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Old Feb 23rd 2017, 05:15 PM   #1
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Determine the natural frequency

Determine the natural frequency in Hz of the system sketched below given the following values:
Mass m=10kg
Spring stiffness k=18kN/m
Length a=114mm
Length b=120mm

I know that ω=(k/m)^0.5
So probably I need to calculate m.
Should I write m=10*9.81*(0.114+0.12)?
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Old Feb 24th 2017, 12:36 PM   #2
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I assume you know that the natural frequency for a simple spring mass system comes from the solution to the differential equation:

$\displaystyle m\ddot x + k x = 0$

The solution of which gives the natural frequency $\displaystyle \omega = \sqrt {k/m}$, right?

The trick in this problem is that the lever arm serves to "magnify" the spring force acting on m by a factor of (a/b). If we set up variables x and y where x = left-right displacement of the mass and y = up-down displacement of the spring, you have:

x = (b/a) y

Spring force = ky,

Force acting on m = (a/b) times the spring force, or (a/b) ky.

The differential equation for motion of mass m is:

$\displaystyle m \ddot x + (a/b) k y = 0$

Sub y = (a/b) x to get:

$\displaystyle m \ddot x + (a/b)^2 kx = 0$

Can you take it from here?
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Last edited by ChipB; Feb 24th 2017 at 01:14 PM.
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