# Natural Frequency Question SI Units

1. Jul 1, 2011

### yanaibarr

i have a basic question about the natural frequency of a system.
for a mass (M), spring (k constant) undamped system the natural frequerncy is:
w_n=sqrt(k/M)
the units of w_n according to a lot of resources i found on the internet & textbooks are [rad/sec], my question is why?
if i use the k constant units divided by the mass i get [Hz]:
[k]/[M]=[N/m]/[kg]=[kg*m/s^2*m]/[kg]=[1/s^2]
[w_n]=sqrt([k]/[M])=[Hz]

i'll appreciate a clarification in this subject.

thanks.

2. Jul 1, 2011

### Studiot

Hello yanaibarr, welcome to Physics Forums

Radians are used because the solution to the governing differential equation is in terms of angualar measure y = Asin(x-ct) and radians (not degrees), being the natural numbers you obtain from such an expression.

3. Jul 1, 2011

### Redbelly98

Staff Emeritus
Actually you got 1/s, not Hz, for the units. You are assuming that 1/s always means Hz (= cycles/s), but that is not always the case.

Frequency can be measured in rad/s or cycles/s. Both radians and cycles are considered unitless, so both types of frequency can show up as 1/s if you use an equation to figure out the units.

4. Jul 1, 2011

### gsal

Actually, Hz is exactly 1/s, and nothing else.

And radians are units to measure angles and represents a very well defined fraction of a revolution...a radian is the angle that you can measure at the center of a circle between two radial lines (from the center to the circle's circumference) whose distance between the two tips of the radii along the perimeter of the circle is again the length of the radius itself.....I think this came out rather convoluted...google it, I am sure you will get a nice picture, somewhere.

Needless to say, pi (3.1415926) has the definition of how many times the diameter of a circle fits along the circumference of the circle itself...see the relation? the diameter fits 3.1416 times along the circumference, and so the radius fits twice as many times...6.2832, that means that a radian spans an angle of 360/6.2832 = 57.29 degrees

So, when you are talking about angular velocity or angular frequency in radians per second, there is a straight relationship between radians per second and revolutions per second (or 1/s)...so, if you have something going at 1 revolution per second (1/s or Hz) is also going at 6.2832 radians per second (rad/s).

Conversely, if you have something going at 1 radian per second, it is also going at 1/6.2832 revolutions per second or 0.159 Hz.

5. Jul 1, 2011

### Redbelly98

Staff Emeritus
Yes, but 1/s could mean either Hz or rad/s. That was my point. The frequency calculated from the $\sqrt{k/m}$ formula has units of 1/s, but is in rad/s, not Hz.

6. Jul 16, 2011

### yanaibarr

But there is still one thing that bothers me,
if i have a first order system the basic transfer function will be:
K/(tau*s+1)
where K is the Gain, and tau is the system's time constant.
tau's units, according to what i've learned, are [sec].
but aren't the s plane units in [rad/sec] (s=jw+sigma)?
That means that tau should be given in [sec/rad] to match the "1" in the transfer function.
I know that rad can be considered "unitless" but when dealing with actual numbers it matters if the system's time constant is 1 [sec] or 1[sec/rad]= 2*pi [sec].

again, i'll appreciate a clarification.

Thanks

7. Jul 18, 2011

### Bob S

If you go back to the basic physics equations:

1) F = -kx
2) m d2x/dt2 = -kx
3) d2x/dt2 = -(k/m) x
4) Substitute a solution x(t) = A sin(ωt) + B cos(ωt)
5) Find ω2 = k/m

Note that ω has units radians per second.

Bob S

Added: The SI units for one complete rotation through 4 quadrants is 2pi (2 π) radians. 360 degrees is NOT an SI unit.

Last edited: Jul 18, 2011
8. Jul 23, 2011

### olivermsun

In step 4 one could substitute a solution in the form x(t) = A sin(2πft) + B cos(2πft), and then step 5 would have (2πf)^2 = k/m and so on...

It just depends what solution form is substituted and whether one like carrying around factors of 2π in the math. (And it often leads to confusion, at least for me!)