{d x}\left(k A \frac{d T}{d x}\right)+\beta P\left(T_{\infty}-T\right)+f=0

$$

which can also be written as

$$

-\frac{d}{d x}\left(k A \frac{d T}{d x}\right)+\beta P\left(T-T_{\infty}\right)=f

$$

**MY-ASSIGNMENTEXPERT™**可以为您提供ENGINEERING MATH4604 FINITE ELEMENT METHOD有限元方法的代写代考和辅导服务！