I'm likely making a dumb mistake but I'm having issues coming out with the correct units for damping constant in US units. The confusion likely lies in the conversion between lbm and lbf but I can't seem to figure it out.
In a damped vibration system the critical damping constant C[sub]c[/sub] has units lbf-sec/in. If damping ratio ζ = 1.0 then C[sub]c[/sub] = 2*√(J[sub]o[/sub] * K[sub]t[/sub]) where J[sub]o[/sub] is the mass moment of inertia (lbf-in/sec[sup]2[/sup]) and K[sub]t[/sub] is the torsional spring constant (lbf-in/rad).
Using the above formula I get:
C[sub]c[/sub] = √[(lbf[sup]2[/sup] * in[sup]2[/sup])/(sec[sup]2[/sup] * rad)]
Radians disappear and the square root is taken, leaving me with:
C[sub]c[/sub] = lbf-in/sec (should be lbf-sec/in!!!)
To further add to the confusion, I tried the same method with a translational system and got the correct units. The critical damping constant for a translational system is C[sub]c[/sub] = 2*√(K*m) where K is the spring rate (lbf/in) and m is the mass (lbf-sec[sup]2[/sup]/in). This gives me the correct units of Cc = lbf-sec/in.
Can anyone spot the error???
In a damped vibration system the critical damping constant C[sub]c[/sub] has units lbf-sec/in. If damping ratio ζ = 1.0 then C[sub]c[/sub] = 2*√(J[sub]o[/sub] * K[sub]t[/sub]) where J[sub]o[/sub] is the mass moment of inertia (lbf-in/sec[sup]2[/sup]) and K[sub]t[/sub] is the torsional spring constant (lbf-in/rad).
Using the above formula I get:
C[sub]c[/sub] = √[(lbf[sup]2[/sup] * in[sup]2[/sup])/(sec[sup]2[/sup] * rad)]
Radians disappear and the square root is taken, leaving me with:
C[sub]c[/sub] = lbf-in/sec (should be lbf-sec/in!!!)
To further add to the confusion, I tried the same method with a translational system and got the correct units. The critical damping constant for a translational system is C[sub]c[/sub] = 2*√(K*m) where K is the spring rate (lbf/in) and m is the mass (lbf-sec[sup]2[/sup]/in). This gives me the correct units of Cc = lbf-sec/in.
Can anyone spot the error???