Solutions Manual Modeling and
Analysis of Dynamic Systems 3rd
edition ByRamin S. Esfandiari, Bei Lu
1. What are the purposes of modeling, and what can we do with a model of a dynamic
system? - answer-1. Modeling allows us to predict the behavior of a system before it is built.
It allows us to check and see if the system is safe.
2. What is the functional definition of linearity? What is a linear equation? What is a linear
differential equation? - answer-1. evaluation of the function with a scalar multiple of the
independent variable x must equal the same multiple times the function of x; function using the
sum of two values of the independent variable must equal the sum of the separate functions of
the 2 values of the independent variable. 2. A linear equation is formed by a linear combination
of the variables and their derivatives. 3. A linear differential equation is formed by a linear
combination of the derivatives of the system variables.
3. What are the so-called elementary inputs that are used as inputs to dynamic systems? Are
there other typical inputs? - answer-elementary inputs are step input and ramp input; other
typical inputs are the step ramp, pulse, impulse, and harmonic or sinusoidal; these are
derivatives or integrals of the elementary inputs
4. The solution to a non-homogeneous differential equation is composed of two parts. What are
they, and how are they found? - answer-the homogeneous solution and the particular solution.
After summing both, solve for A as usual using initial conditions.
5. Discuss the concept of the time constant of a 1st order differential equation. How can the
time constant be determined from the linear differential equation? How can the time constant
be determined from the time response of the system? - answer-The time constant of the
1
, system relates to the amount of time it will take the system to settle to steady state. Time
constant can be determined by normalizing the equation by the lowest ordered coefficient,
then Tau is equal to the coefficient of the derivative term. The time constant can be determined
by the time response by dividing the steady rate response by 4, because at 4 time constants the
system will reach 98% of its steady state operation.
6. There are two performance factors for a first-order differential equation, what are they?
Explain what each factor means in terms of the response of the system. - answer-The two
factors are; time constant(tau) and static gain. The time constant is a multiple of the settling
time. Static gain is the multiple of the input that represents steady state.
7. There are three performance factors for a second-order differential equation. What are they?
Explain what each factor means in terms of the response of the system. - answer-The three
performance factors are; Static gain, natural frequency, and damping ratio. The static gain is the
multiple of the input that represents steady state. The natural frequency is the frequency of
oscillation, if there were no damping. The damping ratio is a dimensionless value of the ratio
between oscillatory and non-oscillatory behaviors of the system.
8. 8. What are the characteristics of an un-damped, an under-damped, a critically damped, and
an over-damped system? - answer-Un-damped: (zeta=0) Response: no dampening occurs,
system free to oscillate for an infinite time.
Under-damped: (zeta<1) Response: decaying oscillatory response, eventually will die out.
Critically-damped: (zeta=1) Response: Exponential and decreasing; physically impossible to
build.
Over-damped: (zeta>1) Response: Exponential and decreasing, system is stable.
9. Explain what is meant by the systems approach in the treatment of dynamic systems. -
answer-Perception of problem, identification of variable, any assumptions made, combining
component equation onto model, analysis of response.
10a. What is an effort variable? - answer-The effort variable expresses effort placed on a
2
Analysis of Dynamic Systems 3rd
edition ByRamin S. Esfandiari, Bei Lu
1. What are the purposes of modeling, and what can we do with a model of a dynamic
system? - answer-1. Modeling allows us to predict the behavior of a system before it is built.
It allows us to check and see if the system is safe.
2. What is the functional definition of linearity? What is a linear equation? What is a linear
differential equation? - answer-1. evaluation of the function with a scalar multiple of the
independent variable x must equal the same multiple times the function of x; function using the
sum of two values of the independent variable must equal the sum of the separate functions of
the 2 values of the independent variable. 2. A linear equation is formed by a linear combination
of the variables and their derivatives. 3. A linear differential equation is formed by a linear
combination of the derivatives of the system variables.
3. What are the so-called elementary inputs that are used as inputs to dynamic systems? Are
there other typical inputs? - answer-elementary inputs are step input and ramp input; other
typical inputs are the step ramp, pulse, impulse, and harmonic or sinusoidal; these are
derivatives or integrals of the elementary inputs
4. The solution to a non-homogeneous differential equation is composed of two parts. What are
they, and how are they found? - answer-the homogeneous solution and the particular solution.
After summing both, solve for A as usual using initial conditions.
5. Discuss the concept of the time constant of a 1st order differential equation. How can the
time constant be determined from the linear differential equation? How can the time constant
be determined from the time response of the system? - answer-The time constant of the
1
, system relates to the amount of time it will take the system to settle to steady state. Time
constant can be determined by normalizing the equation by the lowest ordered coefficient,
then Tau is equal to the coefficient of the derivative term. The time constant can be determined
by the time response by dividing the steady rate response by 4, because at 4 time constants the
system will reach 98% of its steady state operation.
6. There are two performance factors for a first-order differential equation, what are they?
Explain what each factor means in terms of the response of the system. - answer-The two
factors are; time constant(tau) and static gain. The time constant is a multiple of the settling
time. Static gain is the multiple of the input that represents steady state.
7. There are three performance factors for a second-order differential equation. What are they?
Explain what each factor means in terms of the response of the system. - answer-The three
performance factors are; Static gain, natural frequency, and damping ratio. The static gain is the
multiple of the input that represents steady state. The natural frequency is the frequency of
oscillation, if there were no damping. The damping ratio is a dimensionless value of the ratio
between oscillatory and non-oscillatory behaviors of the system.
8. 8. What are the characteristics of an un-damped, an under-damped, a critically damped, and
an over-damped system? - answer-Un-damped: (zeta=0) Response: no dampening occurs,
system free to oscillate for an infinite time.
Under-damped: (zeta<1) Response: decaying oscillatory response, eventually will die out.
Critically-damped: (zeta=1) Response: Exponential and decreasing; physically impossible to
build.
Over-damped: (zeta>1) Response: Exponential and decreasing, system is stable.
9. Explain what is meant by the systems approach in the treatment of dynamic systems. -
answer-Perception of problem, identification of variable, any assumptions made, combining
component equation onto model, analysis of response.
10a. What is an effort variable? - answer-The effort variable expresses effort placed on a
2
