Theory: In this Exercise you will work with an application of Linear Algebra to solving a problem using Hook's Law from Physics. We will consider an elastic beam which is supported on each end and is subject to forces applied at n points. Let vector f list the forces at the n points and let vector y in " list the amounts of deflection (displacements) of the beam at these points. Hooke's Law from Physics describes the relation between the vector of forces and the deflection vector as y = Df, where D is a flexibility matrix. The matrix S =D is called the stiffness matrix. (For more information, please refer to Section 2.2. Example 3 of the textbook.) The function that you will create calculates the forces applied at the points when the amounts of deflection are measured and the flexibility matrix D is given. If the flexibility is measured in inches of deflection per pound of load, then the stiffness is given in pounds of load per inch of deflection. If the flexibility units are centimeters per newton of force, then the units for stiffness are newtons per centimeter of deflection. **Write a MATLAB function that begins with gunction ($F]=beam (Box) Sali n=length(); **First, the function has to check whether the following two conditions hold: a square matrix D is an nxn matrix and D is invertible. (Use the command rank in your code to determine whether D is invertible.) If it is not the case, the program terminates and returns a message that D is not a flexibility matrix. The empty outputs for S and f will stay. (They do not need to be displayed.) If both conditions hold, you will proceed with the following tasks: (1) calculate and output the stiffness matrix S (see the Theory above) (2) calculate and output the vector f of forces for the given deflection vector y by using the stiffness matrix S (3) verify that the physical meaning of a th column of the stiffness matrix S (for all j =1:n) is that it is numerically equal to the vector of forces that have to be applied to n points to produce a unit deflection at the point and zero deflections at all other points. If your code shows that it is true, output the message that the physical meaning of the stiffness matrix is valid; otherwise, the message could be "I may need to check the code". % Write a comment that would explain a meaning of the negative entries of the matrix S. **Type the function beam in your Live Script. **Run the function (5,5]=beam (D x) on the following sets of variables (display the variables in your Live Script): (a) D=0.1* rand(4), v=randi ([20 50], 3,1) (6) D=0.01*magic14), v=randi.([20 50], 4,1) (C) D=0.01* rando), v=0.1*randi ([10 201, 6, 1) Theory: In this Exercise you will work with an application of Linear Algebra to solving a problem using Hook's Law from Physics. We will consider an elastic beam which is supported on each end and is subject to forces applied at n points. Let vector f list the forces at the n points and let vector y in " list the amounts of deflection (displacements) of the beam at these points. Hooke's Law from Physics describes the relation between the vector of forces and the deflection vector as y = Df, where D is a flexibility matrix. The matrix S =D is called the stiffness matrix. (For more information, please refer to Section 2.2. Example 3 of the textbook.) The function that you will create calculates the forces applied at the points when the amounts of deflection are measured and the flexibility matrix D is given. If the flexibility is measured in inches of deflection per pound of load, then the stiffness is given in pounds of load per inch of deflection. If the flexibility units are centimeters per newton of force, then the units for stiffness are newtons per centimeter of deflection. **Write a MATLAB function that begins with gunction ($F]=beam (Box) Sali n=length(); **First, the function has to check whether the following two conditions hold: a square matrix D is an nxn matrix and D is invertible. (Use the command rank in your code to determine whether D is invertible.) If it is not the case, the program terminates and returns a message that D is not a flexibility matrix. The empty outputs for S and f will stay. (They do not need to be displayed.) If both conditions hold, you will proceed with the following tasks: (1) calculate and output the stiffness matrix S (see the Theory above) (2) calculate and output the vector f of forces for the given deflection vector y by using the stiffness matrix S (3) verify that the physical meaning of a th column of the stiffness matrix S (for all j =1:n) is that it is numerically equal to the vector of forces that have to be applied to n points to produce a unit deflection at the point and zero deflections at all other points. If your code shows that it is true, output the message that the physical meaning of the stiffness matrix is valid; otherwise, the message could be "I may need to check the code". % Write a comment that would explain a meaning of the negative entries of the matrix S. **Type the function beam in your Live Script. **Run the function (5,5]=beam (D x) on the following sets of variables (display the variables in your Live Script): (a) D=0.1* rand(4), v=randi ([20 50], 3,1) (6) D=0.01*magic14), v=randi.([20 50], 4,1) (C) D=0.01* rando), v=0.1*randi ([10 201, 6, 1)
