introduction overheads

CE 804.3

INTRO.DOC 1-1 04/01/01

INTRODUCTION TO STRUCTURAL DYNAMICS

Professor: Bruce SparlingOffice: 3B34Telephone: 966-5366Email: [email protected]

Recommended Texts:

1. “Structural Dynamics: Theory and Computation”, 3rd Edition,by Mario Paz,Van Nostrand Reinhold, 1991.

2. “Dynamics of Structures”, 2nd Edition, by R.W. Clough andJ. Penzien, McGraw-Hill,1975.

3. “Dynamics of Structures”, by W.C. Hurty and M.F.Rubinstein, Prentice-Hall, 1964.

4. “The Dynamical Behaviour of Structures”, 2nd Edition, byG.B. Warburton, Pergammon Press, 1976.

5. “Structural Dynamics by Finite Elements”, by WilliamWeaver Jr. and Paul Johnston, Prentice Hall, 1987.

6. “Vibration of Mechanical and Structural Systems: WithMicrocomputer Applications”, 2nd Edition, by M.L. James,G.M. Smith, J.C. Wolford, and P.W. Whaley, Harper CollinsCollege Publishers, 1994.

7. “Dynamics of Structures: Theory and Applications toEarthquake Engineering”, by Anil K. Chopra, Prentice Hall,1995.

8. “Structural Dynamics: Theory and Applications”, by JosephW. Tedesco, William G. McDougal, C. Allen Ross, Addison-Wesley, 1999.

CE 804 Introduction Page 1-2

Introduction to Structural Dynamics 04/01/01

1. BASIC CONCEPTS AND TERMINOLOGY

• Static structural analysis:

Ø Loads applied slowly

Ø No significant motion of structure

• Time dependence of loads and responses:

Ø Generally invariant with time

Ø May vary slowly

• Static equilibrium:

0,0,0

0,0,0

===

===

∑∑∑∑∑∑

zyx

zyx

MMM

FFF[1.1]

• Structural dynamics:

Ø Time-dependent motion

Ø Significant inertial effects

Ø Nature of motion:

♦ Often oscillatory & periodic

♦ Depends on characteristics of loading and system

• Newton’s Second Law:

α== ∑∑ oIMamF ; [1.2]

where m = Mass of structure [kg]

a = Linear acceleration [m/s2]

Io = Mass moment of inertia [kg-m2]

α = Angular acceleration [rad./s2]



1.1. DYNAMIC FORCES ACTING ON A SIMPLE SYSTEM

Fig. 1.1 Single degree-of-freedom structural system

where m = Mass [kg]

k = Linear spring constant [N/m]

c = Viscous damping coefficient [N-s/m]

F(t) = Time-varying external forceapplied to mass [N]

• Assumptions:

Ø Mass moves in horizontal (x) direction only

Ø Spring is unstretched when x = 0

Ø Mass on frictionless rollers

• Single degree-of-freedom (SDOF) system:

Ø Time-varying position of mass described by single variable

Ø x(t) measured with respect to fixed reference point



• Free Body Diagram (FBD) of mass: (Horizontal forces only)

Fk

FD

( )F t

Fig. 1.2 FBD of SDOF system

where kF = Spring restoring force

DF = Damping force due to dashpot

Ø Note: kF and DF always oppose motion of mass

• Newton’s Second Law:

( ) xmFFtF Dk &&=++ [1.3]

where x&& = Acceleration of mass in x direction

= 2

2

dd

t

x

• Elastic force for linear spring:

xkFk −= [1.4]

Ø Spring acts to return mass to its undeflected position

Ø Assumption of perfect elasticity:

♦ No energy lost to internal friction

♦ Work done by the mass on the spring stored as elasticpotential energy

♦ Recoverable

♦ Path independent



• Dashpot force:

Ø Relies on viscous fluid forces

Ø Proportional to velocity of mass

t

xc

xcFD

dd

−=

−= &

[1.5]

Ø Damping force opposes the motion of mass

Ø Work done by DF :

♦ Dissipative (nonconservative)

♦ Converted to heat or some other form of energy

♦ Can not be recovered by reversing the path of motion

• Dynamic equilibrium equation:

( ) xmxcxktF &&& =−− [1.6]

Ø Static equilibrium exists only when 0=x&&

♦ x& may be non-zero



1.2. D’ALEMBERT’S PRINCIPLE: DYNAMIC EQUILIBRIUM

• Re-statement of Newton’s 2nd Law

• State of dynamic equilibrium defined

• Assume a fictitious inertial force IF :

xmFI &&−= [1.7]

Ø Acts on mass during motion

Ø Direction is opposite to acceleration of the mass

• FBD including the fictitious inertial force:

Fk

FD

( )F tFI

+ + +x x x, & , &&

Fig. 1.3 FBD using D’Alembert’s Principle

• State of dynamic equilibrium:

( ) 0=+++ IDk FFFtF [1.8a]

or ( )tFxkxcxm =++ &&& [1.8b]

Ø Instantaneous sum of elastic, damping, and inertial forcesmust equal external force

Ø Basis for study of structural dynamics

Ø Governing differential equation of motion for SDOF system



1.3. MATHEMATICAL MODELS USED IN STRUCTURAL DYNAMICS

• Two basic types of mathematical models:

Ø Lumped mass models

Ø Distributed mass models

• Lumped mass model:

Ø Simple

Ø More approximate

Ø Mass concentrated at selected locations

♦ Lumped masses treated as particles– no physical size

♦ Remainder of structure is massless

♦ Rotary inertia generally neglected– translation only– no rotation

Ø Easily implemented in Finite Element Method (FEM)

• Distributed mass models:

Ø Mass is continuously distributed along structure

Ø More realistic of physical structure

Ø Generally more accurate

Ø Used in:

♦ Rayleigh-Ritz method

♦ Finite element analysis

• Lumped vs distributed mass models:

Ø Results converge for as number of lumped massesincreases

Ø Refinement depends on required accuracy

m n

m n −1

m 2

m1 ( )η1 t

( )η2 t

( )ηn t−1

( )ηn t



1.4. STRUCTURAL DEGREES OF FREEDOM (DOF)

• The number of independent coordinates required to define theposition of every location on a structure

• Depends on:

Ø Type of model

Ø Degree of refinement

• Lumped mass models

Ø DOF’s associated with lumped masses

Ø Each lumped mass has up to three DOF’s

♦ x, y, z displacements

Ø DOF’s may be reduced by physical constraints or simplifyingassumptions

♦ Example - Shear buildings

δ1

δ2

Fig. 1.4 Two storey shear building



• Structures with distributed mass and stiffness:

Ø Infinite number of degrees of freedom

Ø Position of every point must be specified independently

Ø Analytical simplifications:

♦ Finite number of DOF’s considered

♦ Position specified only at selected locations

♦ Motion of remaining structure inferred using assumeddisplacement patterns

Ø Use of geometric “shape” functions:

♦ DOF’s = Amplitudes of shape functions

Rigid Beam

Mass = [kg / m]m k

c c

(a) Rigid Beam

Flexible Beam: Stiffness =

Mass = [kg / m]

EI

m

(b) Flexible Beam

Fig. 1.5 DOF’s for distributed mass systems.

♦ Part (a): Rigid beam

♦ Part (b): Flexible beam



Ø Consequences of using assumed shape functions:

♦ Approximation errors

♦ Depends on similarity between assumed and actualdeflection pattern

Ø Improved accuracy obtained by expressing deflected shapeby a series of assumed shape functions:

♦ Amplitude of each shape function = 1 DOF

♦ Used in Rayleigh-Ritz analysis method

• Effectiveness of stiffness, damping and mass elements:

Ø Depends on amplitude of motion at element location

Ø A generalized (or effective) stiffness, mass and dampingdefined for each displacement pattern

1.5. TYPES OF DYNAMIC LOADS

• Dynamic response depends on applied loading

Ø Dynamic excitation.

• Dynamic load types:

Ø Free vibration

Ø Periodic excitation

Ø Transient excitation

Ø Random excitation

1.5.1. Free Vibration

• Structural system set in motion initially

• No external dynamic excitation during vibration

Ø F(t) = 0



1.5.2. Periodic Loads

• Load patterns repeated exactly at regular intervals

• Possible sources:

Ø Rotating or reciprocating machinery

Ø Vortex shedding

Ø Waves

• Harmonic loading:

Ø Horizontal or vertical components of a vector rotating at aconstant angular velocity

Ø General form:

( ) ( )φ+ω= tFtF o sin [1.9]

where oF = Amplitude [N]

ω = Circular frequency [rad/s]

φ = Phase angle [rad]

Ø Frequency in Hz, f :

fπ=ω 2 [1.10]

( ) ( )φ+π= tfFtF o 2sin [1.11]

Ø Period of a harmonic function, T :

ωπ

==21

fT [1.12]



1.5.3. Transient Loads

• Non-periodic loading histories of short duration

Ø Known time history

Fig. 1.6 Example transient load history.


Ø Collisions

Ø Impacts & moving equipment

Ø Blast loads

Ø Manufacturing processes

Ø Earthquakes

• Loading histories defined by analytical expression or measureddata

Ø Repeatable & predictable

Ø Deterministic process



1.5.4. Random Loads

• Random processes cannot be predicted accurately

Ø Never repeat themselves exactly

t

F(t

)

F(t)

Fig. 1.7 Example random load due to wind


Ø Wind

Ø Earthquakes

Ø Traffic loads on roadways and bridges

• Quantifying random loads:

Ø Often exhibit well defined characteristics

Ø Quantified on a statistical basis

Ø Characterize all possible loading events with similarstatistical characteristics

♦ Eg. Wind & earthquake loads



1.6. FUNDAMENTAL STRUCTURAL PROPERTIES

1.6.1. Structural Stiffness, k

• Elastic restoring forces:

Ø Strains induced in structural members

Ø Bending, axial contraction or extension, or twisting

Ø Always act to return the structure to undeformed position

• Mathematical model of elastic restoring force:

Ø Spring with stiffness ( spring constant) = k

Ø k = static external force (or moment) required to produce acorresponding unit displacement in the structure

• Elastic restoring force, kF :

Ø Linear behaviour: xFk ∝ for all x

Ø Nonlinear behaviour: ( ) xxkFk −= [1.13]

♦ Hardening spring: kF ↑ at an increasing rate as x ↑

♦ Softening spring: kF ↑ at a decreasing rate as x ↑

k

Fk

x

Fig. 1.8 Linear and nonlinear springs.



Table 1.1 Stiffness constants for common structural elements.



1.6.2. Damping, c

• Damping used to describe all types of energy dissipation

• Energy dissipation:

Ø Reduction in kinetic and potential energy

Ø Non-recoverable

• Sources:

Ø Imperfect elasticity → Hysteresis loops

Fig. 1.9 Schematic of hysteresis loop.

Ø Friction: Depends on construction

♦ Welded steel structures

♦ Bolted steel structures

♦ Reinforced concrete structures

Ø Aerodynamic damping

♦ Drag exerted by air as the structure moves.

♦ Important in wind-induced vibrations of dynamicallysensitive structures

Ø Elastic waves through soil away from vibrating foundations

Ø The generation of heat and sound during impact



• Mathematical model:

Ø Exact nature and magnitude uncertain

Ø Simple model - Dashpot (piston in perfectly viscous fluid)

♦ Damping force directly proportional to velocity

♦ Viscous damping constant c defined as force associatedwith a unit relative velocity between ends of dashpot

♦ Accuracy:

§ Adequate for aerodynamic, hysteresis & radiationdamping

§ Very approximate for friction damping

1.7. COMPLEX NUMBER ARITHMETIC

• Convenient for phase (i.e. time shift) relationships betweenharmonic functions

Ø Eg. Loading & response

• Used extensively in structural dynamics

• Complex number, C:

Ø Sum of real and imaginary components

biaC += [1.14]

where a = Real component

b = imaginary component

i = 1−

• Addition & subtraction: ( biaC += and eidD += )

( ) ( )ebidaDC ±+±=± [1.15]



• Multiplication:

( ) ( )

( ) ( )

( ) ( )eadbiebda

ebieadbida

eidbiaDC

⋅+⋅+⋅−⋅=

⋅+⋅+⋅+⋅=

++=⋅

2 [1.16]

• Division:

eid

bia

D

C

++

= [1.17]

( ) ( )

+

⋅−⋅+

+⋅+⋅

=

+⋅−⋅+⋅+⋅

=

−−

++

=

2222

22

ed

eadbi

ed

ebda

ed

eadbiebda

eid

eid

eid

bia

D

C

• Vector form of complex number:

( )θ+θ= sincos irC [1.18]

where r = Modulus of complex value

= 22 ba + [1.19]and

θtan =a

b



• Relating standard and vector forms:

θ=

θ=

sin

cos

rb

ra[1.19]

Ø Analogy: Rotating vector of length r

♦ Complex plane: Real & imaginary axes

♦ Argand diagram

θ

Fig. 1.10 Example Argand diagram.

♦ Note: +θ denotes counter-clockwise rotation

• Euler’s equation:

θ−θ=

θ+θ=

θ−

θ

sincos

sincos

ie

ie

i

i

[1.20]

Therefore:

θ= ierC [1.22]

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