Difference between revisions of "Tutorial Week 4"

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(Created page with "===Worksheet Questions=== # ## Please rewrite the function $y(t)=c_1\cos(\omega t)+c_2\sin(\omega t),$ in the form $y(t)=D\cos(\omega t-\gamma).$ (Find expressions of $D$ and ...")
 
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===Worksheet Questions===
 
===Worksheet Questions===
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## Please rewrite the function $y(t)=c_1\cos(\omega t)+c_2\sin(\omega t),$ in the form $y(t)=D\cos(\omega t-\gamma).$ (Find expressions of $D$ and $\gamma$ in terms of $c_1$ and $c_2$.)
 
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## Find the steady state periodic solution to the mass spring system described by $x''+cx'+x=10\cos(\omega t),$ where $\omega$ and $c$ are both positive constants. Your answer should look like $x(t)=A(\omega)\cos(\omega t)+B(\omega)\sin(\omega t).$ (Find expressions of $A(\omega)$ and $B(\omega)$.)
 
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## For what angular frequency $\omega$ is the amplitude $D(\omega)$ of the steady state periodic solution you obtained in part b maximized? Hint: use your result from part a. Also, note that $C/g(w)$ has a maximum value when $g(w)$ has a minimum value and $\sqrt{f(w)}$ has a minimum when $f(w)$ has a minimum. These last two facts will simplify the calculations required. Be sure to determine conditions under which your critical points are actually maxima.
 
  
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=== Solutions ===
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#An adjustable mass-spring system has a damping coefficient and spring constant that can be changed with the parameters $\alpha$. The spring constant is $\alpha^4-\alpha^2+2\alpha$, the damping coefficient is $2\alpha^2$, and a mass of 1 kg is attached to the spring. No external forces are applied to the system.
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## Write down a differential equation describing the system.
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##For what values of $\alpha$ does the characteristic equation of the system have complex roots?
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## Assuming complex roots, what is the general solution?
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##At what time will the amplitude have decayed to $e^{-1}$ of its original value?
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#For each of the following differential equations, write the general form of its particular solution.
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##$y''-25y'=2e^{8x}$
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##$y''-y=x^2$
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##$y''-2y'+y=e^{x}$
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===Solutions===
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##$y''+2\alpha^2y'+(\alpha^4-\alpha^2+2\alpha)y=0$ (1 mark)
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##$0<\alpha<2$. Here, they need to write the characteristic equation of the form $ar^2+br+c=0$ (1 mark), remember that complex roots occur when $ b^2-4ac<0$ (1 mark), and then solve for the variable $\alpha$( 2 marks - 1 if they only say $\alpha<2$).
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##$y(t)=e^{-\alpha^2t}(A\sin(\alpha^2-2\alpha)t+B\cos(\alpha^2-2\alpha)t)$. Here they need to find that the roots of the characteristic equation (1 mark) simplify to $-\alpha^2\pm\sqrt{\alpha^2-2\alpha}$ and then plug the real and imaginary parts into the formula correctly (2 marks - 1 for using correct form of general equation and 1 for putting real and imaginary parts in the correct places).
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##$t=\frac{1}{\alpha^2}$ (2 marks - 1 for creating an equation describing amplitude, 1 for solving for the correct $t$)
 
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## $D=\sqrt{c_1^2+c_2^2}$, $\gamma=\arctan(\frac{c_2}{c_1})$.
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##$y_c=c_1e^{25x}+c_2$(1 mark), $y_p=Ae^{8x}$ where $A=\frac{-1}{68}$(2 marks - 1 for general $y_p$ and 1 for solving for constant), $y(x) = c_1e^{25x}+c_2+\frac{-1}{68}e^{8x}$ (1 mark)
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## $A(\omega)=\frac{10(1-\omega^2)}{(1-\omega^2)^2+c^2\omega^2}$, $B(\omega)=\frac{10c\omega}{(1-\omega^2)^2+c^2\omega^2}$.
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##$y_c = c_1e^{-x}+c_2e^x$ (1 mark), $y_p =Ax^2+Bx+C$ where $A=-1$, $B=0$, and $C=-2$(2 marks - 1 for general $y_p$ and 1 for solving for constants), $y(x) = c_1e^{-x}+c_2e^x-x^2-2$ (1 mark)
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## $D(\omega)=\frac{10}{\sqrt{(1-\omega^2)^2+c^2\omega^2}}$. When $D(\omega)$ is maximized the denominator of it($\sqrt{g(\omega)}$) is minimized. As we know $\sqrt{g(\omega)}$ has a minimum when $g(\omega)$ has a minimum, and $g'(\omega)=2\omega(-2+2\omega^2+c^2).$ When $\omega=\sqrt{1-\frac{c^2}{2}}$, $g(\omega)$ obtain its minimum value. So if $1-\frac{c^2}{2}$ is positive, i.e. $0<c<\sqrt{2}$, then $\sqrt{1-\frac{c^2}{2}}$ is the practical resonance frequency. Otherwise, no maximum for $\omega>0$.
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##$y_c = c_1e^x+c_2te^x$ (1 mark), $y_p = Ax^2e^x$ where $A=\frac{1}{2}$ (2 marks - 1 for general $y_p$ and 1 for solving for constant) , $y(x) = c_1e^x+c_2te^x+\frac{1}{2}x^2e^x$ (1 mark)

Revision as of 18:31, 1 February 2016

Worksheet Questions

  1. An adjustable mass-spring system has a damping coefficient and spring constant that can be changed with the parameters $\alpha$. The spring constant is $\alpha^4-\alpha^2+2\alpha$, the damping coefficient is $2\alpha^2$, and a mass of 1 kg is attached to the spring. No external forces are applied to the system.
    1. Write down a differential equation describing the system.
    2. For what values of $\alpha$ does the characteristic equation of the system have complex roots?
    3. Assuming complex roots, what is the general solution?
    4. At what time will the amplitude have decayed to $e^{-1}$ of its original value?
  2. For each of the following differential equations, write the general form of its particular solution.
    1. $y''-25y'=2e^{8x}$
    2. $y''-y=x^2$
    3. $y''-2y'+y=e^{x}$

Solutions

    1. $y''+2\alpha^2y'+(\alpha^4-\alpha^2+2\alpha)y=0$ (1 mark)
    2. $0<\alpha<2$. Here, they need to write the characteristic equation of the form $ar^2+br+c=0$ (1 mark), remember that complex roots occur when $ b^2-4ac<0$ (1 mark), and then solve for the variable $\alpha$( 2 marks - 1 if they only say $\alpha<2$).
    3. $y(t)=e^{-\alpha^2t}(A\sin(\alpha^2-2\alpha)t+B\cos(\alpha^2-2\alpha)t)$. Here they need to find that the roots of the characteristic equation (1 mark) simplify to $-\alpha^2\pm\sqrt{\alpha^2-2\alpha}$ and then plug the real and imaginary parts into the formula correctly (2 marks - 1 for using correct form of general equation and 1 for putting real and imaginary parts in the correct places).
    4. $t=\frac{1}{\alpha^2}$ (2 marks - 1 for creating an equation describing amplitude, 1 for solving for the correct $t$)
    1. $y_c=c_1e^{25x}+c_2$(1 mark), $y_p=Ae^{8x}$ where $A=\frac{-1}{68}$(2 marks - 1 for general $y_p$ and 1 for solving for constant), $y(x) = c_1e^{25x}+c_2+\frac{-1}{68}e^{8x}$ (1 mark)
    2. $y_c = c_1e^{-x}+c_2e^x$ (1 mark), $y_p =Ax^2+Bx+C$ where $A=-1$, $B=0$, and $C=-2$(2 marks - 1 for general $y_p$ and 1 for solving for constants), $y(x) = c_1e^{-x}+c_2e^x-x^2-2$ (1 mark)
    3. $y_c = c_1e^x+c_2te^x$ (1 mark), $y_p = Ax^2e^x$ where $A=\frac{1}{2}$ (2 marks - 1 for general $y_p$ and 1 for solving for constant) , $y(x) = c_1e^x+c_2te^x+\frac{1}{2}x^2e^x$ (1 mark)
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