2-DOF system with translational and rotational motion The diagram sho.pdf

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2-DOF system with translational and rotational motion The diagram shows an inverted pendulum on a cart that is driven by a force F. Here we consider the case where F = 0. Assume that the mass of the pendulum is concentrated at the ball (the rod is massless). Let x denote the position of the cart in an inertial reference frame. The linearized equations of motion for small theta, dtheta/dt, and dx/dt are the following: x=mg/m theta theta = M+m/Ml gtheta (a) Define the state variables of the system and the state vector x. Convert the two 2nd-order ODE\'s above into four 1-order ODE \'s that are functions of the state variables. Write an output equation y = Cx in which there are 2 outputs, theta and x. (b) Suppose that the system starts at rest with the cart position at x = 0 and the pendulum angle at theta = pi/10. Use the MATLAB function ode45 to numerically solve the four 1st-order ODE\'s from time 0 sec to 1 sec with parameters in = 0.5 kg. M = 3 kg. l = 1 m, and g = 9.81 m/s^2. Plot x over time in one figure, and plot theta over time in another figure. Hand in your code with your homework. Solution 1)x^2+3x=98 2)the plot comes to be elliptical.

2-DOF system with translational and rotational motion The diagram shows an inverted
pendulum on a cart that is driven by a force F. Here we consider the case where F = 0. Assume
that the mass of the pendulum is concentrated at the ball (the rod is massless). Let x denote the
position of the cart in an inertial reference frame. The linearized equations of motion for small
theta, dtheta/dt, and dx/dt are the following: x=mg/m theta theta = M+m/Ml gtheta (a) Define
the state variables of the system and the state vector x. Convert the two 2nd-order ODE's above
into four 1-order ODE 's that are functions of the state variables. Write an output equation y =
Cx in which there are 2 outputs, theta and x. (b) Suppose that the system starts at rest with the
cart position at x = 0 and the pendulum angle at theta = pi/10. Use the MATLAB function ode45
to numerically solve the four 1st-order ODE's from time 0 sec to 1 sec with parameters in = 0.5
kg. M = 3 kg. l = 1 m, and g = 9.81 m/s^2. Plot x over time in one figure, and plot theta over
time in another figure. Hand in your code with your homework.
Solution
1)x^2+3x=98
2)the plot comes to be elliptical

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MATLAB sessions: Laboratory 6 MAT 275 Laboratory 6 Forced Equations and Resonance In this laboratory we take a deeper look at second-order nonhomogeneous equations. We will concentrate on equations with a periodic harmonic forcing term. This will lead to a study of the phenomenon known as resonance. The equation we consider has the form d2y dt2 + c dy dt + ω20y = cosωt. (L6.1) This equation models the movement of a mass-spring system similar to the one described in Laboratory 5. The forcing term on the right-hand side of (L6.1) models a vibration, with amplitude 1 and frequency ω (in radians per second = 12π rotation per second = 60 2π rotations per minute, or RPM) of the plate holding the mass-spring system. All physical constants are assumed to be positive. Let ω1 = √ ω20 − c2/4. When c < 2ω0 the general solution of (L6.1) is y(t) = e− 1 2 ct(c1 cos(ω1t) + c2 sin(ω1t)) + C cos (ωt− α) (L6.2) with C = 1√ (ω20 − ω2) 2 + c2ω2 , (L6.3) α = ⎧ ⎨ ⎩ arctan ( cω ω20−ω2 ) if ω0 > ω π + arctan ( cω ω20−ω2 ) if ω0 < ω (L6.4) and c1 and c2 determined by the initial conditions. The first term in (L6.2) represents the complementary solution, that is, the general solution to the homogeneous equation (independent of ω), while the second term represents a particular solution of the full ODE. Note that when c > 0 the first term vanishes for large t due to the decreasing exponential factor. The solution then settles into a (forced) oscillation with amplitude C given by (L6.3). The objectives of this laboratory are then to understand 1. the effect of the forcing term on the behavior of the solution for different values of ω, in particular on the amplitude of the solution. 2. the phenomena of resonance and beats in the absence of friction. The Amplitude of Forced Oscillations We assume here that ω0 = 2 and c = 1 are fixed. Initial conditions are set to 0. For each value of ω, the amplitude C can be obtained numerically by taking half the difference between the highs and the lows of the solution computed with a MATLAB ODE solver after a sufficiently large time, as follows: (note that in the M-file below we set ω = 1.4). 1 function LAB06ex1 2 omega0 = 2; c = 1; omega = 1.4; 3 param = [omega0,c,omega]; 4 t0 = 0; y0 = 0; v0 = 0; Y0 = [y0;v0]; tf = 50; 5 options = odeset(’AbsTol’,1e-10,’RelTol’,1e-10); 6 [t,Y] = ode45(@f,[t0,tf],Y0,options,param); 7 y = Y(:,1); v = Y(:,2); 8 figure(1) 9 plot(t,y,’b-’); ylabel(’y’); grid on; c⃝2011 Stefania Tracogna, SoMSS, ASU 1 MATLAB sessions: Laboratory 6 10 t1 = 25; i = find(t>t1); 11 C = (max(Y(i,1))-min(Y(i,1)))/2; 12 disp([’computed amplitude of forced oscillation = ’ num2str(C)]); 13 Ctheory = 1/sqrt((omega0^2-omega^2)^2+(c*omega)^2); 14 disp([’theoretical amplitude = ’ num2str(Ctheory)]); 15 %---------------------------------------------------------------- 16 function dYdt = f(t,Y,param) 17 y = Y(1); v = Y(2); 18 omega0 = param(1); c = param(2); omega = param(3); 19 dYdt = [ v ; cos(omega ...

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2. A. Define power. B. List five bases for power and explain each means in one or two sentences. Give an what example for each. C. Below are short biographies of two EEos of Intel. Using the biographies compare the power bases of the two men. Bob Noyce was a physicist with a technology background. He served as CEO until 1975. He was a person who tried to be friendly with workers and give them lots of freedom to set laid back without big differences in rewards set by rank. their own agendas. He ran a workplac He did not allow gaudy offices or travel by private jet or reserved parking spots for executives Paul Otellini became CEo in 2005. He was the first chief executive not to have a technical background. He downsized the company and eliminated many jobs for people who were not productive. He has tried to speed up new product development Solution 2. A. Power represents the degree and authority and influence of a person on other people. B. Five bases of power are listed below: Expert power: Expert power is derived by a person who possess considerable expertise in particular area. Employees having expert power are considered to be critical resource by an organization. Medical practitioners and lawyers exhibit expert power. Legitimate power: A person derives this power using his position in the company. For example, managers have the legitimate power to assign duties and responsibilities to its subordinates. Referent power: People gain referent power using his interpersonal relationships with other persons. They derive this power using their influence, trust and respect gained from other people. CEO of a company possesses this power. Coercive power: People gain coercive power by influencing others using threats or punishments. It helps the managers to control the behavior of its employees. Reward power: People gain this power base by influencing others using incentives and rewards. Promotions, appraisal and incentives are the tools used by an organization to exert reward power on its employees. C. Bob Noyce possessed the expert power as he was a physicist with technology background. He utilized the referent power using his interpersonal relationships with his employees. Besides this, he also used the reward power. Contrary to this, Paul Otellini possessed coercive and legitimate power..

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2. A pond is stocked with 100 fish in 1997, and it is observed that the fish population doubles every year for the first couple of years In what year a maximum sustainable yield of 2 fish per day, on the average, will occur from this pond? Solution In 1997 the pond is stocked with 100 fishes As per the condition fishes will be doubled every year 1998 200 fishes in the pond 1999 400 fishes 2000 800 fishes in pond (on an average of 2fish/day)..

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13. Is the following proposition true or false? Justify your conclusion with a proof or a counterexample. If f : S arrow T is a surjection and C subset of T, then f (f^-1 (C)) = C. Solution If f:S to T is a surjection, then all elements in T = f(C) Hence each element in f(C) has a preimage in C But as f is not one to one, the preimage is not unique. Hence there may be repititions. f inverse is not a funciton. We can say f inverse is a mapping for a mapping f f( f inverse C) =C For a funciton f, the above is not true..

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141. Which of the following statements is true? A. U.S non- financial firms tend to use more debt than equity financing B. Bondholders are generally granted voting rights equal to those of common shareholders . C. Debt increased the possibility of financial distress D. Payments of both interest and dividends are tax deductible as business expenses E. Unpaid common stock dividends can force a firm into liquidation 141. Which of the following statements is true? A. U.S non- financial firms tend to use more debt than equity financing B. Bondholders are generally granted voting rights equal to those of common shareholders . C. Debt increased the possibility of financial distress D. Payments of both interest and dividends are tax deductible as business expenses E. Unpaid common stock dividends can force a firm into liquidation A. U.S non- financial firms tend to use more debt than equity financing B. Bondholders are generally granted voting rights equal to those of common shareholders . C. Debt increased the possibility of financial distress D. Payments of both interest and dividends are tax deductible as business expenses E. Unpaid common stock dividends can force a firm into liquidation Solution Answer is A. U.S non- financial firms tend to use more debt than equity financing Other statements are false as below: B. Bondholders are lenders to the company and they are not treated at par with common shareholders. C. Debts are an important of the capital structure. An all equity firm is very risky as compared to an firm with optimal blend of debt and equity. D. Dividends are not deductible expense for tax purpose E. Common stock dividends cannot lead liquidation as common stock dividend is not an obligation for company..

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12. Find the transition matrix from the basis B to the basis c where B t and C (0.(0) for R2. In other words find the matrix P with the property that for all UE R2, P(v B Uc. 13 Find the matrix representation with respect to the ordered basis 11, z, T2 H, of the linear transformation S P2 P2 defined by S f(z)) a 1. Jo f(t) dt. 14. Find all real values of z for which (zI A) fails to be invertible if 0 -1 3 A 1 0 2 15. Suppose that T V V is diagonalizable. Explain why the composite To T V V is also diagonalizable. What are the eigenvalues of To T? Solution 12.

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1. What is the purpose of criminal law? 2. What are the various categories and classes of crimes? 3. What is the nature of an act according to the meaning of criminal liability? 4. What are the four mental states that can be found in the criminal code? Solution 1. Porpuse of criminal law to keep order in society and lessen the harm individuals may cause to one another Reference: http://enlightenme.com/criminal-laws/ 2. What are the various categoires and classes of crimes? Personal Crimes.

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