Forced vibration with decreasing force amplitude

In summary, the problem discussed is a damped forced vibration problem with an applied force that decreases at a constant rate over 11 seconds. The system is critically damped and its differential equation can be solved using the method of variation of parameters. The particular solution can be expressed as a function of the complimentary solution and its derivatives, and the constants can be determined by substituting in the initial conditions. The complete solution to the differential equation can then be written as the sum of the complimentary solution and the particular solution.
  • #1
abrandt
10
0
I have a damped forced vibration problem. Most problems I have dealt with have a constant amplitude sinusoidal force such as F_0sin(wt). Let's say you have an applied force that decreases each second at a constant rate over 11 seconds. It looks like this:

F=0.5*[(11-t)sin(2*pi*t)+(11-t)]; (0<= t >=11)

The graph of this force has been attached.

The system is also critically damped so the system's differential equation is:

m[d^2/dt^2](x) + c(dx/dt) +kx = 0.5*[(11-t)sin(2*pi*t)+(11-t)]

From here I'm not too sure how to proceed. Normally with forced vibration the solution would include both a complimentary and particlular soluntion since it is a nonhomogenous linear second order differential equation. So

x = x_c + x_p

If the force had a constant magnitude (steady state vibration) and assuming critical damping x_c would look like:

x_c=(A+B*t)e^(-W_n*t)

but since the vibration is not steady state does this equation hold true? I assume it does only because the complimentary solution is found by setting the force to zero and assuming only free vibration. A and B are then found using the initial conditions where:

A= x_c(0) or is it A=x(0)
B=Dx(0)/dt + w_n*x(0)

Then x_p needs to be found and this I am having trouble with. how would one solve this part of the differential? The variables such as k and m should be left as variables. Do i need an equation solver that can handle solving with variables such as Maple?
 

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  • #2
Any help is appreciated. The solution for the problem can be found by solving the non-homogeneous linear differential equation using the method of variation of parameters. This means expressing the particular solution as a function of the complimentary solution and its derivatives, and then finding the constants in the particular solution by substituting in the initial conditions. The general complimentary solution can be written as: x_c = c_1e^(r_1t) + c_2e^(r_2t)where r_1 and r_2 are the two solutions to the characteristic equation r^2 + c/m r + k/m = 0. The particular solution can then be expressed as: x_p = v_1*x_c + v_2*(dx_c/dt) + v_3*(d^2x_c/dt^2) where v_1, v_2, and v_3 are constants that need to be determined. To find these constants, substitute in the initial conditions. In this case, the initial position and velocity are both zero, so we have: x(0) = x_c(0) + x_p(0) = 0 v(0) = dx_c(0)/dt + dx_p(0)/dt = 0 a(0) = d^2x_c(0)/dt^2 + d^2x_p(0)/dt^2 = F(0)/m = 0.5/m Solving these equations for v_1, v_2, and v_3 gives us the particular solution. The complete solution to the differential equation can then be written as: x = x_c + x_p where x_c = c_1e^(r_1t) + c_2e^(r_2t) x_p = (0.5/m)((r_2 - r_1)e^(r_1t) + (r_1 - r_2)e^(r_2t))*[c_1e^(r_
 

1. What is forced vibration with decreasing force amplitude?

Forced vibration with decreasing force amplitude is a phenomenon where a system or structure is subjected to an external force that decreases in amplitude over time, causing the system to vibrate with varying frequencies and amplitudes.

2. What are the causes of forced vibration with decreasing force amplitude?

Forced vibration with decreasing force amplitude can be caused by various factors such as wind gusts, seismic activity, machinery vibrations, and other external forces acting on a structure.

3. How does forced vibration with decreasing force amplitude affect structures?

Forced vibration with decreasing force amplitude can cause fatigue and stress on structures, leading to potential damage or failure if not properly accounted for in design and maintenance.

4. What are some techniques for analyzing forced vibration with decreasing force amplitude?

Some techniques for analyzing forced vibration with decreasing force amplitude include frequency domain analysis, time domain analysis, and modal analysis, which can help identify the resonant frequencies and responses of a structure to external forces.

5. How can forced vibration with decreasing force amplitude be mitigated?

Forced vibration with decreasing force amplitude can be mitigated through various methods such as damping, isolation, and structural modifications to reduce the effects of external forces on a structure.

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