MATHEMATICAL REPRESENTATIONS
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
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(¨r,r,˙ 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
M¨r(t)+Kr(t)=Bff(t), (3.8) where M is called themass matrixfor 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
M¨r(t)+Dvr(t)˙ +Kr(t)=Bff(t), (3.9) where Dvis theviscous damping matrixfor 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
mr(t¨ )+kr(t)= fof(t), (3.10) where forepresents the amplitude of the time-dependent force f(t). The solution is generally found after normalizing the system to the mass,
¨ r(t)+ k
mr(t)= fo
m f(t). (3.11)
The ratio of the stiffness to the mass is denoted k
m =ωn2 (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)=Asin(ωnt+φ), (3.13)
where
A= 1 ωn
ωn2r(0)2+r˙(0)2
(3.14) φ=tan−1ωnr(0)
r(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−cosωnt) r(0)=r(0)˙ =0 Harmonic: r(t)=r(0)˙
ωn
sinωnt+
r0(0)− f0m ωn2−ω2
cosωnt+ f0/m ω2n−ω2 cosωt
(3.15) The general solution to a forcing input is defined in terms of theconvolution integral,
r(t)= f0
mωn
t 0
f(t−τ) sinωnτdτ. (3.16) When viscous damping is present in a system, the mass normalized equations of motion are
¨
r(t)+2ζωnr(t˙ )+ω2nr(t)= fo
m f(t). (3.17)
The variable ζ is the damping ratioof 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 ofζ. 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−ζωntsin(ωdt+φ) (3.18) where
ωd =ωn
1−ζ2
A=
(˙r(0)+ζωnr(0))2+(r(0)ωd)2 ωd2
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 terme−ζωnt. The rate of decay will increase asζ becomes larger. Asζ →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−ζ2e−ζ ωntcos(ωdt−φ) φ=tan−1 ζ 1−ζ2 Harmonic: r(t)=xcos(ωt−θ)
x= f0m
(ω2n−ω2)2+(2ζωnω)2
θ=tan−1 2ζωnω
ωn2−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−ζωnτsinωdτdτ. (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)=Vejωt, (3.21)
whereVis a vector of unknown coefficients. Substituting equation (3.21) into equa- tion (3.8) yields
K−ω2M
Vejωt=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 asymmetric eigenvalue problem, which yieldsNveigenvaluesω2niand corresponding eigenvectorsVi. Gen- erally, the eigenvalues are ordered such that ωn1< ωn2<· · ·. The solution to the
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 transformationr(t)=Pη(t) into equation (3.8). The result is
MP ¨η(t)+KPη(t)=Bff(t). (3.25) If the eigenvectors are normalized such thatViMVj =δi j, we can premultiply equa- tion (3.25) by P:
PMP ¨η(t)+PKPη(t)=PBff(t). (3.26) Due to the normalization of the eigenvectors,
PMP=I
PKP==diag
ω2n1, ωn22 , . . .
PBf = (3.27)
and the equations of motion can be written as a set ofuncoupledsecond-order equa- tions:
η¨i(t)+ωni2η(t)=
Nf
j=1
i jfj(t). (3.28)
The term i jis 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 forri(t). Once this is completed, the coordinate transformationr(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
PDP=diag(2ζiωni), (3.29)
the decoupled equations of motion are
η¨i(t)+2ζiωniη˙i+ωni2η(t)=
Nf
j=1
i jfj(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 transformationr(t)=Pη(t).