Systems with One Degree of Freedom-Applications
3-7 TRANSIENT VIBRATION-SHOCK SPECTRUM
where
tan-' -n2rz
By superposition, the total response due to the excitation is
For the given data, we have
Thus, the response of the system is
where
3-7 TRANSIENT VIBRATION-SHOCK SPECTRUM
The design of equipment to withstand shock is of concern to the engineer. Vibrations induced by the steady-state operation of a machine are generally periodic. This was discussed in the last two sections.
Vibrations due to shock and transients usually originate from sources outside the machine or from a sudden change in the machine operation.
The transient will die out, but the machine may be damaged or may malfunction momentarily, both of which should be well considered.
A shock is a transient excitation, the duration of which is short compared with the natural period (reciprocal of natural frequency) of oscillation of the system. The transient response due to a transient excitation was discussed in 2-7.The recording from a vibration test, in the form of a "wavering line" versus time, cannot be used directly by the designer. The shock spectrum is a common method to reduce the test data to a more usable form.
SEC.3-7 Transient Vibration-Shock Spectrum 117
A shock spectrum (response spectrum) is a plot of the peak response versus frequency due to the applied shock. The peak response is that of a number of one-degree-of-freedom systems, each tuned to a different natural frequency. The frequency is that of natural frequency of the individual systems. The response may be expressed in units of accelera-tion, velocity, or displacement.* For example, a vibrating reed shown in Fig. 3-3 is a simple mass-spring system.A reed gage consists of a number of reeds of different natural frequencies. Using a reed gage in a shock test, the maximum displacements of the tips of the reeds give the maximum response for the various natural frequencies. The reed gage can be replaced by a single accelerometer and computers employed to simu-late the reeds.? The one-degree-of-freedom system is variously called a
an oscillator, or a simple structure.
The objective of shock spectrum is to describe the effect of shock rather than the shock itself. Shocks are difficult to characterize and a specific pulse shape is difficult to obtain in a test machine. It is necessary to correlate test data from different laboratories. The shock spectrum a
"common denominator" on the assumption that shocks having the same spectrum would produce similar effect. The spectrum may be regarded as indicative of the potential for damage due to the shock. For example, the peak relative displacement between the mass m and its base for the system in Fig. is related to the stress in the spring. In other words, the envelope of the spectrum establishes an upper bound of the stress induced, or the damage potential, by a specific shock on the equipment under test.
Types of are usually categorized using the undamped resonator as the standard system. Types of shocks and methods of data reduction can be found in the It is paradoxical that a complex study like shock is treated in a seemingly simple manner. The reason is that the time and expense for a detail study must be justified by the past experience of the engineer. An undamped resonator is used, since the largest excursion occurs within the first cycle of the transient and the error introduced by neglecting damping tends provide a
safety. Note that a system under test commonly has more than one degree of freedom. It will be shown in later chapters that a complex system has discrete modes of vibration and a natural frequency is associated with
*C.E.Crede,Shock and Vibration Concepts in Engineering Design, Book Co., New York, 1965, p. 138.
C.T.Morrow,Shock and Vibration Engineering,John Wiley and Sons, Inc., New York, 1963,p. 111.
S. Ayre, "Transient Response to Step and Pulse Functions," Chap.8of Shock and Vibration Handbook,vol.1 , C. M. Harris and C. E.Crede (eds.) McGraw-Hill Book, Co., New York, 1961.
S. "Concepts in Shock Data Analysis," Chap. 23 of Shock and Vibration Handbook,vol.2,C. M. Harris andC. E.Crede (eds.), McGraw-Hill Book Co., New 1961.
118 Systems with One Degreeof Freedom-Applications CHAP. 3
each of the modes. Hence a complex system can be described in terms of equivalent one-degree-of-freedom systems. Thus, a shock will excite all the modes of a system.
If a shock is due to a sudden change in the machine m in Fig.
the equation of motion of m is identical to Eq. (3-1). Assuming zero initial conditions, the response from Eq. (2-71) is
where
as defined in Eq. (2-69). On the other hand, if the excitation is applied to the base of the machine as shown in Fig. the equation for the relative displacement between m and its base is identical to Eq.
(3-39).
mi!
+ +
-where and and are the absolute motions
indicated in the figure. Applying Eq. (2-71) yields
where is as defined above. The maximum response and the corres-ponding shock spectrum can be obtained from Eq. (3-66) or (3-69) depending on the application. Computers can be used for the calcula-tions.*
To illustrate a shock spectrum by hand calculation, let a one-half sine pulse shown in 3-39 be applied to the mass m of the system in
FIG. 3-39. A half-sinepulse.
*See, forexample, J.B.Vernon: Linear Vibration and Control System Theory, John Wiley Sons, Inc., New York,1967, pp.
SEC.3-7 Transient Vibration--Shock
Fig. 3-23. Assume the system is undamped. is described by the
sin for t
= for
For t the system response from Eq.(3-66) is
which can be integrated to yield
= (sin
Equating = to find we get
COS -COS =
where is the time when is a maximum or minimum. The roots of this equation is deduced using the identity
cos -cos =-2 sin
+
sin-Thus,
=--2 for n=integer
Consider =
+
Defining = the terms in Eq. (3-70) can be expressed assin sin =sin 2
sin
Hence sin =-sin Recalling k l m , from Eq. we get
=
( :)
1+- 2-=-1 sin
Systemswith DegreeofFreedom-Applications CHAP. 3
Similarly,if = - we obtain
Comparing the last two equations, it is evident a value of n can be selected to have occur at =
+
Equation (3-71) is plotted in Fig. 3-40. The initial shock spectrum, defined by Eq.gives the response within the duration of the shock pulse for
Note that there is no solution for 1, since the maximum response does not occur during the pulse if the natural frequency is smaller than the pulse frequency.
For t the system response from Eq. (3-66) is
The upper limit of the integration is because = for t Performing the integration, substituting = and simplifying, we get
convenient to define t- a new origin for the time axis.
Recalling = defining r and = the equation can be
U n d a m p e d natural frequency Pulse frequency
FIG.3-40. Shock spectra for half-sine pulse applied to m in Fig.
Eq. ( 3 - 7 1 ) shows initial spectrum;Eq.( 3 - 7 2 ) residual spectrum (Crede).
SEC.3-7 Transient Spectrum
simplified to
The maximum value of can be expressed as r2- 1
-This gives the residual ,shock spectrum, as it occurs after the shock has terminated. The equation is plotted in Fig. 3-40 and shown as a dash line.*
Now consider the shock spectrum due to a one-half sine pulse applied to the base of the system in Fig. Assume the system is undamped. The equation of is
for for
For t the equation for the absolute motion is
+
= (t)Since the excitation is sin wt, the response can be obtained from Eq. (3-70) by substituting for that is,
Hence the initial shock spectrum is deduced directly from Eq. (3-71):
-- sin. l - r l + r
The relative motion between m and its base is given by the relation
where is as defined in Eq. (3-73). The expression for can be maximized to give the corresponding spectrum.
The system is unforced for The residual shock spectrum can be calculated as before. shock spectra for the system shown in Fig.
excited by a half sine pulse is illustrated in Fig.
*C.E. Crede,op. p. 85.
L.S.Jacobsen and R. S. Ayre, Engineering Vibrations, Book New York, 1958,p. 163.
Systems with One of Freedom-Applications CHAP.
Pulse duration Natural period of system T
FIG.3 - 4 1 Shock spectra for half-sine pulse applied to the base in Fig.
X is relative displacement; the initial spectrum; the residual spectrum (Jacobsen A y r e ) .