Duration: 23:55

11 views

1

The discussion of vibration theory usually begins with the analysis of a simple mass, spring and damper system. This is because once you analyze the vibration process for this system you can then apply the results to the most complicated vibrating structure. It will be shown later how complex structures vibrate in a superposition of a unique set of different deformation patterns (called mode shapes). Each mode shape has its own resonance frequency and responds to vibratory forces in a manner described by the same differential equations as are used to describe single mass, spring and damper vibration response.

Figure 3 represents a mass, m1, supported by a spring of stiffness, k1, and a damper having a viscous damping value, c1. Consider a vibratory force, f1, applied to the mass as indicated in the free body diagram. The spring and damper react with forces, fk and fc, and the algebraic expressions for these forces are given as Hook’s Law for the spring and the viscous force formula for the damper. The motion resulting from the applied force and reaction forces is given by Newton’s Second Law as shown in the figure.



Figure 3.  Vibration motion of a mass supported by a spring and a damper and responding to an applied vibratory force is analyzed using Newton’s Second Law.

Expanding the Newton’s Law equation of Figure 1 and rearranging gives

 

(1)

 

Since we are interested in the way the system vibrates at the various frequencies across the spectrum it is convenient to perform the Fourier Transform on equation 1), expressing all variables as a function of frequency, ω (radians/sec), rather than time.

 

(2)

 

Note that frequency in Hz, ν, is equal to (1/2π)ω.

 

This second order differential equation with constant coefficients may be expressed as an algebraic equation if we take note that at any frequency of the spectrum there are simple algebraic relationships between displacement, velocity and acceleration:

(3)

 

(4)

 

where we have used  the imaginary number, I, to represent the 90-degree phase shift between the cosine and sine functions.

 

(5)

 

Since these equations apply at all frequencies across the spectrum we see that they provide us with the Fourier Transforms so that equation 2) may now be written as an algebraic equation (no derivatives) using displacements in place of velocity and acceleration.

 

(6)

 

Now we can factor out the displacement:

 

(7)

 

At this point we will not focus on the displacement solution of the equation, but rather we wish to find the ratio, displacement divided by force. This is known as the Frequency Response Function (FRF), h(ω):

 

(8)

 

The FRF may be presented in a more useful form after rationalizing the denominator and introducing a couple of new definitions for β and ζ.

 

(9)

 

β = ω/ωr   (ratio of frequency to resonance frequency)

ζ = c/cc     (ratio of damping to critical damping)

 

With a damping value, c, equal to zero the mass-spring-damping system, if released from a displaced position, would vibrate indefinitely. The damping is said to be underdamped if, when released from a displaced position, the mass-spring-damping system vibrates at resonance but decays with time. if the damping value has a value known as the critically damped value, cc, there will be barely enough damping to avoid oscillation.

Consider the process in which the system of Figure 3 is excited into free vibration using an impact force applied with a hammer. The impact force (during the brief time the hammer is in contact with the mass) has the form of a half-sine pulse that is very short compared to one cycle of vibration.



Figure 4.  The force-time pulse resulting from  a hammer impact applied to the mass of Figure 3.

 

At a time very close to zero time (force pulse has terminated), the displacement of the mass is approximately zero, the velocity has some value, v0, and the acceleration is zero.  Now, the system vibrates at its resonance frequency, decaying in accordance with the solution of the differential equation 1):



10)

 

 

Where V0 is the initial velocity, ν is the system resonance frequency, and ζ is the damping factor.  This solution is graphed in Figure 4.



 

Figure 5.  Damped oscillations from solution to the  mass-spring-damper system differential equation.  The dashed curve envelopes the positive peaks.

 

Now, getting back to the frequency domain solution of equation 9), will consider this solution from a different point of view. Having derived the FRF from the differential equation, we now consider the way in which we may develop the FRF experimentally. Con
sider now that we have acquired the force vs. time Figure 4 by measuring the force. The hammer used for the impact force includes a force transducer in the hammer tip and the force-time signal is captured into our computer. A Fourier Transform is used !