Vibrations Fundamentals and Practice Appa

Vibrations Fundamentals and Practice Appa Maintaining the outstanding features and practical approach that led the bestselling first edition to become a standard textbook in engineering classrooms worldwide, Clarence de Silva''s Vibration: Fundamentals and Practice, Second Edition remains a solid instructional tool for modeling, analyzing, simulating, measuring, monitoring, testing, controlling, and designing for vibration in engineering systems. It condenses the author''s distinguished and extensive experience into an easy-to-use, highly practical text that prepares students for real problems in a variety of engineering fields.

de Silva, Clarence W “Appendix A”

Vibration: Fundamentals and Practice

Clarence W de Silva

Boca Raton: CRC Press LLC, 2000

Appendix A Dynamic Models and Analogies

A system may consist of a set of interacted components or elements, each possessing an input-output (or cause-effect, or causal) relationship A dynamic system is one whose response variables are functions of time, with non-negligible rates of changes A more formal mathematical definition can be given, but it is adequate here to understand that vibrating systems are dynamic systems

A model is some form of representation of a practical system An analytical model (or mathematical model) comprises a set of equations, or an equivalent, that approximately represents the system Sometimes, a set of curves, digital data (table) stored in a computer, and other numerical data — rather than a set of equations — can be termed an analytical model if such data represent the system

of interest A model developed by applying a suitable excitation to a system and measuring the resulting response of the system is called an experimental model In general, then, models can be grouped into the following categories:

1 Physical models (prototypes)

2 Analytical models

3 Computer (numerical) models

4 Experimental models (using input/output experimental data) Mathematical definitions for a dynamic system are given with reference to an analytical model of the system; for example, a state-space model In that context, the system and its analytical model are synonymous In reality, however, an analytical model, or any model for that matter, is an idealization of the actual system Analytical properties that are established and results that are derived would be associated with the model rather than the actual system, whereas the excitations are applied to and the output responses are measured from the actual system This distinction should

be clearly recognized

Analytical models are very useful in predicting the dynamic behavior (response) of a system when it is subjected to a certain excitation (input) Vibration is a dynamic phenomenon and its analysis, practical utilization, and effective control require a good understanding of the vibrating system A recommended way to achieve this is through the use of a suitable model of the system

A model can be employed for designing a mechanical system for proper vibration performance

In the context of vibration testing, for example, analytical models are commonly used to develop test specifications, and also the input signal applied to the shaker, and to study dynamic effects and interactions in the test object, the shaker table, and their interfaces In product qualification by analysis, a suitable analytical model of the product replaces the test specimen In vibration control,

a dynamic model of the vibrating system can be employed to develop the necessary control schemes (e.g., model-based control)

Analytical models can be developed for mechanical, electrical, fluid, and thermal systems in a rather analogous manner because some clear analogies are present among these four types of systems In practice, a dynamic system can exist as a combination of two or more of the these various types, and is termed a mixed system In view of the analogy, then, a unified approach can

be adopted in analysis, design, and control of these different types of systems and mixed systems

In this context, understanding the analogies in different system types is quite useful in the devel-opment and utilization of models This appendix outlines some basic concepts of modeling Also, fundamental analogies that exist among different types of dynamic systems are identified

A.1 MODEL DEVELOPMENT

There are two broad categories of models for dynamic systems: lumped-parameter models and continuous-parameter models Lumped-parameter models are more commonly employed than continuous-parameter models, but continuous-parameter elements sometimes are included in oth-erwise lumped-parameter models in order to improve the model accuracy

In lumped-parameter models, various characteristics in the system are lumped into represen-tative elements located at a discrete set of points in a geometric space A coil spring, for example, has a mass, an elastic (spring) effect, and an energy-dissipation characteristic, each of which is distributed over the entire coil In an analytical model, however, these individual distributed char-acteristics can be approximated by a separate mass element, a spring element, and a damper element, which are interconnected in some parallel-series configuration, thereby producing a lumped-param-eter model

Development of a suitable analytical model for a large and complex system requires a systematic approach Tools are available to aid this process Signals (excitations and response) can be repre-sented either in the frequency domain or in the time domain A time-domain model consists of a set of differential equations In the frequency domain, a model is represented by a set of transfer functions (or frequency-response functions) Frequency-domain transfer function considerations also lead to the concepts of mechanical impedance, mobility, and transmissibility

The process of modeling can be made simple by following a systematic sequence of steps The main steps are summarized below:

1 Identify the system of interest by defining its purpose and system boundaries

2 Identify or specify the variables of interest These include inputs (forcing functions or excitations) and outputs (responses)

3 Approximate (or model) various segments (or processes or phenomena) in the system

by ideal elements, suitably interconnected

4 Draw a free-body diagram for the system with suitably isolated components

5 a Write constitutive equations (physical laws) for the elements

b Write continuity (or conservation) equations for through variables (equilibrium of forces at joints; current balance at nodes, etc.)

c Write compatibility equations for across (potential or path) variables These are loop equations for velocities (geometric connectivity), voltages (potential balance), etc

d Eliminate auxiliary (unwanted) variables that are redundant and not needed to define the model

6 Express system boundary conditions and response initial conditions using system vari-ables

These steps should be self-explanatory or integral with the particular modeling techniques

A.2 ANALOGIES

Analogies exist among mechanical, electrical, hydraulic, and thermal systems The basic system elements can be divided into two groups: energy-storage elements and energy-dissipation elements

Table A.1 shows the linear relationships that describe the behavior of translatory mechanical, electrical, thermal, and fluid elements These relationships are known as constitutive relations In particular, Newton’s second law is considered the constitutive relation for the mass element The analogy used in Table A.1 between mechanical and electrical elements is known as the force–current analogy This analogy appears more logical than a force–voltage analogy, as is clear from Table A.2 This follows from the fact that both force and current are through variables, which are analogous

to fluid flow through a pipe; and both velocity and voltage are across variables, which vary across

the flow direction, as in the case of pressure The correspondence between the parameter pairs given in Table A.2 follows from the relations in Table A.1 Note that the rotational mechanical elements possess constitutive relations between torque and angular velocity, which can be treated

as a generalized force and a generalized velocity, respectively In fluid systems as well, basic elements corresponding to capacitance (capacity), inductance (fluid inertia), and resistance (fluid friction) exist Constitutive relations between pressure difference and mass flow rate can be written for these elements In thermal systems, generally only two elements — capacitance and resistance

— can be identified Constitutive relations exist between temperature difference and heat transfer rate in this case

Proper selection of system variables is crucial in developing an analytical model for a dynamic system A general approach that can be adopted is to use across variables of the A-type (or across-type) energy storage elements and the through variables of the T-type (or through-type) energy

TABLE A.1 Some Linear Constitutive Relations

System Type

Constitutive Relation for

Energy Storage Elements

Energy Dissipating Elements

A-Type (Across) Element

T-Type (Through) Element

D-Type (Dissipative) Element

Translatory mechanical:

v = Velocity

f = Force

Mass:

(Newton’s second law)

m = Mass

Spring:

(Hooke’s law)

k = Stiffness

Viscous damper:

f = bv

b = Damping constant

Electrical:

v = Voltage

i = Current

Capacitor:

C = Capacitance

Inductor:

L = Inductance

Resistor:

Ri = v

R = Resistance

Thermal:

T = Temperature difference

Q = Heat transfer rate

Thermal capacitor:

C t = Thermal capacitance

None Thermal Resistor:

R t Q = T

Rt = Thermal resistance

Fluid:

P = Pressure difference

Q = Volume flow rate

Fluid capacitor:

C f = Fluid capacitance

Fluid inertor:

I f = Inertance

Fluid resistor:

R f Q = P

R f = Fluid resistance

TABLE A.2 Force–Current Analogy

System Type Mechanical Electrical

System-response variables:

Through variables Force f Current i

Across variables Velocity v Voltage v

m dv

dt =kv

C dv

dt=v

C dT

dt Q

C dP

dt Q

dt P

storage elements as system variables (state variables) Note that if any two elements are not inde-pendent (e.g., if two spring elements are directly connected in series or parallel), then only a single state variable should be used to represent both elements Independent variables are not needed for

D-type (dissipative) elements because their response can be represented in terms of the state variables

of the energy storage elements (A-type and T-type)

A.3 MECHANICAL ELEMENTS

Here, one uses velocity (across variable) of each independent mass (A-type element) and force (through variable) of each independent spring (T-type element) as system variables (state variables) The corresponding constitutive equations form the “shell” of an analytical model These equations will directly lead to a state-space model of the system

A.3.1 M ASS (I NERTIA ) E LEMENT

Constitutive equation (Newton’s second law):

(A.1) Since power = fv, the energy of the element is given by

or

(A.2) This is the well-known kinetic energy Now, integrating equation (A.1),

(A.3)

By setting t = 0+, one sees that

(A.4) unless an infinite force f is applied to m Hence, one can state the following:

1 Velocity can represent the state of an inertia element This is justified first because, from equation (A.3), the velocity at any time t can be completely determined with the knowl-edge of the initial velocity and the applied force, and because, from equation (A.2), the energy of an inertia element can be represented in terms of v alone

2 Velocity across an inertia element cannot change instantaneously unless an infinite force/torque is applied to it

3 A finite force cannot cause an infinite acceleration A finite instantaneous change (step)

in velocity will need an infinite force Hence, v is a natural output (or state) variable and

f is a natural input variable for an inertia element

m dv

dt = f

Energy E=1mv

2 2

t

( )= ( )− +

−

∫

0

v( )0+ =v( )0−

A.3.2 S PRING (S TIFFNESS ) E LEMENT

Constitutive equation (Hook’s law):

(A.5)

Note that the conventional force-deflection Hooke’s law has been differentiated in order to be consistent with the variable (velocity) that is used with the inertia element

As before, the energy is

or

(A.6)

This is the well-known (elastic) potential energy

Also,

(A.7)

and hence,

(A.8) unless an infinite velocity is applied to the spring element In summary,

1 Force can represent the state of a stiffness (spring) element This is justified because the force of a spring at any general t can be completely determined with the knowledge of the initial force and the applied velocity, and also because the energy of a spring element can be represented in terms of f alone

2 Force through a stiffness element cannot change instantaneously unless an infinite veloc-ity is applied to it

3 Force f is a natural output (state) variable and v is a natural input variable for a stiffness element

A.4 ELECTRICAL ELEMENTS

Here, one uses voltage (across variable) of each independent capacitor (A-type element) and current (through variable) of each independent inductor (T-type element) as system (state) variables

A.4.1 C APACITOR E LEMENT

Constitutive equation:

(A.9)

df

dt =kv

k

df

dt dt k fdf

Energy E f

k

=1

2 2

t

( )= ( )− +

−

∫

0 0

f( )0+ = f( )0−

C dv

dt =i

Since power is iv, the energy is

or

(A.10)

This is the electrostatic energy of a capacitor

Also,

(A.11)

Hence, for a capacitor,

(A.12) unless an infinite current is applied to a capacitor In summary,

1 Voltage is an appropriate response variable (or state variable) for a capacitor element

2 Voltage across a capacitor cannot change instantaneously unless an infinite current is

applied

3 Voltage is a natural output variable and current is a natural input variable for a capacitor

A.4.2 I NDUCTOR E LEMENT

Constitutive equation:

(A.13)

(A.14)

This is the electromagnetic energy of an inductor

Also,

(A.15)

Hence, for an inductor,

(A.16) unless an infinite voltage is applied In summary,

Energy E= 1Cv

2 2

C idt

t

( )= ( )− +

−

∫

0

v( )0+ =v( )0−

L di

dt=v

Energy E=1Li

2 2

i t i

t

( )= ( )− +

−

∫

0

i( )0+ =i( )0−

1 Current is an appropriate response variable (or state variable) for an inductor.

2 Current through an inductor cannot change instantaneously unless an infinite voltage is applied

3 Current is a natural output variable and voltage is a natural input variable for an inductor

A.5 THERMAL ELEMENTS

Here, the across variable is temperature (T) and the through variable is the heat transfer rate (Q) The thermal capacitor is the A-type element There is no T-type element in a thermal system The

reason is clear There is only one type of energy (thermal energy) in a thermal system, whereas there are two types of energy in mechanical and electrical systems

A.5.1 T HERMAL C APACITOR

Consider a material control volume V, of density ρ, and specific heat c Then, for a net heat transfer rate Q into the control volume, one obtains

(A.17) or

(A.18)

where C t = ρvc is the thermal capacitance of the control volume.

A.5.2 T HERMAL R ESISTANCE

There are three basic processes of heat transfer:

1 Conduction

2 Convection

3 Radiation

There is a thermal resistance associated with each process, given by their constitutive relations as given below

where

k = conductivity

A = area of cross section of the heat conduction element

∆x = length of heat conduction with a temperature drop of T.

The conductive resistance is

(A.20)

dt

= ρ

C dT

t =

x T

=

∆

kA

k = ∆

Convection: (A.21)

where

h c = convection heat transfer coefficient

A = area of heat convection surface with temperature drop of T.

The convective resistance is

(A.22)

where

σ = Stefan-Boltzmann constant

F E = effective emmisivity of the radiation source (of temperature T1)

F A = shape factor of the radiation receiver (of temperature T2)

A = effective surface area of the receiver

This corresponds to a nonlinear thermal resistor

A.6 FLUID ELEMENTS

Here, one uses pressure (across variable) of each independent fluid capacitor (A-type element) and volume flow rate (through variable) of each independent fluid inertor (T-type element) as system

(state) variables

A.6.1 F LUID C APACITOR

The heat transfer rate is

(A.24)

Note that a fluid capacitor stores potential energy (a “fluid spring”) unlike the mechanical A-type

element (inertia), which stores kinetic energy

For a liquid control volume V of bulk modulus β, the fluid capacitance is given by

(A.25)

For an isothermal (constant temperature, slow-process) gas of volume V and pressure P, the fluid

capacitance is

(A.26)

Q=h AT c

R

h A

c c

Q=σF F A T E A ( 1 −T )

4 2 4

f =

Cbulk=V β

P

comp=

For an adiabatic (zero heat transfer, fast process) gas, the capacitance is

(A.27)

where

(A.28)

which is the ratio of specific heats at constant pressure and constant volume

For an incompressible fluid in a container of flexible area A and stiffness k, the capacitance is

(A.29)

Note: For a fluid with bulk modulus, the equivalent capacitance would be

For an incompressible fluid column with an area of cross section A and density ρ, the capacitance is

(A.30)

A.6.2 F LUID I NERTOR

(A.31)

This represents a T-type element However, it stores kinetic energy, unlike the mechanical T-type

element (spring), which stores potential energy For a flow with uniform velocity distribution across

an area A and over a length segment ∆x, the fluid inertance is given by

(A.32)

For a non-uniform velocity distribution,

(A.33)

where a correction factor α has been introduced For a flow of circular cross section with a parabolic velocity distribution, use α = 2.0

kP

comp=

c

p v

=

k

elastic= 2

Cbulk+Celastic

g

grav = ρ

I dQ

f =

A

f = ρ ∆

A

f = αρ ∆