We would like to determine the vibratory response of the elastic structure shown below. The simple structure consists of a beam, made of steel, and two concentrated masses, free to bend laterally only, and m1 and m2 are located as shown below. Assume the dimensions of masses are 40x40x80mm and made of a material that has a ρ = 2700 kg/m^3. The length of the beam is 400mm, width is 40mm, height is 2mm, and is clamped as shown in Figure 1 (Be careful with the usage of the width and height values, check with the physical structure given in the figure). The mass 1 is located at the center of the beam.
a) Find the equivalent mass of the structure as it is seen at the free end of the beam (i.e., at coordinate x2).
b) Find the equivalent stiffness of the structure as it is seen at the free end of the beam (i.e., at coordinate x2).
c) Build a single-degree of freedom lumped parameter model of the structure for the coordinate x2. Write the equation of motion of the system.
d) Plot the deformation profile of the beam if the free end is displaced by 2mm.
e) Assume that there is no damping of the system. Find out the displacement response when initial conditions of the following are applied to the free end and plot displacement versus time (between 0 and 0.2 seconds) for each case in a single graph. Which of the initial conditions is dominant to the response for case 3? Why? Also, draw separately the velocity and acceleration of the system for each case. For Case 1, draw the displacement, velocity, and acceleration response in a single graph.
a. Case 1: x1(0) = 2mm, x1'(0) = 0mm/s
b. Case 2: x1(0) = 0mm, x1'(0) = 2mm/s
c. Case 3: x1(0) = 2mm, x1'(0) = 2mm/s
d. Case 4: x1(0) = 2mm, x1'(0) = 500mm/s
We would like to determine the vibratory response of the elastic structure shown below. The simple structure consists of a beam made of steel, E = 69x10^9 N/m^2, and two concentrated masses, free to bend laterally only, and m1 and m2 are located as shown below. Assume the dimensions of masses are 40x40x80 mm and made of a material that has a ρ = 2700 kg/m^3. The length of the beam is 400 mm, width is 40 mm, height is 2 mm, and is clamped as shown in Figure 1 (Be careful with the usage of the width and height values, check with the physical structure given in the figure). The mass 1 is located at the center of the beam.
a) Find the equivalent mass of the structure as it is seen at the free end of the beam (i.e., at coordinate x2).
b) Find the equivalent stiffness of the structure as it is seen at the free end of the beam (i.e., at coordinate x2).
c) Build a single-degree of freedom lumped parameter model of the structure for the coordinate x. Write the equation of motion of the system.
d) Plot the deformation profile of the beam if the free end is displaced by 2 mm.
e) Assume that there is no damping of the system. Find out the displacement response when initial conditions of the following are applied to the free end and plot displacement versus time (between 0 and 0.2 seconds) for each case in a single graph. Which of the initial conditions is dominant to the response for case 3? Why? Also, draw separately the velocity and acceleration of the system for each case. For Case 1, draw the displacement, velocity, and acceleration response in a single graph.
a. Case 1: x0 = 2mm, x0' = 0mm/s
b. Case 2: x0 = 0mm, x0' = 2mm/s
c. Case 3: x0 = 2mm, x0' = 2mm/s
d. Case 4: x0 = 2mm, x0' = 500mm/s
