with masses declared as grams and static stiffness as N/mm, how do I dimension them for Hz ?
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$\begingroup$ Homework? What have you done so far? If you write out the differential equation for a vibrating assembly, how do the mass and stiffness show up? $\endgroup$– TimWescottNov 19, 2019 at 16:22
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$\begingroup$ What are the units in N ? $\endgroup$– Solar MikeNov 19, 2019 at 16:39
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$\begingroup$ Mike I don't understand your question $\endgroup$– FabioSpaghettiNov 19, 2019 at 20:23
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$\begingroup$ What do you multiply to get Newtons... $\endgroup$– Solar MikeNov 19, 2019 at 22:49
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$\begingroup$ multiply from what to Newtons Mike ? $\endgroup$– FabioSpaghettiNov 20, 2019 at 15:16
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2 Answers
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Units of N, mm (for the stiffness k), and g (for the mass m) are not consistent. You need to convert these values to use in the basic equation for the frequency: $$f=\frac{1}{2*\pi} \sqrt \frac{k}{m}$$ Knowing that $1 \, N = 1 \, kg*m/s^2$, $1 \, m = 1000 \, mm$, and $1 \, kg = 1000 \, g$, you get the following: $$f=\frac{1}{2*\pi} \sqrt \frac{N/mm*((1 kg*m/s^2)/N)*(1000 mm/m)}{g*(1 kg/1000 g)}$$ All of the units cancel except for $\sqrt {1/s^2}$. Doing the math gives the following result: $$f = \frac{1000}{2*\pi} \sqrt \frac{k \; in \; N/mm}{m \; in \; g}$$ where f is in units of cycles/sec = Hz.
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$\begingroup$ You don't have to convert everything into basic SI units, any consistent set of units will do. If the OP wants to use N, mm, and Hz (i.e. seconds) then in "F=ma" you increase the measure of acceleration by a factor of 1000 changing from m to mm, so you need to decrease the measure of mass by 1000, i.e. measure it in Mg not kg. The engineering company I work for uses exactly that set of units, which conveniently give pressure in MPa. Of course density is in the slightly weird units of megagrams per cubic millimeter, but it's easy to remember that just means "specific gravity $\times10^{-9}$" $\endgroup$ Nov 20, 2019 at 22:35
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$\begingroup$ … and forgetting the $10^{-9}$ gives answers that are so completely wrong that nobody would ever believe them, which is the best sort of error to make! $\endgroup$ Nov 20, 2019 at 22:36
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One Newton is equal to 101.971grams and one gram is 0.00980665N. In free undamped vibration we have to convert grams to Newtons,
$$ \omega_n = \sqrt \frac{k}{0.00980665*m_{grams}}$$
$$ T= \frac{2\pi}{\omega_n} $$ $$ f_n=\frac{1}{T}= \frac{\omega_n}{2\pi}$$
Edit
Going over my answer after @John answer, I realize that I was wrong. I leave my answer should it be an example of how you can be wrong.
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$\begingroup$ Force is not mass. This is just nonsense. $\endgroup$ Nov 20, 2019 at 2:34
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$\begingroup$ @alephzero, I don't know what you mean. I am just using a constant. $\endgroup$– kamranNov 20, 2019 at 3:33
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$\begingroup$ Thank you Kamran, I was rather searching to dimension analysis for this formula $\endgroup$ Nov 20, 2019 at 15:16
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$\begingroup$ @FabioSpaghetti, I'm glad to be able to help. $\endgroup$– kamranNov 20, 2019 at 16:19
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$\begingroup$ @Kamran I agree with alephzero that your equation is incorrect. I am not sure where you got 1 N = 101.971 g. I have posted a new answer. $\endgroup$ Nov 20, 2019 at 18:39