Vibration (Single Degree of Freedom)

Consider the single degree of freedom (SDOF) system shown in the figure below:

Equation of Equilibrium

The equation of equilibrium governing the horizontal movement of this system, ignoring damping effects, is as follows:

m ¨ x (t) + k x (t) = P (t)

Where:

$x$	is displacement of the node (m);
$¨ x$	is the second derivative of displacement, $x$ , (m/s²);
$m$	is the mass at the node (kg);
$k$	is the stiffness of element (N/m);
$P$	is the applied forcing function (N);

Element Stiffness

The stiffness of the element,

k

, is given by:

k = \frac{3 E I}{L^{3}}

Where:

$E$	is Young's modulus for the material (Pa);
$I$	is the second moment of area (m⁴);
$L$	is the length of the element (m);

Nodal Mass

The mass can be assumed to be made up of point mass at the end of the element plus half of the distributed mass of the element:

m = m_{1} + \frac{m^{'} L}{2}

Where:

$m_{1}$	is the concentrated mass at the end of the element (kg);
$m^{'}$	is the distributed mass of the element (kg/m);

Free Vibration

The natural circular frequency (radians/second) of this undamped, single degree of freedom system is as follows:

ω = \sqrt{\frac{k}{m}}

The natural frequency in Hertz (cycles/second) is:

f = \frac{ω}{2 π}

The natural period of vibration (seconds) is the inverse of this:

T = \frac{1}{f} = \frac{2 π}{ω}

Forced Vibration Magnification

If we define a system with a harmonically varying load

P (t)

of sine-wave form, amplitude

p_{0}

and circular frequency

¯ ω

m ¨ x (t) + k x (t) = p_{0} sin ¯ ω t

Then define a ratio of the applied loading frequency to the natural free-vibration frequency as above:

β = \frac{¯ ω}{ω}

Plotting the steady state response amplitude against this ratio we can see that as the applied load frequency approaches the natural frequecy of the system this response amplitude approaches an asymptote. See the figure below.

Figure 2: Forced Vibration Magnification