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Home Question Bank Engineering Archive 2024 December 30

Engineering Questions from Dec 30,2024

Browse the Engineering Q&A Archive for Dec 30,2024, featuring a collection of homework questions and answers from this day. Find detailed solutions to enhance your understanding.

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SINAV SORULARI (EXAM Q(ESTIONS) Question 3: Modeling a DC Motor (40 Marks) Censidert a DC motor driven by a voltage source \( V \). The motor's armature curnelt \( 1(t) \) and the angular yelocity \( \omega(t) \) of the motor atre telited by the following equetions: \[ \begin{array}{c} L \frac{d i}{d t}+R i=V-k_{e} \omega \\ J \frac{d \omega}{d t}+b \omega=k_{t} i \end{array} \] Where: - \( L \) is the inductance of the motor, - \( \beta \) is the resistance, - \( \mathrm{F}_{8} \) is the back EMF constant, - If is the torque constant, - \( f \) is the moment of inertie, - \( b \) is the damping coefficient Assume the fallowing parmoter values: - \( \mathrm{L}=0.5 \mathrm{H} \), - \( P=10 \), - \( k_{e}=0.01 \mathrm{~V} \cdot \mathrm{~s} / \mathrm{rad} \) - \( k_{\mathrm{z}}=0.01 \mathrm{Nm} / \mathrm{A} \), - \( J=0.01 \mathrm{~kg} \mathrm{~m}^{2} \), - \( b=0.1 \mathrm{~N}-\mathrm{m} \cdot \mathrm{s} / \mathrm{rad} \). - \( \mathrm{V}=10 \mathrm{~V} \) (conth A. Write down the system of fins-order differential equations for the ourtemt \( A(t) \) med angular velocity \( \omega(t) \). B. Splve this system using Pyrhon and the odeint function foun the scing imequate modite. Write the Python code that: 1. Defines the system of differential equations. 2. Solves the system for \( i(t) \) and \( \omega(t) \). 3. Extracts the answers at specific time points (for example, \( t=0,1,2,3,4,5 \) 4. Plots the solution for \( i(t) \) and \( \omega(t) \) versus time. SINAV SORULARI (EXAM Q(ESTIONS) Question 3: Modeling a DC Motor (40 Marks) Censidert a DC motor driven by a voltage source \( V \). The motor's armature curnelt \( 1(t) \) and the angular yelocity \( \omega(t) \) of the motor atre telited by the following equetions: \[ \begin{array}{c} L \frac{d i}{d t}+R i=V-k_{e} \omega \\ J \frac{d \omega}{d t}+b \omega=k_{t} i \end{array} \] Where: - \( L \) is the inductance of the motor, - \( \beta \) is the resistance, - \( \mathrm{F}_{8} \) is the back EMF constant, - If is the torque constant, - \( f \) is the moment of inertie, - \( b \) is the damping coefficient Assume the fallowing parmoter values: - \( \mathrm{L}=0.5 \mathrm{H} \), - \( P=10 \), - \( k_{e}=0.01 \mathrm{~V} \cdot \mathrm{~s} / \mathrm{rad} \) - \( k_{\mathrm{z}}=0.01 \mathrm{Nm} / \mathrm{A} \), - \( J=0.01 \mathrm{~kg} \mathrm{~m}^{2} \), - \( b=0.1 \mathrm{~N}-\mathrm{m} \cdot \mathrm{s} / \mathrm{rad} \). - \( \mathrm{V}=10 \mathrm{~V} \) (conth A. Write down the system of fins-order differential equations for the ourtemt \( A(t) \) med angular velocity \( \omega(t) \). B. Splve this system using Pyrhon and the odeint function foun the scing imequate modite. Write the Python code that: 1. Defines the system of differential equations. 2. Solves the system for \( i(t) \) and \( \omega(t) \). 3. Extracts the answers at specific time points (for example, \( t=0,1,2,3,4,5 \) 4. Plots the solution for \( i(t) \) and \( \omega(t) \) versus time. Question 2: Modeling a Spring-Mass System (30 Marks) Consider a spring-mass system where a mass mm is attached to a apring with spriag costant \( k \), and damping coefficient \( c \). The displacement of the mass is denoted as \( x(1) \). The sywem is subject to a constant extemal force \( F_{0} \). The equation of motion for this system is given by: \[ m \frac{d^{2} x}{d t^{2}}+c \frac{d x}{d t}+k x=F_{0} \] A. Derive the equation of motion, explaining each step clearly, including assumptions made (such as damping, external force, etc.). Show all steps involved in obtaining the differential equation. B. Solve the differential equation for \( x(t) \), assuming \( m=1 \mathrm{~kg}, c=0.5 \mathrm{Ns} / \mathrm{m}, \mathrm{k}= \) \( 10 \mathrm{~N} / \mathrm{m} \), and \( F 0=5 \mathrm{~N} \). Use initial conditions \( x(0)=0 \) and \( \frac{d x}{d t}(0)=0 \). \[ \left\{\begin{array}{l}\dot{\boldsymbol{x}}=\left[\begin{array}{cc}1 & 3 \\ -2 & 4\end{array}\right] \boldsymbol{x}+\left[\begin{array}{l}2 \\ 1\end{array}\right] \boldsymbol{u} \\ y=\left[\begin{array}{ll}-1 & 1\end{array}\right] \boldsymbol{x}\end{array}\right. \] \[ \begin{array}{l}\text { Determine the controllability by Rank Criteria and Diagon: } \\ \text { Criteria, respectively; }\end{array} \] Determine the observability by Rank Criteria and Diagon 5. (20 Points) Consider the following state-space equation of a \[ \left\{\begin{array}{l}\dot{x}=\left[\begin{array}{cc}0 & 6 \\ 1 & -1\end{array}\right] x+\left[\begin{array}{l}1 \\ 1\end{array}\right] u \\ y=\left[\begin{array}{ll}1 & 0\end{array}\right] x\end{array}\right. \] lease analyze the system's BIBO stability and Internal stability.
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