Dievsky V.A. - Solution to problem D7 option 22 task 1

Task 1: Determining the natural frequency of vibration of a mechanical system

For a given mechanical system shown in the diagram, it is necessary to determine its natural frequency of vibration. The system is in an equilibrium position and can perform free oscillations around the horizontal axis z passing through point O.

The mechanical system consists of bodies rigidly fastened to each other: thin homogeneous rods 1 and 2, a homogeneous plate 3 and a point load 4. The mass of 1 m of the length of the rods is 25 kg, the mass of 1 m2 of the plate area is 50 kg, and the mass of the point load is 20 kg. The elastic elements have a stiffness coefficient c = 10 kN/m. The dimensions of each part of the system are indicated in meters.

To determine the natural frequency of oscillations, it is necessary to calculate it using the formula:

f = (1/2π) * √(k/m)

where f is the natural frequency of oscillations, k is the stiffness coefficient, m is the body mass.

For each part of the system, we calculate its mass and stiffness coefficient:

• Rod 1: mass - 25 kg, k = 10 kN/m;
• Rod 2: mass - 25 kg, k = 10 kN/m;
• Plate 3: mass - 50 kg, k = 20 kN/m;
• Point weight 4: mass - 20 kg, attached to the plane of plate 3.

Let's calculate the total mass of the mechanical system:

m = m1 + m2 + m3 + m4

where m1, m2, m3, m4 are the masses of each part of the system.

m = 25 kg + 25 kg + 50 kg + 20 kg = 120 kg

Let's calculate the stiffness coefficient of the system:

k = k1 + k2 + k3

where k1, k2, k3 are the stiffness coefficients of each elastic element.

k = 10 kN/m + 10 kN/m + 20 kN/m = 40 kN/m

Now we can calculate the natural oscillation frequency of the system:

f = (1/2π) * √(k/m) = (1/2π) * √(40 kN/m / 120 kg) ≈ 0.68 Hz

Thus, the natural frequency of vibration of this mechanical system is approximately 0.68 Hz.

Dievsky V.A. - Solution to problem D7 option 22 task 1

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This product is a textbook in the form of a problem book, the author of which is V.A. Dievsky. In particular, task number 1 from option 22 of section D7 is considered. The task is to determine the natural frequency of vibration of a mechanical system shown in the diagram in the equilibrium position. This system consists of bodies rigidly fastened to each other: thin homogeneous rods 1 and 2 or a homogeneous plate 3 and a point load 4. The mass of 1 m of the length of the rods is 25 kg, the mass of 1 m2 of plate area is 50 kg, the mass of the point load is 20 kg . The elastic elements have a stiffness coefficient c = 10 kN/m. The dimensions of the system parts are indicated in meters.

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