Dievsky V.A. - Solving problem D7 option 4 task 2

D7-04 (Task 2) The mechanical system that needs to be investigated is equipped with a weightless viscous damper. The damper is installed at point B parallel to the axis of the elastic element and creates a resistance force that is proportional to the speed of point B. The resistance coefficient is b = 20 Ns/m. In addition, a driving force F = F0 sin pt begins to act on the system, where F0 = 60 N, and p = 25 s-1 is the amplitude and frequency of the driving force. The driving force is applied at point B and acts parallel to the axis of the elastic element. If point B coincides with point A, the diagram indicates B = A. It is necessary to determine the amplitude of purely forced oscillations of the mechanical system.

To determine the amplitude of purely forced oscillations, it is necessary to consider the behavior of the system in the mode of steady oscillations, when the resistance force of the damper is equal to the external driving force. In this case, we can write the equation of motion of the system:

mx'' + bx' + kx = F0 sin pt,

where m is the mass of the system, x is the coordinate of point B, k is the stiffness coefficient of the elastic element.

A particular solution to this equation can be sought in the form x = A sin(pt - φ), where A is the amplitude of oscillations, φ is the initial phase of oscillations.

Substituting this expression into the equation of motion, we get:

-Amp^2 m sin(pt - φ) + Abp m cos(pt - φ) + Abp b sin(pt - φ) + Ak A sin(pt - φ) = F0 sin pt,

где Amp = p^2 m + k, bp = p b.

To determine the amplitude of purely forced oscillations, it is necessary to perform calculations and obtain the value of amplitude A.

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Description of the product "Dievsky V.A. - Solution of problem D7 option 4 task 2":

This digital product represents the solution to problem D7-04 (task 2) from a mechanics textbook, authored by V.A. Dievsky. The problem considers a mechanical system that is equipped with a viscous damper and begins to be subjected to a driving force. The product provides a complete solution to the problem, including the process of solving it and obtaining the values ​​necessary to determine the amplitude of purely forced oscillations of the system.

The product is designed in a convenient and beautiful HTML format, which provides easy access to information and ease of use. The solution to the problem is intended for students and teachers studying mechanics and wishing to deepen their knowledge in this area. By purchasing this product, you receive a high-quality and complete solution to the problem, which will help you successfully master the material and prepare for exams.

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This product is a solution to problem D7 option 4 task 2, which describes a mechanical system equipped with a weightless viscous damper and subject to a driving force. The task gives the system parameters, such as the resistance coefficient and the amplitude and frequency of the driving force, and also requires determining the amplitude of purely forced oscillations of the system. Solving a problem involves analyzing forces, differential equations, and applying appropriate formulas and methods to arrive at the final answer.

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