Simp,e Harmonic Motion with Damping

[ga400man]

Hey, I have a simple question to ask. Some sources I have seen develop the solution for the underdamped case with an exponential term multiplied by a Cosine term. This is due to the application of the Euler identity to the solution with two exponentials each with one of the complex conjugate roots. The sin terms cancel leaving just the cosine. Other sources show a solution with the exponential multiplied by both a sine and a cosine term (Wolfram Mathematics for example). Are these solutions the same essentially or am I missing something?

 

[DrFurious]

I'm not 100% sure what you're asking, but solutions to differential equations (like the damped harmonic oscillator) are best solved by use of complex variables.. the complex exponential in this case. Read the Wikipedia page on "Harmonic oscillator" and it explains it. A sin or a cos is only really half the right answer, but will do in a lot of cases.

 

[timthereaper]

DrFurious is right. The complete general solution to the differential equation involves both sine and cosine terms. However, the specific solution depends on the boundary conditions. There might be specific boundary conditions that make the coefficient for the sine part zero, leaving only the cosine term. That depends on the problem itself.

I guess you could also reason that the sine and cosine functions differ only by a phase shift of Pi/2, so you could just express the solution in terms of a sum of cosines with phase shifts.

 

[AlephZero]

If you only want to study the general behavior of the solution, then it doesn't really matter whether you take the sin or cos option.

On the other hand if you want to specify some boundary conditons to look at the response in a particular physical problem, then in general you have two boundary conditions (e.g. x and x' at time = 0) and you need both the sin and cos terms to satisfy them.

Alternatively, you can say the solution is the real part of $A e^{st}$ where s is complex and A is an arbitrary complex constant. The two boundary conditions can be satisfied by taking suitable values of the real and imaginary parts of A.

 

[ga400man]

Thanks to everyone who replied. I guess I was too hasty in stating my concern and should have been less vague. The textbook solution that I saw stated that the Euler identity could be used to remove the two terms in e with complex exponents to be replaced with a 2cos(wt+p) term (p being the phase shift) but I didn't see how that was possible since the the two terms of e have different constant multipliers and therefore I am forced to retain the sin terms. Perhaps using the difference and sum of angles trig identities might show me they are the same but the general solution with both sine and cosine terms seems to be the best. As you have stated the boundary condition of the dependent variable (distance in this case) at t(0) and also dx/dt at t(0) should resolve the value of the constants. However, the solution with just the cosine term and the phase shift seems to satisfy any initial conditions too.
I guess that the general solution with the sine and cosine terms should be the one used since it is the most rigourous.