Show that the function given in Equation 15-42 actually satisfies 15-41 -bv - kx

Question

Show that the function given in Equation 15-42 actually satisfies 15-41 -bv - kx = ma. Substituting dx/dt for v and d2x/dt2 for a and rearranging give us the differential equation m d2x / dt2 + b dx / dt + kx = 0. Damped Harmonic Motion The mechanical energy E in a real oscillating system decreases during the oscillations because external forces, such as a drag force, inhibit the oscillations and transfer mechanical energy to thermal energy. The real oscillator and its motion are then said to be dumped. If the damping force is given by Fvector4 = -bvvector, where vvector is the velocity of the oscillator and b is a damping constant, then the displacement of the oscillator is given by x(t) = xm e-te2m cos(omega't + phi).

Explanation / Answer

I am attempting to derive the equation for dampened harmonic motion from the differential equation:
md2x/dt2+bdx/dt+kx=0


It would be painful to type the whole derivation out, but there is one point relevant to the problem I am having. In order to get SHM, we assume that the dampening force is small compared to the restoring force, so I assumed:
b24km<0

Which allowed me to write:
(b24km)=i(4kmb2)


Which aided in my analysis. Now, after a lot of algebra I get something that looks like this:

x(t)=Aexp(bt2m)4km/4kmb2Cos((1/2m(4kmb2)tArcTan(b/(4kmb2))


The correct formula looks like this:
x(t)=Aexp(bt/2m)Cos((12m4kmb2)t+)

you can compare this eqn with tha eqn 15.42

Related Questions

Navigate

• Browse (All)
• Browse S
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.