Dievsky V.A. - Solving problem D7 option 6 task 2

For the mechanical system described by task D7-06 (Task 2) by Dievsky, a weightless viscous damper is installed at point B parallel to the axis of the elastic element, which creates a resistance force proportional to the speed of point B: R = -bvB, where b = 20 Ns/m - resistance coefficient. In addition, the system is subject to a driving force F = F0 sin pt, where F0 = 60 N, p = 25 s-1 - 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 system.

To determine the amplitude of purely forced oscillations of the system, it is necessary to consider the equation of motion of the system. Taking into account the action of the resistance force and the driving force, it will look like:

m d^2x/dt^2 + b dx/dt + kx = F0 sin(pt)

where m is the mass of the system, k is the rigidity of the elastic element, x is the displacement of point B from the equilibrium position, t is time.

For purely forced oscillations of the system, it is necessary to set x = Acos(pt), where A is the amplitude of oscillations. Substituting this expression into the equation of motion, we get:

-mAp^2cos(pt) - bAp^2sin(pt) + kAcos(pt) = F0sin(pt)

Considering that cos(pt) and sin(pt) are orthogonal to each other, we can write two equations to determine the amplitude A:

-mAp^2 + kA = 0

bAp^2 = F0

Solving these equations, we get:

A = F0 / (bp^2)

Thus, the amplitude of purely forced oscillations of the system is equal to F0 / (bp^2) = 0.096 m.

Dievsky V.A. - Solving problem D7 option 6 task 2

We present to your attention the solution to problem D7 option 6 task 2 from the author V.A. Dievsky. This digital product is suitable for students and teachers who are learning mechanics and need help solving problems.

Solving this problem will help you understand how to calculate the amplitude of purely forced oscillations of a system, taking into account the action of the resistance force and the driving force. The author provided a detailed description of all stages of solving the problem, as well as step-by-step instructions that will help you understand solving similar problems in the future.

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Dievsky V.A. - Solution to problem D7 option 6 task 2 describes a mechanical system that is equipped with a weightless viscous damper installed at point B parallel to the axis of the elastic element. The damper creates a drag force proportional to the speed of point B, where the drag coefficient is 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. It is required to determine the amplitude of purely forced oscillations of the system.

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