Solving Problems in Dynamics and Vibrations Using MATLAB ...

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2 Contents I. An Introduction to MATLAB…………………………………………………(3) 1. Spring-Mass-Damper system…………………………………………………(11) 2. Simple pendulum……………………………………………………………..(21) 3. Coulomb friction……………………………………………………..……….(26) 4. Trajectory motion with aerodynamic drag……………………………..……..(30) 5. Pendulum with aerodynamic and viscous damping……………………..……(34) 6. A half cylinder rolling on a horizontal plane……………………………..…..(38) 7. A bar supported by a cord…………………………………………………….(41) 8. Double pendulum……………………………………………………………..(50) 9. Frequency response of systems having more than one degree of freedom…...(56) 10. A three bar linkage problem…………………………………………………..(63) 11. Slider-Crank mechanism……………………………………………………....(70) 12. A bar supported by a wire and a horizontal plane……………………………..(77) 13. Two bar linkage assembly supported by a pivot joint and a horizontal plane…(84) 14. Valve Spring Model...........................................................................................(92)
3 An Introduction to MATLAB Purpose of the Handout This handout was developed to help you understand the basic features of MATLAB and also to help you understand the use of some basic functions. This is not a comprehensive tutorial for MATLAB. To learn more about a certain function, you should use the online help. For example, if you want to know more about the function ‘solve’, then type the following command in the command window at the prompt: help solve Introduction MATLAB is a high performance language for technical computing. The name MATLAB stands for matrix laboratory. Some of the typical uses of MATLAB are given below: • Math and Computation • Algorithm Development • Modeling, Simulation and Prototyping M-Files Files that contain code in MATLAB language are called M-Files. You create a M-File using a text editor and then use them as you would any other MATLAB function or command. But before using the user defined functions always make sure that the ‘path’ is set to the current directory. There are two kinds of M-Files: • Scripts, which do not accept input arguments or return output arguments. • Functions, which can accept input and output arguments. When you invoke a script MATLAB simply executes the commands found in the file. For example, create a file called ‘SumOfNumbers’ that contains these commands: % To find the sum of first 10 natural numbers n = 0; i = 1; for i=1:10; n = n + i; i = i + 1; end n Go to the command window and set the path to the current directory. Then typing
Page 1: Solving Problems in Dynamics and Vi
Page 5 and 6: 5 x + 2y = 6 x − y = 0 There are
Page 7 and 8: 7 p = ‘x + 2*y = a + 6’ q = ‘
Page 9 and 10: 9 Suppose one wants to change the c
Page 11 and 12: 11 1. Spring Mass Damper System - U
Page 13 and 14: 13 MATLAB Code In order to apply th
Page 15 and 16: 15 Assignment Solve for six cycles
Page 17 and 18: 17 For our convenience, put x = y (
Page 20 and 21: 20 Assignment Plot the response of
Page 22 and 23: 22 When the values of ‘theta’ a
Page 25 and 26: 25 Assignment a) Compute and plot t
Page 27: 27 tspan=[0 2] y0=[0;0] [t,y]=ode45
Page 30 and 31: 30 4. Trajectory Motion with Aerody
Page 33 and 34: 33 Assignment Solve the following d
Page 35 and 36: 35 Open a new m-file and type the f
Page 37 and 38: 37 Assignment a) Solve the followin
Page 39 and 40: 39 y0=[0.5233;0]; [t,y]=ode45('tofr
Page 41 and 42: 41 7. A bar supported by a cord Der
Page 43 and 44: 43 MATLAB Code In this type of a pr
Page 45: 45 tspan=[0 30] y0=[0.5233;0;1.0467
Page 50 and 51: 50 8. Double Pendulum Derive the go
Page 52 and 53:
52 yp(1)=y(2); yp(2)=-(w1+w2)*sin(y
Page 55 and 56:
55 Assignment 1.) For the double pe
Page 57 and 58:
57 2 ⎡−ω [ M ] + [ K] ω[ C]
Page 59:
59 figure(1) plot(f,x(1,:));grid xl
Page 63 and 64:
63 10. A Three Bar Linkage Problem
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65 xlabel('alpha') ylabel('gamma do
Page 70 and 71:
70 11. Slider-Crank Mechanism Examp
Page 72:
72 To calculate the angular acceler
Page 77 and 78:
77 12. A Bar supported by a wire an
Page 79 and 80:
79 end To store the right hand side
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81 t1=t'; figure(3) plot(t1,theta);
Page 85 and 86:
85 The above model involves second
Page 87 and 88:
87 l=1; L=1.5; m=5; M=5; g=9.81; w=
Page 91 and 92:
91 Assignment: Three bar linkage as
Page 93 and 94:
93 where ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
Page 95 and 96:
95 ⎡kx1 ⎤ ⎢ ⎥ ⎢ 0 ⎥ ⎢
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97 m = 0.002; k = 175000; M1 = m*ey
Page 99 and 100:
99 cnt = cnt +1; end end end % To p
Page 101 and 102:
101 plot(vect3(:,i),c(i)) hold on;
Page 103 and 104:
103
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