- #1
grilo
- 6
- 0
Trying to solve the ODE
mx''(t) + bx'(t) + kx(t) = F(t)
with m measured in Kg, b in Kg/s and Kg/s^2, F(t) in Kgm/s^2 and x(t) in m with initial conditions x(0) = 0 and x'(0) = 0, i got the following Green's function
G(t,t') = \frac{1}{m\omega} e^{-\omega_1(t-t')}\sinh\left[\omega(t-t')\right]
for t \geq t', with
\omega=\sqrt{\omega_1^2+\omega_0^2}
\omega = \sqrt{\frac{k}{m}}
\omega_1=\frac{b}{2m}.
Adding up all the responses to unit impulses i got
x(t) = \int_0^t F(t')G(t,t')dt'
So far, everything was ok.
But when i tried to get the response of the ODE for a unit impulse of amplitude f_0 at t = t_0 (F(t) = f_0\delta(t-t_0)) i integrated the equation above but the dimension of the expression i got for x(t) is m/s instead of m.
If F(t) is any other function it works fine, but with F(t) = f_0\delta(t-t_0) I get this dimensional error.
Can someone help me, please?
mx''(t) + bx'(t) + kx(t) = F(t)
with m measured in Kg, b in Kg/s and Kg/s^2, F(t) in Kgm/s^2 and x(t) in m with initial conditions x(0) = 0 and x'(0) = 0, i got the following Green's function
G(t,t') = \frac{1}{m\omega} e^{-\omega_1(t-t')}\sinh\left[\omega(t-t')\right]
for t \geq t', with
\omega=\sqrt{\omega_1^2+\omega_0^2}
\omega = \sqrt{\frac{k}{m}}
\omega_1=\frac{b}{2m}.
Adding up all the responses to unit impulses i got
x(t) = \int_0^t F(t')G(t,t')dt'
So far, everything was ok.
But when i tried to get the response of the ODE for a unit impulse of amplitude f_0 at t = t_0 (F(t) = f_0\delta(t-t_0)) i integrated the equation above but the dimension of the expression i got for x(t) is m/s instead of m.
If F(t) is any other function it works fine, but with F(t) = f_0\delta(t-t_0) I get this dimensional error.
Can someone help me, please?
