It Can Be Shown That . . .

Recall that the spring-mass-dashpot system is governed by the following 2nd Order Ordinary Differential Equation:

(1")     .

The free vibration response is based upon the homogeneous solution to Equation 1". The response is induced by displacing the system an initial amount and then allowing it to reachieve equilibrium. Thus there will be no additional, external forces applied to the system after the initial displacement. So substituting F(t)=0 into Equation 1" yields:

(1)     .

Guess a solution of the form:

(2")     .

Based on the Theorem of the Existence and Uniqueness of Solutions to Ordinary Differential Equations, the solution is valid if it will satisfy the initial (boundary) conditions.

The Boundary Conditions for a cantiliver beam, fixed at one end are:

(2)     .

Taking the first derivative of Equation 2" with respect to time yields:

(3:)     .

Taking the second derivative of Equation 2" with respect to time yields:

(4")     .

Substitute Equation 2", Equation 3", and Equation 4" into Equation 1 yields:

(5")     .

Equation 5" can be reduced to the following for non-zero , A, and t:

(6")     .

Rearranging Equation 6" yields:

(7")     .

Using the Quadratic Equation to solve for yields:

(8")     

and

(9")     .

Simplifying Equation 8" and Equation 9" yields:

(10")     

and

(11")     .

Using the principles of superposition and proportionality that state for [[lambda]]1 and [[lambda]]2 that are solutions to the differential equation, a linear combination of the two solutions is also a solution to the differential equation.

Thus:

(12")     .

Equation 12" can be simplified to:

(13")     

where:

is Over Damped,

is Critically Damped, and

is Under Damped.

For the Under Damped case which applies to our apparatus, Equation 13" is given by:

(14")     .

Then using the identity:

(15")     ,

Equation 14" is given by:

(16")     .

Defining the Ringing Frequency (or damped natural frequency) as:

(17")     

and subsituting into Equation 16" yields:

(18")     .

To confirm the solution, use the Theorem on the Existence and Uniqueness of Solutions to Linear Ordinary Differential Equations.

Applying the Boundary Condition:

(19")     

to Equation 18" yields:

(20")     .

After algebraic manipulation, Equation 20" becomes:

(21")     .

Comparing Equation 19" and Equation 21",

(22")     .

Applying the other Boundary Condition:

(23")     

to Equation 18" yields:

(24")     .

After algebraic manipulation, Equation 24" becomes:

(25")     

and

(26")     .

Substituting Equation 22" and Equation 26"

(3)     .