Systems with One Degree of Freedom-Applications
Case 5. Seismic Instruments
A schematic sketch of a seismic instrument is shown in Fig. 3-29. It consists of a mass m attached to the base by means of springs and dampers, that is, the base of the instrument is securely attached to a vibrating body, the motion of which is be measured. Let be the motion of the base and the motion of The relative motion
is used to indicate As illustrated, is recorded by means of a pen and a rotating drum.
Since this system is essentially the same as that shown in Fig.
its equation of motion is identical to Eq. Let x,=X , sin
Defining = - and substituting = into Eq.
(3-35) yields
m f
+ +
kx=- = sinot (3-39)which is of the same form as Eq. (3-22) with
Using the impedance method 2-6, Eq. (3-39) gives
02m
-k - 02m + 1
-where =tan-' r2) , = and = The: amplitude ratio the equation above is
The right side of this equation is identical to Eqs. (3-28) and Hence the characteristic of the equation is also portrayed in Fig. 3-16.
The relative motion between the mass m and its in a seismic instrument, or vibration pickup, can be measured mechanically as illus-trated in Fig. 3-29. For high speed operations and convenience, this motion is often converted into an electrical signal. The schemes illustrated in Figs. and ( b )for this conversion are self-evident. A typical piezoelectric accelerometer is illustrated in Fig. The piezoelectric
Rotating
Record
I Base
FIG. Schematic sketch of seismic instrument.
Systemswith One Degree of Freedom-Applications CHAP. 3
Piezoelectric discs
Frame Mounting stud [a) Pickup with strain ( b ) Variable-reluctance (c) Piezoelectric accelerometer
sensitive element pickup
FIG.3-30. Vibration pickups with electrical output.
elements are sandwiched between the mass and the frame. The voltage output of the device is due to the cyclic deformation of the piezoelectric crystals. The effective spring, damping, and mass in this accelerometer, however, are not self-evident. Since vibration measurement is a separate study, we shall not pursue the subject further.
Vibrometer: Referring to Fig. 3-16, if or r the ratio in Eq.(3-40)approaches unity regardless of the value of In other words, the relative displacement is equal to the displacement which is the motion to be measured. The phase angle is approximately 180". An instrument for displacement measurement is called a vibrometer.Since there is no advantage in introducing damping in the system, a vibrometer is designed with damping only to minimize the transient vibration.
The performance characteristics of a typical inductive type velocity pickup is illustrated in Fig. 3-31. The useful range is marked off in the figure with bold lines. Displacement and acceleration can be obtained from the electrical output of the velocity pickup by integration and differen-tiation. The natural frequency of this pickup is 8Hz. The displacement amplitude ranges from 0.0025 to 1 0 mm.
A seismic instrument to measure acceleration is called an accelerometer.Due to their small size and high sensitivity, most vibration measurements today are made with accelerometers. The veloc-ity and displacement can be obtained from the electrical output of the accelerometer by integration. For example, the motion of the piston of an internal combustion engine can be indicated in this manner.
If the motion to be measured is sin the amplitude of the acceleration is 02X,. From Eq. we obtain
SEC. Damped Forced Vibration-Harmonic 103
The quantity is a constant, since is a property of the system. The relative motion is proportional to the acceleration if the magnification factor is constant for all ranges sf operation.
A periodic vibration generally has a number of harmonic components, each of which gives a corresponding value of r ( = in Eq. (3-41).
Amplitude distortion occurs if the magnification factor -
+
changes with the harmonic components. In other words, the magnification of each harmonic component must be identical in order to reproduce the input waveform. Since R 1 when r approaches zero, an accelerome-ter is constructed such that or The percent amplitude distortion is defined as
Amplitude distortion=( R-1 ) 100% (3-42)
FIG. 3-31. characteristics of a typical inductive velocity pickup.
Systems with One Degree of Freedom-Applications CHAP.3
Frequency ratio
FIG. 3-32. Amplitude distortion in accelerometer.
This is plotted in Fig. 3-32. Note that (1)an accelerometer should be built with in order to minimize the amplitude distortion and (2) the usable range is 0.6.
Phase distortion occurs if there is a shift in the relative phase between the harmonic components in a periodic signal. Assume a periodic signal
in Fig. has two harmonic components. The recorded signal in Fig. consists of the same components without amplitude distor-tion. The relative phase between the components, however, has changed.
Evidently the distortion in the wave-form of the recorded signal is due to the phase distortion. Phase distortion is secondary for applications.
It is important for the applications in which the wave-form must be preserved.
For zero phase distortion, the phase shift of each of the harmonic components in the signal must increase linearly with frequency. Consider the equations
= +
-4,)
+
-= sin sin
-+
sin-Harmonic components
(a) Original signal Recorded signal
FIG.3-33. Phase distortion.
SEC.3-5 Forced 105
Frequency ratio
FIG. 3-34. Phase distortion in accelerometer.
The quantities and denote the delay or the shift of the signal along the positive time axis. If each harmonic component of a signal is shifted by the same amount of time, the wave-form is preserved. This requires that = =constant, that is, = =constant. In other words, the phase angle varies linearly with the frequency
It is observed in Fig. 2-9 that, for the phase angle vary linearly with frequency over an acceptable range the phase shift is
r).Hence the phase distortion of an accelerometer is defined as Phase distortion = -9 0 r )deg (3-43) This is plotted in Fig. 3-34. Again, it is seen that an appropriate damping in an accelerometer is necessary in order to minimize the phase distor-tion.
Example 20
A machine component is vibrating with the motion
Y= +
= sin
+
sin=0.10sin
+
0.05sinDetermine the vibration record that would be obtained with an accelerome-ter. Assume =0.65 and = =1,500H z .
Solution:
From the equation of motion is
Systems with One of Freedom-Applications CHAP.3 The harmonic response due to of the components in the input can be obtained from Eq. (3-40). By superposition, we have
where
=tan-'- and =tan-'
1
-The frequency ratios are = = =1/50 and =
=2/50. The values of the magnification factors and are almost unity. Thus,
=tan-' 1
-Hence the acceleration record is
The output of an accelerometer is usually converted to an electrical signal and amplified. Hence the value of the recorded acceleration depends on the amplification used in the data processing.If the equation above is integrated twice to give a displacement, the measured value of the given motion is
y =
-(measured)
+
0.05Note that the measured value of has practically no amplitude distortion and only a slight shift in the phase angles. There is no phase distortion, however, because the phase shift is linear with the frequency of the harmonic components. There is a slight time delay between the input and its measured value. The time delay is For the given values, we have time delay =1.49' 0.14 ms.The results above are due to the high natural frequency of the accelerometer.
Case 6. Elastically Supported Damped Systems
T h e damping of a real system can b e considerably m o r e complex than a simple d a m p e r shown in Fig. 3-23. Equivalent viscous damping will b e discussed in 3-8.W e shall consider t h e elastically supported d a m p e r as illustrated in Fig. 3-35.
From t h e free body sketch, t h e equation of motion for t h e mass m is
SEC. 3-5 Damped Excitation Fsinw t
t
Fsinw t
Free-body. sketches
FIG. 3-35. Elastically supported damper.
Since the damper c and the spring are in series, the damping force is equal to the spring force.
-Since the motions and are harmonic, the equations above can be solved by the mechanical impedance method cliscussed in 2-6.Substituting for F s i n for the time derivatives, and using phasor notation, the equations become
The phasors and can be solved for by
Defining the stiffness ratio of the springs as =
r= and
we obtain
=tan-'
-tan-'
-1 - r2
(3-48)
=tan-'
1-r2 2
Systems with One Degree of Freedom-Applications CHAP.3
Example21. Vibration Isolation*
A machine of mass m is mounted on shown in
Fig. 3-35. Define = = r= and N= ( a )
De-rive the equation for the transmissibility T R of the system. (b) If r 0.6,
=2 , and 0.4, find the transmissibility TR.(c) Repeat partbif r 10 and the other parameters remained unchanged. (d) Compare the values of TR from partb and part with that expressed in Eq. ( 3 - 3 2 ) .
Solution:
( a ) The force transmitted to the foundation is the sum of the forces transmitted through the springs k and
T h e phasors and are given in Eq. Using vectorial addition, the phasor of the transmitted force is
k Z +
+
jwc)+
-
+
-where is the phase angle of the transmitted force relative to the excitation and is defined in Eq. ( 3 - 4 6 ) . The transmissibility T R is the ratio of the magnitude of the transmitted force relative to that of the excitation; that is,
(b) For =0.6,N=2 , and =0.4, we obtain
For r=10, N=2, and 0.4, we have
(d) Substituting the corresponding values in Eq. the values of T R for the system in Fig. are
for
for r - 1 0
*Adetailedanalysisisshown inJ.C . Vibration and Shock in Damped Mechani-cal Systems,John Sons, Inc., New York,1968, pp. 33-38.
SEC. Forced Excitation 109 Comparing parts (b),(c), and (d), the transmissibility for the two isolation systems are approximately the same for r=0.6.When operat-ingat r 10,the system in Fig.3-35seems to be superior to that in Fig.
3-6 DAMPED FORCED