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3
MATHEMATICAL
REPRESENTATIONS
OF SMART
MATERIAL SYSTEMS
In Chapter 2 we saw that there are alternative methods for deriving governing equa-
tions for smart material systems. We considered the combined effects of mechanical
and electrical forces and determined that the model for systems in static equilibrium
or for dynamic systems could be derived from the governing equations or through
variational methods based on the concepts of work and energy. In either case the
model for a system in static equilibrium was found to be a set of algebraic equations
in the generalized state variables. In the case of a dynamic system, the governing
equations are a set of differential equations in time.
In this chapter we discuss methods of solving for the time and frequency response
of the equations that govern the response of smart material systems. We first discuss
solutions of equations that are described by algebraic equations that model the static
response of a system. Next, we discuss the solution of the differential equations that
describe a dynamic system. Both first- and second-order equations of motion are
studied. Finally, we describe impedance methods used to solve for system response.
3.1 ALGEBRAIC EQUATIONS FOR SYSTEMS IN STATIC EQUILIBRIUM
The equations that govern the response of a system in static equilibrium are a set
of algebraic expressions that relate the generalized state variables to the generalized
forces that act on a system. In Chapter 2 we saw that the generalized state variables
consist of generalized coordinates that describe the mechanical response and charge
coordinates that describe the electrical response. To generalize the result for discussion
in this chapter, we define the vector of generalized state variables as r. The generalized
forces are denoted f even though they could represent both mechanical and electrical
forces that act on the system.
The most general relationship for the set of governing equations is a functional
relationship between the generalized forces f and generalized state variables r,
g (r) = f. (3.1)
91Engineering Analysis of Smart Material Systems. Donald J. Leo
Copyright © 2007 John Wiley & Sons, Inc.
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92 MATHEMATICAL REPRESENTATIONS OF SMART MATERIAL SYSTEMS
The term g denotes a set of functions that represent the governing equations for the
system. In most of the book we study systems whose governing equations are linear.
In this case the system of equations is reduced to
Kr = B f f, (3.2)
where K is called the stiffness matrix and B f the input matrix or influence matrix of
the system. If the stiffness matrix is nonsingular, the matrix inverse of the stiffness
matrix exists and the generalized state variables can be computed from the expression
r = K−1B f f. (3.3)
For low-order models (e.g., models with two or three generalized states), equa-
tion (3.3) can be solved by hand. For higher-order systems there are a number of
efficient computer algorithms for solving matrix inverses for systems with hundreds
and possibly thousands of generalized states.
In many problems it is useful to define an observation matrix or output matrix that
relates the generalized states to an observed set of outputs. For example, it is often the
case in smart material systems that certain sensors are used to measure, for example,
the displacement of a device or at a particular location on a structure. The outputs of
the system are some function of the generalized state variables, and the observation
matrix defines this functional relationship. In the most general form, the relationship
between the outputs observed, y, and the generalized state variables is
y = h (r) . (3.4)
In most cases in this book we assume that the outputs can be written as a linear
combination of the generalized states; therefore, the outputs can be written in matrix
form as
y = Hdr. (3.5)
Combining equations (3.3) and (3.5), we can write the input–output relationship
between the applied forces and the outputs observed:
y = HdK−1B f f. (3.6)
In a majority of cases, the number of outputs is smaller than the number of generalized
states; therefore, the observation matrix Hd has more columns than rows.
3.2 SECOND-ORDER MODELS OF DYNAMIC SYSTEMS
Models of dynamic systems are represented as differential equations in time as dis-
cussed in detail in Chapter 2. In this book, models for dynamic systems are derived
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SECOND-ORDER MODELS OF DYNAMIC SYSTEMS 93
from application of the governing laws of mechanics or through the application of
the variational principle for dynamic systems. As we saw in Chapter 2, these models
result in the definition of a set of second-order equations that represent the balance of
applied forces (or equivalently, work) with the forces due to the stored potential and
kinetic energy of the system. A general relationship for a dynamic system is of the
form
f = g (r̈, ṙ, r) . (3.7)
If the equations are linear and there are no terms due to the first derivative of the
generalized states, the equations of motion can be written
Mr̈(t) + Kr(t) = B f f(t), (3.8)
where M is called the mass matrix for the system. The mass matrix arises from the
kinetic energy terms and represents forces due to the time derivative of the momentum.
In many instances there are also forces due to viscous damping. These forces are
represented as a force that is proportional to the first derivative of the generalized
states and can be added into equation (3.8) as
Mr̈(t) + Dv ṙ(t) + Kr(t) = B f f(t), (3.9)
where Dv is the viscous damping matrix for the system.
Before solving the matrix set of equations, let’s consider the case where we have
only a single generalized state to illustrate the fundamental results associated with
second-order systems. In the case in which there is no damping, the equations of
motion are written as
mr̈ (t) + kr (t) = fo f (t), (3.10)
where fo represents the amplitude of the time-dependent force f (t). The solution is
generally found after normalizing the system to the mass,
r̈ (t) + k
m
r (t) = fo
m
f (t). (3.11)
The ratio of the stiffness to the mass is denoted
k
m
= ω2n (3.12)
and is called the undamped natural frequency of the system. The importance of
the undamped natural frequency is evident when we consider the solution of the
homogeneous differential equation. Setting f (t) = 0, the solution is
r (t) = A sin(ωnt + φ), (3.13)
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94 MATHEMATICAL REPRESENTATIONS OF SMART MATERIAL SYSTEMS
where
A = 1
ωn
√
ω2nr (0)2 + ṙ (0)2
(3.14)
φ = tan−1 ωnr (0)
ṙ (0)
.
Thus, the solution to an unforced second-order dynamic system is a harmonic function
that oscillates at the undamped natural frequency. The amplitude and phase of the
system is defined by the initial displacement and initial velocity.
The solution to a forced system depends on the type of forcing input. Typical
forcing inputs are step functions and harmonic functions. The solutions to these two
types of inputs are defined as
Step: r (t) = f0
k
(1 − cos ωnt) r (0) = ṙ (0) = 0
Harmonic: r (t) = ṙ (0)
ωn
sin ωnt +
(
r0(0) − f0m
ω2n − ω2
)
cos ωnt + f0/m
ω2n − ω2
cos ωt
(3.15)
The general solution to a forcing input is defined in terms of the convolution integral,
r (t) = f0
mωn
∫ t
0
f (t − τ ) sin ωnτ dτ. (3.16)
When viscous damping is present in a system, the mass normalized equations of
motion are
r̈ (t) + 2ζωnṙ (t) + ω2nr (t) =
fo
m
f (t). (3.17)
The variable ζ is the damping ratio of the system and is related to the amount of
viscous damping. For most systems we study in this book, the damping ratio is a
positive value. The form of the solution for a damped system depends on the value
of ζ . In this book we study systems that have a limited amount of viscous damping,
and generally the damping ratio will be much less than 1. For any system in which
ζ < 1, the homogeneous solution is
r (t) = Ae−ζωn t sin(ωd t + φ)(3.18)
where
ωd = ωn
√
1 − ζ 2
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SECOND-ORDER MODELS OF DYNAMIC SYSTEMS 95
A =
√
(ṙ (0) + ζωnr (0))2 + (r (0)ωd )2
ω2d
φ = tan−1 r (0)ωd
f (0) + ζωnr (0)
in terms of the initial conditions. As we see from the solution, a damped second-order
system will also oscillate at the damped natural frequency. The primary difference
is that the amplitude of the system will decay with time due to the term e−ζωn t . The
rate of decay will increase as ζ becomes larger. As ζ → 0, the solution will approach
the solution of the undamped system. The solutions for common types of forcing
functions are (for zero initial conditions)
Step: r (t) = f0
k
− f0
k
√
1 − ζ 2
e−ζωn t cos(ωd t − φ) φ = tan−1 ζ√
1 − ζ 2
Harmonic: r (t) = x cos(ωt − θ )
x = f0m√
(ω2n − ω2)2 + (2ζωnω)2
θ = tan−1 2ζωnω
ω2n − ω2
(3.19)
and the general solution is written in terms of the convolution integral:
r (t) = f0
mωd
∫ t
0
f (t − τ )e−ζωnτ sin ωdτ dτ. (3.20)
Return now to second-order systems with multiple degrees of freedom as modeled
by equation (3.8). The solution for the homogeneous undamped multiple-degree-of-
freedom (MDOF) case is obtained by assuming a solution of the form
r(t) = Ve jωt , (3.21)
where V is a vector of unknown coefficients. Substituting equation (3.21) into equa-
tion (3.8) yields
(
K − ω2M)Ve jωt = 0. (3.22)
The only nontrivial solution to equation (3.22) is the case in which
∣∣K − ω2M∣∣ = 0. (3.23)
Solving for the determinant is equivalent to the solution of a symmetric eigenvalue
problem, which yields Nv eigenvalues ω2ni and corresponding eigenvectors Vi . Gen-
erally, the eigenvalues are ordered such that ωn1 < ωn2 < · · ·. The solution to the
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96 MATHEMATICAL REPRESENTATIONS OF SMART MATERIAL SYSTEMS
MDOF problem can be cast as the solution to a set of SDOF problems by forming
the matrix
P = [V1 V2 · · · VNv ] , (3.24)
and substituting the coordinate transformation r(t) = Pη(t) into equation (3.8). The
result is
MPη̈(t) + KPη(t) = B f f(t). (3.25)
If the eigenvectors are normalized such that Vi MV j = δi j , we can premultiply equa-
tion (3.25) by P′:
P′MPη̈(t) + P′KPη(t) = P′B f f(t). (3.26)
Due to the normalization of the eigenvectors,
P′MP = I
P′KP = � = diag (ω2n1, ω2n2, . . .)
P′B f = 	 (3.27)
and the equations of motion can be written as a set of uncoupled second-order equa-
tions:
η̈i (t) + ω2niη(t) =
N f∑
j=1
	i j f j (t). (3.28)
The term 	i j is the (i, j)th element of the matrix 	.
Decoupling the equations of motion is a significant result because it allows the
solution of the multiple-degree-of-freedom system to be obtained by applying the re-
sults for single-degree-of-freedom systems. Once the MDOF system has been written
as a set of decoupled equations, as in equation (3.28), the results discussed previously
in this section can be applied to solve each of the equations separately for ri (t). Once
this is completed, the coordinate transformation r(t) = Pη(t) is applied to obtain the
complete solution in the coordinates of the original system.
Models that incorporate viscous damping as shown in equation (3.9) can also be
decoupled if the viscous damping matrix is decoupled by the eigenvectors of the
undamped system. For systems with light damping this assumption is often made
because it greatly simplifies the analysis. Additionally, the model of viscous damping
is often added into the decoupled equations because an exact model of damping is
not available or there are experimental data that allow one to estimate the damping
coefficient. Under the assumption that
P′DP = diag(2ζiωni ), (3.29)
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FIRST-ORDER MODELS OF DYNAMIC SYSTEMS 97
the decoupled equations of motion are
η̈i (t) + 2ζiωni η̇i + ω2niη(t) =
N f∑
j=1
	i j f j (t). (3.30)
As in the case of the undamped system, the equations for each of the transformed
coordinates are solved separately, and the total result can be obtained by applying the
coordinate transformation r(t) = Pη(t).
3.3 FIRST-ORDER MODELS OF DYNAMIC SYSTEMS
Many of the systems that we discuss are amenable to being modeled as second-order
differential equations, as discussed in Section 3.2. Modeling a dynamic system as a
set of second-order equations is often desirable because it provides insight into the
vibrational characteristics of the system. Another common way to represent models
of dynamic systems is in first-order or state variable form. First-order form is often
desirable when analyzing the control of smart material systems.
A general representation of a first-order model is
dz(t)
dt
= g(z(t), w(t), t), (3.31)
where z(t) is a vector of states and w(t) is a vector of inputs. If the right-hand side
of equation (3.31) can be written as a linear combination of the states and the inputs,
we can write
dz(t)
dt
= A(t)z(t) + B(t)w(t), (3.32)
where the notation in equation (3.32) implies that the coefficients on the right-hand
side are explicit functions of time. The matrix A(t) is the state matrix and B(t) is the
input matrix.
If the coefficients on the right-hand side of equation (3.32) are independent of
time, the state variable equations are written
dz(t)
dt
= Az(t) + Bw(t). (3.33)
Equation (3.33) is the linear time-invariant (LTI) form of the state equations.
The LTI state equations for a dynamic system can be solved to obtain an expression
for the states as a function of time:
z(t) = eAt z(0) +
∫ t
0
eA(t−λ)Bu(λ) dλ, (3.34)
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98 MATHEMATICAL REPRESENTATIONS OF SMART MATERIAL SYSTEMS
where
eAt = L−1 (sI − A)−1 (3.35)
is the state transition matrix. As shown in equation (3.35), this is computed from the
inverse Laplace transform of a matrix constructed from (sI − A)−1.
3.3.1 Transformation of Second-Order Models to First-Order Form
Second-order models of dynamic systems can be transformed to first-order form
by defining a relationship between the states, z, of the first-order system and the
generalized states of the second-order model, r. Defining two sets of states as
z1 = r
z2 = ṙ (3.36)
and substituting into equation (3.9) yields
Mż2 + Dvz2 + Kz1 = B f f. (3.37)
Premultiplying equation (3.37) by the inverse of the mass matrix and solving for ż2
produces the expression
ż2 = −M−1Kz1 − M−1Dvz2 + M−1B f f. (3.38)
Equation (3.38) is one of the two first-order equations that are required to model the
second-order system. The remaining equation is derived from the definition of the
state variables:
ż1 = z2. (3.39)
The first-order form of the equations is written by combining equations (3.38) and
(3.39). The expressions are written in matrix form as
(
ż1
ż2
)
=
[
0 I
−M−1K −M−1Dv
](
z1
z2
)
+
[
0
M−1B f
]
f. (3.40)
Comparing equation (3.40) with equation (3.33), we see that
A =
[
0 I
−M−1K −M−1Dv
]
B =
[
0
M−1B f
]
(3.41)
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FIRST-ORDER MODELS OF DYNAMIC SYSTEMS 99
is the relationship between the matrices that represent the second-order system and
state and input matrices of first-order form.
3.3.2 Output Equations for State Variable Models
Equations (3.31), (3.32), and (3.33) represent expressions for the states of a dynamic
system. It is rare that all of the internal system states can be observed directly at
the output of the system; therefore, we must define a second set of equations that
expresses the internal states that can be measured at the output. For a general state
variable model, the output equations can again be represented in general function
form:
y(t) = h (z(t), w(t), t) . (3.42)
If the observed outputs can be written as a linear combination of the states and the
inputs, we can write the output expression as
y(t) = C(t)z(t) + D(t)w(t), (3.43)
where C(t) and D(t) are called the observation or output matrix and the direct trans-
mission matrix, respectively. If the coefficient matrices are time invariant, the output
expressions are written as
y(t) = Cz(t) + Dw(t). (3.44)The dimension and definitions of matrices C and D are equivalent to the case of
time-varying systems.
Expressions for the outputs as a function of the states for an LTI system are obtained
by combining equations (3.34) and (3.44):
y(t) = CeAt z(0) +
∫ t
0
CeA(t−λ)Bw(λ) dλ + Dw(t). (3.45)
Example 3.1 Derive the state equations for a mass–spring–damper oscillator with
a force applied to the mass as shown in Figure 3.1. Write the state equations and the
output equations assuming that the position and input force are the observed variables.
Solution Drawing a free-body diagram and applying Newton’s second law yields
mü(t) = −ku(t) − cu̇(t) + f (t). (3.46)
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100 MATHEMATICAL REPRESENTATIONS OF SMART MATERIAL SYSTEMS
m
c k
u
f
Figure 3.1 Mass–spring–damper oscillator with a force applied.
Dividing by the mass yields
ü(t) = − k
m
u(t) − c
m
u̇(t) + 1
m
f (t). (3.47)
Defining the state variables as
z1(t) = u(t) (3.48)
z2(t) = u̇(t) (3.49)
and substituting into equation (3.47) produces an expression for the first derivative of
z2(t):
ż2(t) = − k
m
z1(t) − c
m
z2(t) + 1
m
f (t). (3.50)
The equation for the derivative of the first state is obtained from the state definitions:
ż1(t) = z2(t). (3.51)
Combining equations (3.50) and (3.51) into a matrix expression yields
(
ż1(t)
ż2(t)
)
=
[
0 1
−k/m −c/m
](
z1(t)
z2(t)
)
+
[
0
1/m
]
f (t). (3.52)
The output equations are written as
y1(t) = z1(t) (3.53)
y2(t) = f (t). (3.54)
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INPUT–OUTPUT MODELS AND FREQUENCY RESPONSE 101
Combining these expressions into one matrix expression produces
y(t) =
[
1 0
0 0
](
z1(t)
z2(t)
)
+
[
0
1
]
u(t). (3.55)
The state matrices are determined from equations (3.52) and (3.55) as
A =
[
0 1
−k/m −c/m
]
(3.56)
B =
[
0
1/m
]
(3.57)
C =
[
1 0
0 0
]
(3.58)
D =
[
0
1
]
. (3.59)
3.4 INPUT–OUTPUT MODELS AND FREQUENCY RESPONSE
First and second-order models of a dynamic system contain information about the
internal states of the system. Earlier we have shown how these models are derived
from the governing equations or through application of a variational principle. In
some cases a model that includes information about the internal states of the system
is not necessary or not possible. In these instances we can derive an input–output
model that contains only information about how the outputs of a system will change
as a function of the inputs. Such input–output models are generally expressed in the
Laplace or frequency domain. In this section we derive a general formulation for an
input–output transfer function of an LTI system, and in Section 3.5 we transform the
result into the frequency domain.
Consider again the state variable equations for an LTI system and the associated
output equations (3.33) and (3.44). Transforming equation (3.33) into the Laplace
domain assuming zero initial conditions yields
sz(s) = Az(s) + Bw(s). (3.60)
Combining like terms on the left-hand side and premultiplying by (sI − A)−1 produces
z(s) = (sI − A)−1Bw(s). (3.61)
Transforming equation (3.44) into the Laplace domain yields
y(s) = Cz(s) + Dw(s). (3.62)
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102 MATHEMATICAL REPRESENTATIONS OF SMART MATERIAL SYSTEMS
Substituting equation (3.61) into equation (3.62) and combining terms yields
y(s) = [C(sI − A)−1B + D]w(s). (3.63)
Equation (3.63) represents the general expression for a matrix of transfer functions
between the inputs w(t) and the outputs y(t). Note that variables related to the internal
states are not contained explicitly within the equation for the transfer functions. The
number of internal states does influence this equation, though, through the size of the
matrix (sI − A)−1.
Any relationship between the i th output and the j th input can be determined from
the expression
yi (s) =
[
Ci (sI − A)−1B j + Di j
]
w j (s), (3.64)
where Ci is the i th row of C, B j is the j th column of B, and Di j is the (i, j)th element
of D. Since the input–output relationship in equation (3.64) is a single function, we
can write yi (s)/w j (s) as a ratio of Laplace polynomials:
yi (s)
w j (s)
= Ci (sI − A)−1B j + Di j = bos
m + b1sm−1 + · · · + bm−1s + bm
sn + a1sn−1 + · · · + an−1s + an .
(3.65)
The roots of the denominator polynomial are called the poles of the system, and the
roots of the numerator polynomial are the zeros. There are two important properties
to note about the single input–output transfer function expressed as a ratio of Laplace
polynomials. First, the number of terms in the denominator will be equal to or less
than the number of internal states of the dynamic system. When there are no pole–
zero cancellations between the numerator and denominator, n = k, where k is the
number of internal states. If pole–zero cancellations occur, n = k − pz, where pz is
the number of pole–zero cancellations. Second, the relative order of the numerator
and denominator is related directly to the existence of the direct transmission term
in the state variable model. A nonzero direct transmission term, Di j , will produce
a transfer function in which the order of the numerator is equal to the order of the
denominator (i.e., m = n). In this case the transfer function is called proper. In the
case in which Di j = 0, the order m will be less than the order n and the transfer
function is called strictly proper.
Example 3.2 Determine the transfer function matrix for the mass–spring–damper
system introduced in Example 3.1.
Solution The state matrices for the mass–spring–damper system discussed in Ex-
ample 3.1 are shown in equations (3.56) to (3.59). The matrix of transfer functions is
determined from the expression C (sI − A) B + D. The matrix expression sI − A
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INPUT–OUTPUT MODELS AND FREQUENCY RESPONSE 103
is
sI − A =
[
s −1
k/m s + c/m
]
. (3.66)
The matrix inverse is
(sI − A)−1 = 1
s2 + (c/m)s + k/m
[
s + c/m 1
−k/m s
]
. (3.67)
The transfer function is obtained from
C (sI − A) B + D = 1
s2 + (c/m)s + k/m
[
1 0
0 0
] [
s + c/m 1
−k/m s
] [
0
1/m
]
+
[
0
1
]
.
(3.68)
Multiplying out the expression on the right-hand side yields
C (sI − A) B + D =


1/m
s2 + (c/m)s + k/m
1

 . (3.69)
The dimensions of the transfer function matrix are 2 × 1, which matches the fact that
there are two outputs and one input to this system. The transfer function matrix can
be placed into a form that is familiar to vibration analysis through the substitutions
c
m
= 2ζωn (3.70)√
k
m
= ωn, (3.71)
where ζ is the nondimensional damping ratio and ωn is the natural frequency in rad/s.
With these definitions, the solution can be written as
C (sI − A) B + D =


1/m
s2 + 2ζωns + ω2n
1

 . (3.72)
3.4.1 Frequency Response
An input–output representation of an LTI dynamic system leads to the concept of the
frequency response. Consider an LTI dynamic system modeled as a matrix of transfer
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104 MATHEMATICAL REPRESENTATIONS OF SMART MATERIAL SYSTEMS
functions,
y(s) = H(s)w(s), (3.73)
where H(s) is obtained from equation (3.63):
H(s) = C(sI − A)−1B + D. (3.74)
The matrix H(s) is the matrix of input–output transfer functions as defined by the
state variable representation of the system. Consider a harmonic input of the form
w j (t) = W j sin ωt, (3.75)
where W j is an amplitude of the j th input and ω is the frequency of the harmonic
excitation. All other inputs are assumed to be equal to zero. The expression for the
i th output can be written as
yi (s) = Hi j (s)w j (s). (3.76)
Assuming that the system is asymptotically stable, the steady-state output to a har-
monic excitation can be written as
yi (t → ∞) = |Hi j ( jω)|W j sin(ωt + � Hi j ( jω)). (3.77)
The term Hi j ( jω) is a complex-valued expression that can be written in real and
imaginary terms as
Hi j ( jω) = �
{
Hi j ( jω)
}+ � {Hi j ( jω)} . (3.78)
The magnitude and phase can then be determined from
|Hi j ( jω)| =
√�{Hi j ( jω)}2 + �{Hi j ( jω)}2 (3.79)
� Hi j ( jω) = tan−1 �{Hi j ( jω)}� {Hi j ( jω)} . (3.80)
Equation (3.77) illustrates three important results in linear system theory:
1. The steady-state response of an asymptotically stable system oscillates at the
same frequency as the frequency of the input.
2. The amplitude of the output is scaled by the magnitude of the transfer function
evaluated at s = jω.
3. The phase of the output is shifted by the phase of the transfer function evaluated
at s = jω.
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INPUT–OUTPUT MODELS AND FREQUENCY RESPONSE 105
These results emphasize the importance of the magnitude and phase of the input–
output frequency response. Evaluating the input–output transfer function at s = jω
allows us to determine the amplitude and phase of the output relative to the input. For
an asymptotically stable system, this explicitly determines the steady-state response
of the system.
Example 3.3 Show that the frequency response ku( jω)/ f ( jω) in Example 3.2 can
be written as a nondimensional function of the damping ratio and the frequency ratio
� = ω/ωn .
Solution The solution to Example 3.2 can be written in the Laplace domain as
(
y(s)
f (s)
)
=


1/m
s2 + 2ζωns + ω2n
1

 f (s). (3.81)
The input–output transfer function u( jω)/ f ( jω) is equal to the first row of the transfer
function matrix:
u(s)
f (s)
= 1/m
s2 + 2ζωns + ω2n
. (3.82)
The frequency response is obtained by substituting s = jω into expression (3.82) and
combining terms:
u( jω)
f ( jω)
= 1/m
ω2n − ω2 + j2ζωnω
. (3.83)
Substituting the parameter � = ω/ωn into the expression and dividing through by ω2n
yields
u( jω)
f ( jω)
= 1
mω2n
1
1 − �2 + j2ζ�. (3.84)
Recalling that ω2n = k/m, we can multiply both sides by the stiffness to produce the
nondimensional expression
ku( jω)
f ( jω)
= 1
1 − �2 + j2ζ�. (3.85)
This result demonstrates that the right-hand side of the expression is a nondimensional
function of the damping ratio and the frequency ratio.
ch03 JWPR009-LEO July 19, 2007 18:44
106 MATHEMATICAL REPRESENTATIONS OF SMART MATERIAL SYSTEMS
Example 3.4 Plot the magnitude and phase of the frequency response ku( jω)/
f ( jω) for ζ = 0.01, 0.05, 0.10, 0.30, and 0.707. Discuss the nature of the response
for � << 1, � ≈ 1, and � >> 1.
Solution Plotting the frequency response requires that we obtain an expression for
the magnitude and phase of the transfer function. This is obtained by first writing
the transfer function in real and imaginary components as shown in equation (3.78).
Multiplying the transfer function by the complex conjugate of the denominator yields
ku( jω)
f ( jω)
= 1 − �
2(
1 − �2)2 + 4ζ 2�2 − j
2ζ�(
1 − �2)2 + 4ζ 2�2 . (3.86)
The magnitude of the frequency response is obtained from equation (3.79) and the
phase is obtained from equation (3.80):
∣∣∣∣ku( jω)f ( jω)
∣∣∣∣ = 1√(
1 − �2)2 + 4ζ 2�2 (3.87)⌊
ku( jω)
f ( jω)
= −2ζ�
1 − �2 . (3.88)
A plot of the frequency response magnitude and phase is shown in Figure 3.2.
The frequency response can be separated into three distinct regions. At excitation
frequencies well below the resonance frequency of the system (� � 1), the magnitude
of the frequency response function is flat and the phase is approximately equal to 0◦.
10
–2
10
–1
10
0
10
1
10
2
10
–4
10
–3
10
–2
10
–1
10
0
10
1
10
2
Ω = ω/ωn
|k
x(
jω
)/
u
(j
ω
)|
10
–2
10
–1
10
0
10
1
10
2
–180
–160
–140
–120
–100
–80
–60
–40
–20
0
(a) (b)
Ω = ω/ωn
ph
as
e 
(d
eg
re
es
)
ζ = 0.01
ζ = 0.05
ζ = 0.10
ζ = 0.30
ζ = 0.707
ζ = 0.01
ζ = 0.05
ζ = 0.10
ζ = 0.30
ζ = 0.707
Figure 3.2 (a) Magnitude and (b) phase of a mass–spring–damper system as a function of
nondimensional frequency and damping ratio.
ch03 JWPR009-LEO July 19, 2007 18:44
INPUT–OUTPUT MODELS AND FREQUENCY RESPONSE 107
In this frequency region, ∣∣∣∣ku( jω)f ( jω)
∣∣∣∣ ≈ 1 � � 1; (3.89)
therefore, the amplitude u( jω)/ f ( jω) ≈ 1/k when the excitation frequency is much
less than the natural frequency. Furthermore, the input and output waveform are
approximately in phase at these frequencies. Notice that the damping ratio, and hence
the damping coefficient and mass, do not influence the amplitude of the response at
frequencies well below resonance. The magnitude of the response becomes amplified
when the excitation frequency approaches the natural frequency of the system. The
magnitude and phase at this frequency are∣∣∣∣ku( jωn)f ( jωn)
∣∣∣∣ = 12ζ (3.90)⌊
ku( jωn)
f ( jωn)
→ ±∞. (3.91)
These results demonstrate that the damping ratio strongly influences the amplitude at
resonance. Smaller values of ζ will produce a larger resonant amplitude.
The relationship between damping and the response at resonance is due to the com-
peting physical processes within the system. At resonance, the force due to the spring
stiffness and the force due to the inertial acceleration of the mass cancel one an-
other out. Thus, the only force that resists motion at the resonance frequency is the
force associated with the damping in the system. This result is general and highlights
an important relationship between system response and damping. Energy dissipa-
tion strongly influences the response of the system near resonance frequencies. The
smaller the energy dissipation, the larger the resonant amplification.
As the excitation frequency becomes much larger than the resonant frequency
(� 
 1), the magnitude and phase approach∣∣∣∣ku( jω)f ( jω)
∣∣∣∣ → 1�2 (3.92)⌊
ku( jω)
f ( jω)
→ −2ζ−� . (3.93)
The expression for the magnitude approaches a small number as the frequency ratio
becomes large, indicating that the displacement amplitude will decrease as the exci-
tation frequency becomes large with respect to the natural frequency. As shown in
Figure 3.2b, the phase approaches −180◦ as the frequency ratio becomes much larger
than 1. (Note that the signs in the phase expression have been retained to emphasize
quadrant associated with the inverse tangent.) The phase response illustrates that the
input and output will be of opposite sign at frequencies much higher than the resonant
frequency.
ch03 JWPR009-LEO July 19, 2007 18:44
108 MATHEMATICAL REPRESENTATIONS OF SMART MATERIAL SYSTEMS
A second interpretation of the frequency response of an LTI system is obtained
by considering an impulse input applied to the system at time zero. In this case, the
input to the system is modeled as a delta function
w j (t) = δ(t). (3.94)
The Laplace transform of an impulse input at time zero is
L{δ(t)} = 1. (3.95)
Substituting this result into equation (3.76) yields the result
yi (s) = Hi j (s). (3.96)
The time response yi (t) can be determined from the inverse Laplace transform of
equation (3.96),
yi (t) = L−1{yi (s)} = L−1{Hi j (s)}. (3.97)
This result demonstrates that the impulse response of an LTI system is equivalent
to the inverse Laplace transform of the input–output transfer function. This result is
used continuously; therefore, we generally assign the symbol
hi j (t) = L−1{Hi j (s)} (3.98)
to designate the impulse response between the i th output and j th input.
The importance of this result can be understood by examining the inverse Laplace
transform of equation (3.76). If we apply the inverse Laplace transform to equa-
tion (3.76), we obtain the following expression for yi (t) through the convolution
theorem:
yi (t) = L−1{yi (s)} = L−1{Hi j (s)w j (s)} =
∫ t
0
hi j (t − λ)w j (λ)dλ. (3.99)
Note that equation (3.99) applies to any deterministic function w j (t). Thus, the re-
sponse to any deterministic input can be determined by convolving the impulse re-
sponse and the input function. As discussed above, the impulse response is related to
the frequency response of the dynamic system; therefore, having an expression for the
frequency response is equivalent to saying that the response to any deterministic input
can be obtained from convolution. This is a very powerful result for LTI systems.
Comparing this result to equation (3.45),we see that the impulse response can be
determined directly from the state variable representation as
hi j (t) = Ci eAt B j . (3.100)
ch03 JWPR009-LEO July 19, 2007 18:44
IMPEDANCE AND ADMITTANCE MODELS 109
This result provides a link between the state variable representation and the impulse
response of an LTI system.
3.5 IMPEDANCE AND ADMITTANCE MODELS
One type of input–output model that bears additional attention is an impedance model
of a dynamic system. This type of model will be useful for the purpose of gaining
insight into the concepts of power and energy transfer associated with smart material
systems. As highlighted in Chapter 2, certain analogies can be made between the
generalized state variables that model electrical and mechanical systems. For example,
the concept of mechanical force and stress in a mechanical system is analogous to the
applied voltage or applied electric field in an electrical system. Similarly, displacement
and strain in a mechanical system have analogous relationships to charge or electric
displacement in an electrical system.
The concept of analogous state variables come together when discussing the work
or energy of a system. As discussed in Chapter 2, the work or stored energy associated
with a system is related to an integral of the force and displacement for a mechanical
system or equivalently, charge and voltage for an electrical system. The time rate of
change of work is the power associated with a system. It has units of joules per second
or watts (W). The instantaneous mechanical power is defined as
�M (t) = f (t)u̇(t), (3.101)
and the instantaneous electrical power is defined as
�E (t) = v(t)q̇(t) = v(t)i(t), (3.102)
where i(t) = q̇(t) is the current. The average power over a defined time interval T is
defined as
< �M > = 1
T
∫ t+T
t
f (τ )u̇(τ ) dτ
< �E > = 1
T
∫ t+T
t
v(τ )i(τ ) dτ. (3.103)
The average power at the location of an input to a system can be related to the
input–output response of an LTI model. The analogy between force and voltage and
velocity and current can be extended by considering an LTI model which has as the
input a generalized force, which we denote φ, and generalized flux, which we denote
ψ . The force could be either a mechanical force or an applied voltage, and the flux
can be either a velocity or current. The input–output relationship of an LTI model
is expressed in equation (3.73) as a matrix of Laplace domain transfer functions. If
we assume that the input to this system is the generalized force, φ, and the output is
ch03 JWPR009-LEO July 19, 2007 18:44
110 MATHEMATICAL REPRESENTATIONS OF SMART MATERIAL SYSTEMS
the generalized flux term, ψ , the input–output response is written in the frequency
domain as
ψ = Z−1( jω)φ, (3.104)
where Z−1( jω) is the inverse of the impedance matrix Z( jω). The matrix of transfer
functions Z−1( jω) is also called the admittance matrix. Multiplying both sides of
equation (3.104) by the impedance matrix produces an expression between the applied
forces and the flux response,
φ = Z( jω)ψ. (3.105)
The physical significance of the impedance matrix is analyzed by considering the
case of a harmonic input to the i th flux input:
ψi (t) = �i sin ωt. (3.106)
The steady-state output of the force at the i th location to this input (assuming that the
system is asymptotically stable) is
φi (t) = |Zi i | �i sin(ωt + � Zi i ). (3.107)
The instantaneous power is the product of the flux and the force:
�(t) = ψi (t)φi (t) = |Zi i | �2i
[
1 − cos 2ωt
2
cos � Zi i + 1
2
sin 2ωt sin � Zi i
]
= �2i
[
1 − cos 2ωt
2
�(Zi j ) + 1
2
sin 2ωt�(Zi j )
]
. (3.108)
This form of the instantaneous power provides insight into the meaning of the
impedance. The term that multiplies the real part of the impedance is bounded between
0 and 1, whereas the term that multiplies the imaginary part of the impedance has a
mean of zero and is bounded by ±1. The first term in brackets in equation (3.108) is
called the real impedance; the second term is called the reactive impedance. The real
impedance is proportional to the real component of Zi j , and the reactive impedance
is proportional to the imaginary component. Computing the average power over a
single period, we see that
< � >= 1
T
∫ T
0
ψi (t)φi (t)dt = 1
2
�2i �(Zi i ). (3.109)
Thus, the average power is proportional to the real part of the impedance function Zi i .
As shown in equation (3.108), the reactive component of the impedance is proportional
ch03 JWPR009-LEO July 19, 2007 18:44
IMPEDANCE AND ADMITTANCE MODELS 111
to the imaginary component of Zi i , and the contribution of the reactive component to
the average power is zero.
The notion of real and reactive impedance is often related to the power flow at
the input–output location. The impedance function allows us to analyze the power
flow directly. A real-valued impedance indicates that the system is dissipative at that
frequency, whereas an impedance that is purely imaginary indicates that the system
exhibits energy storage and the power will oscillate between the source and the
system at the terminal location. For an impedance that consists of real and imaginary
components, the relative amplitude of the terms is directly related to the amount of
energy dissipation and energy storage at the terminals.
Example 3.5 Derive the admittance and impedance functions for the mass–spring–
damper oscillator shown in Figure 3.1. Nondimensionalize the expressions in terms
of the frequency ratio ω/ωn . Plot the magnitude and phase of the impedance function
and discuss the physical signficance of regions in which the impedance is low.
Solution As shown in equation (3.104), the admittance is the ratio of the flux to the
force. In a mechanical system, this reduces to the ratio of the velocity of the mass,
ẋ(t), to the input force, u(t). The ratio of velocity to force can be derived using the
state variable representation shown in equation (3.52) and the output equations
y(t) = [0 1] (z1(t)
z2(t)
)
. (3.110)
Applying equation (3.63) yields
L {u̇(t)} = su(s) = s/m
s2 + (c/m)s + k/m f (s). (3.111)
The admittance function is then written as
A(s) = su(s)
f (s)
= s/m
s2 + (c/m)s + k/m . (3.112)
Note that the subscripts have been dropped for convenience since there is only a
single input and a single output [i.e., A(s) = A11(s)]. Substituting the definitions
c/m = 2ζωn and k/m = ω2n , we can write the frequency response of the admittance
function as
A( jω) = jω/m
ω2n − ω2 + j2ζωωn
. (3.113)
The expression can be nondimensionalized through the substitution � = ω/ωn as
A( jω) = 1
mωn
j�
1 − �2 + j2ζ�. (3.114)
ch03 JWPR009-LEO July 19, 2007 18:44
112 MATHEMATICAL REPRESENTATIONS OF SMART MATERIAL SYSTEMS
10
–2
10
–1
10
0
10
1
10
2
–100
–50
0
50
100
Ω = ω/ω n
ph
as
e 
(d
eg
)
10
–2
10
(a) (b)
–1
10
0
10
1
10
2
10
–2
10
–1
10
0
10
1
10
2
Ω = ω/ωn
|Z
(j
ω
)|
/m
ω
n
ζ = 0.707
ζ = 0.30
ζ = 0.10
ζ = 0.05
ζ = 0.01
ζ = 0.707
ζ = 0.30
ζ = 0.10
ζ = 0.05
ζ = 0.01
Figure 3.3 (a) Magnitude and (b) phase of the impedance function for a mass–spring–damper
system as a function of damping ratio and nondimensional frequency.
Checking units, we see that the units of mωn are s/kg, which is correct for a ratio of
velocity to force.
The impedance function is simply the inverse of the admittance; therefore,
Z ( jω) = A( jω)−1 = mωn
2ζ� + j (�2 − 1)
�
. (3.115)
This result is obtained by multiplying the numerator and denominator by j and rear-
ranging terms. The magnitude and phase can be determined by writing the impedance
as a sum of real and imaginary terms:
1
mωn
Z ( jω) = 2ζ + j �
2 − 1
�
(3.116)
and computing the magnitude and phase using equations (3.79) and (3.80). The plots
are shown in Figure 3.3, where we see that the impedance is low in the frequency
range in which ω ≈ ωn . This frequency range corresponds to the frequency range of
the mechanical resonance of the system. The impedance at this frequency is only a
function of the damping ratio.This can be quantified by computing
1
mωn
Z ( jωn) = 2ζ. (3.117)
This expression illustrates that when the excitation frequency is equivalent to the
natural frequency, the impedance is only a function of the energy dissipation in the
system. Once again this is due to the fact that the forces due to the spring stiffness and
the inertial motion cancel one another at this frequency. A low impedance implies
that it would be easy to move this system at this frequency (i.e., the system is soft
when the excitation frequency is equal to the natural frequency).
ch03 JWPR009-LEO July 19, 2007 18:44
IMPEDANCE AND ADMITTANCE MODELS 113
3.5.1 System Impedance Models and Terminal Constraints
The concept of impedance can be applied to the analysis of systems with multiple
inputs and outputs. The expression for a system impedance, equation (3.105), can be
written in matrix notation as

φ1
...
φn

 =


Z11 · · · Z1n
...
. . .
...
Zn1 · · · Znn




ψ1
...
ψn

 . (3.118)
The system-level impedance model is a convenient framework for analyzing the re-
lationships between forces and fluxes in a dynamic system. When a terminal constraint
exists between a force and a flux, the impedance model expressed in equation (3.118)
can be modified to determine the effect of this terminal constraint on the remain-
ing force and flux terms. We define a terminal constraint as an explicit relationship
between a single force–flux term,
φk = −Zcψk, (3.119)
where the constraint is expressed as an impedance −Zc. The negative sign is chosen
for convenience. Expanding equation (3.118) to include the terminal constraint, we
have
φ1 = Z11ψ1 + · · · + Z1kψk + · · · + Z1nψn
...
φk = Zk1ψ1 + · · · + Zkkψk + · · · + Zknψn (3.120)
...
φn = Zn1ψ1 + · · · + Znkψk + · · · + Znnψn.
Substituting equation (3.119) into equation (3.121) and solving the kth equation for
ψk yields
ψk = − Zk1
Zc + Zkk ψ1 − · · · −
Zkn
Zc + Zkk ψn. (3.121)
Substituting this expression into the remaining n − 1 equations produces
φ1 =
(
Z11 − Z1k Zk1
Zc + Zkk
)
ψ1 + · · · +
(
Z1n − Z1k Zkn
Zc + Zkk
)
ψn (3.122)
...
φn =
(
Zn1 − Znk Zk1
Zc + Zkk
)
ψ1 + · · · +
(
Znn − Znk Zkn
Zc + Zkk
)
ψn. (3.123)
ch03 JWPR009-LEO July 19, 2007 18:44
114 MATHEMATICAL REPRESENTATIONS OF SMART MATERIAL SYSTEMS
Examining equation (3.122), we see that a general expression for the i th force when
a constraint exists at the kth location is
φi =
n∑
m=1
(
Zim − Zik Zkm
Zc + Zkk
)
ψm m �= k. (3.124)
This result indicates that, in general, every force–flux relationship is affected by a
terminal constraint at a single location in the system.
Two particular types of terminal constraints are zero-force constraints and zero-
flux constraints. A zero-force constraint can be determined by setting Zc = 0 in
equation (3.119) and rewriting the remaining n − 1 transduction equations. The result
is
φ
φk
i =
n∑
j=1
(
Zi j − Zik Zk j
Zkk
)
ψ j j �= k. (3.125)
The superscript on the force term indicates that a zero-force constraint exists at loca-
tion k. We can rewrite equation (3.125) as
φ
φk
i =
n∑
m=1
Zi j
(
1 − Zik Zk j
Zi j Zkk
)
ψ j j �= k (3.126)
and denote
Zik Zk j
Zi j Zkk
= K (3.127)
as the generalized coupling coefficient. With this definition we can rewrite equa-
tion (3.126) as
φ
φk
i =
n∑
j=1
Zi j (1 − K) ψ j j �= k. (3.128)
This definition makes it clear that the coupling coefficient describes how much the
impedance function changes upon the introduction of a zero-force constraint at loca-
tion k.
Zero-flux constraints are imposed by letting Zc → ∞ and substituting the result
into equation (3.124). Letting the constraint impedance approach infinity produces
the result
φ
ψk
i =
n∑
j=1
Zi jψ j j �= k. (3.129)
ch03 JWPR009-LEO July 19, 2007 18:44
IMPEDANCE AND ADMITTANCE MODELS 115
v1
i1
v2
i2
R1
R2
Figure 3.4 Voltage divider with input voltage and current and output voltage and current.
This result demonstrates that the impedance terms at the remaining n − 1 locations
are not changed by the introduction of a zero-flux constraint at location k.
Another interpretation of the coupling coefficient is related to the change in the
impedance from a zero-force to a zero-flux constraint at location k. The coupling
coefficient is also equal to the difference in the zero-flux and zero-force impedance
divided by the original impedance,
φ
ψk
i − φφki
Zim
= Zim − Zim(1 − K)
Zim
= K. (3.130)
We will find in future chapters that the concept of coupling coefficients is related
directly both to material properties and to the interaction of an active material with
the external system.
Example 3.6 Consider the voltage divider shown in Figure 3.4 with input voltage,
v1, and input current, i1, and output voltage and current, v2 and i2, respectively.
Determine the impedance model of this circuit.
Solution The voltage divider has two forces and two flux terms; therefore, the
impedance model will be a 2 × 2 system. The individual impedance terms can be ob-
tained by setting one of the flux terms equal to zero and determining the corresponding
force terms.
The impedance terms Z11 and Z21 can be obtained by setting the current i2 equal
to zero and determining the voltage as a function of the current i1. When i2 = 0, the
voltage terms are
v1 = (R1 + R2) i1
v2 = R2i1
and the impedance terms can be written as
Z11 = R1 + R2
Z12 = R2.
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116 MATHEMATICAL REPRESENTATIONS OF SMART MATERIAL SYSTEMS
Setting the current i1 equal to zero produces a zero voltage drop over R1 and the two
voltages are equivalent. Thus,
v1 = R2i2
v2 = R2i2
and
Z12 = R2
Z22 = R2
Combining the impedance terms yields the matrix
Z =
[
R1 + R2 R2
R2 R2
]
. (3.131)
Example 3.7 Determine (a) the coupling coefficients of the system introduced in
Example 3.5 and (b) the impedance Z11 when there is a zero-force constraint at
terminal 2.
Solution (a) The voltage divider discussed in Example 3.5, has two force–flux
terms; therefore, n = 2. For i = 1, k = 2, and m = 1, equation (3.127) is reduced to
K = Z12 Z21
Z11 Z22
= R
2
2
R2(R1 + R2) =
R2
R1 + R2 . (3.132)
The coupling coefficient when i = 2, k = 1, and m = 2 is
K = Z21 Z12
Z22 Z11
= R
2
2
R2(R1 + R2) =
R2
R1 + R2 . (3.133)
This is consistent with the result that there are n − 1 coupling coefficients for a system
that has n terminals.
(b) The impedance Z11 when a zero-force constraint exists at terminal 2 can be
computed from equation (3.128):
Zφ211 = Z11 (1 − K )
= (R1 + R2)
(
1 − R2
R1 + R2
)
= (R1 + R2) R1
R1 + R2
= R1. (3.134)
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IMPEDANCE AND ADMITTANCE MODELS 117
v1
i1
v2 = 0
R1
R2
Figure 3.5 Voltage divider with a zero-force (voltage) constraint at terminal 2.
This result is reasonable because a zero-force constraint at terminal 2 is equivalent to
connecting the terminal to ground. Therefore, the impedance of the resulting circuit
is simply the resistance R1 (see Figure 3.5).
Example 3.8 Terminal 2 of the voltage divider is connected to a digital voltmeter
that draws no current, as shown in Figure 3.6. Determine the ratio of the output voltage
to the input voltage with this constraint at terminal 2.
Solution The digital voltmeter is assumed to draw no current; therefore, it imposes
a zero-flux constraint at terminal 2. With a zero-flux constraint at the terminal, the
transduction equations reduce to
v1 = (R1 + R2)i1
v2 = R2i1 (3.135)
Solving for ψ1 from the first expression and substituting it into the second expression
yields
v2 = R2
R1 + R2 v1, (3.136)
i1 = 0
v2
i2 = 0
R1
R2
v1
digital
volt-
meter
Figure 3.6 Voltage divider with a measurement device that draws zero flux (current).
ch03 JWPR009-LEO July 19, 2007 18:44
118 MATHEMATICAL REPRESENTATIONS OF SMART MATERIAL SYSTEMS
which can be solved to yield the ratio
v2
v1
= R2
R1 + R2 . (3.137)
This is the common result for a voltage divider, except that this analysis highlights
the fact that the typicalequation for a voltage divider implicitly assumes that zero
current is being drawn at the output terminal.
3.6 CHAPTER SUMMARY
Several approaches to modeling systems were presented in this chapter. The static
response of systems is obtained through the solution of a set of algebraic equations.
Dynamic systems are modeled as a set of differential equations that must be solved
to determine the response of the system as a function of time. The basic properties of
a second-order mass–spring–damper system were analyzed. Second-order equations
of motion were presented as a means of representing structural material systems.
A discussion of eigenvalue analysis was presented and related to the solution of
systems modeled with second-order equations of motion. Systems modeled in second-
order form can also be transformed to first-order form, also called state variable
form. State variable representations of the differential equations enable alternative
methods of solving for the time response of the system. Although the two methods will
yield identical results when solved properly, it is generally agreed that second-order
representations yield physical insight into solution of the differential equations, while
state variable methods are generally superior when analyzing feedback control. In the
final section of the chapter we discussed impedance methods for system analysis.
Although impedance methods are limited by the fact that they are used only for
frequency-domain analysis, they do yield physical insight into the solution of a number
of problems relevant for analysis of smart material systems.
PROBLEMS
3.1. The equations of equilibrium for a system in static equilibrium are


10 −1 −2
−1 15 −4
−2 −4 12




v1
v2
v3

 =


1 0
0 1
1 −1


(
f1
f2
)
.
(a) Compute the static response v for this system of equations.
(b) Compute the input–output relationship for the output y = [1 −1 2]v.
ch03 JWPR009-LEO July 19, 2007 18:44
PROBLEMS 119
3.2. The state equations of equilibrium for the system studied in Example 2.8 is
(
f1
f2
)
= k
[
1 + α −α
−α 1 + α
](
u1
u2
)
.
Determine the expressions for the static response of the system and analyze the
response as α → 0 and α → ∞. Relate this analysis to the system drawing in
Figure 2.18.
3.3. The equations of motion for a mass–spring system are
[
2 0
0 1
](
ü1(t)
ü2(t)
)
+
[
5 −2
−2 3
](
u1(t)
u2(t)
)
=
[
1
2
]
f.
(a) Compute the eigenvalues and eigenvectors of the system.
(b) Compute the response u1(t) and u2(t) to a unit step input at time zero.
(c) Plot the results from part (b).
3.4. The equations of motion for a mass-spring system are


2 0 0
0 3 0
0 0 2




ü1(t)
ü2(t)
ü3(t)

+


10 −1 −2
−1 5 −1
−2 −1 3




u1(t)
u2(t)
u3(t)

 =


0
0
0

 f (t).
(a) Compute the eigenvalues and eigenvectors of the system.
(b) Compute the response of each coordinate to an initial condition u(0) =
[1 2 −1]′.
3.5. Repeat Problem 3 including a damping matrix in which each modal coordinate
has a damping ratio of 0.01, 0.05, and 0.30. Plot the results.
3.6. Repeat Problem 4 including a damping matrix in which each modal coordinate
has a damping ratio of 0.01, 0.05, and 0.30. Plot the results.
3.7. Transform the second-order equations from Problem 3 into first-order form.
3.8. The first-order equations that govern a system are
(
ż1(t)
ż2(t)
)
=
[
−3 1
0 −1
](
z1(t)
z2(t)
)
+
[
−2
1
]
f (t).
(a) Solve for the response to the initial condition z(0) = [1 −1]′ when f (t) =
0.
(b) Solve for the state response when the input is a unit step at time zero and
the initial conditions are zero.
ch03 JWPR009-LEO July 19, 2007 18:44
120 MATHEMATICAL REPRESENTATIONS OF SMART MATERIAL SYSTEMS
v1
i1
v2C
R
+
–
i2
RL
Figure 3.7 Passive resistive–capacitive circuit.
(c) Solve for the frequency response when y(t) = [1 1].
(d) Solve for the impulse response function of the output.
3.9. The first-order equations that govern a system are
(
ż1(t)
ż2(t)
)
=
[
0 1
−10 −1
](
z1(t)
z2(t)
)
+
[
0
1
]
f (t).
(a) Solve for the response due to the initial conditions z(0) = [1 −1]′ when
f (t) = 0.
(b) Solve the the state response when the input is a unit step at time zero and
the initial conditions.
(c) Solve for the frequency response when y(t) = [1 1].
(d) Solve for the impulse response function of the output.
3.10. (a) Determine the impedance matrix for the system shown in Figure 3.7.
(b) Determine the impedance v1/ i1 when the resistor RL is connected across
the terminal at v2.
(c) Determine the impedance v1/ i1 for the limits RL → 0 and RL → ∞. Re-
late these conditions to the zero-force and zero-flux conditions.
3.11. The impedance function for a mechanical system is
Zm( j�) = 1 − �
2 + j2ζ�
j�
,
where � = ω/ωn is the ratio of the driving frequency to the excitation
frequency.
(a) Determine the average power at � = 0.1, 1 and 10.
(b) Determine the magnitude of the reactive power at � = 0.1, 1 and 10.
ch03 JWPR009-LEO July 19, 2007 18:44
NOTES 121
NOTES
References on second-order systems are described in any number of textbooks on
vibrations and vibration analysis. A general text on the engineering analysis of
vibrating systems is that of Inman [17]. This book includes derivations of the response
functions for single oscillators with various damping mechanisms and a discussion
of multiple-degree-of-freedom systems. Details on the mathematical foundations of
vibration analysis can also be found in a number of texts, such as that of Meirovitch
[18] and his more recent text [19]. These books derive the relationships between
eigenvalue problems and the solution of systems of equations that represent vibrating
systems.
The foundations of first-order analysis are also discussed in a number of textbooks
on systems theory and control theory. A good overview of systems theory is that of
Chen [20], which contains the mathematical foundations of eigenvalue analysis for
first-order systems. Others who discuss linear system theory as it relates to control
are Frieldland [21] and Franklin et al. [22]. Their books include examples of trans-
forming equations of motion into first-order form for dynamic analysis. Research on
impedance analysis is fading fast since the inception of modern computer-aided design
techniques, but Beranek [23] presented very clear examples of how to use systems
theory and impedance analyses to understand the response of dynamic systems.