- 7

- 0

## Main Question or Discussion Point

Trying to solve the ODE

[tex]mx''(t) + bx'(t) + kx(t) = F(t)[/tex]

with [tex]m[/tex] measured in [tex]Kg[/tex], [tex]b[/tex] in [tex]Kg/s[/tex] and [tex]Kg/s^2[/tex], [tex]F(t)[/tex] in [tex]Kgm/s^2[/tex] and [tex]x(t)[/tex] in [tex] m[/tex] with initial conditions [tex]x(0) = 0[/tex] and [tex]x'(0) = 0[/tex], i got the following Green's function

[tex]G(t,t') = \frac{1}{m\omega} e^{-\omega_1(t-t')}\sinh\left[\omega(t-t')\right][/tex]

for [tex]t \geq t'[/tex], with

[tex]\omega=\sqrt{\omega_1^2+\omega_0^2}[/tex]

[tex]\omega = \sqrt{\frac{k}{m}}[/tex]

[tex]\omega_1=\frac{b}{2m}[/tex].

Adding up all the responses to unit impulses i got

[tex]x(t) = \int_0^t F(t')G(t,t')dt'[/tex]

So far, everything was ok.

But when i tried to get the response of the ODE for a unit impulse of amplitude [tex]f_0[/tex] at [tex]t = t_0[/tex] ([tex]F(t) = f_0\delta(t-t_0)[/tex]) i integrated the equation above but the dimension of the expression i got for [tex]x(t)[/tex] is [tex]m/s[/tex] instead of [tex]m[/tex].

If [tex]F(t)[/tex] is any other function it works fine, but with [tex]F(t) = f_0\delta(t-t_0)[/tex] I get this dimensional error.

Can someone help me, please?

[tex]mx''(t) + bx'(t) + kx(t) = F(t)[/tex]

with [tex]m[/tex] measured in [tex]Kg[/tex], [tex]b[/tex] in [tex]Kg/s[/tex] and [tex]Kg/s^2[/tex], [tex]F(t)[/tex] in [tex]Kgm/s^2[/tex] and [tex]x(t)[/tex] in [tex] m[/tex] with initial conditions [tex]x(0) = 0[/tex] and [tex]x'(0) = 0[/tex], i got the following Green's function

[tex]G(t,t') = \frac{1}{m\omega} e^{-\omega_1(t-t')}\sinh\left[\omega(t-t')\right][/tex]

for [tex]t \geq t'[/tex], with

[tex]\omega=\sqrt{\omega_1^2+\omega_0^2}[/tex]

[tex]\omega = \sqrt{\frac{k}{m}}[/tex]

[tex]\omega_1=\frac{b}{2m}[/tex].

Adding up all the responses to unit impulses i got

[tex]x(t) = \int_0^t F(t')G(t,t')dt'[/tex]

So far, everything was ok.

But when i tried to get the response of the ODE for a unit impulse of amplitude [tex]f_0[/tex] at [tex]t = t_0[/tex] ([tex]F(t) = f_0\delta(t-t_0)[/tex]) i integrated the equation above but the dimension of the expression i got for [tex]x(t)[/tex] is [tex]m/s[/tex] instead of [tex]m[/tex].

If [tex]F(t)[/tex] is any other function it works fine, but with [tex]F(t) = f_0\delta(t-t_0)[/tex] I get this dimensional error.

Can someone help me, please?