Johnson Alex

We can follow these steps:

Step 1: Define the system and coordinate system:

- We have a two-degree-of-freedom system consisting of two masses connected by springs.

- Assign a coordinate system to the system, such as x1 and x2, representing the displacements of mass m1 and m2, respectively.

Step 2: Write the equations of motion:

- Write the equations of motion for each mass using Newton's second law.

- Apply the principle of equilibrium to each mass separately.

For mass m1:

m1 * x1'' + k1 * (x1 - 0) + k2 * (x1 - x2) = 0 -- (Equation 1)

For mass m2:

m2 * x2'' + k2 * (x2 - x1) + 0 = 0 -- (Equation 2)

Step 3: Express the equations of motion in matrix form:

- Rewrite equations (1) and (2) in matrix form using the mass and stiffness matrices.

The mass matrix [M] and stiffness matrix [K] are defined as follows:

[M] = [m1 0; 0 m2]

[K] = [k1 + k2 -k2; -k2 k2]

The equations of motion in matrix form are:

[M] * {x''} + [K] * {x} = {0} -- (Equation 3)

where {x} = [x1; x2] and {0} = [0; 0]

Step 4: Solve the eigenvalue problem:

- Solve the eigenvalue problem to find the natural frequencies and mode shapes of the system.

- Substitute {x} = {V} * {q} into equation (3), where {V} is the matrix of eigenvectors and {q} is the vector of generalized coordinates.

The eigenvalue problem becomes:

[M] * {V} * {q}'' + [K] * {V} * {q} = {0}

Simplifying, we get:

([V^T] * [M] * {V}) * {q}'' + ([V^T] * [K] * {V}) * {q} = {0}

The resulting equation is in the form:

[M'] * {q}'' + [K'] * {q} = {0}

where [M'] = [V^T] * [M] * {V} and [K'] = [V^T] * [K] * {V}

Step 5: Calculate the natural frequencies and mode shapes:

- Solve the generalized eigenvalue problem [K'] * {q} = lambda * [M'] * {q} to obtain the eigenvalues (natural frequencies squared) and eigenvectors (mode shapes).

Step 6: Calculate the mass matrix, stiffness matrix, natural frequencies, and mode shapes using the given values:

- Substitute the given values m1 = m/3, m2 = 2m/5, k1 = 11k/3, k2 = 7k/2, m = 10 kg, and k = 1000 N/m into the derived equations to obtain the specific values.

To perform these calculations in MATLAB, we can use symbolic math tools and solve the eigenvalue problem numerically.

```matlab

syms m1 m2 k1 k2 m k

% Given values

m = 10; % kg

k = 1000; % N/m

m1 = m/3;

m2 = 2*m/5;

k1 = 11*k/3;

k2 = 7*k/2;

% Mass matrix

M = [m1 0; 0 m2];

% Stiffness matrix

K = [k1 + k2 -k2; -k2 k2];

% Eigenvalue problem

[V, D] = eig(K, M);

% Natural frequencies (square root of eigenvalues)

omega = sqrt(diag(D));

% Mode shapes (eigenvectors)

mode_shapes = V;

% Display results

disp("Mass matrix:");

disp(M);

disp("Stiffness matrix:");

disp(K);

disp("Natural frequencies:");

disp(omega);

disp("Mode shapes:");

disp(mode_shapes);

```

Running this MATLAB code will provide the mass matrix, stiffness matrix, natural frequencies, and mode shapes for the given system using the provided values.

Good luck: partial credit AI