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CRANFIELD
INSTITUTE OF TECHNOLOGY
ACTIVE DAMPING IN STRUCTURES
by
CRANFIELD INSTITUTE OF TECHNOLOGY
ACTIVE DAMPING IN STRUCTURES by
-John Roorda
Visiting member: Structural and Aerospace Dynamics Group, Cranfield Institute of Technology.
Associate Professor: Department of Civil Engineering, University of Waterloo, Canada.
S U M M A R Y
The modern trend toward lighter and more flexible structural designs brings with it increasing demands for ways and means of dissipating excess energy
transferred to such structures by the action of dynamic external forces. The present study concerns the use of active feedback control mechanisms as a means of damping out excess vibrations. The basic active damping scheme involves the introduction of a control force on the structure through a feedback network whose input is dictated by the motion of the structure. The feedback network
is composed of a sensor, an electro-hydraulic servo-mechanism and a forcer, and its action in connection with a simple spring-mass system is studied. In all nine different control schemes are investigated and their relative merits discussed. An application of the active damping concept is worked out in the Appendix.
SUMMARY
1. INTRODUCTION 1 2. THE BASIC STRUCTURE 2
3. ACTIVE DAMPERS 3 4. STABILITY CONSIDERATIONS 9
5. FREQUENCY RESPONSE FUNCTIONS 11 6. ENERGY DISSIPATION PER CYCLE 15
7. WHITE NOISE EXCITATION 18 8. OPTIMIZATION OF PARAMETERS 21 9. AMPLITUDE AND PHASE CURVES 22
ACKNOWLEDGEMENTS 23
REFERENCES 23 APPENDIX Al FIGURES
1. INTRODUCTION
A structure may take up troublesome, and at times damaging vibrations when it is subjected to alternating loads whose frequency is at or near the resonance frequency. Examples of this problem are for instance; (a) tall masts or
chimneys subjected to random "gusting" pressures due to a turbulent wind; (b) building vibrations in response to earthquake action; (c) vibrations of structures due to moving loads such as trains; and (d) resonances induced by machinery supported by the structure.
Vibrations resulting from such inputs may, if unchecked, cause serious damage in the way of inadequate fatigue life, or extensive cracking of
architectual finishes. Secondly, excessive vibrations can potentially interfere with the operation of equipment supported by the structure. A case in point
is a tower supporting radio or television transmission equipment. Lastly there is also the problem of annoyance to the occupants of a vibrating structure, who may be upset by excessive motion. These, and other, problems are more
fully discussed in Ref.l.
In order to improve the situation one might first of all attempt to eliminate or reduce the source of the vibration. Although in some cases this approach has been successful, an example being the addition of strakes or perforated
screens on chimneys to reduce or eliminate vortex action, it is often impractical and may not even be possible. Secondly, steps might be taken to isolate the structure from the source but in most cases this is also impractical. Thirdly, one might attempt to reduce the response of the structure by ensuring that the energy transferred to the structure from the input disturbance is rapidly dissipated. Sources of energy dissipation are inherently present in all structural systems. To name but a few, one thinks of internal damping of structural materials, aerodynamic damping forces, and energy dissipation in slip joints. In many structural schemes this type of inherent damping is sufficiently large to prevent excessive response. In others additional
dampers must be introduced to achieve safe response characteristics. Rubber mountings or more sophisticated dynamic vibration absorbers (2) employing
auxiliary masses are frequently installed to excellent advantage. Ref, 1 contains a successful application of the dynamic vibration absorber to obtain improved performance of a tall chimney structure.
The forms of damping so far discussed are all "passive" in nature, their performance being purely dependent upon the interaction of material
properties, structural and response components. In such dampers, no additional external source of energy is needed. Another class of dampers, whose performance relies upon the availability of an external power or energy supply, may also be used. Such dampers may be termed "active", and generally constitute a
feedback control system which is designed to sense the structural motions and to generate a corrective control force on the structure to alleviate its response characteristics.
Active control mechanisms find wide use in the aero-space industry to improve the ride quality and to lengthen the fatigue life of flexible air and space craft (3,5). Further applications of such active mechanisms as dampers or vibration suppressors may be found in Ref. 4, 6, 7, 8.
The trend toward lighter and more flexible structural designs in civil engineering will no doubt induce future designers to consider the use of active control to augment energy dissipation in such structures. Although more
literature in this area is available in the aerospace and automatic control fields and to some extent in the machine design field, relatively little has
appeared in civil engineering and structural design journals. It is therefore the purpose of this report to bring down to earth some of the basic concepts involved and to present some simple active damping schemes which may find useful applications in the world of civil engineering structures.
To preserve clarity of concept, the report deals primarily with a single-degree-of-freedom spring-mass system which may represent a variety of different structural systems. In total, nine different schemes are considered, all of which are schematically similar. The relative merits of these active damping
schemes are discussed and a sample application of one or two of these schemes in the vibration problem of a uniform cantilever beam subjected to a distributed, time varying transverse pressure, is presented in the Appendix. In the
development of the results, much of the background material in control systems engineering is drawn from References 9, 10, 11,12,
2. THE BASIC STRUCTURE
Consider a structural system which can be represented by the single-degree-of-freedom, linear spring-mass-dashpot system shown in Fig. 1. The spring stiffness is given by k-^, the damping constant by c-^ and the mass by mj^. The force F acting on the mass is a function of time and is positive in the direction shown. The equation of motion for this system is given by
m X + e x + k X = F(t) , (1)
where x is the displacement of the mass from its rest position and the dots denote differentiation with respect to time. The theory of Laplace transforms offers a convenient method for the solution of linear differential equations such as eq. (1). The formulation of this method is particularly simple if it is required to find the normal response of the system to the input excitation F(t), The Laplace transformation
00
i(s) = £[g(t)] = ƒ g(t)e"^''dt (2)
o _
associates a unique result function g(s) of the complex variable (frequency) s = a + iu with every single-valued object function g(t) (t real) such that the improper integral (2) exists. The function g(s) is called the Laplace transform of g(t). Sufficient conditions for the existence of the Laplace transforms (13) will, in what follows, be assumed to be satisfied.
Equating the Laplace transforms of the two sides of eq. (1), and making use of the properties of the transforms, yields
2 -(m^s + c^s + k^)x(s) = F(8) . (3) Writing G'(s) = ^ , (A) m^s +c^s+k one obtains x(s) = G'(s)F(s), (5) Here G'(s) represents the transfer function of the system. Eq. (5) may be
represented in block diagram form as shown in Fig, 2(a),
Alternatively, if one considers the damping term in eq. (1) to be a control force fj(t) which modifies the external exciting force F(t) as dictated by the instantaneous velocity, a simple feedback control system results. Expanding eq, (1) into two separate equations as
m^^x + kj^x = F(t) + f^(t), (a)
f^(t) = - c^i, (b)
and changing to Laplace notation, yields
(m^s^+k^)x(s) = F(s) +f^is), (a)
f^(s) = - Cj^s x(s). (b) (6)
(7)
The block diagram for this alternative system is depicted in Fig. 2(b). The forward link transfer function is
d m^ s +K^ and the feedback link transfer function is
- ^ ) - ^d(^> = - '^l^- (5> The transfer function of the total system is obtainable from inspection of
Fig. 2(b) or by elimination of fj(s) from eq. (7). It takes the form x(s) _ G(s)
^d'
%(^> = m = I-G:(:)G(S)- ^^^^
Substitution from eq, (8) and eq. (9) into eq. (10) gives the anticipated equality
G^(s) = G ' ( S ) , (11)
The basic system described above is one in which the velocity dependent damping force fj(t) is passive in nature. The question arizes whether or not it is possible to replace or supplement the passive feedback link in Fig, 2(b) by a similar link which will produce a force that is actively dependent upon the motion of the system. In short, it is desired to introduce a measure of artificial damping into the system by use of an active feedback system, 3, ACTIVE DAMPERS
(a) Forcers
In an active damping scheme the damping (or control) force can be realized by use of any one of the three basic structural elements, namely, through an auxilliary spring of stiffness k„, an auxilliary dashpot with constant C2 or an auxilliary mass mo. The physical arrangement of these three types of "forcers"
in conjunction with the basic structural system is shown in Figs, 3(a), (b) and (c), respectively. The displacement u(t) in Fig, 3 is controlled by a servo-mechanism which in turn is controlled by the motion of the main mass m^. More will be said about this later,
The forces (positive to the right) exerted by the three forcing elements on the main system are given by
f(t) = k2(u(t) - x(t)), (a)
f(t) = C2(u(t) - x(t)), (b) (12) f(t) = m2(ü(t) - x(t)), (c)
respectively for the spring, dashpot and mass "forcers". Equating the Laplace transforms of the two sides of eq, (12) yields
f(s) = k2(n(s) - x(s)), (a)
f(s) - C2s(ü(s) - x(s)), (b) (13) f(s) " m2S (u(s) - x(s)), (c)
. . 2
m which the coefficients k„, c„s and m„s may be regarded as the transfer functions of the three different forcers, respectively,
(b) Electro-hydraulic servo-mechanism
Servo-mechanisms can be classified in a number of ways (9), They can be classified (1) as to use, (2) by their motive characteristics, or (3) by their control characteristics. Space does not permit a more detailed general
description here. Suffice it to say that in general all types of servo-mechanisms are mathematically similar. Considerations as to the choice of a particular type of motive power depend on local circumstances and on the charact-eristics of the equipment under consideration. In the current application certain loads have to be moved quickly and accurately in response to inputs
from the motion of the main structural system. These inputs will be in the form of electrical signals which originate in a transducer that senses the motion of the main mass, thence are amplified and, through the servo-mechanism itself, converted into the desired displacement u(t). For this type of application the electro-hydraulic servo-mechanism offers a convenient solution.
A block diagram of a simple electro-hydraulic servo-system is depicted in Fig, 4(a). Only a brief description of the components of this type of device will be given here. An in-depth study of electro-hydraulic servo-systems can be found in Ref. 10,
The input voltage, •vt(t) , originating at the sensing device which monitors the motion of the main mass, is combined with the feedback voltage, V£(t), in a mixing network to produce an error voltage v.(t) - V£(t) , This error voltage in turn is converted into a signal (valve) current with no change in phase but some gain in level according to the relationship
where KA is the gain of the servo-amplifier. The output is regulated power which reproduces the input at a higher energy level and becomes the input to
the servo-valve. Taking Laplace transforms of eq, (14) yields
i(s) = K^(Vj(s) - v^(s)) (15)
The servo-valve transforms the current i(t) into hydraulic power in two stages. First an intermediate transformation to mechanical motion is made by means of an electro-magnetic torque motor and this is then used to stroke the mechanical control element of the valve. Most two-stage electro-hydraulic servo-valves use a nozzle flapper valve for the first stage which is used as a pressure
controller with a spring loaded spool as a second stage. The control signal progresses through the valve in the following fashion: The electrical signal i(t) generates a magnetic force in the torque motor. This force creates a
differential pressure, or control pressure, which is applied across the spool, Because of this the spool is displaced a distance y(t) against springs or some
feedback arrangement.
The dynamic characteristics of the valve may be derived by writing the equations around its component parts. This has been done in detail in Ref.10 for a typical two stage valve and only the final result in Laplacian notation will be given here. The transfer relationship for the servo valve appears as
in which K^ is the gain of the servo-valve and + the valve time constant. The actuator is the motion producing half of the servo-mechanism. The linear actuator which is required in the present context takes the form of a hydraulic ram. The flow into the actuator is proportional to the valve spool displacement and is also proportional to the velocity of the ram. Thus
u(t) = Kpy(t), (17) or transformed
K
ü(s) = -^^(8) (18) where K is the actuator gain,
P ^
Electro-hydraulic servo systems require a position feedback to be stable. The position feedback element may be mechanical or electrical. Most electro-hydraulic servos utilize a linear motion electrical feedback transducer which
converts the piston displacement u(t) into a voltage vf(t) , characterized by the transformed equation
v^(s) = Kpü(s), (19)
where Kp is the gain of the feedback transducer. The transfer properties of the individual components of the servo-mechanism are brought together in convenient form in Fig, 4(b),
The overall transfer function of the servo-mechanism is required and is given by • ^ • ^ v ^ r r^\ - "(s) _ s(l+ts) v^(s) F A V P '• s(l+ts)
This expression can be simplified by setting K = K K^C,K , The final result is
G^(s) = h - T ^ - (21)
Fts +S+K c
If the servo-valve has a small time constant t the squared term in s in the denominator of eq, (21) can usually be ignored, leaving the first order closed loop transfer function
%^'^=i;7^-
(22)
It should be noted that the above analysis does not include such factors as internal damping (due to leakage for instance) and compliance of the load and support structure. These factors could be included and have the effect of introducing higher order terms in s, both in the numerator and the denominator of eq, (22). For simplicity and to preserve clarity of concepts, these factors are omitted from the present analysis,
(c) Sensors
The input v,(t) of the servo-mechanism originates in a transducer which senses the motion of the main structure (i,e., the mass m ^ ) , Although in theory any one of the quantities x, x, x, *x*, .,, could be used as the controlled variable, this report will concern itself with the three fundamental variables, namely
displacement x, velocity x and acceleration x. The equations governing the action of a displacement transducer, a velocity transducer and an acceleration transducer respectively, are
Vj(t) = T^ x(t), (a) Vj(t) = T^ i(t), (b) (23) Vj(t) = T^ x(t), (c) or transformed, Vj.(s) = T^ x(s), (a) Vj(s) = T^s x(s) (b) (24) Vj(s) = T^s^ x(s), (c) 2
where T,, T s, and T s are the respective transfer functions,
d) Basic control scheme
The formation of an active feedback link on the main systi;m is now merely a matter of patching together eqs. (13), (22) and (24) along with eq. (4) or (10) for the main system. Some further simplification is achieved by setting
G^j^(s) = k2, G^^is) = C2S, G^^is) = m2s2 in eq. (13) and G^^(s) = T^, G^2(s) = T s, G^^is) = T^s2 in eq. (24).
A block diagram which combines all the components of the active damping scheme can now be drawn and appears in Fig, 5 along with details of the
individual transfer functions involved. Both the passive and the active damping links have been drawn in to form a basis for comparison. The overall transfer characteristic for the total system now remains to be evaluated. This can readily be done by inspection of Fig, 5. Summing the contributions from these inputs to the basic undamped structure yields
x(s) = F(s)G(s) + x(s)G^(s)G(s)
+ X(S)[G^.(S)G^(S) - l]G^j(s)G(s)
(25)
in which i and j can take the values 1,2 and 3, The total transfer function is then
r (a\ = ^(s) _ G(s)
^ ^ F(s) ~ l-Gd(s)G(s) - [Gj..(s)Gjs) - l^jG^^ (s) G(s) ^^^^ With active damping only, i,e., if G,(s) is zero, the system transfer function
reduces to
^ij^^^ ^ T - [Gj..(s)Gj[s) - l-]G^.(s)G(s) ^^7) Substitution in eq. (27) for the individual transfer components from Fig. 5 yields
after some algebraic manipulation, the following nine different expressions; s+K G ( s ) = S , ( a ) m s +K m^s + ( k , + k „ ) s + K (k +k -rH<-T) 1 c l - 1 2 C 1 2 K 2 r S+K G ( s ) = £ , ( b ) 1 9 1 m, s +(K m i + c _ ) s + ( k - -jr- K c„+K c . ) s + K k, 1 c l 2 l K „ c 2 c 2 c l F s+K G ( s ) = , ( c ) o H 9 ( m , + m „ ) s +K ( m , + m „ - T r - « u ) s + k , s + K k , 1 2 c l 2 K ^ 2 1 c l F s+K G 2 , ( s ) = — - - ^ , ( d ) m s +K m . s + ( k , - -rr^C k , + k „ ) s + K ^ ( k , + k „ ) i c 1 1 K c 2 2 c 1 2 s+K G 2 2 ( s ) = — ^ - ^ ^ , ( e ) m , s +(K m - -rz^ K c „ + c _ ) s + ( k , + K C-)s+K k. i C i K e l l \. Q. I C i F
s+K G..is) = £ , (f) V 3 2 (m^ - ~ K m„+m„)s +K (mT+m„)s +kTS+K k. 1 K c 2 2 c i 2 1 c l r S+K G3,(s) = — ^— , (g) "^1^ ^ ' ^ c ^ ^ - K ; h'>' -(k,+k2)s+K^(k^+k2) (28) s+K G32(s) = ^ ^— , (h) Cm - -Ac c „ ) s +(K m^+c„)s +(k^+K C-)s+K k, i K c 2 c l l l e l c i r S+K 033(3) = - ^ - £^ ^ ^ . ( i ) - rr- K m„s +(m,+m„)s +K (m,+m„)s +k^s+K k, K^ c l I I c L l l e l r
In order to simplify these expressions and at the same time to allow a broader interpretation of the subsequent results, the following dimensionless parameters are introduced:
^ = ?
m^ K K = - ^ , ^d Y = M = % ""2 2/kj^mj^ s •^1 . T w, V 1 > 2 " 1 6 = 2 ' ^ l ' = m , . T . 2 a 1 c - ~~^— (a) (b) (29) (c)The parameters 3, y and 6 represent the useful ratios of the constants associated with the forcing element of the damper to the corresponding elements in the basic
structure. The terra 2/k m is in fact the critical damping of the spring-mass system. The natural frequency Wi of the basic structure is used to nondimen-sionalize K leaving K and p to represent the constant associated with the servo-mechanism and the complex frequency, respectively. The remaining parameters ot^j, a and a govern, in a sense, the servo-ram displacement for displacement, velocity or acceleration input, respectively. More details regarding the exact ram
displacement relationship appear in Section 6.
Cast in terms of these parameters, the dimensionless transfer functions for the nine possible active damping schemes take the forms below.
Q,Au)=-. j - ^ ^ , (a) y +KM +(l+6)y+K(l-6(a--l)) d (30) 6,„(y)=-^ HlK ^ (^^) y^+(K+2Y)y +(l-2YK(a^-l))y+K
Here, Q.Av) = :r^ Ö , (c) ^•^ (1+B)y +(l-B(a-l))K/+y+K p +Ku +(l-ö(a K-l))p+(l+ö)K V eoo^iJ) = -ï ^^ 2 ' (e> (30) y''+(K-2Y(a K-l))p^+(l+2YK)y+K
e
(^)
.
HlK
^
(f)
(l-e(a K-l))p''+(l+e)Kp +P+K e o i ( y ) - - ö ^^^ p , (g) p-^+K(l-a 6)p +(l+6)y+K(l+6) S O J C P ) = ^^^ 5 . (h) (l-2Ya K)p''+(K+2Y)y + ( 1 + 2 Y K ) P + K 633(11) ^ 3 2 • (^^-a gKp +(l+3)y +K(l+3)u +y+K
e.j(y) = m^üij G.j(8), i,j = 1,2,3, (31)
are the dimensionless transfer functions.
It is noted that each transfer function contains three parameters which characterize the feedback link, one associated with the sensor, one with the servo-mechanism, and one with the forcer. These must be chosen in such a way that the total damped system is stable and has the desired damping properties, Factors such as size limitations, costs, availability and the particular
application will of course introduce certain constraints on the choice of these parameters.
4) STABILITY CONSIDERATIONS
Returning for a moment to eq. (27) it is readily seen that the equation of motion, in Laplace transform notation, is
The values of the roots of the characteristic equation
1 - | G (s)G (S)-1|G (s)G(s) = 0 L l S IJ
(33)
determine the stability of the system, which is stable if, and only if, eq. (33) has no roots with a non-negative real part. The classical Routh-Hurwitz
criterion(l-^)provides a means of establishing the stability of the system without actually solving for the roots of eq, (33), It is noted from the denominators in eqs, (3)) that the characteristic equation (or its equivalent) appears in its final form as a polynomial equation of the type
n a y + a.
o
n-1 n-2
y + a„y +
^ Vl^
+ a = 0, (34)where the coefficients are real. The Routh-Hurwitz criterion then states that all the roots have negative real parts if, and only if,
a > 0 o (a) a > 0
(b)
> 0 (c) ^3 ^5 a a. a, o 2 4h
^3
> 0 (d) (35) ^ = > 0 (e) V = n > 0 (n)in which any term a., whose subscript j > n, is set equal to zero,
These criteria can now be used to obtain stability boundaries on the values of K and the values of a^, n and a for the respective systems i-j (i,j=l,2,3), Before preceding it is noted that the parameters 3, Y and 6 must, of necessity, always be positive to give a physically realizable system. Application of the Routh-Hurwitz criteria to the nine systems in turn yields the following stable ranges for K and a:
1 +(^ System 1-1: K > O, O < a, < -— (a) 2 System 1-2: K > O, a, < ^ •^2YK+1 ^^^ K + 2 Y K System 1-3: K > O, a, < O (c) System 2-1: K > O, a < O (d) 2 System 2-2: K > O, a < ^ -^2YK+1 ^^^ ^3^^ ^ 2 Y K +K System 2-3: K > O, O < a < ^ (f) V pK. System 3-1: K > O, a < O (g) K 2 + 2 Y K + 1 1 . . SysteiM 3-2: K > O, " ^2 "^ "a " 2 Y K ^"^^ K.
System 3-3: no stable combinations of K and a exist in
this case, (i)
It seems that System 3-3, which makes use of an acceleration sensor and
has the control force derived from the acceleration of a mass, is the only totally unstable case and for this reason will be omitted from further consideration, Stable combinations of servo and sensor constants can, however, be found in all other cases. These require further study to gain some insight into their relative merits as active structural dampers.
5, FREQUENCY RESPONSE FUNCTIONS
To this point the emphasis has been on the transfer functions of the active damping systems, the transfer function being the ratio of the Laplace transforms of any normal response and the input that produces it. The transfer function is defined for both stable and unstable systems and is written in terms of the general complex values of the argument s = a + io).
Attention is now turned to a special type of input - the pure sinusoidal input - and the related frequency response function. The response of a stable linear system to a pure sinusoidal input function is also sinusoidal with the
same frequency but generally different amplitude and phase. The frequency response function expresses the relative amplitude and phase of input and output as
functions of frequency. It is defined only for stable systems, since a pure
sinusoidal input must start indefinitely far in the past and can thus be considered only in connection with a stable system. The importance of the frequency
response function rests on the fact that any input function can normally be written as the sum of sinsuoidal oscillations, and hence the response of a linear
system can be expressed as a similar sum of responses to the sinusoidal input components by means of the frequency response function, which relates corres-ponding input and output components,
For stable systems the frequency response function can be derived from the transfer function by setting s = iw; i.e., the values of the frequency response function are the values of the transfer function along the imaginary axis of the s plane. Alternatively, in the non-dimensional form, one sets y = iv, where V = ü)/ü)^ ,
For the eight stable systems in question the frequency response functions are respectively, from eq. (30),
IT / • \ iv+K , , H (iv) = ^ = , (a) -iv -Kv +(l+6)iv+K(l-6(aj-l)) II /• \ i v + K , , , H^jCi^') = Ö Ö , (b) -iv - ( K + 2 Y ) V +(l-2YK(a^-l))iv+K II /• • \ iv+K , . H,^(iv) = ^ 5 , (c) -(l+3)iv -(l-3(a,-l))Kv +iv+K
d
(37)
II /• \ i v + K , ,.
H j ^ d v ) = r j , (d)
-iv -KV +(l-ó(a K-l))iv+(l+6)K
II /• \ iv+K , . H -(iv) = ^ -z , (e) -iv-(K-2Y(o;yK-l))v^+(l+2YK)iv+K ,1 /• • \ iv+K , c\ H,^(lV) = r , (f) -(l-3(ayK-l)iv -(l+3)Kv^+iv+K II II \ iv+K , V H-, (iv) = T 5 , (g) •'^ -iv-K(l-a 6)v'' + (l+6)iv+K(l+6) a H (iv) = r , (h) ^ -(l-2Ya K)iv - ( K + 2 Y ) V ' ' + ( 1 + 2 Y K ) I V + K
in which H^.(iv),(i,j=l,2,3) represent the complex frequency responses. In formal terms the amplitude of the response is the input amplitude nultiplied by the factor
and the phase difference between the input and output is
= tan
•1 j^ rH(iv)-H(-iv)]
i [H(iv)+H(-iv)J'
(39)where H(-iv) denotes the complex conjugate. The subscripts ij have been left out for simplicity. A more useful approach, and one which allows the response
amplitude and phase to be written directly from the complex frequency response function, is to write H(iv) in terms of its numerator ^(iv) and its denominator D(iv), each of which are complex functions themselves. Thus
H(iv) N(iv))
D(iv)' (40)
in which each term can be expressed in terms of polar co-ordinates in the form re , where r is the modulus and \\) is the argument. After some algebra it then becomes clear that the amplitude of the response is
H = •(ReN)2+(iniN)2T2 (ReD)2 + (InüD)2
(41)
and the phase angle is
^ -1 ImN -1 ImD ij» = tan -jT—- — tan •;r~F:t
ReN ReD' (42)
where "Re" and "Im" stand for "real part of" and "imaginary part of", respectively. Making use of the identity
^ -I ^ -1, , -1 a-b tan a - tan b = tan
1+ab'
the phase angle finally becomes, after some more algebra, •1 (ImN) (ReD) - (ReN) (ImD)
(fi = tan
(ReN) (ReD) + (ImN) (ImD)
(43)
(44)
Using eq. (41) the eight amplitude functions can be directly read from eqs, (37), They are |H^^(iv)|=)-2 2"^^^ 2-~2 2~2f ' ^^^ '•^ ( K ^ ( 1 - V -6(a,-l)) +v^(l+6-v^)^J d
|Hi2(iv)|=j' *" ''2 2 2
-2]' ^^^
i
(K-(K+2Y)v'^)'^+v'^(l-2YK(a^-l)-v ) J
2 2
i
|H
3(iv)
1= J-2
^^^
5-2—^ ^-yl ,(C)
< K^(l-(l-3(a^-l))v^)^+V (l-(l+3)v^)^J IK (1+6-v ) +v (l-6(a K-l)-v ) J (d) (45)2 2 li O-M J '
1^22(^^)1=1
^-^
2 2 2 2~2f ' ^^^
* (K-(K-2Y(a K-l))v ) +vV ( 1 + 2 Y K - V ) J,2 2 .i
|H(iv) 1= j
—
^ "^2 2 2 T T I ^^^
(K^(l-(l+3)v ) +v (l-(l-3(a K-l))v ) J
V .2..2 ,i | H 3 i ( i ^ ) | = i ^ ^^~^ 2 2 2 T T ! ' (Ê) IK (l+6-(l-a 6)v )+v (1+6-v ) J f K 2 + 2 il|H32(iv)|=j
TT~2
T T T • (^)
( ( K - ( K + 2 Y ) V ) +v (l+2YK-(l-2Ya K)v ) JThe corresponding phase angles are read from eqs. (37) using eq. (44). The
result, after some simplification, is
-1 -K6a ,v (j) (v) = tan -= T = =-, (a) *-*• K''(l-ó(a.-l)-v)+v (1+6-v ) d _^ -2Yv(v2-K2(a^-l)) <t).„(v) = tan 5 X 5-, (b) K ( K - ( K + 2 Y ) V )+v (l-2YK(aj-l)-v ) 3 -I K 3 a j V 41 (v) = tan -5 5 5 5-, (c) K^(l-(l-3(aj-l))v^)+v^(l-(l+3)v^) , K26a V <^ (v) = tan"^ -5 —^ =-, (d) (46) ^ K^(l+6-v )+v (l-6(a K-l)-v ) _^ -2Yv(K2-(a K-l)v2) ^jj(\>) = tan ^~ = r- (e) K(K-(K-2Y(a K-l))v )+v ( 1 + 2 Y K - V ) 2 3 _^ - K 3a V <t>oo(v) = tan -= j ^ — 5 5- (f) K^(l-(l+3)v )+v (l-(l-3(a K-l))v^) V _^ +K6ct v"^ (|) (v) = tan —^ — 5 5 5-, (g) ^ K (l+6-(l-a 6)v)+v (1+6-v ) a _j - 2 Y V ( V 2 + K 2 ( 1 + O I ^ V 2 ) ) (|).„(v) = tan T r s-, (h) K ( K - ( K + 2 Y ) V )+v (l+2YK-(l-2Ya K)v )
It now remains to settle upon some suitable choice for the three control parameters in each system. In this effort one must take accoun". of the stability conditions set forth in section 4. Within the stability boundaries the choice of parameters must be such as to give the total system optimum damping character-istics. As mentioned earlier, cost and space considerations and the particular application of the active damper will of course impose certain constraints on this optimization problem. Some guidance as to the relative effects of changes in the three control parameters can be obtained by studying two fairly simple situations.
The most direct approach is to calculate the energy dissipated by the damper during one cycle of sinusoidal motion of the main structure, A second approach is to assume an ideal white noise excitation and to calculate the mean square
response of the system. Both of these approaches will yield certain relationships between the control parameters, in some cases allow certain optimality criteria to be established. The next few sections will deal with these two problems,
6, ENERGY DISSIPATION PER CYCLE
For linear systems it is known that a harmonic disturbing force F(t) induces a response which is also harmonic at the same frequency but lagging in phase, For an assumed response
x(t) = 0, t < 0
(47) x(t) « sin w^vt, t > 0
it is desired to find the energy dissipated by the active damping mechanisms. It is therefore necessary to determine the control force f(t) for this response and to calculate the work done by the basic structure on the control force during one cycle of steady state motion. Clearly only the feedback link in the damped system is involved in this calculation because the response has been assumed a priori, This simplifies the work considerably.
» In order to acquire some more detailed insight into the behaviour of the feedback link, the control force will be calculated in three stages. First the displacement u(t) of the servo-ram is of some interest. From Fig. 5 it is readily seen that the displacement u(t), in Laplace motion, is given by
ü.(s) = Gj..(s)G^(s)x(s), i = l,2,3. (48)
The Laplace transform of the assumed response can be obtained from standard tables(13)ancj^ in dimensionless terms, may be written as
cü^x(s) = ~ - 2 (^^^
y +v
Substituting in eq. (48) from eq. (49), using the appropriate transfer functions for the three different sensors (as given in Fig. 5 ) , and introducing the dimensionless parameters defined by eqs. (29), one obtains for
i = 1: (jj^u (s) = 5 — ^ , (a) (y+K)(y%v^) _ Ka vy i = 2: (O U2(s) = "" 2 2 ' ^^^ (^°) (y+K)(y +v ) 2 Ka vy i = 3: lOj^u (s) = 2 — 2 ~ * ('^^ (y+K)(y +v )
These Laplace transforms are in standard form and their inverses
uijU(t) = j!l''^{w^ü(s)}, t > 0 (51)
may readily be found from tables. In each case the result will include a transient term of the form e~^^ and a sinusoidal term of the form sin(a) vt+il») , where ijj is a phase angle. Interest in this section is focussed on the steady state response and hence the transient term may be disregarded. Expressing the final result as a sum of two harmonics yields for
Ka
i = 1: u (t) = —^—=-(-vcosu)^ vt+Ksinw vt) , (a) K +v
Ka V
i = 2: U2(t) = ^ ^ (Kcosto vt+vsino) v t ) , (b) (52) K +v
Ka v2
i = 3: u (t) = —^—=-(vco8a)j^vt-Ksinco vt) . (c) K +v
It is clear from eqs. (52) that the parameters a., a^, and a^ play a
primary role in determining the ram displacement relative to the displacement of the main mass. The phase lag of the ram motion is dependent on both the response frequency and the parameter K which characterizes the electro-hydraulic servo-mechanism.
The next quantity of interest is the displacement of the forcer (i.e., the spring, dashpot, or mass which ultimately produces the control force). This requires the displacement x(t) to be subtracted from the servo-ram displacement (see Fig. 3 ) . Thus, if z(t) is the displacement of the forcer,
z.(t) = u.(t) - x(t), i = 1,2,3 (53) For t h e t h r e e s e n s o r s the r e s u l t i n g f o r c e r d i s p l a c e m e n t a r e Ka^ K2(a - l ) - v ) 2 i = 1: z ^ ( t ) = —X—s-(-vcosa) v t + — sinco vt) , ( a ) K +v d Ka V v2(Ka - 1 ) - K 2 i = 2 : Z 2 ( t ) = - ^ ^ ( K c o s w ^ v t + —j^ sintü^vt) , (b) K +v V
Ka v2 K2(v2a^+l)+v2 J
i = 3: z„(t) = —^—j(vcos(jjj^vt ^ sinco, vt) . (c) (54) K +v Ka V
a
The control force, f(t), is now readily obtainable from eqs, (12), (53) and (54), Disregarding the unstable case, namely system 3-3, the eight
remaining possibilities are
\ I
(55) f^^(t) = k 2 Z j ( t ) , ^12^^^ " C2Z^(t), fi3(t) = in2Z^(t) (a)
f2i(t:) = k 2 Z 2 ( t ) , ^22^*^^ = C2Z2(t) , f23(t) = m2Z2(t) (b)
f3j(t) « k 2 Z 3 ( t ) , f32(*=) = C2^3(*^)» ('^^
The work done by the structure on the control force per cycle of oscillation is
W.. = - §f..(t) x(t)dt, (56)
Using eq, (47) this can be rewritten as
W. . =-a),v § f. .(t)co8ü),vt dt. (57) ij 1 ij 1
The quantity W^^ is in fact the energy dissipated by the active damper during one steady-state cycle. In the integral (57) , the two standard integrals
§ cos w,vt dt = , (a) I Ü), V
^ (58) § cosü), vt sinco-iVt dt = 0, (b)
appear, which simplify the result to a considerable extent. Substitution of eqs, (54) in eqs, (55) and thence in eq. (57), and evaluating the integral for the eight stable systems, yields
6K(x ,v
^11
= -r^ '
^*>
K +v 2 Y ( V ^ - K 2 ( C I ^ - 1 ) ) V^2
=
X ~ 2 ~'
^^^
K +v SKa^v"^ ^ 3 = - -2-T-' <^> K +v 6K2a V ^21 = " " T ~ 2 ~ ' ^^'^ ^^^^ K +v 2Y(K2-v2(Ka -l))v A22 = 2 2 ^ • ^^^ K +v^11 = ^12 = ^ 3 = ^21 =
r =
22 ^23 ^31 = ^32 = In these or 3 appear. of constant F^ dissipation pe and 3 = 0.2. indicated by s K2+l-6(a,-l) Ka^6(l-6(aj-l)) ^^^ K 2 + 2 Y K + 1 . . r 2l ^ ' 2Y[l-2Y(a^-l)K-(a^-l)K^J K2(l-3(ct -1))+1 '^ (c) -Ka,3 ^'^^ K 2 + 1 + 6 ,.. -K^a 6(1+6) V (69) « K (l-2Ya ) + 2 Y K + 1 2Y[K2(l-2Ya^)+K(2Y-a^)+l] K 2 ( 1 + 3 ) + 1 ^,J K a^3 K2(l-a 6)+1+6 ^ (o^ 9 ^*' -Ka 6(1+6) a K 2 + 2 Y K + 1 ... 7 2 Y K (1+a ) + 2 Y K + 1 ] ^ a "^expressions four parameters, namely T^i, K, a^ ^ and one of 6, Y The dependency on v has been eliminated by integration. Curves 4 can be plotted in much the same way as was done for the energy r cycle. These appear in Figures 9, 10 and 11 for 6 = 0.5, Y = G.l
Where appropriate, unstable combinations of K and a^ ^ ^ are tiaded curves. These are of course obtained from eqs! (36). Comparison of Figures 6, 7 and 8 with Figures 9, 10 and 11, respectively, shows the curves in all systems but one to be very similar in form. System 1-1 is the exception. More will be said about this particular system later. The overall si
, expected since frequencies at
Tiilarity betx^jeen the two sets of curves is, in a way, to be the major contribution to the mean squared response comes from , or near, the resonant frequency (i.e. the natural frequency of the basic structure). The energy dissipated per cycle at this frequency will therefore largely determine the mean squared response value.
8. OPTIMIZATION OF PARAMETERS
A study of the results in the preceding two sections indicates that for some of the systems the parameter K may be chosen so as to optimize the damping performance for given values of a^j, a^ or a . This possibility exists in systems 1-1, 1-3, 2-2 and 3-1 when the dissipation per cycle of sinusoidal
response is taken as a measure of damping effectiveness. When the mean squared response to white noise excitation is used as a measure of effectiveness the same systems allow optimization with respect to K, and for system 1-1 optimization with respect to aj as well,
a) Damping effectiveness given by A^-:
Partial differentiation of Afj (as given by eqs. (59a, c, e and g)) with respect to K and setting the resulting expressions equal to zero yields in each case the simple result,
% t = ^ (70)
The line given by eq, (70) (dashed in figures 6, 7 and 8) represents a trough in the three dimensional A^j , K, a^ ^ ^ space and combinations of K and a^j ^ on this line optimize the energy dissipation per cycle for a given value of a^ ^ ^. In words, eq. (70) means that the constant associated with the servo-mechanism should be equal to, or near, the exciting frequency to give the best performance in these four systems.
b) Damping effectiveness given by r..
A similar exercise can be done here. Partial differentiation of r^i (as given by eqs. (69a, c, e and g)) with respect to K and equating to zero yields for
System 1-1: K 2 ^ = 1 - 6(a,-l), (a)
opt d ' system 1-3: ^l^^ = ^-^-—, (b) (71) System 2-2: K J ^ ^ = ^ - i ^ , (c) System 3-1: K 2 = ii|—.. (d) opt l-6a a
The curves represented by eqs. (71) are shown as dashed lines in figures 9, 10 and 11. Combinations of K and aj^^^a on these lines optimize the mean squared response of the white noise excited structure for given values of a, ,
d ,v, a
For system 1-1, it is possible to minimize Tn with respect to a^j as well, Partial differentiation of Tn with respect to aj and equating to zero yields the expression
,2 ^ [^("dopt-^>-^l' (72)
^(2«dopt-l>-^ '
which is represented by a dashed curve in Fig. 9(a). Equations (71a) and (72) have a common solution when
2(1+6) r \
"d = "dopt = — 3 6 - ' ^^^
(73)
""
= %t = f F ^ (b)
Back substitution for K * K ^ ^ and a^j = a^^ j. into eq. (69a) gives the optimum mean squared response for System 1-1 as
3/2
^l(opt) = ^1T6^ (7^)
9. AMPLITUDE AND PHASE CURVES
Results of the type shown in Figures 6, 7 and 8 or in Figures 9, 10 and 11 allow an intelligent choice to be made for the system parameters a^ ^ g, given the energy dissipation level required and given the constant associated with the forcing element in the system (i.e. 6, Y or 3). Having settled upon some values for the three basic feedback parameters of the active damping link it is possible to plot the amplitude, |H(iv) | , and phase, (J>(v) , relationships given by Eqs. (45) and (46). Typical amplitude and phase curves for the three
possible forcing elements (i.e., a spring, a dashpot, or a mass) are plotted in Figures 12(a,b), 13(a,b) and 14(a,b), respectively. In these figures the solid curve is for displacement input, the dashed curve for velocity input and the long-short dashed curve for acceleration input. The values chosen for the feedback parameters are indicated in each case.
It is noted that in all cases the peak amplitude magnification occurs when the phase angle between the input and response equals approximately 90°. This familiar behaviour is of course also present in passively damped systems.
A spring "forcer" has the tendency to shift the peak amplitude toward higher frequencies, a result which is not unexpected because the basic structure is partially stiffened by the additional auxilliary spring. The increase in total natural frequency appears to be affected by the feedback input used. The mass "forcer", on the other hand, tends to decrease the total natural frequency by virtue of the increased mass of the system, the amount of decrease again being dependent upon the input to the feedback loop. Peaks for the dashpot "forcer" occur on both sides of the basic natural frequency.
The relationship between the peak amplitude response and the parameter associated with the forcing element of the active damper is indicated in Figures 12(c), 13(c) and 14(c), respectively, for the spring, dashpot and mass forcers. It is clear that substantial reductions in amplitude response can be achieved in each case by a suitable increase of the parameters 6, Y or 3.
As noted in the preceding section. System 1-1, when subjected to white noise excitation, admits optimum values for both K and a^ as functions of the spring stiffness ratio 6 (see eqs. (73) and (74)). This is an interesting and potentially useful property. The "white noise optimized" amplitude and phase curves for this sytem appear in Fig. 15 for three values of 6, The limiting case (indicated by the dashed curve) for 6 = 0 is physically unattain-able because the servo-ram displacement approaches infinity as the spring stiffness ratio approaches zero. The curious increase from the standard undamped response at low frequencies is due to the additional positive force provided by the auxilliary spring as it is activated by the positive displace-ment x(t) resulting from the application of a positive external force F(t).
The rapid initial increase in phase angle, coupled with the relatively low slope of the phase curve as it passes through 90°, must be viewed as the basic reason for the excellent damping characteristics of this optimized system.
10. SAMPLE PROBLEM
In the Appendix an application of the above concepts is presented. The example, a uniform cantilever subjected to a distributed transverse exciting force, can be viewed as a gross idealization of a tall chimney or mast subjected to "gusting" forces caused by turbulent wind conditions. Such structures often exhibit troublesome vibrations, particularly when their fundamental freqencies lie within the range of frequencies present in a turbulent wind.
In this example problem the time variation of the exciting force is assumed to vary identically for all points along the length of the cantilever. Its magnitude distribution along the length is left in functional form. With these assumptions, the analysis becomes much simpler than that for an actual turbulent wind situation in which cross correlations with height, aerodynamic damping and virtual mass terms must be taken into account.
The geometry of this type of problem immediately suggests that, in the absence of any guy wire arrangement, an active damping force to be introduced must be derived from the acceleration of a mass. The damping force will be most effective when the auxiliary mass is situated near the tip of the cantilever where the displacements are largest.
The mass "forcer" is therefore studied in connection with this problem and the relative merits of two possible inputs, displacement and velocity, are investigated. The auxiliary mass takes the form of a pendulum suspended from some point on the cantilever.
Although the analysis takes into consideration all the mode shapes in its initial stages, in the final results of the appendix only the first and second modes are retained to allow uncoupling of the governing equations. It is found that, as far as the first mode response is concerned, the results obtained from the general analysis of the single-degree-of-freedom spring-mass system in this report are directly applicable to the cantilever problem.
ACKNOWLEDGEMENTS
Financial support for this research project was provided by the Science Research Council, U.K.
REFERENCES
1. Waller, R.A.
2, Den Hartog, J.P.
3, Edinger, L.D.
"The design of tall structures with particular reference to vibration"
Institution of Civil Engineers, Paper 7361, Vol. 48, p.303, 1971.
Mechanical Vibrations, McGraw-Hill Book Company, Inc., New York, 1956.
"Design of elastic mode suppression systems for ride quality improvement",
Journal of Aircraft, Vol. 5, No. 2, p. 161, 1968.
4. Comstock, T.R., Tse, F.S. and Lemon, J,R. 5. Swain, R,L, 6. Schubert, D,W, and Ruzicka, J,E. 7. Berkman, F, and Kamopp, D, 8. Rockwell, T.H. and Lawther, J,M, 9. James, H.M., Nichols, N.B. and Phillips, R,S, 10. Morse, A.C.
11. Seifert, W,W, and Steeg, Jr.C.W,
12. Greensite, A,L,
13. Korn, G.A. and Korn, T.M.
14. Crandall, S,H. and Mark, W,D.
15. Warburton, G.B.
"Application of controlled mechanical impedance for reducing machine tool vibrations",
ASME Journal of Engineering for Industry, Vol. 91, p, 1057, 1969,
"Control system synthesis for launch vehicles with severe mode interactions",
IEEE Transactions, Automatic Control, Vol. 14, No. 5, p. 517, 1969.
"Theoretical and experimental investigation of electro-hydraulic vibration isolation systems",
ASME, Journal of Engineering for Industry, Vol. 91, p. 981, 1969.
"Complete response of distributed systems controlled by a finite number of feedback loops",
ASME, Journal of Engineering for Industry, Vol. 91, p. 1063, 1969.
"Theoretical and experimental results on active vibration dampers",
Journal of the Acoustical Society of America, Vol. 36, No, 8, p, 1507, 1964, Theory of Servo-mechanisms,
Dover Publications, Inc., New York, 1965, Electrohydraulic Servomechanisms,
McGraw-Hill Book Company, Inc,, New York, 1963,
Control Systems Engineering,
McGaw-Hill Book Company, I n c , New York, 1960,
"Analysis and design of space vehicle flight control systems". Vol, III, Linear Systems, NASA CR-822, 1967. Mathematical Handbook for Scientists and Engineers,
McGraw-Hill Book Company, Inc., New York, 1961,
Random Vibrations in Mechanical Systems, Andemic Press, New York, 1963.
The Dynamical Behaviour of Structures, Pergamon Press, Oxford, 1964.
APPENDIX
Continuous Cantilever with Controlled Pendulum Damper
Consider a uniform cantilever subjected to a distributed disturbing force which is a function of time. At any point y along the length of the cantilever the force per unit length is given by p(y)F(t); i,e., the disturbing force varies with time in the same way for every point on the cantilever. The equation of motion for this system takes the familiar form (15),
CA - ^ + EI i x ^ p^y^p^j.^^ (Al)
at 3y
where C. A, E, I and x are, respectively, the density of the material, the
cross-sectional area of the beam. Young's modulus, moment of inertia and lateral displacement. Note that x = x(y,t) is a function of both position and time,
A pendulum of length L and mass m, driven by an actuator attached a distance d from the pivot, is attached to the cantilever at height a from the base. It is assumed that the pivot point is frictionless. The actuator reacts transversely against the cantilever at distance b = a - d above the base,
A diagram of the basic cantilever, its load and its pendulum damper is shown, in deflected form, in Fig, Al, Deflections are assumed to be small enough so that linear analysis can be used,
The mass, m, of the pendulum as it is moved transversely by the actuators (which is controlled by either the displacement or the velocity at some point on the cantilever), will generate an inertial force that is transmitted to the cantilever through forces at the pivot point and the bearing point of the actuator. These forces will have the effect of adding terms to the right hand side of Eq, (Al),
The displacement of the mass can by similarity of triangles be inferred from the equation
z(t) + x(a,t) ^ L , u(t) - x(b,t) + x(a,t) d' ^ ^
where z(t) is the displacement of the mass and u(t) the actuator displacement. Finally,
z(t) = i[u(t) - x(b,t) + (1 - •^)x(a,t)], (A3)
If f(t) is the compressive force in the actuator and R(t) the reaction at the pivot, then the equations of equilibrium for the pendulum are readily obtained
as
f(t) = i 2 2(t) (a)
(A4)
R(t) = - (1 - J)f(t) (b)
Inclusion of f(t) and R(t) in the right hand side of eq, (Al) yields, after elimination of R(t), the equation of motion for the actively damped system as follows:
2 4
a ^ + EI - ^ = F(t)p(y) + f(t)rö(y-b) - (1 - •^)6(y-a)J, (A5) 3t^ 3y^ ^
in which the concentrated control forces are represented as delta functions multiplied by f(t), The solution of eq, (A5) will be sought in terms of the normal modes, e'(y), each of which is associated with a natural frequency (Hi. Making the assumption that the mode shapes will not be affected to any great extent by the introduction of damping, it becomes possible to write the solution to eq. (A5) as
x(y,t) = I 6 (y)q (t), (A6)
j -^
where q'(t) represents the j generalized co-ordinate. Substituting for x from eq, (A6; in Eq, ( A 5 ) , multiplying by 6i(y) , integrating with respect to y over the length •£, and introducing the well-known orthogonality conditions(15), yields an infinite set of ordinary differential equations of motion in terms of the generalized co-ordinates q £ ( t ) . After some algebraic manipulation these take the form 2 /oP(y)öi(y)dy q. + tü.q. = F(t) a / ; 9 ^ ( y ) d y
It [«(y-b)-(i4)^(y-a)]e-(y)dy
+ -2 • „ ^ i f (t) . (A7)CA
Jl
e^(y)dy
For normalized mode shapes
fA .2 c
/^ e2(y)dy = Z. (A8)
It is recognized that 5AJI = M = the total mass of the cantilever, and after setting
"i = i fp(y)ei(y)'Jy. (^)
oAi = /^['5(y-b)-(l-^)6(y-a)]e.(y)dy (A9)
= e.(b) - (1-^)9. (a), (b)
The coefficients fi£ and A^ represent certain weighting functions which associate the force term F(t) and f(t) with the ith natural mode and which can be evaluated for known mode shapes. Finally, introducing the transformation (A6) into
eq. (A3), differentiating twice with respect to time, and using eq. (A4a) to eliminate f(t) from eq. (AlO), yields,
2
q. + ü)^q. = Q. 4 ^ + A- \ ^ [ ü - l A-q-l- (AH) ^1 1^1 1 ?A •"- H J J
A Laplace transformation changes this equation to 2
(s2+a)?)q. = n. I - + A. ^ ^[s^ü - Y A.s^q.l, (A12) 1 ^1 1 ?A 1 ,2 ML '^ 2 J
The servo-ram displacement u is controlled by either the displacement or the velocity at some point y = c on the cantilever (recall that a mass damper controlled by an acceleration produces an unstable system, and hence this case will be omitted). From Fig. 5 it is readily verified that the transfer relationship for displacement control is
"^^> =K7fTrT^(^'^> (Ai3>
and for velocity control
Substitution for u(s) in eq. (A12) from eq. (A13) or (A14) and writing x(c,s) in terms of the generalized variables q^ using eq, (A6), allows the total system transfer equation to be calculated. For displacement control the transfer equation is
2 rK T,s2
/ 2 2,- ^ F . L m c d v^ / s~
(S +..)q. = «i ^ ^ A. .-^ -• Ij-^-^^^ |e.(c)q.
i=l,2,3,.,. . (A15)
For velocity control the transfer equation is , 2^ 2.- „ F . L ni (s +oj.)q.= J2. —T- + A. - Ö w 1 ^t 1 CA 1 2 M K T s c \f 3
i c r ^ ) ^'j^'^^^j
, 2-A.s q J JJ F ^ c' J i=l,2,3 (A16)The second term on the right hand side of these equations represents the effect of the active damper on the system. In general, the equations (A15) or (A16) are coupled across all modes as a result of the action of the damping mechanism. A complete study of the interaction of all modes appears to be intractable. For this reason, and to keep the presentation as simple as possible, only the first two modes will be considered here. This means that
q = O for k > 2, The remaining two transfer equations can then be uncoupled by judicious choice of the parameters, b, c, d and L, If d = L (i,e,, the
feedback actuator is exerted directly onto the mass m ) , and if b = c = ynoje ~ y* (i.e., the mass and the sensor are both located at the node of the second natural mode shape), then the equations will be uncoupled. With these simplifications eq. (A9b) yields
A^ = 6^(y ) A2 = 0. * ^1' (a) (b) (A17) Also * e^(c) 62(0) = 0. (a) (b) (A18)
By use of Eqs. (A17) and (A18) the uncoupled transfer equations for displacement control can be deduced from Eq. (A15) as
r 2 ^ 2 ,.*,2 m ^ c d^ 2^1 - „ F , ,
[
s +0)2 2 2^12 = "2 ü '
(A19) (b)
Eq, (A16), valid for velocity control, reduces in a similar manner to 3 1 — (a) 2 2 ,„*.2 m r^c^v^ 2-, \ I ^ I ,c^.i m /• c V i-\
[«
*\
- Oi) M ^(s+K^) -
' \
\
= "1 ü'
2 ^ 2 s +0)2^2 = "2
IA-(A20) (b)
Clearly the second mode remains undamped in both cases. The transfer function for the first mode is the ratio of the Laplace transform of the output to that of the input. In terms of dimensionless variables the displacement controlled, first mode transfer function is:
^ ^ ^ ^ ^ y+K
" 1 ^ (l+3*)y-^+K(l-3*(a -l))y2+y+K'
(A21)
and the velocity controlled, first mode transfer function is
y+K
2-CAü) q
"1 (l-3*(Ka -l))y-^+K(l+3*)y^+y+K
(A22)
The dimensionless parameters y, K, a, and a are defined by Eq, (29). The parameter 3* is the effective mass ratio defined by
e*
=
i^yi-
(A23)
The forms of eqs. (A21) and (A22) are identical to eqs. (30c) and (30f), respectively, and hence the results obtained from the simple mass-spring system which the latter two represent can be used directly in the present application. The frequency response curves shown in Fig, 14(a) are of particular interest in this respect. For 3* = 0.2, K = l,and oid = - 3 the amplitude response for the displacement controlled system has a maximum of 5,35 when the frequency is about 0,8 times the natural frequency of the undamped cantilever. This is achieved when the ratio of the mass of the pendulum to the total cantilever mass is equal to 0.2/(e-|^)2. A good approximation for 9^ can be found by assuming the fundamental mode to be the static deflection curve for a cantilever under uniformly distributed load; i.e.,
e^(y) = a(6£2y2 _ 4^y3 + y^). (^24) The normality condition (A8) requires that a = 5/6jl . It is known that the
node point of the second mode occurs at y = 0,783)1, Hence 9? = 1.78 from which it can be shown that m/M must be 0.063 at 3* = 0.2. Thus a pendulum mass of 6,3% of the cantilever mass is required in the above situation. For a velocity controlled system, the same pendulum will give a maximum amplitude response of 3,15 at about the natural frequency of the basic cantilever when K = 1 and a = + 3.
V
If the disturbing force is in the form of white noise then it is readily shown that the dimensionless mean squared response
.2
1.1 J„ r 2 T
? A Wj^E[qj^J
r = — Y—- , ikib)
is given by eq. (69c) for displacement input and by eq, (69f) for velocity input. Here S^ represents the intensity of the input and E|q^| is the mean square expectancy of the first mode response. Using eq, (A6) and eq, (A25) the mean square displacement at y = y* can be written as
E [ ( X * ) 2 ] = (9*) 2 E[q2], (A26) or ^j^2g
E [ ( X * ) 2 ]
=^JL£_(e*)2r. (A27)
C A Wj^
Assuming the fundamental mode 9^(y) to be given by eq, (A24) and making use of the known fundamental frequency for uniform cantilevers, i.e.,
3,516fEI^è /AooN
it is possible to simplify eq, (A27) to
E [ ( X )^1 = 0.229 -!-^T = . (A29)
12 I2
(EI) ^^(a) ^
For a known force distribution ü-^ can be calculated from eqs, (A24) and (A9a) , and the quantity F may be obtained from eqs. (69c) and (69f) for displacement input and velocity input, respectively.
p
p
In Oo
a. „XI O in OINPUT VOLTAGE - > ERROR ^ O U T A G E SERVO AMPLIFIER
f
ENERGY SOURCE I FEEDBACK VOLTAGE VALV^ SERVO V A L V E f ENERGV SOURCE FEEDBACK TRANSDUCER SPOOL^ DISPL: ACTUATOR ENERGY SOURCE PISTON^ IDISPL. • (a)5^^±m
i^fW i fs) I + t s K, y{s)J ^
Ü W FIG. 4. (b)U^}
f'C») JkR*) + ^dW + fW -H+; ^ H W f (s) G (s) 5E (s) G , J . ) a(s)-ir(s) 4 ïï{s) v,W ^ t , f'^ PASSIVE DAMPING X ( i ) BASIC UNDAMPED SYSTEM ACTIVE DAMPING G (s) = !, m, s2 t K | G H (S) = - C, S G fs1 - — ^'^ G t i ( s ) . T j , G ^ i (s) . k 2 G {s) = T g 5 2 , G . (s) = m 2 s 2 FIG. 5• ^ 2 (c) SYSTEM 1-3, ( v ' l ) - I - 3 O 2 (b) SYSTEM 1-2, (» * O FIG. 6. ^21 ' - 0 - 2 5 Ct (o) SYSTEM 2 - 1 , ( T » i l ) (b) SYSTEM 2 - 2 . (V = I ) FIG. 7
- 2 (o) SYSTEM 3-1, ( y = l ) (b) SYSTEM 3-2, (V ° l) FIG.8. //7f7///yyfyy/yy'y'yyyf/yy7/7//7 2 (a) SYSTEM l-l I 2 _ - I -3 - 2 - 3 'Jitmttfftm 5 O (b) SYSTEM 1 2 (C) SYSTEM 1-3 FIG. 9
- 2 (a) SYSTEM 2 - t 1^3 = 2 0 . 0 I 2 3 (C) SYSTEM 2 - 3 (b) SYSTEM 2 - 2 FIG. 10 (b) SYSTEM 3 - 2 FIG. II
-I20 -ieo,<#)(>>i 8 6 4 2 \ \ l \ \ \ \ \
- K ^ . ^
(c) I ^ 1 0 2 5 0 5 0-75 I-O SYSTEM l-l (K = l.o«<i = l ) SYSTEM 2-1 f K = l , « v ' - l ) SYSTEM 3-1 ( K = I , « Q = - I ) FIG. 12. ,t|H(iy)(* (c) o 2 0 - 4 0-6 SYSTEM 1-2 ( K - l . "«d = l ) SYSTEM 2-2 [ K - I . « « - = - 1 ) ( K - I . ««• = SYSTEM 3 2 ( K = l , o<g= l ) F I G . j iIO Cl -120 - I 8 0 SYSTEMI-3 (K=l,"<5,=-3) SYSTEM2-3 (K = l,«^= 3 )
FIG. J4_
(<^('')-<3-
rrJDtoinimïïit.
K < O 8"° i<39
o
O <o
ó
r\4 Ó ca in Ó 00 Ó O Oï
^
