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Introduction to Mechanical Vibrations
Introduction to Mechanical Vibrations
1.1
Introduction
All mechanical systems composed of mass, stiffness and damping elements exhibit vibratory response when subject to time-varying disturbances. The prediction and control of these disturbances is fundamental to the design and operation of mechanical equipment. In particular, the use of secondary, active inputs to the system in order to modify the system response in a controllable way is the topic of this book. The analysis of controlled systems is founded on the same analytical approaches used to study the vibrations of elastic structures. A brief review of the main concepts of vibration analysis and the associated techniques of solution is necessary to set the foundation for the following chapters. In this chapter we begin by defining terminology and the mathematical methods for describing the linear response of vibrating systems. The equations of motion and linear behaviour of single-degree-of-freedom systems are outlined for both free and forced response. The use of the Laplace transform to solve for transient response is reviewed. The extension to multi-degree-of-freedom systems and then the use of finite element analysis are briefly introduced. These approaches are valid for lightly damped structures or elements that are small relative to the wavelength of motion. For more detail the reader is referred to the texts of Thomson (1993), Meirovitch (1967) and Inman (1994).
1.2 Terminology The following is a brief list of the main terminology and definitions used in analysing the vibratory response of mechanical systems. Mechanical system. A mechanical system is composed of distributed elements which exhibit characteristics of mass, elasticity and damping. Degrees of freedom. The number of degrees of freedom of a system is equal to the number of independent coordinate positions required to completely describe the motion of the system. System response. All mechanical systems exhibit some form of vibratory response when excited by either internal or external forces. This motion may be irregular or may repeat itself at regular intervals, in which case it is called periodic motion. Period. The period T is the time taken for one complete cycle of motion.
2
ACTIVE CONTROLOF VIBRATION
Harmonic motion is the simplest form of periodic motion whereby the actual or observed motion can be represented by oscillatory functions such as the sine and the cosine functions. Motion that can be described by a continuous sine or cosine function is called steady state. For example one may write the actual displacement w ( t ) in the form
w(t) = [A I cos(~ot + q~),
(1.2.1)
where w ( t ) and IAI are real, a~ is circular frequency in radians per second and q~ is an arbitrary phase angle in radians. Equation (1.2.1) can also be expressed as a superposition of a sine and a cosine function as w ( t ) - AR cos cot- At sin cot,
(1.2.2)
where AR and At are real numbers such that AR = I Alcos ¢,
(1.2.3a)
A, = I alsin ¢,
(1.2.3b)
where the phase angle is specified by qb = tan - ~(AJAR).
(1.2.4)
The constant I AI in equation (1.2.1) is related to the constants At and AR in equation (1.2.2) by
[al
= ( a z + a l z)1/2.
(1.2.5)
Frequency is the number of cycles per second (also called hertz) of the motion and is the reciprocal of the period. Therefore frequency is specified by
1 f= --
T
(1.2.6)
and circular frequency co (radians per second) is given by o9 = 2z~f. Amplitude is the measure of For example if the motion is corresponds to the amplitude of The mean square amplitude response. Thus, for example
(1.2.7)
the maximum response of the system during a period. specified by equation (1.2.1) then the constant I AI the motion. is defined as the time average of the square of the
= lim --1 [ r w2(t) dt. T--->~* Z J°
(1.2.8)
The root mean square (rms) amplitude is the positive square root of the mean square amplitude. For the harmonic oscillation of equation (1.2.1) the rms amplitude is independent of phase and is equal to Ial/~. Free vibrations are the motions of the system in the absence of external disturbances and as a result of some initial conditions. Forced vibrations are the motions of the system produced by external, persistently applied disturbances.
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INTRODUCTION TO MECHANICAL VIBRATIONS
Natural frequencies are those frequencies at which response exists during free vibration. The lowest natural frequency is called the fundamental frequency. Transient motion is motion other than steady state response. If damping is present, transient response will decay with increasing time. Phasor. A phasor is a rotating vector representation of the harmonic motion of the system. The periodic motion of equations (1.2.1) and (1.2.2) can be represented in a complex form, which is more convenient for mathematical manipulations and is given by w(t) = A e j°~t, (1.2.9)
where A and w(t) are complex with the complex amplitude specified as (1.2.10)
A= AR + jAI,
The phasor representation of equation (1.2.9) is shown in Fig. 1.1. The length of the vector, I AI, is the real amplitude of motion. As the vector rotates with angular velocity to in a counterclockwise direction, its projection on the real and imaginary axis of the complex w plane varies harmonically with time t. A rotation of the vector through 360 ° corresponds to a cycle of motion. In this text the convention used is that the real component of the phasor or the complex description of the motion corresponds to the actual, observed or measured motion. Therefore the actual motion is given by w(t) = Re[A eJ°'t].
(1.2.11)
Using the relationship eJ°"=cos tot + j sin tot and substituting A = AR +jAI into the above expression yields equation (1.2.2). The phase of the motion, q~, is thus retained through the ratio of the imaginary and real components of the phasor as specified in +lm
AE J I I I I I
AR
+Re
ines phasor position at t = O)
Fig. 1.1 Phasor diagram representation of harmonic motion.
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ACTIVE CONTROLOF VIBRATION
equation (1.2.4). Note that a negative sign has been used in equation (1.2.2) in contrast to many texts dealing with vibration. This choice ensures that the phase q~ is positive and since At is a constant to be determined by boundary conditions, the choice of negative sign does not affect the result. Since the phasor is a vector, any number of harmonic motions of the same frequency can be added vectorially. For linear motions written in complex notation, since the principle of superposition holds, this simply means separately summing the real and imaginary components of the individual motions. In this text the majority of the response equations are written using complex notation as this is the most convenient form for analysing systems where responses are superimposed (i.e. as is the case in active control simulations). The actual motion can be directly recovered by taking the real part of the complex description. Where the motion is directly described in the actual form, it is indicated in the text.
1.3 Single-degree-of-freedom (SDOF) systems Consider a mass M supported by a massless spring as shown in Fig. 1.2(a). As the displacement of the system can be completely specified at all times by a single variable w, the system is said to possess a single degree of freedom (SDOF). By appropriately w
T (a)
M
(b)
M
(d)
M
K
//
/
/ / / /
/
/ //'
1" (c)
M
K<
////I Fig. 1.2
G
// ///
SDOF systems and free body diagrams: (a) and (b), undamped; (c) and (d), damped.
INTRODUCTIONTO MECHANICALVIBRATIONS
5
choosing the origin of the coordinates at the rest position (i.e. in static equilibrium), the constant force due to gravity can be ignored. When the mass is displaced an amount w from its equilibrium position, the spring will exert a restoring force - K w due to being elongated (for positive w) as shown in the free body diagram of Fig. 1.2(b). On release of the mass, the spring will attempt to accelerate the mass and the restoring force and acceleration are related by Newton's second law of motion which shows that d2w M
dt 2
= -Kw.
(1.3.1)
Rearranging terms, we obtain the differential equation describing the motion of this simple SDOF system. This is given by dt 2 +
w-0.
(1.3.2)
Equation (1.3.2) is a second-order ordinary differential equation and therefore must have a solution which is specified in terms of two unknown constants or amplitudes of motion. Although the above analysis is straightforward, it does illustrate the basic process by which elastic systems are generally analysed. The system is first broken into elements (or blocks). For some initial conditions the restoring and inertial forces are balanced, thus providing the differential equation describing the motion of the system.
1.4
Free motion of SDOF systems
Based upon the observation that mechanical systems respond harmonically in free motion, the solution of equation (1.3.2)can be assumed to be of the form given by equation (1.2.2). Therefore we assume that the actual motion can be described as w ( t ) = AR cos oJt- AI sin tot.
(1.4.1)
where AR and A/are real amplitudes of motion. Substitution of equation (1.4.1) into equation (1.3.2), differentiating with respect to time and eliminating common terms, provides a relation for the frequency ton at which the system will naturally vibrate. This is given by oJn = I ~
(1.4.2)
and thus the solution of equation (1.3.2) becomes w ( t ) = AR cos to, t - A~ sin ~o,t.
(1.4.3)
In order to specify the motion completely, the unknown constants AR and A~ need to be determined and these are found by applying given boundary or initial conditions to the system. The frequency oJ, is called the natural or resonance frequency and is a very important characteristic of the system as will be shown in Section 1.6. Note that the natural frequency of the SDOF system increases with stiffness K and decreases with increasing mass M. These observations are in general true for all linear elastic systems.
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ACTIVE CONTROL OF VIBRATION
To specify the motion of the system completely, one needs to apply initial conditions. For example, if at t = 0 the system has an initial real displacement w(0) and a real velocity ~¢(0) then the unknown constants in equation (1.4.3) can be determined from w(0) = AR,
(1.4.4a)
w(0) = - o9~A,,
(1.4.4b)
where the use of the overdot is a compact notation for differentiation with respect to time. The observed response can be obtained by solving for the constants AR and At from equations (1.4.4a) and (1.4.4b) and substituting these into equation (1.4.3). The actual response of the system to arbitrary initial conditions is then given by
w(t) = w(0) cos tOnt+
w(O)
sin (Ont.
(1.4.5)
This motion can also be written as w(t)= [A lcos(O)nt + ~9),
(1.4.6)
where the phase angle q~ is specified from equation (1.2.4) by q~= tan_l( -
tOnW(O)l~(O)
(1.4.7)
and the amplitude of motion that results from equation (1.2.5) is given by IAI=
[w(O)] ~ +
(1.4.8) (.On
Thus the response of the SDOF can be seen to be simple harmonic motion at the natural frequency ~o, with an amplitude [A[ and a phase angle q~ given by equation (1.4.7) and (1.4.8) respectively.
1.5 Damped motion of SDOF systems All vibrations in realistic systems occur with some form of damping mechanism, where the energy of vibration is dissipated during a cycle of motion. The simplest form of damping is when the resisting force associated with the damping is proportional to, and acts in an opposite direction to, the velocity of the element. Thus the damping force is specified by dw F~ - - C ~ ,
(1.5.1)
dt where C is the damping coefficient. Figure 1.2(c) shows an SDOF system with this form of damping which is called viscous damping. Including the additional damping force into the force balance of the new system, as shown in the free body diagram of
INTRODUCTION TO MECHANICALVIBRATIONS
7
Fig. 1.2 (d), leads to a new differential equation for an SDOF system given by
M
d2w
dt 2
+C
dw
dt
+ Kw = 0.
(1.5.2)
It is now more convenient to use a complex description of the motion. Thus a solution is assumed of the form
w(t) = A e "/',
(1.5.3)
where w(t) is now a complex variable. Substituting equation (1.5.3) into equation (1.5.2) provides the values of ~, for which a solution exists. These are given by 7=
2~/±
-
.
(1.5.4)
It is convenient to express C in terms of the critical damping Cc = 2M~on. The damping ratio is then defined by ~ = C/Cc. Equation (1.5.4) then reduces to 7 = -ogn~ ±jogJ 1 - ~2,
(1.5.5)
where co, is the undamped natural frequency given by equation (1.4.2). When ¢ > 1, both terms in equation (1.5.5) will be real and this implies a steadily decaying response with no oscillation. This is termed an overdamped system. When = 1, the system is said to be critically damped. This value of C represents the smallest possible damping required to prevent oscillatory motion and ensures that the system returns to its rest position in the shortest time as shown in Fig. 1.3. When ~ < 1 then the square root term will be real, positive and 7 in this case will be complex with a negative real part. Thus the response will oscillate at a damped natural frequency oJa = oJ~x/1 - ¢2
(1.5.6)
and decay in amplitude with increasing time. This is called light damping and it is exhibited by most structural systems which are thus described as underdamped. The observed response to the specified initial conditions defined in Section 1.4 is obtained by using the real part of equation (1.5.3) and the initial conditions to solve for the constants A ! and AR as described in Section 1.4. The actual displacement is then given by
w(t) = e -~,.¢t[w(0) cos COdt+ vi,(0) + ¢co,w(0)sin (Oat] .
(1.5.7)
(.,0d
This equation can also be written in simple harmonic form as -~o,¢t W(t) = IAle cos(~oat+ q~),
(1.5.8)
where the phase angle q~ is now given by q~= t a n _ l ( w ( 0 ) + ~ w ( 0 ) ) -
w(O)~oa
(1.5.9)
and the real amplitude by
I AI = {[w(0)]2 + [w(0)+ ~c0~w(0)]2/~03} 1/2.
(1.5.10)
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ACTIVE CONTROL OF VIBRATION
Wo
light damping
~t)
~'<1
Wo
criticaldamping
w(t)
~r=l
Wo
~
heavydamping t
~'>1
Fig. 1.3 Response of an SDOF system with various values of damping. Initial displacement w(0) = w0 while initial velocity r0(0)= 0. Equation (1.5.8) reveals that the response consists of harmonic motion at a frequency of rod and with an amplitude given by IAle -~.~' which decays with increasing time. Note that from equation (1.5.6) it is apparent that the damped frequency is lower than the natural frequency. Obviously the inclusion and characteristics of damping are very important to active control methods since it represents a process by which the response of a system can also be reduced by passive means. Figure 1.3 shows typical response versus time curves for an SDOF system with light, critical and heavy damping. As predicted by the above equations, light damping leads to a response that oscillates at rod and is slowly decreased in amplitude with time. With critical damping the response moves towards the equilibrium position but does not cross it. Heavy damping leads to almost complete suppression of the oscillatory type motion; the damping force is such that it significantly slows the motion towards the equilibrium position and the system takes longer to return to the equilibrium position than with critical damping.
INTRODUCTIONTO MECHANICALVIBRATIONS
1.6
9
Forced response of SDOF systems
Many systems are excited by persistently applied disturbances rather than being initially excited as in the cases of free motion discussed above. Let us assume that the input disturbance is specified by a harmonic force of constant amplitude written in complex form and given by f ( t ) = F e TM,
(1.6.1)
where F is a complex number defining the amplitude and relative phase of the applied force. The homogeneous differential equation of equation (1.3.2) thus has to be modified to take account of the disturbance force and is written as d2w M ~
dt 2
+ K w = F e ~'.
(1.6.2)
Since it is assumed that the disturbance has been applied for all time, t, the transient response component is zero and it is reasonable to assume the steady state response will be again harmonic and specified in complex form by w ( t ) = A e TM,
(1.6.3)
where w ( t ) and A are again, in general, complex. On substitution of this assumed solution into equation (1.6.2) we obtain -~o 2 +
A = ~. M
(1.6.4)
The unknown complex response amplitude A can be obtained by rearranging equation (1.6.4) and is given by ElK
A=
.
(1.6.5)
1 - (O.)/O)n) 2
Equation (1.6.5) proviaes immediate insight into the response of elastic systems to a steafly state harmonic flisturbance. When ~o = a,. the flisplacement amplitufle will theoreticany be infinite for continuous steafly state excitation. Thus when systems are flriven at a frequency iclentical to or very close to their natural frequency a very large response will result. Under these conditions the system is ~escribefl as being flriven o n anti is a very important conflition from a control point of view. resonance
Extension of the above approach to more complex systems becomes increasingly difficult. In this case, it is often easier to use the impedance method. We define mechanical input impedance as the ratio of the complex amplitudes of input force to the velocity at the drive point. Thus for the harmonic displacement of equation (1.6.3) the velocity is given by vi,(t) and is specified by ¢v(t) = ja)A e TM.
(1.6.6)
The velocity thus has complex amplitude javA and input impedance is given by Z~ =
F jooA
(1.6.7)
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ACTIVE CONTROL OF VIBRATION
evaluated at the disturbance location. The input impedance of the system in Fig. 1.2(a) can be shown to be
-jK[ 1 -
Z~ =
(co/co,) 2]
.
(1.6.8)
co
Once the input impedance of a system is determined or measured, the response of the system to a steady state harmonic disturbance force can be directly calculated using equation (1.6.7). Inclusion of damping into the relationship for the forced response modifies the resulting amplitude to
F/K
A= 1
-
.
(1.6.9)
(co/con)2 + j2~(CO/CO,)
(a) --
dctJ tO
~' = 0.05
6
~'= o.~
5
Orj
-tO
"~ t,-E
4
~= o.~5
3 ~"= 0.25
2
t.--
o
z
~'= 0.5 1 0.5
1.0 1.5 2.0 Frequency ratio, co~con
2.5
3.0
(b)
o~ o
d c'-
-90
03 t-
13_
~ ' ~ 0 ~ ~ -180
Fig. 1.4
0
1.0
2.0 3.0 4.0 Frequency ratio, o~/con
n 5.0
Forced response of an SDOF system: (a) magnitude, (b) phase.
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INTRODUCTION TO MECHANICAL VIBRATIONS
Figure 1.4(a) presents the non-dimensional displacement response amplitude equal to
IAK/FI versus the non-dimensional input frequency, co/~o,, for various values of damping ratio ~. The first important feature apparent from Fig. 1.4(a) is that the inclusion of small amounts of damping has led to large reductions of response at or near resonance but has small effect away from the resonance condition. Secondly, damping leads to the response of the system being bounded at resonance. Increasing damping also causes the frequency of maximum response to move to lower values. The corresponding phase response is plotted in Fig. 1.4(b). For very light damping the phase of the displacement response relative to the excitation force of the system flips through nearly a 180 ° phase change as the excitation frequency is increased through the resonance frequency. Increasing the amount of damping leads to a decrease in sharpness of the phase transition. These observations will be seen in later sections to have important implications for the performance and stability of actively controlled elastic systems, particularly when they have many degrees of freedom.
1.7
Transient response of SDOF systems and the use of the Laplace transform
We first consider the response of an SDOF system to an impulse excitation as this case provides the basis for the study of more general forms of transient excitation. An impulse is generally thought of as a disturbance force acting on a system over a very short time. We define an impulse as the time integral of the force I = I ~f(t) dt, 0
( 1.7.1 )
where f(t) is a real function describing the variation of the force with time and r is the duration of the impulse. Since fdt = Mdw where w is the velocity of the mass we can rewrite equation (1.7.1) in the form I = I:(~)M dw.
(1.7.2)
If we make the further approximation that the impulse acts over such a small time that v~(r) -- w(0), or in other words the system reaches a velocity ~¢(0) instantaneously, then I = My0(0),
(1.7.3)
which is known as the impulse-momentum relation. Thus the initial conditions for the impulsively excited system are w(0) =0,
(1.7.4a)
¢v(O) = I/M.
(1.7.4b)
Applying these initial conditions to the damped SDOF system, the response to a unit impulse can be determined from equation (1.5.7) with I = 1 and is given by -~%t
w(t) =
e
Mood
sin cojt,
(1.7.5)
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ACTIVE CONTROLOF VIBRATION
where w(t) is the actual motion of the mass. Equation (1.7.5) is known as the unit impulse response function of the system generally denoted h(t), and is a very useful parameter. Note that h(t) is a purely real function. The knowledge of the unit impulse response function allows us to derive expressions for the response of a mechanical system to any general excitation, including transients, and the response due to specified initial conditions using Duhamel's integral (Meirovitch, 1967; Newland, 1984). The actual response of the SDOF system to a general real forcing function f(t) is then given by -~tOn(t- r)
w(t) = I~f(r) e
MtOd
sin[tOd(t- r)] dr
[w w(O)+~tOnw(O) q sin rod • +e -¢'°"' (0) cos tOdt+ rod
(1.7.6)
A more specific form of equation (1.7.6) is called the convolution integral and is written as
w(t)= I~f(r)h(t- r) dr,
(1.7.7)
where h(t) is the unit impulse response function of the system. Equation (1.7.7) essentially expresses the response of the system to any general excitation f(t) by using the principle of superposition, that is, the total response consists of the sum of a series of impulses appropriately weighted and positioned with the magnitude of the forcing function f ( r ) at various times, r. We now examine the behaviour of the impulse function for decreasing time durations. Figure 1.5 shows a rectangular impulse function of length r, centred on t = 0 and of impulsive magnitude I. If we take I to be a constant, then by examining equation (1.7.1) it is apparent that as we progressively reduce r, then f(t) will increase. In the limiting case of r tending to zero we have a function which is infinite at t = 0 and has a value of zero at all other values of time t. If we further assume that the product f(t)
-d2
d2
Fig. 1.5 General impulse function.
t
INTRODUCTION TO MECHANICALVIBRATIONS
13
r F is equal to unity (i.e. I = 1) then we have defined the Dirac delta function described by the pair of relationships iS(t) = 0,
t ¢ 0,
I ~ d(t)dt = 1.
(1.7.8a) (1.7.8b)
If we assumed that the rectangular pulse was centred at time to then we could equivalently define the function in the form d(t - to) = 0,
t ¢ to,
I?= d(t- to)dt = 1.
(1.7.9a) (1.7.9b)
A very useful characteristic of the Dirac delta function is called the 'sifting' property given by
I? f(t)d(t- to) dt=f(to).
(1.7.10)
Other properties and uses of generalised functions such as the Dirac delta function are described by Farassat (1994). We now turn our attention to the use of the Laplace transform in analysing the response of mechanical systems to transient disturbances. The Laplace transform is a generalisation of the Fourier transform and is generally useful for disturbances which do not have a convergent Fourier transform (Nelson and Elliott, 1992). The Laplace transform can often be used to transform complicated disturbances associated with impulsive and transient excitation into a form which, in conjunction with the transformed equation for the response of the system, are more easily manipulated. The Laplace transform of f(t) is defined to be (see, e.g. Kuo, 1966; Meirovitch, 1967)
.~f(t) = F(s) = I : f ( t ) e -s' dt,
(1.7.11 )
where s is a complex variable defined by
s= cr + jw.
(1.7.12)
The inverse Laplace transform is defined by (Meirovitch, 1967) ~ - l F (s) = f ( t ) =
1 ~7-~ [,/+~ F(s) es' ds, 2Jrj
(1.7.13)
where the path of integration is a line parallel to the imaginary axis crossing the real axis at Re{ s} =~, and extending from -,,,, to +,,,,. As an example, the equation describing the motion of an SDOF system can be written in the form
MfC(t) + C~(t) + Kw(t)= f(t).
(1.7.14)
Applying the Laplace transform to both sides of the above equation, we obtain
M[sZW(s)- w(O)s- w(0)] + C[sW(s) - w(0)] + KW(s) = F(s).
(1.7.15)
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ACTIVE CONTROL OF VIBRATION
Collecting terms and rearranging leads to the s-domain relation for the system response described by
F(s)
W(s) =
M s 2 + Cs +
K
+
(Ms + C)w(O) + MCv(O) Ms 2+
Cs + K
,
(1.7.16)
where the first term in equation (1.7.16) describes the forced response and the second term, the motion due to initial conditions w(0) and vi,(0). If we restrict our attention to the forced response, the system transfer function for an SDOF system can be written in the form
H(s) = M s 2 + Cs +
K
.
(1.7.17)
It is often useful in the analysis of active control systems to represent the input-output relationship using a block diagram as shown in Fig. 1.6. Note that if we choose to consider only the imaginary axis of the Laplace domain s, then the system transfer function H(s) is equivalent to a frequency domain transfer function H(jto), as discussed in Nelson and Elliott (1992). Meirovitch (1967) also demonstrates that the system transfer function can be obtained from the system impulse response function by the relationship
H(s)=~h(t).
(1.7.18)
We now illustrate the use of the Laplace transform by solving for the transient response of an SDOF system excited by a harmonic forcing function turned on at t =0. The input disturbance is specified by f(t) = O, f ( t ) = F sin tot,
t O.
(1.7.19)
Taking the Laplace transform of the damped inhomogeneous SDOF equation with the forcing function specified by equation (1.7.19) we obtain, with zero initial conditions
(Ms 2 + Cs + K)W(s)= F(s),
(1.7.20)
where F(s) is the Laplace transform of the input disturbance and is specified for the function described in equation (1.7.19) by 09
F(s) = ~ .
(1.7.21)
$ 2 + 0) 2
Transfer function Input
.I
-I
H(s)
Output
Fig. 1.6 System block diagram.
W(s)
15
INTRODUCTION TO MECHANICAL VIBRATIONS
Solving for W(s) we find
W(S)
"-"
M
S 2 + a)
2
X
$2
1)
22,
2 "
+ 2~a).s + o9~
Using the method of partial fractions and tables of known inverse Laplace transforms (Meirovitch, 1967), we find the solution for the actual response of the system to be given by
w(t)
=
1 M
)[
(a) 2 - 0)2) 2 - 4~2(.020) 2
-2~09 n COS a ) t -
(09 2 - 092n)+ 2~209n
-¢%t
2 ~ n COS OJat +
+e
sin cot O)
sin a~a •
(1.7.23)
O.)d
Hence the response of the system to a sine-wave forcing function turned on at t = 0 consists of a damped oscillation at the natural frequency of the system and a steady state response at the forcing frequency. If the damping is small or the excitation is close to the natural frequency of the system then the vibrations associated with the natural frequency of the system can extend for long times. This type of analysis and the results produced have important implications in terms of performance and stability of actively controlled mechanical systems.
1.8 Multi-degree-of-freedom (MDOF) systems When N independent coordinates are required to completely specify the system response, the system is said to have N degrees of freedom. Such a system is also said to have multi-degrees of freedom (MDOF). When a mechanical system has many degrees of freedom it is more convenient to use a matrix representation to describe and analyse the motion. In this section we formulate the equations of motion in matrix form. The reader is referred to the Appendix of the companion text (Nelson and Elliott, 1992) for a review of vector and matrix theory. Consider, for example, a simple N-degree-of-freedom system as shown in Fig. 1.7 (a) which is fixed at one end and free at the other. Excitation forces are directly applied to each mass. For a particular mass M,, the free body diagram is shown in Fig. 1.7 (b) and the corresponding equation of motion can be written as d2w M n
dt 2
=Kn(Wn_l-Wn)-Kn+l(Wn-Wn+l)Jrfn
.
(1.8.1)
This equation can then be manipulated to take the form
M~
d2wn
dt 2
+ K~w~_, + (K~ + K~+ ~)w~- K~+ ,w~+, =f~,
(1.8.2)
where fn is the forcing function applied to the nth mass. Similar equations for the motion of all the masses can be written except for n = 1, where w0 =0, and n = N,
16
ACTIVE CONTROL OF VIBRATION
where K u + 1=0, which correspond to the fixed and free boundary conditions of the system of Fig. 1.7 (a) respectively. The resulting simultaneous equations for each mass can be written in matrix form as M d2w
+ K w = f,
(1.8.3)
dt 2
where M and K are the mass and stiffness matrices. In the following text, matrices and vectors will be indicated by upper- and lower-case boldface symbols respectively. For typical linear structures, the matrices appearing in equation (1.8.3) are real and symmetric. The mass matrix M is specified by
M=
M1 0 0 0 M: 0 0 M~ 0
(1.8.4) 0
M~ which is a diagonal matrix. The stiffness matrix K is given by
(K, +K2)
K
._.
-K:
o
(/<2 + K3)
-K3
0
(1.8.5) 0
-K~_, (K~_,+ K~) 0 -KN which is a symmetric matrix. The vectors w and f are given by
1
w=
,
f=
Lw~(t)]
1.9
"
.
(1.8.6a,b)
f~(t)
Free motion of MDOF systems
If the forcing functions f are set to zero and the system is given some energy by an initial condition, then the MDOF system described by equation (1.8.3) will vibrate indefinitely due to lack of damping. The vector differential equation describing the motion can be written with matrix coefficients as d2w
M
dt 2
+ Kw = 0.
(1.9.1)
INTRODUCTION TO MECHANICALVIBRATIONS Wl I I
~
w2 I I
Wn
~
I I
17
WN- 1
,~ v
1 I
4
, i
WN
,.._ v
I I
'
(a)
Kn+l (Wn- Wn+l)
K n (wn_l- Wn)
~n (b) Fig. 1.7 Example MDOF system: (a) system configuration; (b) free body diagram of the nth mass. The solution of equation (1.9.1) will provide the undamped free vibrations of the system. We again assume a harmonic solution of the complex form w, (t) - A n
e TM,
n = 1,2 ..... N,
( 1.9.2)
where Wn is the displacement and A, is the complex amplitude (i.e. represents both amplitude and phase) of the motion of the nth mass. Substituting equation (1.9.2) into equation (1.9.1) we find the relation - 2 M A + Ka = 0,
(1.9.3)
where 2 = w 2 and the amplitude vector a is given by A1 A2 a=
AN.
(1.9.4)
By premultiplying equation (1.9.3) by M - ~ and rearranging we obtain the expression 2Ia - Ha = 0,
(1.9.5)
where I is the identity matrix and the dynamic matrix H is specified by H=M-~K.
(1.9.6)
Equation (1.9.5) can be further manipulated in the form [2I - H ] a = 0,
(1.9.7)
18
ACTIVE CONTROLOF VIBRATION
For a non-trivial solution a s 0 and setting the determinant of equation (1.9.7) to zero provides the characteristic equation of the MDOF system which can be written as 12I-
H I = 0,
(1.9.8)
The roots 2, of the characteristic equation are called the eigenvalues which provide the natural frequencies of the system. The expansion of equation (1.9.8) leads to an Nth order polynomial equation in 2. The solution of this polynomial equation provides N roots of 2 which are real and positive when the matrices M and K are positive definite (Cook, 1981). The positive square root of the N values of ;t will provide N resonance frequencies for the system with the lowest value, n = 1, corresponding to the fundamental resonance frequency, i.e. to, = a/r-~, n = 1, 2 . . . . . N.
(1.9.9)
If the number of degrees of freedom is small, the equation (1.9.8) can be solved algebraically. However, if the number of degrees of freedom of the system is large, then the solution of the polynomial obtained by expanding equation (1.9.8) is difficult and it is more appropriate to use numerical methods as discussed in Rao (1990) or Inman (1994). Once the eigenvalues are found, the mode shapes or modal eigenvector ¥, can be determined by substituting 2, into equation (1.9.7). The relative variation in the amplitude of motion of each element provides the mode shape ¥ , corresponding to the nth eigenvalue as [in I - H ] ¥ , = 0,
n = 1,2, ..., N,
(1.9.10)
where the mode shape for the nth mode is given by the vector of resulting amplitudes A1]
v.-
I .
.
(1.9.11)
[A~vJ The form of equation (1.9.10) is such that only the relative values of the mode vector components can be found. A constraint is imposed such as a unity value for one of the modal components, and then equation (1.9.10) is solved for the remaining components. Since the mode shape functions are arbitrary to a constant multiplier, it is convenient to normalise the modal vector. Two normalisation techniques are traditionally used in structural dynamics. The first technique is to set the maximum absolute component of the modal vector to one. That is max IV,, I = 1.
(1.9.12)
The second approach is to scale the mode shape with respect to the mass matrix as follows: ¥ ~ M ¥ , = 1.
(1.9.13)
The mode shapes also satisfy the following orthogonality conditions with respect to the mass and stiffness matrices
yTMym = 5,mM,,
(1.9.14a,b)
INTRODUCTION TO MECHANICAL VIBRATIONS
19
where Kn and M, are called the modal stiffness and mass coefficients, the indices m and n refer to the modal order and 6nm is the Kronecker delta function, defined as 6~m= 1 if n=mand6,m=0if n¢m. If the normalisation scheme of equation (1.9.13) is used, the coefficients in equation (1.9.14a,b) reduce to K~ = co,2 and M~ = 1. Also from the orthogonality conditions of equation (1.9.14a,b) we find 2
a)~ =
.
(1.9.15)
M.
The mode shapes form a set of linearly independent vectors, and thus they can be used as a basis in which to expand the solution vector w. Then, equation (1.9.2) which describes the individual motion of an element of the system, can be expanded in matrix form to describe the motion of the complete system by w(t) = ~ q ( t ) ,
(1.9.16)
where • is the modal matrix whose columns are the modal vectors, and q ( t ) = {ql(t) . . . . . qu(t)} T is referred as the modal displacement or generalised coordinate vector. Substituting equation (1.9.16) into (1.9.1), we obtain d2q MR/
dt 2
+ KRJq = 0.
(1.9.17)
We now premultiply equation (1.9.17) by the transpose of the mode shape matrix to obtain ~IJTM~I/ d2q + ~IJTK~IJq = 0. dt 2
(1.9.18)
Using equations (1.9.14a,b) and (1.9.15) we can then describe the original equation of motion of the MDOF system as a set of N second-order differential equations dZqn(t) dt 2
2
+ co~q,(t) = 0,
n = 1,2 . . . . . N.
(1.9.19)
This set of differential equations have the same form as an SDOF oscillator with a unit mass and a spring constant equal to ~on. 2 Thus the actual time history response of the nth modal coordinate to initial conditions can be written as (see equation (1.5.7) with
~=o) qn(t) = qn(0) cos %t +
4.(0)
sin %t,
(1.9.20)
(-On
where q,(0) and 4,(0) are the initial displacement and velocity of the nth modal coordinate. These are obtained from the initial conditions of the system w(0) and ";v(0) as follows"
q~ (0) = ~P~Mw (0),
O~(0) = ~P~ Mw (0).
(1.9.21)
20
ACTIVE CONTROL OF VIBRATION
Once the response of the modal coordinates are found, they are substituted into equation (1.9.16) to solve for the physical vector of coordinates w(t). Thus, the response of an MDOF system has been reduced to the linear combination of the response of a set of independent SDOF oscillators. Equation (1.9.20) can also be written in a form similar to equation (1.4.6) using equations (1.4.7) and (1.4.8).
1.10
Forced response of MDOF systems
The equation of motion of an MDOF system under an external force disturbance is given by M d2w ----- + Kw = fit). dt 2
(1 10.1)
To solve equation (1.10.1) it is first necessary to solve for the eigenvalues of the free system, as described in the previous section. This will provide a set of N natural frequencies, wl, w2 ... ton and associated mode shape vectors ¥1, Y2 .... YN- We can then write the solution vector w as a linear combination of modes by again separating the response vector w into spatial and time varying components:
w(t)=Wq(t),
(1.10.2)
where W is the mode matrix described previously. Substituting equation (1.10.2) into equation (1.10.1) and premultiplying the resulting equation by WT we obtain the following set of N uncoupled differential equations: d2wn dt 2
2
+ to,w~ = F~(t),
(1.10.3)
where F,(t) is the nth generalised force associated with the external force vector f(t) that is given by (1.10.4)
Fn(t)=wTf(t).
The solution of equation (1.10.3) for the actual motion can be obtained from Duhamel's integral as (Rao, 1990) 1 q,(O =
r| ' F,(r) sin [to,(t- r)] dr
Mn(.On J o
+q,(O) cos w,t +
q,(O)
sin to~t,
(1.10.5)
ton where F,(t) is the actual component of the generalised force.
1.11 Damped motion of MDOF systems With the inclusion of damping, equation (1.9.1) is modified to d2w M
dt 2
dw +C
, + K w = 0,
dt
(1.11.1)
INTRODUCTION TO MECHANICAL VIBRATIONS
21
where C is the damping matrix. We again replace the modal expansion of w(t) from equation (1.9.16) into (1.11.1) and premultiply by WT. In general, the matrix product w T c w , denoted as the modal damping matrix, will not result in a diagonal matrix. Thus, the matrix derivative equation in (1.11.1) will not be uncoupled by the modal matrix. The modal differential equations are said to be coupled by the damping matrix C. There are special cases that will result in a diagonal modal damping matrix, and therefore decouple the equations of motion. For example, this will occur when the damping matrix is given by (Thomson, 1993) C = a M + ilK,
(1.11.2)
which is a linear combination of mass and stiffness matrices. This condition leads to what is known as proportional damping. The purpose of choosing such damping is that the procedure outlined above will still produce a set of N uncoupled equations as in the undamped case. However, it should be remembered that this is not generally the case for real systems. Under these ideal circumstances it is straightforward to show that the differential equation of motion for the nth mode becomes (Rao, 1990)
Mn d2q"+ Cn dq. dt 2
- ~ t + K.x = F.(t),
(1.11.3)
where (1.11.4)
C. = a M . + ilK..
Equation (1.11.3) can also be written in the more convenient form as 2
F.(t)
tT.(t) + 2~.tO.q. + q.tO. = ~ ,
(1.11.5)
Mn where ~, is the damping ratio for the nth mode. The solution of this equation for the actual motion is again provided by Duhamel's integral and is given by
f
1 t F,(r) e q.(t) = M.tO. o +e
-~nwn(t-r) sin [tO,~(t- r)] dr
(0) cos to,~t +
~,to,~q,(0) sin to,~ , tOnd
(1.11.6)
where 09,d is the damped natural frequency of the nth mode defined as ~o~ = ~o,~/1 - ~2.
1.12
Finite element analysis of vibrating mechanical systems
The derivation of the equations of motion for the structural system of Fig. 1.7 was based on writing the equilibrium equation for each of the masses in the system using Newton's second law. This approach is applicable to simple one-dimensional systems. However, it quickly becomes cumbersome and impractical for complex structures. The finite element method (Zienkiewicz, 1977) is a technique developed in order to
ACTIVECONTROLOFVIBRATION
22
overcome these difficulties. The general steps involved in the finite element method (FEM) can be described as follows: (i) The basic concept is to subdivide the structure into a finite number of elements, a process which is known as the discretisation of the continuum. (ii) The displacement field of each element is then approximated by interpolation functions and the displacement is calculated at a reduced number of discrete points or nodes, N~. (iii) The equilibrium equations for each one of the nodes is derived to form the equations of motion for the free-free element that defines the element mass and stiffness matrices M e and K e, respectively. (iv) These element matrices are used to build the matrix differential equation for the complete structure. This process consists of assembling the global mass and stiffness matrices by bringing the contribution of each element mass and stiffness coefficients to the proper nodal point. (v) Finally, the boundary conditions are imposed on the system by either constraining the corresponding nodal displacements or applying nodal forces. The formulation described above will be illustrated for a simple one-dimensional uniform bar in longitudinal motion, as shown in Fig. 1.8(a). All motion will be described in terms of the actual or observed motion of the system. We first discretise the continuum into finite elements, as depicted in Fig. 1.8 (b). The elements will have a node at each end, and the axial displacement field will be approximated by linear interpolation functions as follows and as shown in Fig. 1.8 (c) (Zienkiewicz, 1977):
(112.1)
u(x,t) = Un(t)(l _ ~n) +Un+l(t) --x ln,
where x is a local coordinate, l~ is the element length, and u,(t) and u, +~(t) are the unknown actual axial displacements at the nodes n and n + 1, respectively. To derive the equations of motion of the element, it is convenient to use Lagrange's equation (Meirovitch, 1967) given by d
dr~ +
dt ~--~r]
r n,n + 1,
=0,
(1.12.2)
dur
where T n and V, are the kinetic and strain energy of the nth element, respectively. The kinetic energy, Tn, and strain energy, Vn, of the element are defined by 1
t,
T, = 2 Io pSu2(x) dx,
1
I,
V~= 2 Io ESu2(x) dx,
(1.12.3a,b)
where p is the mass density, E is the elastic modulus and S is the cross-sectional area which is assumed to be uniform. Substituting equation (1.12.1) into (1.12.3), performing the integrations and substituting the results into equation (1.12.2) yields
l, pS 6
]f } S[l l][U t f00/
1 an + 2 //,÷1 ~
=
-1
,
(1.12.4)
1 U,+l
which is the matrix differential equation for the nth element. The matrices in equation (1.12.4) are the element mass and stiffness matrices given by
/nf)S[2l l] Ue-T 2'
Ke WS[ 1 -1] ----~n--1 1 "
(1.12.5)
23
INTRODUCTION TO MECHANICAL VIBRATIONS
~1"- I
Bar
I -I~
(a)
Node
\
/-.I. v
A
,~
n
nth finite element A v
.L v
(b)
A v
n+l n
1- x/~n
(c)
X/~n~ / i
Fig. 1.8 Finite element analysis of the motion of a bar: (a) uniform bar in extension; (b) discretisation into nodes; (c) linear interpolation functions.
The global mass and stiffness matrices are next built by adding the influence of each element matrix coefficient. Then, the equation of motion of the complete structure is given by Mii + Ku = f,
(1.12.6)
where
M = p__S_S 6
211
l]
0
0
...
0
l]
2(/1 +/2)
12
0
...
0
0
12
2(/2+/3)
13
...
0
•
:
:
"..
0
0
0
In
0
...
0
(1.12.7)
2(In+In+]) ln+l ...
lu
0 2/u+ 1
24
ACTIVE CONTROLOF VIBRATION
is the mass matrix of the discretised system and
1/ll -I/I,
K = ES
I
-1/l, (lfl,) + (1/12)
0 -1/12
-1/12
(1//2) + (1//3/
0 0 -1/13
°,° °°° .°°
°°.
l
-1/l=
(1//,,) + (1//,,+,) -1//~+, °..
-1/lN
°°.
I/IN+, (1.12.8)
is the stiffness matrix, and U --'- { / , / 1 , / , / 2 ,
...,
U N } T,
f=
{f,,f2 . . . . .
fN} T
(1.12.9)
are the displacement and force vectors respectively. We will now, for illustrative purposes, assume that the beam is clamped at one end and driven by an oscillating in-plane force f = FR cos tot at the other end. This configuration would be implemented as follows. The clamped boundary condition is imposed by setting u~ = 0. In order to solve the linear system of equations represented by equation (1.12.6), the first row of vectors u and f is eliminated and the first row and column of matrices M and K are removed. The force boundary condition is imposed by setting at node N, fN = FR and f~ = 0 for n = 1 . . . . . N - 1. Since the in-plane motion of this one-dimensional beam model is governed by a second-order differential equation (see Section 2.2), the model will have N degrees of freedom for N nodes. This brief description of the FEM is only intended as an introduction for the reader not familiar with the technique. For a more detailed description of the methods for differing motions, structural shapes and boundary conditions the reader is referred to texts such as those by Zienkiewicz (1977), Cook (1981) and Petyt (1990).
1.1
Introduction
All mechanical systems composed of mass, stiffness and damping elements exhibit vibratory response when subject to time-varying disturbances. The prediction and control of these disturbances is fundamental to the design and operation of mechanical equipment. In particular, the use of secondary, active inputs to the system in order to modify the system response in a controllable way is the topic of this book. The analysis of controlled systems is founded on the same analytical approaches used to study the vibrations of elastic structures. A brief review of the main concepts of vibration analysis and the associated techniques of solution is necessary to set the foundation for the following chapters. In this chapter we begin by defining terminology and the mathematical methods for describing the linear response of vibrating systems. The equations of motion and linear behaviour of single-degree-of-freedom systems are outlined for both free and forced response. The use of the Laplace transform to solve for transient response is reviewed. The extension to multi-degree-of-freedom systems and then the use of finite element analysis are briefly introduced. These approaches are valid for lightly damped structures or elements that are small relative to the wavelength of motion. For more detail the reader is referred to the texts of Thomson (1993), Meirovitch (1967) and Inman (1994).
1.2 Terminology The following is a brief list of the main terminology and definitions used in analysing the vibratory response of mechanical systems. Mechanical system. A mechanical system is composed of distributed elements which exhibit characteristics of mass, elasticity and damping. Degrees of freedom. The number of degrees of freedom of a system is equal to the number of independent coordinate positions required to completely describe the motion of the system. System response. All mechanical systems exhibit some form of vibratory response when excited by either internal or external forces. This motion may be irregular or may repeat itself at regular intervals, in which case it is called periodic motion. Period. The period T is the time taken for one complete cycle of motion.
2
ACTIVE CONTROLOF VIBRATION
Harmonic motion is the simplest form of periodic motion whereby the actual or observed motion can be represented by oscillatory functions such as the sine and the cosine functions. Motion that can be described by a continuous sine or cosine function is called steady state. For example one may write the actual displacement w ( t ) in the form
w(t) = [A I cos(~ot + q~),
(1.2.1)
where w ( t ) and IAI are real, a~ is circular frequency in radians per second and q~ is an arbitrary phase angle in radians. Equation (1.2.1) can also be expressed as a superposition of a sine and a cosine function as w ( t ) - AR cos cot- At sin cot,
(1.2.2)
where AR and At are real numbers such that AR = I Alcos ¢,
(1.2.3a)
A, = I alsin ¢,
(1.2.3b)
where the phase angle is specified by qb = tan - ~(AJAR).
(1.2.4)
The constant I AI in equation (1.2.1) is related to the constants At and AR in equation (1.2.2) by
[al
= ( a z + a l z)1/2.
(1.2.5)
Frequency is the number of cycles per second (also called hertz) of the motion and is the reciprocal of the period. Therefore frequency is specified by
1 f= --
T
(1.2.6)
and circular frequency co (radians per second) is given by o9 = 2z~f. Amplitude is the measure of For example if the motion is corresponds to the amplitude of The mean square amplitude response. Thus, for example
(1.2.7)
the maximum response of the system during a period. specified by equation (1.2.1) then the constant I AI the motion. is defined as the time average of the square of the
= lim --1 [ r w2(t) dt. T--->~* Z J°
(1.2.8)
The root mean square (rms) amplitude is the positive square root of the mean square amplitude. For the harmonic oscillation of equation (1.2.1) the rms amplitude is independent of phase and is equal to Ial/~. Free vibrations are the motions of the system in the absence of external disturbances and as a result of some initial conditions. Forced vibrations are the motions of the system produced by external, persistently applied disturbances.
3
INTRODUCTION TO MECHANICAL VIBRATIONS
Natural frequencies are those frequencies at which response exists during free vibration. The lowest natural frequency is called the fundamental frequency. Transient motion is motion other than steady state response. If damping is present, transient response will decay with increasing time. Phasor. A phasor is a rotating vector representation of the harmonic motion of the system. The periodic motion of equations (1.2.1) and (1.2.2) can be represented in a complex form, which is more convenient for mathematical manipulations and is given by w(t) = A e j°~t, (1.2.9)
where A and w(t) are complex with the complex amplitude specified as (1.2.10)
A= AR + jAI,
The phasor representation of equation (1.2.9) is shown in Fig. 1.1. The length of the vector, I AI, is the real amplitude of motion. As the vector rotates with angular velocity to in a counterclockwise direction, its projection on the real and imaginary axis of the complex w plane varies harmonically with time t. A rotation of the vector through 360 ° corresponds to a cycle of motion. In this text the convention used is that the real component of the phasor or the complex description of the motion corresponds to the actual, observed or measured motion. Therefore the actual motion is given by w(t) = Re[A eJ°'t].
(1.2.11)
Using the relationship eJ°"=cos tot + j sin tot and substituting A = AR +jAI into the above expression yields equation (1.2.2). The phase of the motion, q~, is thus retained through the ratio of the imaginary and real components of the phasor as specified in +lm
AE J I I I I I
AR
+Re
ines phasor position at t = O)
Fig. 1.1 Phasor diagram representation of harmonic motion.
4
ACTIVE CONTROLOF VIBRATION
equation (1.2.4). Note that a negative sign has been used in equation (1.2.2) in contrast to many texts dealing with vibration. This choice ensures that the phase q~ is positive and since At is a constant to be determined by boundary conditions, the choice of negative sign does not affect the result. Since the phasor is a vector, any number of harmonic motions of the same frequency can be added vectorially. For linear motions written in complex notation, since the principle of superposition holds, this simply means separately summing the real and imaginary components of the individual motions. In this text the majority of the response equations are written using complex notation as this is the most convenient form for analysing systems where responses are superimposed (i.e. as is the case in active control simulations). The actual motion can be directly recovered by taking the real part of the complex description. Where the motion is directly described in the actual form, it is indicated in the text.
1.3 Single-degree-of-freedom (SDOF) systems Consider a mass M supported by a massless spring as shown in Fig. 1.2(a). As the displacement of the system can be completely specified at all times by a single variable w, the system is said to possess a single degree of freedom (SDOF). By appropriately w
T (a)
M
(b)
M
(d)
M
K
//
/
/ / / /
/
/ //'
1" (c)
M
K<
////I Fig. 1.2
G
// ///
SDOF systems and free body diagrams: (a) and (b), undamped; (c) and (d), damped.
INTRODUCTIONTO MECHANICALVIBRATIONS
5
choosing the origin of the coordinates at the rest position (i.e. in static equilibrium), the constant force due to gravity can be ignored. When the mass is displaced an amount w from its equilibrium position, the spring will exert a restoring force - K w due to being elongated (for positive w) as shown in the free body diagram of Fig. 1.2(b). On release of the mass, the spring will attempt to accelerate the mass and the restoring force and acceleration are related by Newton's second law of motion which shows that d2w M
dt 2
= -Kw.
(1.3.1)
Rearranging terms, we obtain the differential equation describing the motion of this simple SDOF system. This is given by dt 2 +
w-0.
(1.3.2)
Equation (1.3.2) is a second-order ordinary differential equation and therefore must have a solution which is specified in terms of two unknown constants or amplitudes of motion. Although the above analysis is straightforward, it does illustrate the basic process by which elastic systems are generally analysed. The system is first broken into elements (or blocks). For some initial conditions the restoring and inertial forces are balanced, thus providing the differential equation describing the motion of the system.
1.4
Free motion of SDOF systems
Based upon the observation that mechanical systems respond harmonically in free motion, the solution of equation (1.3.2)can be assumed to be of the form given by equation (1.2.2). Therefore we assume that the actual motion can be described as w ( t ) = AR cos oJt- AI sin tot.
(1.4.1)
where AR and A/are real amplitudes of motion. Substitution of equation (1.4.1) into equation (1.3.2), differentiating with respect to time and eliminating common terms, provides a relation for the frequency ton at which the system will naturally vibrate. This is given by oJn = I ~
(1.4.2)
and thus the solution of equation (1.3.2) becomes w ( t ) = AR cos to, t - A~ sin ~o,t.
(1.4.3)
In order to specify the motion completely, the unknown constants AR and A~ need to be determined and these are found by applying given boundary or initial conditions to the system. The frequency oJ, is called the natural or resonance frequency and is a very important characteristic of the system as will be shown in Section 1.6. Note that the natural frequency of the SDOF system increases with stiffness K and decreases with increasing mass M. These observations are in general true for all linear elastic systems.
6
ACTIVE CONTROL OF VIBRATION
To specify the motion of the system completely, one needs to apply initial conditions. For example, if at t = 0 the system has an initial real displacement w(0) and a real velocity ~¢(0) then the unknown constants in equation (1.4.3) can be determined from w(0) = AR,
(1.4.4a)
w(0) = - o9~A,,
(1.4.4b)
where the use of the overdot is a compact notation for differentiation with respect to time. The observed response can be obtained by solving for the constants AR and At from equations (1.4.4a) and (1.4.4b) and substituting these into equation (1.4.3). The actual response of the system to arbitrary initial conditions is then given by
w(t) = w(0) cos tOnt+
w(O)
sin (Ont.
(1.4.5)
This motion can also be written as w(t)= [A lcos(O)nt + ~9),
(1.4.6)
where the phase angle q~ is specified from equation (1.2.4) by q~= tan_l( -
tOnW(O)l~(O)
(1.4.7)
and the amplitude of motion that results from equation (1.2.5) is given by IAI=
[w(O)] ~ +
(1.4.8) (.On
Thus the response of the SDOF can be seen to be simple harmonic motion at the natural frequency ~o, with an amplitude [A[ and a phase angle q~ given by equation (1.4.7) and (1.4.8) respectively.
1.5 Damped motion of SDOF systems All vibrations in realistic systems occur with some form of damping mechanism, where the energy of vibration is dissipated during a cycle of motion. The simplest form of damping is when the resisting force associated with the damping is proportional to, and acts in an opposite direction to, the velocity of the element. Thus the damping force is specified by dw F~ - - C ~ ,
(1.5.1)
dt where C is the damping coefficient. Figure 1.2(c) shows an SDOF system with this form of damping which is called viscous damping. Including the additional damping force into the force balance of the new system, as shown in the free body diagram of
INTRODUCTION TO MECHANICALVIBRATIONS
7
Fig. 1.2 (d), leads to a new differential equation for an SDOF system given by
M
d2w
dt 2
+C
dw
dt
+ Kw = 0.
(1.5.2)
It is now more convenient to use a complex description of the motion. Thus a solution is assumed of the form
w(t) = A e "/',
(1.5.3)
where w(t) is now a complex variable. Substituting equation (1.5.3) into equation (1.5.2) provides the values of ~, for which a solution exists. These are given by 7=
2~/±
-
.
(1.5.4)
It is convenient to express C in terms of the critical damping Cc = 2M~on. The damping ratio is then defined by ~ = C/Cc. Equation (1.5.4) then reduces to 7 = -ogn~ ±jogJ 1 - ~2,
(1.5.5)
where co, is the undamped natural frequency given by equation (1.4.2). When ¢ > 1, both terms in equation (1.5.5) will be real and this implies a steadily decaying response with no oscillation. This is termed an overdamped system. When = 1, the system is said to be critically damped. This value of C represents the smallest possible damping required to prevent oscillatory motion and ensures that the system returns to its rest position in the shortest time as shown in Fig. 1.3. When ~ < 1 then the square root term will be real, positive and 7 in this case will be complex with a negative real part. Thus the response will oscillate at a damped natural frequency oJa = oJ~x/1 - ¢2
(1.5.6)
and decay in amplitude with increasing time. This is called light damping and it is exhibited by most structural systems which are thus described as underdamped. The observed response to the specified initial conditions defined in Section 1.4 is obtained by using the real part of equation (1.5.3) and the initial conditions to solve for the constants A ! and AR as described in Section 1.4. The actual displacement is then given by
w(t) = e -~,.¢t[w(0) cos COdt+ vi,(0) + ¢co,w(0)sin (Oat] .
(1.5.7)
(.,0d
This equation can also be written in simple harmonic form as -~o,¢t W(t) = IAle cos(~oat+ q~),
(1.5.8)
where the phase angle q~ is now given by q~= t a n _ l ( w ( 0 ) + ~ w ( 0 ) ) -
w(O)~oa
(1.5.9)
and the real amplitude by
I AI = {[w(0)]2 + [w(0)+ ~c0~w(0)]2/~03} 1/2.
(1.5.10)
8
ACTIVE CONTROL OF VIBRATION
Wo
light damping
~t)
~'<1
Wo
criticaldamping
w(t)
~r=l
Wo
~
heavydamping t
~'>1
Fig. 1.3 Response of an SDOF system with various values of damping. Initial displacement w(0) = w0 while initial velocity r0(0)= 0. Equation (1.5.8) reveals that the response consists of harmonic motion at a frequency of rod and with an amplitude given by IAle -~.~' which decays with increasing time. Note that from equation (1.5.6) it is apparent that the damped frequency is lower than the natural frequency. Obviously the inclusion and characteristics of damping are very important to active control methods since it represents a process by which the response of a system can also be reduced by passive means. Figure 1.3 shows typical response versus time curves for an SDOF system with light, critical and heavy damping. As predicted by the above equations, light damping leads to a response that oscillates at rod and is slowly decreased in amplitude with time. With critical damping the response moves towards the equilibrium position but does not cross it. Heavy damping leads to almost complete suppression of the oscillatory type motion; the damping force is such that it significantly slows the motion towards the equilibrium position and the system takes longer to return to the equilibrium position than with critical damping.
INTRODUCTIONTO MECHANICALVIBRATIONS
1.6
9
Forced response of SDOF systems
Many systems are excited by persistently applied disturbances rather than being initially excited as in the cases of free motion discussed above. Let us assume that the input disturbance is specified by a harmonic force of constant amplitude written in complex form and given by f ( t ) = F e TM,
(1.6.1)
where F is a complex number defining the amplitude and relative phase of the applied force. The homogeneous differential equation of equation (1.3.2) thus has to be modified to take account of the disturbance force and is written as d2w M ~
dt 2
+ K w = F e ~'.
(1.6.2)
Since it is assumed that the disturbance has been applied for all time, t, the transient response component is zero and it is reasonable to assume the steady state response will be again harmonic and specified in complex form by w ( t ) = A e TM,
(1.6.3)
where w ( t ) and A are again, in general, complex. On substitution of this assumed solution into equation (1.6.2) we obtain -~o 2 +
A = ~. M
(1.6.4)
The unknown complex response amplitude A can be obtained by rearranging equation (1.6.4) and is given by ElK
A=
.
(1.6.5)
1 - (O.)/O)n) 2
Equation (1.6.5) proviaes immediate insight into the response of elastic systems to a steafly state harmonic flisturbance. When ~o = a,. the flisplacement amplitufle will theoreticany be infinite for continuous steafly state excitation. Thus when systems are flriven at a frequency iclentical to or very close to their natural frequency a very large response will result. Under these conditions the system is ~escribefl as being flriven o n anti is a very important conflition from a control point of view. resonance
Extension of the above approach to more complex systems becomes increasingly difficult. In this case, it is often easier to use the impedance method. We define mechanical input impedance as the ratio of the complex amplitudes of input force to the velocity at the drive point. Thus for the harmonic displacement of equation (1.6.3) the velocity is given by vi,(t) and is specified by ¢v(t) = ja)A e TM.
(1.6.6)
The velocity thus has complex amplitude javA and input impedance is given by Z~ =
F jooA
(1.6.7)
10
ACTIVE CONTROL OF VIBRATION
evaluated at the disturbance location. The input impedance of the system in Fig. 1.2(a) can be shown to be
-jK[ 1 -
Z~ =
(co/co,) 2]
.
(1.6.8)
co
Once the input impedance of a system is determined or measured, the response of the system to a steady state harmonic disturbance force can be directly calculated using equation (1.6.7). Inclusion of damping into the relationship for the forced response modifies the resulting amplitude to
F/K
A= 1
-
.
(1.6.9)
(co/con)2 + j2~(CO/CO,)
(a) --
dctJ tO
~' = 0.05
6
~'= o.~
5
Orj
-tO
"~ t,-E
4
~= o.~5
3 ~"= 0.25
2
t.--
o
z
~'= 0.5 1 0.5
1.0 1.5 2.0 Frequency ratio, co~con
2.5
3.0
(b)
o~ o
d c'-
-90
03 t-
13_
~ ' ~ 0 ~ ~ -180
Fig. 1.4
0
1.0
2.0 3.0 4.0 Frequency ratio, o~/con
n 5.0
Forced response of an SDOF system: (a) magnitude, (b) phase.
11
INTRODUCTION TO MECHANICAL VIBRATIONS
Figure 1.4(a) presents the non-dimensional displacement response amplitude equal to
IAK/FI versus the non-dimensional input frequency, co/~o,, for various values of damping ratio ~. The first important feature apparent from Fig. 1.4(a) is that the inclusion of small amounts of damping has led to large reductions of response at or near resonance but has small effect away from the resonance condition. Secondly, damping leads to the response of the system being bounded at resonance. Increasing damping also causes the frequency of maximum response to move to lower values. The corresponding phase response is plotted in Fig. 1.4(b). For very light damping the phase of the displacement response relative to the excitation force of the system flips through nearly a 180 ° phase change as the excitation frequency is increased through the resonance frequency. Increasing the amount of damping leads to a decrease in sharpness of the phase transition. These observations will be seen in later sections to have important implications for the performance and stability of actively controlled elastic systems, particularly when they have many degrees of freedom.
1.7
Transient response of SDOF systems and the use of the Laplace transform
We first consider the response of an SDOF system to an impulse excitation as this case provides the basis for the study of more general forms of transient excitation. An impulse is generally thought of as a disturbance force acting on a system over a very short time. We define an impulse as the time integral of the force I = I ~f(t) dt, 0
( 1.7.1 )
where f(t) is a real function describing the variation of the force with time and r is the duration of the impulse. Since fdt = Mdw where w is the velocity of the mass we can rewrite equation (1.7.1) in the form I = I:(~)M dw.
(1.7.2)
If we make the further approximation that the impulse acts over such a small time that v~(r) -- w(0), or in other words the system reaches a velocity ~¢(0) instantaneously, then I = My0(0),
(1.7.3)
which is known as the impulse-momentum relation. Thus the initial conditions for the impulsively excited system are w(0) =0,
(1.7.4a)
¢v(O) = I/M.
(1.7.4b)
Applying these initial conditions to the damped SDOF system, the response to a unit impulse can be determined from equation (1.5.7) with I = 1 and is given by -~%t
w(t) =
e
Mood
sin cojt,
(1.7.5)
12
ACTIVE CONTROLOF VIBRATION
where w(t) is the actual motion of the mass. Equation (1.7.5) is known as the unit impulse response function of the system generally denoted h(t), and is a very useful parameter. Note that h(t) is a purely real function. The knowledge of the unit impulse response function allows us to derive expressions for the response of a mechanical system to any general excitation, including transients, and the response due to specified initial conditions using Duhamel's integral (Meirovitch, 1967; Newland, 1984). The actual response of the SDOF system to a general real forcing function f(t) is then given by -~tOn(t- r)
w(t) = I~f(r) e
MtOd
sin[tOd(t- r)] dr
[w w(O)+~tOnw(O) q sin rod • +e -¢'°"' (0) cos tOdt+ rod
(1.7.6)
A more specific form of equation (1.7.6) is called the convolution integral and is written as
w(t)= I~f(r)h(t- r) dr,
(1.7.7)
where h(t) is the unit impulse response function of the system. Equation (1.7.7) essentially expresses the response of the system to any general excitation f(t) by using the principle of superposition, that is, the total response consists of the sum of a series of impulses appropriately weighted and positioned with the magnitude of the forcing function f ( r ) at various times, r. We now examine the behaviour of the impulse function for decreasing time durations. Figure 1.5 shows a rectangular impulse function of length r, centred on t = 0 and of impulsive magnitude I. If we take I to be a constant, then by examining equation (1.7.1) it is apparent that as we progressively reduce r, then f(t) will increase. In the limiting case of r tending to zero we have a function which is infinite at t = 0 and has a value of zero at all other values of time t. If we further assume that the product f(t)
-d2
d2
Fig. 1.5 General impulse function.
t
INTRODUCTION TO MECHANICALVIBRATIONS
13
r F is equal to unity (i.e. I = 1) then we have defined the Dirac delta function described by the pair of relationships iS(t) = 0,
t ¢ 0,
I ~ d(t)dt = 1.
(1.7.8a) (1.7.8b)
If we assumed that the rectangular pulse was centred at time to then we could equivalently define the function in the form d(t - to) = 0,
t ¢ to,
I?= d(t- to)dt = 1.
(1.7.9a) (1.7.9b)
A very useful characteristic of the Dirac delta function is called the 'sifting' property given by
I? f(t)d(t- to) dt=f(to).
(1.7.10)
Other properties and uses of generalised functions such as the Dirac delta function are described by Farassat (1994). We now turn our attention to the use of the Laplace transform in analysing the response of mechanical systems to transient disturbances. The Laplace transform is a generalisation of the Fourier transform and is generally useful for disturbances which do not have a convergent Fourier transform (Nelson and Elliott, 1992). The Laplace transform can often be used to transform complicated disturbances associated with impulsive and transient excitation into a form which, in conjunction with the transformed equation for the response of the system, are more easily manipulated. The Laplace transform of f(t) is defined to be (see, e.g. Kuo, 1966; Meirovitch, 1967)
.~f(t) = F(s) = I : f ( t ) e -s' dt,
(1.7.11 )
where s is a complex variable defined by
s= cr + jw.
(1.7.12)
The inverse Laplace transform is defined by (Meirovitch, 1967) ~ - l F (s) = f ( t ) =
1 ~7-~ [,/+~ F(s) es' ds, 2Jrj
(1.7.13)
where the path of integration is a line parallel to the imaginary axis crossing the real axis at Re{ s} =~, and extending from -,,,, to +,,,,. As an example, the equation describing the motion of an SDOF system can be written in the form
MfC(t) + C~(t) + Kw(t)= f(t).
(1.7.14)
Applying the Laplace transform to both sides of the above equation, we obtain
M[sZW(s)- w(O)s- w(0)] + C[sW(s) - w(0)] + KW(s) = F(s).
(1.7.15)
14
ACTIVE CONTROL OF VIBRATION
Collecting terms and rearranging leads to the s-domain relation for the system response described by
F(s)
W(s) =
M s 2 + Cs +
K
+
(Ms + C)w(O) + MCv(O) Ms 2+
Cs + K
,
(1.7.16)
where the first term in equation (1.7.16) describes the forced response and the second term, the motion due to initial conditions w(0) and vi,(0). If we restrict our attention to the forced response, the system transfer function for an SDOF system can be written in the form
H(s) = M s 2 + Cs +
K
.
(1.7.17)
It is often useful in the analysis of active control systems to represent the input-output relationship using a block diagram as shown in Fig. 1.6. Note that if we choose to consider only the imaginary axis of the Laplace domain s, then the system transfer function H(s) is equivalent to a frequency domain transfer function H(jto), as discussed in Nelson and Elliott (1992). Meirovitch (1967) also demonstrates that the system transfer function can be obtained from the system impulse response function by the relationship
H(s)=~h(t).
(1.7.18)
We now illustrate the use of the Laplace transform by solving for the transient response of an SDOF system excited by a harmonic forcing function turned on at t =0. The input disturbance is specified by f(t) = O, f ( t ) = F sin tot,
t
(1.7.19)
Taking the Laplace transform of the damped inhomogeneous SDOF equation with the forcing function specified by equation (1.7.19) we obtain, with zero initial conditions
(Ms 2 + Cs + K)W(s)= F(s),
(1.7.20)
where F(s) is the Laplace transform of the input disturbance and is specified for the function described in equation (1.7.19) by 09
F(s) = ~ .
(1.7.21)
$ 2 + 0) 2
Transfer function Input
.I
-I
H(s)
Output
Fig. 1.6 System block diagram.
W(s)
15
INTRODUCTION TO MECHANICAL VIBRATIONS
Solving for W(s) we find
W(S)
"-"
M
S 2 + a)
2
X
$2
1)
22,
2 "
+ 2~a).s + o9~
Using the method of partial fractions and tables of known inverse Laplace transforms (Meirovitch, 1967), we find the solution for the actual response of the system to be given by
w(t)
=
1 M
)[
(a) 2 - 0)2) 2 - 4~2(.020) 2
-2~09 n COS a ) t -
(09 2 - 092n)+ 2~209n
-¢%t
2 ~ n COS OJat +
+e
sin cot O)
sin a~a •
(1.7.23)
O.)d
Hence the response of the system to a sine-wave forcing function turned on at t = 0 consists of a damped oscillation at the natural frequency of the system and a steady state response at the forcing frequency. If the damping is small or the excitation is close to the natural frequency of the system then the vibrations associated with the natural frequency of the system can extend for long times. This type of analysis and the results produced have important implications in terms of performance and stability of actively controlled mechanical systems.
1.8 Multi-degree-of-freedom (MDOF) systems When N independent coordinates are required to completely specify the system response, the system is said to have N degrees of freedom. Such a system is also said to have multi-degrees of freedom (MDOF). When a mechanical system has many degrees of freedom it is more convenient to use a matrix representation to describe and analyse the motion. In this section we formulate the equations of motion in matrix form. The reader is referred to the Appendix of the companion text (Nelson and Elliott, 1992) for a review of vector and matrix theory. Consider, for example, a simple N-degree-of-freedom system as shown in Fig. 1.7 (a) which is fixed at one end and free at the other. Excitation forces are directly applied to each mass. For a particular mass M,, the free body diagram is shown in Fig. 1.7 (b) and the corresponding equation of motion can be written as d2w M n
dt 2
=Kn(Wn_l-Wn)-Kn+l(Wn-Wn+l)Jrfn
.
(1.8.1)
This equation can then be manipulated to take the form
M~
d2wn
dt 2
+ K~w~_, + (K~ + K~+ ~)w~- K~+ ,w~+, =f~,
(1.8.2)
where fn is the forcing function applied to the nth mass. Similar equations for the motion of all the masses can be written except for n = 1, where w0 =0, and n = N,
16
ACTIVE CONTROL OF VIBRATION
where K u + 1=0, which correspond to the fixed and free boundary conditions of the system of Fig. 1.7 (a) respectively. The resulting simultaneous equations for each mass can be written in matrix form as M d2w
+ K w = f,
(1.8.3)
dt 2
where M and K are the mass and stiffness matrices. In the following text, matrices and vectors will be indicated by upper- and lower-case boldface symbols respectively. For typical linear structures, the matrices appearing in equation (1.8.3) are real and symmetric. The mass matrix M is specified by
M=
M1 0 0 0 M: 0 0 M~ 0
(1.8.4) 0
M~ which is a diagonal matrix. The stiffness matrix K is given by
(K, +K2)
K
._.
-K:
o
(/<2 + K3)
-K3
0
(1.8.5) 0
-K~_, (K~_,+ K~) 0 -KN which is a symmetric matrix. The vectors w and f are given by
1
w=
,
f=
Lw~(t)]
1.9
"
.
(1.8.6a,b)
f~(t)
Free motion of MDOF systems
If the forcing functions f are set to zero and the system is given some energy by an initial condition, then the MDOF system described by equation (1.8.3) will vibrate indefinitely due to lack of damping. The vector differential equation describing the motion can be written with matrix coefficients as d2w
M
dt 2
+ Kw = 0.
(1.9.1)
INTRODUCTION TO MECHANICALVIBRATIONS Wl I I
~
w2 I I
Wn
~
I I
17
WN- 1
,~ v
1 I
4
, i
WN
,.._ v
I I
'
(a)
Kn+l (Wn- Wn+l)
K n (wn_l- Wn)
~n (b) Fig. 1.7 Example MDOF system: (a) system configuration; (b) free body diagram of the nth mass. The solution of equation (1.9.1) will provide the undamped free vibrations of the system. We again assume a harmonic solution of the complex form w, (t) - A n
e TM,
n = 1,2 ..... N,
( 1.9.2)
where Wn is the displacement and A, is the complex amplitude (i.e. represents both amplitude and phase) of the motion of the nth mass. Substituting equation (1.9.2) into equation (1.9.1) we find the relation - 2 M A + Ka = 0,
(1.9.3)
where 2 = w 2 and the amplitude vector a is given by A1 A2 a=
AN.
(1.9.4)
By premultiplying equation (1.9.3) by M - ~ and rearranging we obtain the expression 2Ia - Ha = 0,
(1.9.5)
where I is the identity matrix and the dynamic matrix H is specified by H=M-~K.
(1.9.6)
Equation (1.9.5) can be further manipulated in the form [2I - H ] a = 0,
(1.9.7)
18
ACTIVE CONTROLOF VIBRATION
For a non-trivial solution a s 0 and setting the determinant of equation (1.9.7) to zero provides the characteristic equation of the MDOF system which can be written as 12I-
H I = 0,
(1.9.8)
The roots 2, of the characteristic equation are called the eigenvalues which provide the natural frequencies of the system. The expansion of equation (1.9.8) leads to an Nth order polynomial equation in 2. The solution of this polynomial equation provides N roots of 2 which are real and positive when the matrices M and K are positive definite (Cook, 1981). The positive square root of the N values of ;t will provide N resonance frequencies for the system with the lowest value, n = 1, corresponding to the fundamental resonance frequency, i.e. to, = a/r-~, n = 1, 2 . . . . . N.
(1.9.9)
If the number of degrees of freedom is small, the equation (1.9.8) can be solved algebraically. However, if the number of degrees of freedom of the system is large, then the solution of the polynomial obtained by expanding equation (1.9.8) is difficult and it is more appropriate to use numerical methods as discussed in Rao (1990) or Inman (1994). Once the eigenvalues are found, the mode shapes or modal eigenvector ¥, can be determined by substituting 2, into equation (1.9.7). The relative variation in the amplitude of motion of each element provides the mode shape ¥ , corresponding to the nth eigenvalue as [in I - H ] ¥ , = 0,
n = 1,2, ..., N,
(1.9.10)
where the mode shape for the nth mode is given by the vector of resulting amplitudes A1]
v.-
I .
.
(1.9.11)
[A~vJ The form of equation (1.9.10) is such that only the relative values of the mode vector components can be found. A constraint is imposed such as a unity value for one of the modal components, and then equation (1.9.10) is solved for the remaining components. Since the mode shape functions are arbitrary to a constant multiplier, it is convenient to normalise the modal vector. Two normalisation techniques are traditionally used in structural dynamics. The first technique is to set the maximum absolute component of the modal vector to one. That is max IV,, I = 1.
(1.9.12)
The second approach is to scale the mode shape with respect to the mass matrix as follows: ¥ ~ M ¥ , = 1.
(1.9.13)
The mode shapes also satisfy the following orthogonality conditions with respect to the mass and stiffness matrices
yTMym = 5,mM,,
(1.9.14a,b)
INTRODUCTION TO MECHANICAL VIBRATIONS
19
where Kn and M, are called the modal stiffness and mass coefficients, the indices m and n refer to the modal order and 6nm is the Kronecker delta function, defined as 6~m= 1 if n=mand6,m=0if n¢m. If the normalisation scheme of equation (1.9.13) is used, the coefficients in equation (1.9.14a,b) reduce to K~ = co,2 and M~ = 1. Also from the orthogonality conditions of equation (1.9.14a,b) we find 2
a)~ =
.
(1.9.15)
M.
The mode shapes form a set of linearly independent vectors, and thus they can be used as a basis in which to expand the solution vector w. Then, equation (1.9.2) which describes the individual motion of an element of the system, can be expanded in matrix form to describe the motion of the complete system by w(t) = ~ q ( t ) ,
(1.9.16)
where • is the modal matrix whose columns are the modal vectors, and q ( t ) = {ql(t) . . . . . qu(t)} T is referred as the modal displacement or generalised coordinate vector. Substituting equation (1.9.16) into (1.9.1), we obtain d2q MR/
dt 2
+ KRJq = 0.
(1.9.17)
We now premultiply equation (1.9.17) by the transpose of the mode shape matrix to obtain ~IJTM~I/ d2q + ~IJTK~IJq = 0. dt 2
(1.9.18)
Using equations (1.9.14a,b) and (1.9.15) we can then describe the original equation of motion of the MDOF system as a set of N second-order differential equations dZqn(t) dt 2
2
+ co~q,(t) = 0,
n = 1,2 . . . . . N.
(1.9.19)
This set of differential equations have the same form as an SDOF oscillator with a unit mass and a spring constant equal to ~on. 2 Thus the actual time history response of the nth modal coordinate to initial conditions can be written as (see equation (1.5.7) with
~=o) qn(t) = qn(0) cos %t +
4.(0)
sin %t,
(1.9.20)
(-On
where q,(0) and 4,(0) are the initial displacement and velocity of the nth modal coordinate. These are obtained from the initial conditions of the system w(0) and ";v(0) as follows"
q~ (0) = ~P~Mw (0),
O~(0) = ~P~ Mw (0).
(1.9.21)
20
ACTIVE CONTROL OF VIBRATION
Once the response of the modal coordinates are found, they are substituted into equation (1.9.16) to solve for the physical vector of coordinates w(t). Thus, the response of an MDOF system has been reduced to the linear combination of the response of a set of independent SDOF oscillators. Equation (1.9.20) can also be written in a form similar to equation (1.4.6) using equations (1.4.7) and (1.4.8).
1.10
Forced response of MDOF systems
The equation of motion of an MDOF system under an external force disturbance is given by M d2w ----- + Kw = fit). dt 2
(1 10.1)
To solve equation (1.10.1) it is first necessary to solve for the eigenvalues of the free system, as described in the previous section. This will provide a set of N natural frequencies, wl, w2 ... ton and associated mode shape vectors ¥1, Y2 .... YN- We can then write the solution vector w as a linear combination of modes by again separating the response vector w into spatial and time varying components:
w(t)=Wq(t),
(1.10.2)
where W is the mode matrix described previously. Substituting equation (1.10.2) into equation (1.10.1) and premultiplying the resulting equation by WT we obtain the following set of N uncoupled differential equations: d2wn dt 2
2
+ to,w~ = F~(t),
(1.10.3)
where F,(t) is the nth generalised force associated with the external force vector f(t) that is given by (1.10.4)
Fn(t)=wTf(t).
The solution of equation (1.10.3) for the actual motion can be obtained from Duhamel's integral as (Rao, 1990) 1 q,(O =
r| ' F,(r) sin [to,(t- r)] dr
Mn(.On J o
+q,(O) cos w,t +
q,(O)
sin to~t,
(1.10.5)
ton where F,(t) is the actual component of the generalised force.
1.11 Damped motion of MDOF systems With the inclusion of damping, equation (1.9.1) is modified to d2w M
dt 2
dw +C
, + K w = 0,
dt
(1.11.1)
INTRODUCTION TO MECHANICAL VIBRATIONS
21
where C is the damping matrix. We again replace the modal expansion of w(t) from equation (1.9.16) into (1.11.1) and premultiply by WT. In general, the matrix product w T c w , denoted as the modal damping matrix, will not result in a diagonal matrix. Thus, the matrix derivative equation in (1.11.1) will not be uncoupled by the modal matrix. The modal differential equations are said to be coupled by the damping matrix C. There are special cases that will result in a diagonal modal damping matrix, and therefore decouple the equations of motion. For example, this will occur when the damping matrix is given by (Thomson, 1993) C = a M + ilK,
(1.11.2)
which is a linear combination of mass and stiffness matrices. This condition leads to what is known as proportional damping. The purpose of choosing such damping is that the procedure outlined above will still produce a set of N uncoupled equations as in the undamped case. However, it should be remembered that this is not generally the case for real systems. Under these ideal circumstances it is straightforward to show that the differential equation of motion for the nth mode becomes (Rao, 1990)
Mn d2q"+ Cn dq. dt 2
- ~ t + K.x = F.(t),
(1.11.3)
where (1.11.4)
C. = a M . + ilK..
Equation (1.11.3) can also be written in the more convenient form as 2
F.(t)
tT.(t) + 2~.tO.q. + q.tO. = ~ ,
(1.11.5)
Mn where ~, is the damping ratio for the nth mode. The solution of this equation for the actual motion is again provided by Duhamel's integral and is given by
f
1 t F,(r) e q.(t) = M.tO. o +e
-~nwn(t-r) sin [tO,~(t- r)] dr
(0) cos to,~t +
~,to,~q,(0) sin to,~ , tOnd
(1.11.6)
where 09,d is the damped natural frequency of the nth mode defined as ~o~ = ~o,~/1 - ~2.
1.12
Finite element analysis of vibrating mechanical systems
The derivation of the equations of motion for the structural system of Fig. 1.7 was based on writing the equilibrium equation for each of the masses in the system using Newton's second law. This approach is applicable to simple one-dimensional systems. However, it quickly becomes cumbersome and impractical for complex structures. The finite element method (Zienkiewicz, 1977) is a technique developed in order to
ACTIVECONTROLOFVIBRATION
22
overcome these difficulties. The general steps involved in the finite element method (FEM) can be described as follows: (i) The basic concept is to subdivide the structure into a finite number of elements, a process which is known as the discretisation of the continuum. (ii) The displacement field of each element is then approximated by interpolation functions and the displacement is calculated at a reduced number of discrete points or nodes, N~. (iii) The equilibrium equations for each one of the nodes is derived to form the equations of motion for the free-free element that defines the element mass and stiffness matrices M e and K e, respectively. (iv) These element matrices are used to build the matrix differential equation for the complete structure. This process consists of assembling the global mass and stiffness matrices by bringing the contribution of each element mass and stiffness coefficients to the proper nodal point. (v) Finally, the boundary conditions are imposed on the system by either constraining the corresponding nodal displacements or applying nodal forces. The formulation described above will be illustrated for a simple one-dimensional uniform bar in longitudinal motion, as shown in Fig. 1.8(a). All motion will be described in terms of the actual or observed motion of the system. We first discretise the continuum into finite elements, as depicted in Fig. 1.8 (b). The elements will have a node at each end, and the axial displacement field will be approximated by linear interpolation functions as follows and as shown in Fig. 1.8 (c) (Zienkiewicz, 1977):
(112.1)
u(x,t) = Un(t)(l _ ~n) +Un+l(t) --x ln,
where x is a local coordinate, l~ is the element length, and u,(t) and u, +~(t) are the unknown actual axial displacements at the nodes n and n + 1, respectively. To derive the equations of motion of the element, it is convenient to use Lagrange's equation (Meirovitch, 1967) given by d
dr~ +
dt ~--~r]
r n,n + 1,
=0,
(1.12.2)
dur
where T n and V, are the kinetic and strain energy of the nth element, respectively. The kinetic energy, Tn, and strain energy, Vn, of the element are defined by 1
t,
T, = 2 Io pSu2(x) dx,
1
I,
V~= 2 Io ESu2(x) dx,
(1.12.3a,b)
where p is the mass density, E is the elastic modulus and S is the cross-sectional area which is assumed to be uniform. Substituting equation (1.12.1) into (1.12.3), performing the integrations and substituting the results into equation (1.12.2) yields
l, pS 6
]f } S[l l][U t f00/
1 an + 2 //,÷1 ~
=
-1
,
(1.12.4)
1 U,+l
which is the matrix differential equation for the nth element. The matrices in equation (1.12.4) are the element mass and stiffness matrices given by
/nf)S[2l l] Ue-T 2'
Ke WS[ 1 -1] ----~n--1 1 "
(1.12.5)
23
INTRODUCTION TO MECHANICAL VIBRATIONS
~1"- I
Bar
I -I~
(a)
Node
\
/-.I. v
A
,~
n
nth finite element A v
.L v
(b)
A v
n+l n
1- x/~n
(c)
X/~n~ / i
Fig. 1.8 Finite element analysis of the motion of a bar: (a) uniform bar in extension; (b) discretisation into nodes; (c) linear interpolation functions.
The global mass and stiffness matrices are next built by adding the influence of each element matrix coefficient. Then, the equation of motion of the complete structure is given by Mii + Ku = f,
(1.12.6)
where
M = p__S_S 6
211
l]
0
0
...
0
l]
2(/1 +/2)
12
0
...
0
0
12
2(/2+/3)
13
...
0
•
:
:
"..
0
0
0
In
0
...
0
(1.12.7)
2(In+In+]) ln+l ...
lu
0 2/u+ 1
24
ACTIVE CONTROLOF VIBRATION
is the mass matrix of the discretised system and
1/ll -I/I,
K = ES
I
-1/l, (lfl,) + (1/12)
0 -1/12
-1/12
(1//2) + (1//3/
0 0 -1/13
°,° °°° .°°
°°.
l
-1/l=
(1//,,) + (1//,,+,) -1//~+, °..
-1/lN
°°.
I/IN+, (1.12.8)
is the stiffness matrix, and U --'- { / , / 1 , / , / 2 ,
...,
U N } T,
f=
{f,,f2 . . . . .
fN} T
(1.12.9)
are the displacement and force vectors respectively. We will now, for illustrative purposes, assume that the beam is clamped at one end and driven by an oscillating in-plane force f = FR cos tot at the other end. This configuration would be implemented as follows. The clamped boundary condition is imposed by setting u~ = 0. In order to solve the linear system of equations represented by equation (1.12.6), the first row of vectors u and f is eliminated and the first row and column of matrices M and K are removed. The force boundary condition is imposed by setting at node N, fN = FR and f~ = 0 for n = 1 . . . . . N - 1. Since the in-plane motion of this one-dimensional beam model is governed by a second-order differential equation (see Section 2.2), the model will have N degrees of freedom for N nodes. This brief description of the FEM is only intended as an introduction for the reader not familiar with the technique. For a more detailed description of the methods for differing motions, structural shapes and boundary conditions the reader is referred to texts such as those by Zienkiewicz (1977), Cook (1981) and Petyt (1990).