Mechanical Vibrations Differential Equation

In summary, the general solution for the given differential equation is C1cos(4t) + C2sin(4t) = 0, and the particular solution may need to be solved using variation of parameters due to the discontinuities in the tangent function.
  • #1
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Homework Statement



Find the general solution for the differential equation

Homework Equations



y'' + 16y = tan(4t)

The Attempt at a Solution



I get C1cos(4t) + C2sin(4t) = 0 for my homogeneous equation.

I did the usual method of undetermined coefficients (I think I took all of the derivatives correctly). However I'm getting something strange for yp. I can't really make sense of the solution. Does this have something to do with the fact that the tangent function has discontinuities? Would this be some kind of resonance?
 
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  • #2
There is no sort of 'fixed' form for the PI of tan(4t), you might need to try variation of parameters to solve it.
 
  • #3
Oh yeah! Thanks for the help!
 

Related to Mechanical Vibrations Differential Equation

What is a mechanical vibrations differential equation?

A mechanical vibrations differential equation is a mathematical equation that describes the motion of a mechanical system that is subject to periodic disturbances or oscillations. It takes into account the mass, stiffness, and damping of the system to determine the displacement, velocity, and acceleration of the system at any given time.

What are the applications of mechanical vibrations differential equations?

Mechanical vibrations differential equations are used in various fields such as engineering, physics, and mathematics to analyze and design mechanical systems. Some common applications include analyzing the vibrations of bridges, buildings, and other structures, designing and optimizing suspension systems in vehicles, and predicting the behavior of machinery and equipment.

How is a mechanical vibrations differential equation solved?

The solution to a mechanical vibrations differential equation depends on the type of equation and the specific system being analyzed. In general, the equation is solved using techniques from calculus, such as separation of variables or Laplace transforms. Computer simulations and numerical methods are also commonly used to solve these equations.

What is the role of damping in a mechanical vibrations differential equation?

Damping refers to the dissipation of energy in a mechanical system, typically through friction or resistance. In a mechanical vibrations differential equation, damping is represented by a damping coefficient that affects the rate at which the system's energy decreases. Damping is important because it can affect the stability and behavior of a system, and must be taken into account when solving these equations.

How do mechanical vibrations differential equations relate to harmonic motion?

Harmonic motion is a type of periodic motion that can be described by a sinusoidal function. Mechanical vibrations differential equations are often used to model systems that exhibit harmonic motion, such as a mass on a spring or a pendulum. By solving these equations, we can determine the amplitude, frequency, and phase of the harmonic motion of a system.

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