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me5180_numerical/hw2_work/hw2_work.ipynb

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"![Whirling Roller-coaster](./figures/roller_coaster.png)\n",
"\n",
"**1\\.** A roller coaster is being designed on a parabolic track that rotates at a constant\n",
"speed as seen in the figure above. Assume the cart rolls on the track as a frictionless\n",
"point-mass of 100-kg. Determine the equations of motion in terms of the distance from the\n",
"lowest point, $q_1=x_2$.\n",
"\n",
" a. What is the kinetic energy of the cart?\n",
"\n",
" b. What is the virtual work done by gravity on the cart, $F_{q_{1}}$ ?\n",
"\n",
" c. What is the equation of motion for the cart?\n",
"\n",
" d. Plot the $x_{2}$-position over time if $x_2(0)$=10 m, and $\\dot{x}_{2}(0)$=0 m/s, <!---\n",
"_{} end the highlighting\n",
"--> for two cases:\n",
"\n",
" i. $\\omega$=1 rad/s\n",
"\n",
" ii. $\\omega$=5 rad/s\n"
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"g=9.81;\n",
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"xlabel('time (s)')\n",
"ylabel('position (m)')"
]
},
{
"cell_type": "code",
"execution_count": 6,
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"[t5,r5]=ode23(@(t,r) ode1(t,r,5), [0 5], [10 0]);\n",
"plot(t5,r5(:,1))\n",
"xlabel('time (s)')\n",
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"![Rotating pendulum](./figures/rotating_pendulum.png)\n",
"\n",
"**2\\.** A pendulum is attached to a square L-frame of length a. The frame rotates at a constant\n",
"rate, $\\omega$. The length, l=1 m, of the pendulum link does not change.\n",
"\n",
" a. What is the kinetic energy of the system?\n",
"\n",
" b. What is the virtual work done by gravity on the mass?\n",
"\n",
" c. Derive the equations of motion for the system.\n",
"\n",
" d. Use a single plot to compare $\\theta(t)$ for two cases:\n",
"\n",
" i. a=0 m, $\\omega$=1 rad/s\n",
"\n",
" ii. a=10 m, $\\omega$=1 rad/s\n"
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"w=1; \n",
"l=1;\n",
"wnapprox=@(w) sqrt(g/l-w^2);\n",
"ode2= @(t,r,a) [r(2); a*w^2/l*cos(r(1))-(g/l-cos(r(1))*w^2).*sin(r(1))];\n",
"[t0,r0]=ode23(@(t,r) ode2(t,r,0), [0 2*2*pi/wnapprox(w)],[pi/9,0]);"
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"[t10,r10]=ode23(@(t,r) ode2(t,r,10), [0 2*2*pi/wnapprox(w)],[pi/9,0]);\n",
"\n",
"plot(t0,r0(:,1)*180/pi,t10,r10(:,1)*180/pi)\n",
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"![Pendulum bob attached to springs](./figures/f3-6.png)\n",
"\n",
"**3\\.** The pendulum bob of mass in, shown in the figure above, is suspended by an inextensible\n",
"string from the point p. This point is free to move along a straight horizontal line under\n",
"the action of the springs, each having a constant k. Assume that the mass is displaced\n",
"only slightly from the equilibrium position and released. Neglecting the mass of the\n",
"springs, show that the pendulum oscillates with a period of\n",
"\n",
"$P=2\\pi\\sqrt{\\frac{mg+2kr}{2kg}}$\n",
"\n",
"Solve for $\\theta(t)$ if m=0.1 kg, r=1 m, $\\theta(0)$=pi/6 rad, and $\\dot{\\theta}(0)$=0 rad/s for\n",
"2 cases:\n",
"\n",
" a. k=2 N/m\n",
"\n",
" b. k=$\\infty$ N/m\n",
"\n",
" c. Plot the solutions of $\\theta(t)$ for 2 periods on one figure\n"
]
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"%ode3 defined in separate m-file\n",
"l=1; m=0.1; k=2; \n",
"T=2*pi/sqrt(2*k*g/(2*k*l+m*g));\n",
"\n",
"tinf=linspace(0,2*T);\n",
"rinf=30*cos(sqrt(g/l)*tinf);\n",
"f=figure(4);\n",
"plot(tinf,30*cos(2*pi/T*tinf),tinf,rinf)\n",
"xlabel('time (s)')\n",
"ylabel('angle (deg)')\n",
"legend('k=2 N/m','k=inf N/m','Location','Southeast')"
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"## Nonlinear solution (extra information)\n",
"\n",
"ode3 defined in `ode3.m`, for convergence at small angles use the linear ode equations. \n",
"\n",
"```matlab\n",
"function dr=ode3(t,r,l,k,m)\n",
" % first order ODE for pendulum attached to 2 springs \n",
" % l is pendulum length\n",
" % k is spring stiffness of one spring\n",
" % m is mass of pendulum bob\n",
" g=9.81;\n",
" dr=zeros(size(r));\n",
" dr(1)=r(2); % dtheta/dt=dtheta/dt\n",
" dr(3)=r(4); % dy/dt=dy/dt\n",
" if abs(r(1))<=0.002\n",
" dr(2)=-2*k*g/(2*k*l+m*g)*r(1);\n",
" dr(4)=-2*k/m*r(3);\n",
" else\n",
" ddy=1/l*1/sin(r(1))^2*[1 -cos(r(1));-l*cos(r(1)) l]*[-g*sin(r(1));-2*k/m*r(3)];\n",
" dr(2)=ddy(1);\n",
" dr(4)=ddy(2);\n",
"end\n",
"```"
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L456.8,65.3 L456.9,65.4 L456.9,65.5 L457.0,65.5 L457.0,65.6 L457.0,65.7 L457.1,65.7 L457.1,65.8 L457.1,65.9 L457.2,66.0 L457.2,66.1 L457.2,66.2 L457.3,66.2 L457.3,66.3 L457.3,66.4 L457.3,66.5 L457.4,66.5 L457.4,66.6 L457.4,66.7 L457.4,66.8 L457.5,66.9 L457.5,67.0 L457.5,67.1 L457.6,67.2 L457.6,67.3 L457.6,67.4 L457.6,67.5 L457.7,67.6 L457.7,67.7 L457.7,67.8 L457.7,67.9 L457.8,68.0 L457.8,68.1 L457.8,68.2 L457.8,68.3 L457.9,68.4 L457.9,68.5 L457.9,68.6 L457.9,68.8 L458.0,68.9 L458.0,69.0 L458.0,69.1 L458.0,69.2 L458.2,70.0 L458.5,71.6 L458.8,73.1 L459.1,74.7 L459.4,76.2 L459.7,77.8 L460.0,79.4 L460.3,80.9 L460.6,82.4 L460.8,84.0 L461.1,85.5 L461.4,87.1 L461.7,88.6 L462.0,90.1 L462.3,91.7 L462.6,93.2 L462.9,94.7 L463.2,96.2 L463.5,97.8 L463.8,99.3 L464.1,100.8 L464.3,102.3 L464.6,103.8 L464.9,105.3 L465.2,106.8 L465.5,108.3 L465.8,109.9 L466.1,111.4 L466.4,112.9 L466.7,114.4 L466.7,114.5 L466.7,114.6 L466.7,114.7 L466.7,114.8 L466.8,114.9 L466.8,115.0 L466.8,115.1 L466.8,115.2 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"l=1; m=0.1; k=2;\n",
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"T=2*pi/sqrt(2*k*g/(2*k*l+m*g));\n",
"[t2,r2]=ode23(@(t,r) ode3(t,r,l,k,m),[0 4],[pi/6,0,0,0]);\n",
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"xlabel('time (s)')\n",
"ylabel('angle (deg)')\n",
"subplot(2,1,2)\n",
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"source": [
"## Animation of pendulum on elastic support\n",
"\n",
"![pendulum animation](./figures/pend_anim.gif)"
]
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"![Two masses connected by a spring are rotating on a frictionless\n",
"table](./figures/spring-mass.png)\n",
"\n",
"**4\\.** Two 0.1 kg-masses are connected to a linear spring on a frictionless table. The center\n",
"of mass of the 2-particle system is stationary. At the instant shown, the velocity is\n",
"$\\bar{v}_{1}=\\bar{v}_{2}=0.1\\hat{e}_{r}+2.5\\hat{e}_{\\theta}$ m/s and r=0.5 m. The spring\n",
"constant, K=10 N/m, and the spring applies no force when the masses are at the origin, so\n",
"F=2Kr.\n",
"\n",
" a. How many degrees of freedom are in the system?\n",
"\n",
" b. Write the total kinetic energy of the two masses.\n",
"\n",
" c. Write the expression for the virtual work done by the spring\n",
"\n",
" d. Derive the equations of motion for the position of the two particles\n",
"\n",
" e. Plot the path (x,y-coordinates) for one period of oscillation of one of the particles\n"
]
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"[t,r]=ode23(@(t,r) ode4(t,r,10,0.5),[0 5],[0.5 0.1 0 2.5/0.5]);"
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