ME3253 – Linear System Theory
Prof. R. Gao
Course Review (1)
Chapters 1-2. Translational Mechanical Systems
1) Free-body diagrams involving mass, damper, spring, ideal pulley, and junctions;
2) Serial and parallel combinations of dampers and springs;
3) D’Alembert’s law for representing inertial force in a moving object:
 Using a dashed arrow, applied in opposite direction to the assumed direction of body
motion.
4) Choice of reference positions when setting up system models using displacement:
 Fixed, with respect to a non-moving point (e.g. a wall).
 Relative, with respect to another moving body (e.g. z as the relative motion between
two bodies, see Example 2.4).
5) When modeling using relative displacements, note that inertial forces are always
æ·· ··ö
proportional to the absolute acceleration, e.g. çç x + z ÷÷÷ , instead of the relative acceleration
è
ø
with respect to another moving body. In comparison, frictional forces are expressed in
terms of the relative motion of the two bodies involved (see Example 2.4).
6) Treatment of the gravitational force (Mg) and its effect in modeling of systems moving in the
vertical direction:
 Chose static equilibrium position as the reference point for zero displacement, instead
of the position when springs are neither stretched nor compressed. As a result, the
term Mg will not appear in the modeling equations (see Examples 2.5-2.6).
7) The effect of pulley when added to a translational system:
 A pulley only changes the direction of force applied to the bodies that are attached to
the cable around the pulley.
 A cable around a pulley can’t stretch or push apart bodies attached to its ends; it
would buckle and be detached from the pulley (see Example 2.7).
Chapter 3. State Variable and Input-Output Equations for System Models
1) General concept of state variables (SVs):
 Sufficient to describe the effect of past history of the system on its response in the
future  knowing values of SVs at t0 and system inputs at t ≥ t0, we can evaluate
both SVs and system’s outputs for all t ≥ t0.
 SV equations are first-order differential equations, preferably with no differential
terms on the right side of the equations.
2) Choice of state variables:
Prof. R. Gao
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 Sate variables chosen must be independent from each other.
 Usually related to the energy stored in energy-storing elements, e.g. masses (kinetic
energy) and springs (potential energy).
3) Standard form of state variable equations (input and output);
 Multiple SV equations, each of which is a first-order differential equation.
 The SV equations must be solved as a group instead of separately.
 A single output equation, which is an algebraic function of the SVs and inputs.
·
··
·
 Revised free-body diagrams (FBDs): use M v and Bv instead of M x and B x to
express forces in terms of SVs and inputs. (see pp. 48-50).
4) Examples of using SVs for system modeling:
 Examples 3.1 – 3.2: use absolute displacements and velocities as the SVs.
 Example 3.3: use relative displacement and velocity as the SVs.
 Example 3.4: involving nonlinear spring and damper.
 Example 3.5: when # of SVs and # energy-storing elements is not identical. An
additional SV not related to energy storage is needed.
 Example 3.6: when # of SVs is less than # of energy-storing elements.
 Example 3.7: when massless junctions are involved.
 Example 3.8: elimination of derivatives on the right side of the SV equations.
5) General concept of Input-Output equations (I/O Eqs.)
 One single differential equation containing only the inputs and outputs and their
derivatives related to the system being modeled, with all other variables eliminated.
 Of higher order than each of the SV equations (one nth-order vs. n-1st order DEQs).
 Computationally less efficient than SV equations.
 Can be constructed by combining the equations in a SV model.
6) The p-operator method for reducing simultaneous differential equations:
 p denotes the differential operator d/dt.
 p is not a variable or algebraic quantity itself.
 p helps simplify the process of combining multiple DEQs by converting them into
algebraic Eqs.
7) Matrix formulation of state variable equations (refer to recitations and lab sessions).
Chapter 5. Rotational Mechanical Systems
1) Free-body diagrams involving angular displacement/velocity/acceleration, moment of
inertia, friction, stiffness, and torque.
2) Fundamental concepts related to levers and gears (e.g. gear ratio, and the treatment of
internal force fc in the modeling process).
3) D’Alembert’s law for representing inertial torque.
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4) Similarities between translational and rotational systems in terms of setting up systems’
models using SVs (generally use  and ω for rotational systems, similar to using x and v for
translational systems).
5) Modeling of rotational system involving rotating disks and rods (Examples 5.1-5.5).
6) Modeling of systems containing levers and pendulum (Examples 5.6- 5.8).
7) Modeling of systems containing levers and pendulum (Examples 5.9- 5.10).
8) Modeling of systems combining translational and rotational motions, e.g. through the use of
rack and pinion gears (Example 5.11).
9) Modeling of systems combining translational and rotational motions, e.g. through the use of
rack and pinion gears (Example 5.11), or a pulley (Example 5.12).
Chapter 7. Transform Solution of Linear Models
1) Physical significance and definition of the Laplace transform, time and complex-frequency
domains;
2) Laplace transform of several commonly used, basic functions: step, exponential, ramp,
trigonometric, and rectangular functions;
3) Transform properties: superposition, differentiation, multiplication, and integration (page
192-197);
4) General first-order system model, and its solution using Laplace transform;

y
Y ( s) 
1

y  f (t )
y (0)
A

1
1

s
s s  

 
y (t )   A  [ y (0)   A] e  
t
5) The concept of zero-input response yzi(t) and zero-state response yzs(t);
6) The concept of transient response ytr(t) and steady-state response yss(t) (page 200-201, and
Example.7.2-7.3).
7) System solution involving initial conditions (page 203-207).
8) First-order system response to step and impulse inputs (page 207-214);
9) Partial-fraction expansion technique for obtaining transform inversion:
N ( s)
b s m  ...  b0
F ( s) 
 n m n 1
, with D ( s ) expressed as
D ( s ) s  an 1s  ...  a0
D ( s )  ( s  s1 )( s  s2 )...( s  sn )
10) The concept and technique of determining poles: distinct, repeated, and complex (page 221228, Examples 7.9 – 7.12)
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Chapter 8: Transfer Function Analysis
1) Review the general procedure for modeling 2nd-order or higher-order dynamic systems
(Examples 8.1 and 8.4~8.5);
2) Zero-Input response:


Defined as a system’s response y(t) to nonzero initial conditions, under zero input u(t);
General form of the input-output differential equation:

an y ( n )  an 1 y ( n 1)  ...  a1 y  a0 y  0
F (s)
;
P( s)

Laplace transform of the zero-input response : Y ( s ) 

The general expressions of the numerator F(s) and denominator P(s) of Y(s),
respectively;
How the poles s1, s2, … sn are reflected in the Laplace transformed output.

3) Standard expression of 2nd-Order systems, poles s1 and s2, the characteristic equation, and
the corresponding response of the system, y(t), in the time-domain;
4) Analysis of the 2nd-order system’s behavior in terms of stability by examining the specific
location of the roots (of the characteristic equation) in the complex plane (Figs. 8.5~8.8).



Configuration of the complex plane;
Stable, unstable, and marginally stable responses;
Simple harmonic oscillator.
5) Definition of Damping Ratio (), range of its values, what it means physically, and how it is
expressed in the standard form of the characteristic equation (CE) of a 2nd-order system,
given by:
s 2  2n s  n 2  0
and how it is expressed in the roots of the characteristic equation, s1 and s2.
6) Property of stable systems, mathematically and physically (page 256-257);
7) Effect of damping ratio  on the system’s zero-input response for different  values. Critical
damping, undamped system response (Fig. 8.10-8.11).
8) Definition of Undamped Natural Frequency (n), its physical significance, and how it is
related to .
9) Location of the characteristic roots in terms of n and  on the complex plane (Fig. 8.10,
Examples 8.7~8.8).
10) Zero-State response:


Defined as a system’s response y(t) to nonzero input u(t), under zero initial conditions;
General form of the input-output differential equation and its Laplace transform:


an y ( n )  an 1 y ( n 1)  ...  a1 y  a0 y  bm u ( m )  bm 1u ( m 1)  ...  b1 u  b0u (t )
Prof. R. Gao
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(an s n  an 1s n 1  ...  a1s  a0 ) Y ( s )  (bm s m  bm 1s m 1  ...  b1s  b0 )U ( s )

Laplace transform of the zero-response response: Y ( s )  H ( s )  U ( s ) .
bm s m  ...  b0
Y ( s) 
an s n  an 1s n 1  ... a1s  a0

General expression of the system’s transfer function H(s) and its physical
significance.
H ( s) 

Y ( s)
bm s m  ...  b0

U ( s ) an s n  an 1s n 1  ... a1s  a0
How to express the system’s input-output differential equation based on the factored
form of the transfer function H(s) (page 265-266):
H ( s)  K
( s  z1 )( s  z2 ) ... ( s  zm )
( s  p1 )( s  p2 ) ... ( s  pn )

The factored form of transfer function containing the zeros (z1, z2, … zn) and poles (p1,
p2, … pn), in polynomials in s (Example 8.10).

The concept of transient and steady-state response and how to identify the
corresponding components (Example 8.11).

Impulse

Step response: response of a dynamic system to step input Y ( s )  H ( s ) 
response: response of
Y ( s )  H ( s ) (Examples 8.12~13).
a
dynamic
system
to
impulsive
input
1
(Examples
s
8.14~15).
11) The central role that Transfer Function H(s) plays in determining system’s response
(Figure 8.21).
Prof. R. Gao
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Fall 2011