THEORETICAL OUTLINE
The basic mass–spring oscillator, with or without a damper, is the principal subject of inves-tigation in this chapter. Both the translational and the rotational versions are considered.
Physical simplicity of such oscillators is often misleading because in practical applications they not only may represent the behavior of single structural elements, but they often may approximate the responses of entire structures, or at least some aspects of such responses.
E
QUATIONSOFM
OTIONUnderstanding of the response of a simple oscillator to various types of shock loading is fundamental to comprehending more general problems. First of all, any structure with a dominating, single dynamic degree of freedom (SDOF) is reducible to such an oscillator.
Second, complex systems can often be approximated by such an oscillator, which certainly justifi es devoting attention to the subject. To make the relationships more general the infl u-ence of damping needs to be included.
Any of the two arrangements in Figure 3.1 will be referred to as a simple, damped oscilla-tor. In Figure 3.1a, mass M is restrained by a spring of stiffness k and a viscous damper, which offers a resisting force –Cu· , proportional to the magnitude of velocity u· , but opposite to its direction. Displacement u is measured from a position in which the spring is unstretched.
Figure 3.1b illustrates the basic oscillator for a rotational motion. A disk with mass moment of inertia J is restrained by a shaft with rotational stiffness K. The blades attached to the shaft are in contact with a viscous medium, which provides a resisting moment –Cα· .
Newton’s second law applied to the translational oscillator gives ( )
Mu+Cu+ku=W t (3.1)
Not only the displacement u, but also force W(t) is, in general, function of time. When W(t) = 0, we have the so-called free motion and Equation 3.1 may now be written as
2 2 0
u+ ωζ + ω =u u (3.2) where
2
c c
; ; 2 2
k C
C kM M
M C
ω = ζ = = = ω (3.3)
In most technical applications one has C < Cc or ζ < 1 and our interest is therefore limited to this range of moderate damping.* The damping ratio ζ is another practical and convenient
*For C > Cc the motion changes its character to nonperiodic. Cc is called the critical damping.
measure of damping. When ζ = 0, we obtain the expression of free, undamped motion, analogous to the previously quoted Equation 1.1, which can also be written as
2 0
u+ ω =u (3.4)
F
REE, U
NDAMPEDM
OTIONTo solve either Equation 3.2 or 3.4 for u(t), the initial values of displacement u(0) = u0 and/or velocity u· (0) = v0 must be prescribed. The solution of Equation 3.4 is
= 0cosω +(v / )sin0 ω ω
u u t t (3.5)
or, after using some trigonometric identities
cos( )
u=A ω − αt (3.6)
in which
2 1/2 2 0
0
A=⎡⎢⎢⎣u +⎛ ⎞⎜ ⎟⎝ ⎠vω ⎤⎥⎥⎦ (3.7a)
and
0 0
tan v
α = ωu (3.7b)
(Notice that there are two values of α differing by 180° and both giving the same tan α in a 360° angular segment. Only one of those, however, will satisfy the initial conditions.) The motion represented by Equations 3.5 and 3.6 is periodic with the period τ = 2π/ω, which means that every τ seconds the mass begins to repeat its path. Successive differentiation of Equation 3.6 yields velocity v = u· and acceleration a = ü:
sin( )
u= − ωA ω − αt (3.8a)
FIGURE 3.1 Damped oscillators: (a) translatory and (b) rotational.
C W
u
k M
(a)
T
K J α
(b) C΄
2cos( )
u= − ωA ω − αt (3.8b) The relationships between the extreme values of velocity v and acceleration a are
m m
v = ωu (3.9a)
and
m vm
a = ω (3.9b)
The graphical representation of this motion is given in Figure 3.2. When displacement is a sine or cosine function of time, as in Equation 3.6, it is called simple harmonic motion. The displacement at any time point is a projection of vector A, rotating with a constant angular speed ω, on the horizontal axis.
The initial displacement and/or velocity are imposed in the following manner. Suppose we move the mass M in Figure 3.1a from the neutral, unstrained position by a distance u0 and allow it to vibrate freely, setting t = 0 at the instant just before the release. This is what it means to impose an initial displacement. The initial velocity v0 may be induced by impact-ing the mass and lettimpact-ing it vibrate afterward. The time t = 0 is chosen at the end of impact.
If no initial displacement is to be introduced, the impact must last for a short interval of time only.
F
ORCED, U
NDAMPEDM
OTIONWhen a load W is applied to an undamped translational system, Equation 3.1 may be writ-ten as
2 /
u+ ω =u W M (3.10)
FIGURE 3.2 Representing harmonic motion as rotating vectors.
ωt
u/ω v0/ω
A u0
α
u ωt – α
.
which is Equation 3.2 with the damping term missing. The solution has the form
1cos 2sin p( )
u=B ω +t B ω +t u t (3.11a)
which, in essence, differs from Equation 3.5 or 3.6 only by the last term. Velocity and accel-eration are obtained by differentiation:
1 sin 2 cos p( )
u= − ωB ω + ωt B ω +t u t (3.11b)
2 2
1 cos 2 sin p( )
u= − ωB ω − ωt B ω +t u t
(3.11c)
When the term response is used, it may mean any of the variables like displacement, veloc-ity, acceleration, or a spring force resulting from some external action. When an analytical expression for displacement is available, the spring force is obtained after multiplying that displacement by a spring constant. The two remaining variables result from differentiation of displacement.
The coeffi cients B1 and B2 are the integration constants to be determined from the initial conditions. The fi rst two terms in Equation 3.11a are of the same form as a general solu-tion of Equasolu-tion 3.4, while the addisolu-tional funcsolu-tion up(t) is called the particular solution of Equation 3.10. There are several methods of fi nding up(t). The most common one is to assume that up(t) has the same form as W/M, substitute the assumed expression in place of u in Equation 3.10, and use the resulting identity to fi nd the unknown coeffi cients.
Once the calculation is complete and u is known for any instant of time, one can divide the extreme defl ection um by the value ust obtained from static application of the same force W. The ratio of the absolute values of the two will be called the dynamic factor (for defl ections):
m st
DF( ) u
u = u (3.12)
If W changes with time, W = W(t), the peak value Wm should be used for calculating ust. For the mass–spring system, which we are considering at this moment, DF(u) is the same for displacement as well as for the force in the spring, DF(P).*
F
REE, D
AMPEDM
OTIONThe form, which solution of Equation 3.1, with W(t) = 0, will take depends on the magni-tude of the damping ratio ζ. Two ranges of values are of practical interest:
Small damping, or ζ << 1.0. This is the most frequent case with ζ typically not exceeding a few percent.
Moderate damping, or ζ < 1.0. This happens in special situations, including structures under extreme loading.
*Many authors use the term “dynamic magnifi cation factor.” It does not seem purposeful, since a reduction as well as an increase can also take place.
The solutions given here will be valid for both of the above ranges, unless otherwise mentioned. (Large damping, ζ > 1.0, requires different formulas, but has very limited appli-cations.) For Equation 3.1:
The new constant ωd is called damped natural frequency. The main difference between Equations 3.6 and 3.13 is the presence of the exponential term in the latter. Consequently, the amplitude of damped vibrations diminishes with time until it becomes insignifi cantly small. Equation 3.13 may also be written in a more compact form:
d d
(As mentioned in the section “Free, undamped motion,” some caution is needed in deter-mining which of the two possible angles αd will fi t the initial conditions.)
The cosine function is ±1 at the extreme points. This means that the defl ection described by Equation 3.15 has the absolute value not larger than Ae−ωζt for any given t. The motion is not exactly periodic, because it does not repeat itself after each damped period τd = 2π/ωd. Yet, it may be shown that not only zero points, but also the extreme points of u(t) are attained after every τd. For this reason the term pseudoperiodic may be used to describe this type of oscillation. One may also notice that u from Equation 3.15 can also be repre-sented by projection of a rotating vector, as in Figure 3.2, but the length of this vector is now decreasing with time.
Suppose that for a certain time point t1 a local maximum displacement u1 is obtained.
After another cycle of motion, at time t1 + τd another maximum u2 is reached. The ratio of these maxima is (by Equation 3.15)
1 1
This means that the ratio of any two adjacent maximum defl ections is always the same. One can easily show that after n cycles we have
1 d
1 exp( d) n
n
u n e
u+ = ωζτ ≡ ωζτ (3.17b)
The term “ratio of amplitudes” may be used instead of “ratio of maximum displacements”
for the sake of brevity, since the amplitude is not a parameter of motion as it is in case of simple harmonic vibrations.
The logarithm of the displacement ratio in Equation 3.17a is called logarithmic decre-ment Δ:
1 2 d
log u 2
u
Δ = ⎛ ⎞⎜ ⎟⎝ ⎠= ωζτ ≈ πζ (3.18)
(In some texts Δ is used as the basic damping constant.) The near-equality sign used in the above expression is valid for a small damping ratio. In this case τd is only very slightly larger than τ. The case with ζ < 1.0 just discussed is the only situation where vibratory motion is possible. For ζ ≥ 1.0, the motion loses its oscillatory character.
F
ORCED, D
AMPEDM
OTIONThis type of motion is described by Equation 3.1 with all the terms present, or by an alter-native form:
2 2 /
u+ ωζ + ω =u u W M (3.19) When damping is less than critical (i.e., ζ < 1.0), the solution of this equation is
1 d 2 d p
( cos sin ) t ( )
u= B ω +t B ω t e−ωζ +u t (3.20)
The fi rst term is the general solution of Equation 3.19. The second term, up(t), is the par-ticular solution. The method of fi nding up(t) is the same as that described for free, damped motion.
Because of the presence of an exponential expression, which is responsible for the grad-ual diminishing of the amplitude, the fi rst term is also called the transient component.
After some time it becomes insignifi cantly small, and then we have u ≈ up. This is why up is referred to as the steady-state component.
Since up does not contain the integration constants, it is independent of the initial condi-tions and therefore it is a function of forcing and system properties only. When the phrase transient response is used in connection with dynamic analysis, it usually refers to the response just after the application of load, when the transient component is to be included along with the steady-state component. From a physical point of view, the transient vibra-tory component is a disturbance that takes place because of the transition from rest to motion and decays as time goes on, leaving only the steady-state component.
(Having read all this one must remember that the terms “transient” and “steady state”
are meaningful only for prolonged vibrations, while in shock response analysis the distinc-tion has little practical value.)
Equation 3.20 was discussed only for the sake of completeness. It is rarely used in its full form for practical reasons. Once we have determined the formula for up(t), we will pro-ceed to fi nd the integration constants B1 and B2. The resulting response expression is quite lengthy. If we want to know what is happening during the early stage of motion, an alternate procedure is recommended:
1. Calculate the steady-state component up(t) using Equation 3.19.
2. Instead of using Equation 3.20 to fi nd B1 and B2 for the given initial conditions, use Equation 3.11a.
The second step is equivalent to ignoring damping in the transient component and is justi-fi ed only when ζ is small, otherwise the calculated response is excessive. Of all system properties, damping is usually known with the least accuracy, and by ignoring and obtain-ing somewhat larger response the analyst errs on the side of caution.
R
ESPONSEOFO
SCILLATORTOS
TEPL
OADINGThe basic undamped oscillator, initially at rest, is subjected to a step loading shown in Figure 3.3a, which can be expressed as W(t) = W0H(t). The function H(t) is known in math-ematics as the Heaviside step function. It has a value of unity for t > 0 and nil for t ≤ 0. This means that during the time of interest W(t) is a constant, which makes the response calcula-tions quite simple. In determination of the displacement u(t) Equation 3.10 is used and the particular solution is assumed as constant B:
p( )
u t =B (3.21)
Then, after substituting this into Equation 3.10 we obtain
0 0
which is the static displacement. When this is substituted in Equation 3.11b along with the ini-tial condition u· (0) = 0, one gets B2 = 0. Then, from Equation 3.11a: B1 = −ust. The end result is
FIGURE 3.3 Step load (a) and oscillator response (b).
The mass oscillates between u = 0 and u = 2ust, which means the amplitude is ust and the vibration is taking place about the position of static equilibrium, Figure 3.3b.
The above load was implied to be lasting for indefi nitely long time. If it is removed after time t0, as shown in Figure 3.4a, we deal with a truncated step load or a rectangular pulse.
Now the extreme displacement will depend on the magnitude of t0. The pattern may be treated as a superposition of two step loads as shown in Figure 3.4b.
The fi rst step originating at t = 0 gives a defl ection u1 defi ned by Equation 3.20. The second step beginning at t = t0 will result in
2 st[1 cos ( 0)]
u = −u − ω −t t (3.24)
For t < t0, u1(t) prescribed by Equation 3.23 is the defl ection response. For t > t0 both terms are superposed and the total response is
0 0
and the extreme values are
m 2 st
u = u (3.27a)
and
m m
u = ωu (3.27b)
Only the second trigonometric term in displacement and velocity expressions is time-depen-dent and oscillates in the range between +1 and −1. The fi rst term is a function of t0 and the natural period τ. Its absolute value reaches unity for t0 = τ/2, 3τ/2, 5τ/2,…. This means that if the load W0 is removed after any of those values of t0, the amplitude of displacement during the free vibrations that follow will be 2ust. An inspection of Figure 3.3b reveals why this happens. The amplitude has doubled in comparison with what is obtained from the step
t
FIGURE 3.4 Truncated step load (a) as a superposition of two steps in (b).
loading in Figure 3.3a, because there are two shock loads applied here: sudden loading one way followed by sudden unloading when defl ection reaches maximum.
If t0 is different from an odd multiple of τ/2, the amplitude of defl ection will be less than 2ust. In fact, when t0 is equal to a multiple of τ, no vibration takes place for t > t0 so the mass will be at rest thereafter.
The damped response of an oscillator can be found in Case 3.9.
R
ESPONSEOFO
SCILLATORTOI
MPULSIVEL
OADWhen a plot of the applied force vs. time is drawn, the area under the W – t curve in Figure 3.5 represents the impulse S acting on the system. If the load that acts for a very short time, say t0 < 0.1τ or less, will be referred to as impulsive. The peak force may be very large, the duration quite small, but the impulse will have some prescribed value. Mathematically, a function acting in the interval from t to t + dt, with the integral of that function over dt remaining constant (regardless of how small dt becomes), is known as Dirac delta func-tion δ(t). (In fact, an idealized impulse of a specifi ed value S, with infi nitely short durafunc-tion, implies an infi nitely large load.) When this impulse is applied to mass M, the mass acquires its initial velocity v0 instantly. The latter is found by the application of the impulse–momen-tum principle
0 0 v0 or v0 /
S=W t =M =S M (3.28)
assuming of course that the mass is initially at rest. (One may note that δ(t) is a derivative of H(t) defi ned before, but this appears to be of little practical signifi cance.)
With the initial velocity given by Equation 3.28 one can apply Equation 3.5 to get the maximum displacement:
0 m
u = ωv (3.29a)
or
m
u S k
= ω (3.29b)
FIGURE 3.5 Applied force vs. time (a) and a short impulse (b).
t S
W
(a) (b) dt
W0
t S
When the rectangular pulse is of somewhat longer duration, 0.1τ < t0 < 0.25τ, a more accu-rate formula for amplitude, derived from Equation 3.25, must be used:
0 much more diffi cult to use than Equation 3.29, there is little point in using the latter, in spite of it being quite popular in technical literature.
In the load case tabulations to follow, the term short impulse or short pulse is intended to represent a real impulse of magnitude W0t0 and of duration t0, which is so short that it can be treated as instantaneous. This is equivalent to imposing the initial velocity v0 on mass M. If one begins with Equation 3.30 and assumes that a sine of a small angle can be replaced by an angle itself, Equation 3.29, or, in effect, a larger displacement is obtained. In this sense Equation 3.29 gives an upper bound of a result produced by Equation 3.30. This leads to the following rule in response calculations:
If a rectangular load pulse W0t0 is replaced by an instantaneous one, equivalent to the initial velocity v0 = W0 t0/M, the displacement response obtained in this way is the upper bound of the true result.
O
SCILLATORR
ESPONSETOT
RIANGULARP
ULSESThere are three basic types of triangular pulses, as shown in Figure 3.6: symmetrical, increasing (with a sudden drop) and decreasing (after a sudden rise). The second one, of increasing magnitude, is typical of many impact situations. The third, a decreasing func-tion, is often used as an approximation of the positive phase of a blast load. The symmetri-cal pulse displacement response, patterned after Harris and Crede [30], has three phases of motion separately described:
FIGURE 3.6 Basic triangular pulses: (a) symmetric, (b) increasing, and (c) decreasing.
0 0 0
Static displacement, ust, is calculated as if it was induced by the peak loading W0. The response described by the above relationships is not only a function of time, but also depends on the ratio t0/τ. It is easy to determine that for t < t0/2 there is no extremum. For t0/2 < t < t0 one can obtain from Equation 3.31b the peak displacement of um ≈ 1.52ust when t0/τ ≈ 0.92.
Since two variables are involved, it makes the usual extremum search somewhat complex and necessitates the use of numerical methods. Equation 3.31c gives residual response, i.e., history of motion following the entire pulse application. It is a simple sinusoidal motion.
The peak response here is u/ust = 1.45, somewhat less than before, but that can be attained only if t0/τ 0.74. A good graphical display of results can be viewed in TM5 [106].
The response to an increasing and a decreasing pulse is given in Case 3.3. One should remember that replacing any such shape with a symmetrical triangular pulse results in obtaining a higher um/ust ratio. The exception is the decreasing pulse, which gives a higher response when t0 > 0.9τ. With regard to the latter shape, one should also keep in mind that when t0 is extended far beyond the natural period, this loading begins to approximate a load step, rather than a distinct pulse. As the summary Table 3.1 shows, the peak responses to all three triangular pulses are similar in magnitude.
When pulses are short, i.e., when t0/τ = 0.1, the exact shape of the applied pulse is not very important, therefore an equivalent rectangular-shaped pulse (one that has the same impulse S) will give a good approximation of response. The distinctions between the inter-vals of time, as defi ned above, become meaningless and Equation 3.29 may be used. Here we have S = W0t0/2, which leads to
m 0
st 2
u t
u = ω (3.32)
keeping in mind that ust = W0/k. For triangular pulses, the above expression, or its more accurate version, namely Equation 3.30, can be used up to t0/τ = 0.4.
TABLE 3.1
Note: For a triangular decreasing pulse the value of um is for t0 = τ.
S
UMMARYOFR
ESPONSETOS
INGLEP
ULSESThe peak responses divided by static displacements (or DFs) are summarized in Table 3.1.
The rectangular pulse must be at least as long as the half-period, i.e., t0 > τ/2 to achieve its peak response. For the triangular decreasing pulse, the longer the duration, the larger the peak value can be attained. If continued long enough, it tends to DF = 2.0, the value related to the step load. Since our interest is limited to relatively short pulses, the DF corresponding to t0/τ = 1.0 was selected for this pulse. The ratio of applied impulse to that associated with the rectangular pulse of the same duration is also given in Table 3.1.
Damped oscillator responses lead to more complex formulas. Step loading W0H(t) gives
Damped oscillator responses lead to more complex formulas. Step loading W0H(t) gives