1

Dynamics of Structures

I. Introduction 1.1 Dynamic and Dynamics

Dynamic load--- any load of which its magnitude, direction, and/or point of application varies with

the time. If it is applied to an engineering structure, the resulting deflections and stresses

(response) are also time-varying (dynamic).

If the loading is a known function of time, the corresponding analysis of a specified structural

system is called a deterministic analysis. Otherwise, if the time history of the loading is not

completely known (or only known in a statistical sense---random), the structural response analysis

is nondeterministic.

Static, statics

1.2 Investigation of the Dynamical Behaviour of Structures Design, analysis and testing

Fig.1.1 Steps in a dynamic investigation

1.3 Degrees of Freedom The number of independent coordinates necessary to specify the configuration or position of a

system at any time is referred to as the number of degrees of freedom (DOF).

The analytical models in analysis stage fall into two basic categories:

discrete-parameter models

continuous models

Figure 1.2 shows these models of a cantilever beam (small deformation). The discrete-parameter

models here are called lumped-mass models because the mass of the system is assumed to be

represented by a small number of point masses, or particles. They are described by finite DOF systems,

while a continuous model represents an infinite DOF system (Fig.1.2c).

Existing Structure or

Design, drawings and data

Analytical model

(assumption, sketches, etc.)

Dynamic Bahviour

(Solutions of DE's)

Mathematical Model

(Differential equations)

Dynamic Testing

Physical Model

Meet

requirement Stop

Design

Analysis

Testing

Yes

No

2

(a) One DOF model (b) Three DOF model (c) Distributed mass model

Fig.1.2 Discrete-parameter and continuous analytical models of a cantilever beam.

However, in cases where the mass of the system is quite uniformly distributed throughout, an

alternative approach to limiting the number of degrees of freedom may be preferable. This procedure is

based on the assumption that the deflected shape of the structure can be expressed as the sum of a

series of specified displacement patterns. The amplitudes of these patterns may be considered to be the

displacement coordinates of the system, and the infinite number of DOF of the actual beam is

represented by the infinite number of terms included in the series. Generally, the expression for the

displacement of any one-dimensional structure might be

n

nn xZxu )()(

where n(x) represents a geometric shape function which is compatible with the prescribed geometric-

support conditions and maintains the necessary continuity. Its contribution to the resulting shape of the

structure depends upon the amplitude Zn, which will be referred to as generalised coordinates.

1.4 Formulation of the Equations of Motion Various types of force

Elastic force ukf s , (u=u2-u1)

k is called spring constant (N/m)

Viscous force ucf D

c is called coefficient of viscous damping

(damping constant) (N s/m)

Inertial force auu

f mmdt

dm

dt

dI

)(

A force developed by the moving mass is proportional to its acceleration and opposing it----

d'Alembert force. Typical inertia forces are shown in Fig.1.3.

Fig.1.3 Inertia forces

Direct equilibrium (using d'Alembert's principle)

From the mathematical expressions of Newton's second law of motion

uu

p mdt

dm

dt

dt )()(

----suitable for many simple problems

Other methods to form the equation of motion include principle of virtual displacement and

variational approach based on Hamilton's principle (well-known principle of minimum potential

energy in static analysis).

u(t)

x

u(x, t)

x

u3(t)

x

u1(t) u2(t)

fs

fs

u1 u2

u1 u2

fD

fD

max

may

)(12

22 cbm

I o

b

c

max

may

2

2

1rmI o

r

3

II. Single-Degree-of-Freedom Systems (SDOF)

2.1 Basic Dynamic System and its Equation of Motion The essential physical properties of any linearly elastic structural or mechanical system subjected to an

external source of excitation or dynamic loading are its mass, elastic properties and energy-loss

mechanism or damping. In the simplest model of a SDOF system, each of these properties is assumed

to be represented by a single physical element (mass block, spring and dashpot). A sketch of such a

system is shown in Fig. 2.1

(a) basic components (b) free body diagram

Fig.2.1 SDOF model

Derivation of the equation of motion by d'Alembert's principle

(1) select free body

(2) apply external and internal loads

(3) add inertia force

(4) consider the equilibrium of the body

)(tpfff SDI (2.1)

Substituting expressions of the relevant forces, the final equation of motion is

)()()()( tptuktuctum (2.2)

Derivation of the equation of motion by virtual displacement principle

If the mass is given a virtual displacement u compatible with the system's constraints, the total virtual

work done by the equilibrium system of forces must be zero

0)( utpufufuf SDI (2.3)

Considering u is arbitrary and nonzero, the same equation of motion as Eq.(2.2) can be obtained.

Example 1: Influence of weight (gravitational forces)

Rotating the model of Fig2.1(a) clockwise to a vertical position (90 degree) as shown in Fig.2.2,

the influence of weight of the mass must be considered in equilibrium equation of the system.

Fig.2.2 Influence of weight Fig.2.3 Influence of support excitation

m

k

c p(t)

u(t)

p(t)

u(t)

fI(t) fs(t)

fD W

st

ur(t)

u(t)

unstretched

length

ut(t)

u(t)

ug(t)

Fixed

line

c

k/2 k/2

m p(t)

4

)()()()( tpWtuktuctum (2.4)

Because rst uu and kWst / is the static displacement of the mass on the linear spring and

ur is the displacement of the mass measured relative to the static equilibrium position, the equation of

motion after substituting these relation into Eq.(2.4) is simplified to

)()()()( tptuktuctum rrr (2.5)

Example 2: Influence of support movement

Dynamic stresses and deflections can be induced in a structure not only by a time-varying applied

load, but also by motions of its support points such as the motions of a building caused be

earthquake etc. (Fig.2.3).

Since the inertial force in this case is given by

)()( tumtf t

I , (2.6)

the equilibrium equation leads

)()()()( tptuktuctum t (2.4)

Where total displacement ut , the displacement of the structure's base relative to a fixed reference axis

ug and the structure's displacement relative to the support u has a relation of

)()()( tututu g

t . (2.5)

The equation of motion of Eq.(2.4) can more conveniently be written

)()()()()()()( tptptumtptuktuctum effg (2.6)

2.2 Analysis of SDOF System The solution of the general equation (2.2) of motion is

u(t)=up(t)+uc(t) (2.7)

where up(t) is a particular solution related directly to p(t) and uc(t) a complementary solution. To find

up(t) and uc(t), it is convenient to rewrite Eq.(2.2) as

)(22

2 tpk

uuu (2.8)

where 2=k/m and is called the (undamped) circular frequency and is a dimensionless quantity

called the viscous damping factor and

cc

c

m

c

2 . (2.9)

cc is critical damping coefficient. Both and are very important parameters determining the response

of a SDOF system.

2.2.1 Free vibration of undamped SDOF system

The equation of motion 02uu (2.10)

The solution tu

tuu sin)0(

cos)0(

)sin()cos( 1tt (2.11)

where

/)0(

)0(tan and

)0(

/)0(tan ,

)0()0( 1

2

2

u

u

u

uuu

(2.12)

The natural frequency is defined by 2

f , which has a unit of Hz (=1cycle/s). Its reciprocal is the

undamped natural period T=1/f =2 / . The rotating vector representation in complex plane and general

response curve in time domain are shown in Fig.2.4 and Fig.2.5 respectively.

5

Fig.2.4 The vector representation of displacement Fig.2.5 General response curve in time domain

in complex plane

2.2.2 Free vibration of damped SDOF system

The equation of motion 02 2uuu (2.13)

Assume a solution of the form rtCeu , we obtain the characteristic equation

02 22 rr (2.14)

and 12

2

1

r

r (2.15)

The solution depends on the value of which can be used to distinguish three cases:

>1, i.e., c>cc overdamped case

The solution ]11[ 2

2

2

1 tshCtchCeu t

]11

)0()0(1)0([ 2

2

2 tshuu

tchue t (2.16)

=1, i.e., c=cc, critically-damped case

The solution ]))0()0(()0([)( 21 tuuuetCCeu tt (2.17)

Example 3: An example for both nonoscillatory responses is seen in Fig. 2.6

Fig.2.6 Nonoscillatory response systems

<1, i.e., c<cc underdamped case

Diir

r2

2

11 (2.18)

t

R

I

t

u(0)

ù(0)/

/ T=2 /

u(0) ù(0)

t

u(t)

0.8

1.6

=1

=2

u(t) (cm)

0.8 1.6 2.4

t (sec)

srad

scmu

u

/5

/20)0(

0)0(

6

and 21D is called damped circular frequency. The solution of Eq.(2.13) is

]sin)0()0(

cos)0([ tuu

tueu D

D

D

t

)cos( te D

t (2.19)

where

)0(

)0()0(tan and

)0()0()0(

2

2

u

uuuuu

DD

(2.20)

The free vibration response of undercritically damped (underdamped) system with zero initial velocity

is shown in Fig. 2.7.

Fig.2.7 Free vibration response of underdamped system

Logarithmic decrement of damping which is defined by

21

2ln

21n

n

u

u (2.21)

2.2.3 Response of SDOF system to harmonic loading

Assume a SDOF system is subjected to a load varying harmonically with time, saying sine- (or/and

cosine-) wave form. Without loss generality, p(t) in Eq.(2.2) can be assumed a force having an

amplitude p0 and circular frequency and

tptuktuctum sin)()()( 0 (2.22)

A complementary solution uc(t) and the particular solution up(t) of above equation are respectively

t

DDc etBtAu )sincos( (2.23)

]cos2sin)1[()2()1(

1 2

222

0 ttk

pu p (2.24)

where = / is defined as the ratio of the applied loading frequency to the natural vibration

frequency (frequency ratio). The unknown constant A and B should be determined from the initial

condition of the system. For example, if 0)0()0( uu , then

222

22

0

222

0

)2()1(

)1(2 and

)2()1(

2

Dk

pB

k

pA

2 / D 4 / D / D 3 / D

u1

u2

te

)0(uD

u(t)

t

0)0(u

7

(a) Response of undamped system ( =0). From Eqs.(2.23) and (2.24), we have

tk

ptBtAu sin

1

1sincos

2

0 . (2.25)

Considering an initial condition of zero displacement and velocity, the above solution becomes

)sin(sin1

12

0 ttk

pu . (2.26)

Response ratio

)sin(sin1

1

/

)()(

2

0

ttkp

tutR (2.27)

(b) Steady-state harmonic response of damped system

The response is given by Eq.(2.24) because complementary will decay soon (with no interest) in

damped system. Eq.(2.24) can be written as

)sin()( ttu p . (2.28)

Where 2222

0

1

2 tanand

)2()1(

1

k

p (2.29)

Magnification factor

2220 )2()1(

1

/ kpD (2.30)

Fig.2.8 Variation of magnification factor with frequency ratio

0

30

60

90

120

150

180

0 0.5 1 1.5 2 2.5

Ph

ase

angle

Frequency ratio

Fig.2.9 Variation of phase angle with frequency ratio

0

1

2

3

4

5

0 1 2

Mag

nif

icat

ion

fac

tor

D

Frequency ratio

8

(c) Resonant response

It is apparent that from Fig.2.8 the steady-state response amplitude of an undamped system tends

towards infinity as the frequency ratio approaches unity. For some other lower value of damping, it is

seen that the maximum steady-state response amplitude occurs at a frequency ratio slightly less than

unity. To find the maximum or peak value of dynamic magnification factor, the differentiating result of

Eq.(2.30) with respect to will give

221peak (2.31)

and 2

max

12

1D (2.32)

Even so, the response resulting from the unity frequency ratio, i.e., the frequency of the applied loading

equals to the undamped natural vibration frequency, is called resonance. Under this condition,

2

1maxD (2.33)

(the difference between Eqs.(2.32) and (2.33) is only 2 percent for =0.2). At the resonant exciting

frequency ( =1), the displacement response becomes

2

cos)sincos( 0 t

k

petBtAu t

DD (2.34)

If the system starts from rest (zero velocity and displacement initial conditions), then

]cos)sin1

[(cos2

1

2

0 tettk

pu t

DD (2.35)

The response ratio kp

u

/0

has an envelope of 2

1.

(d) Vibration isolation

Two types of isolation problems:

Prevent oscillatory forces of the system from support

A rotating unbalance SDOF system is shown in left figure and its

steady-state displacement response is

)sin()( 0 tk

Dptu p

The force transfer to the foundation can be estimated on

)sin()( 0 tDptkufS

)cos(2)cos()( 0

0 tDptk

DcptucfD

Fig. 2.10

The total maximum force transfer to the foundation is

2

0max

2/122

max )2(1|)( Dpfff Ds , and 2

0max )2(1/ Dpf

Transmissibility

Isolate harmful support movement from a structural system

The governing equation of relative displacement is

)()()()()( tptumtuktuctum effg (2.36)

A harmonic support movement leads a steady-state relative

displacement according to Eqs.(2.24) or (2.28) for the Eq.(2.36)

)sin()sin()( 0 tDk

pttu p

Fig. 2.11

tptp sin)( 0

u(t)

k c

m

f=fS+fD

tuug sin0

ut(t)=ug(t)+u(t)

k c

m

9

]cos2sin)1[()2()1(

2

222

2

0 ttu

The total displacement is )()()( ttt g

tuuu

(Note: the main contribution to u(t) is only from up(t)).

Similarly 2

0max )2(1/ Duu t (2.37)

Fig.2.12 Variation of transmissibilitywith frequency ratio

2.3 Response to Impulsive Loading Property

The load is of short duration

Damping has much less importance in controlling the maximum response of a structure because

the maximum response to a particular impulsive load will be reached in a very short time, before the

damping forces can absorb much energy from the structure.

Two phases: forced and free vibration during and after loading respectively.

Example 4: Response of sine wave impulse

Phase 1

During this phase, the structure is subjected to a single half -

sine wave loading as shown in the right figure. Assuming

that the system starts from rest, the undamped response

ratio, including the transient as well as the steady-state term,

is given by Eq.(2.27)

)sin(sin1

1

/

)()(

2

0

ttkp

tutR (t1 t) (2.38)

where

112

2

tt and

11 2t

T

t Fig.2.13

Phase 2

The free vibration which occurs during this phase (t t1) depends on the displacement u(t1) and velocity

)( 1tu existing at the end of Phase I. In terms of the response ratio, it depends on the values of R(t1) and

)( 1tR . Considering the displacement response of free vibration of undamped system Eq.(2.11), we

have

t-t1

Phase I

p0

t

p(t)

t1

Phase II

0

1

2

3

4

5

0 0.5 1 1.5 2 2.5

Tra

nsm

issi

bil

ity

Frequency ratio

10

)(sin)(

)(cos)()( 11

11 tttR

tttRtR

(t t1) (2.39)

Note that from Eq.(2.38) sin1

)(21tR and )cos1(

1)(

21tR , then

)()2

sin()2

cos(1

2)]sin([sin

1

)]sin()cos1()cos([sin1

)(

122

2

ttttt

tttR

(2.40)

The maximum (displacement) response can occur in either Phase I or Phase II depending on the ratio of

load duration t1 to the period of vibration of the structure.

However, if =1, both Eq.(2.38) and Eq.(2.40) are not suitable. Please derive the similar expressions

for the displacement response (in terms of response ratio).

Exercise: Response for rectangular and/or triangular impulses

Response of undamped system to a very short duration impulse

Let the undamped system be at rest for 0 t. The equation of

motion and initial conditions are

1

1

0

0 )()()(

tt

tttptkutum (2.41)

and 0)0()0( uu (2.42) Fig.2.14

Integrating Eq.(2.41) with respect to time and incorporating the initial conditions, we have

1

01 )]()([)(

t

dttkutptum )0 ( )( 10

1

tdttpt

Hence 1

01 )(

1)(

t

dttpm

tu and 0)( 1tu when t1 0. The impulse response is

)( )(sin)(

)(cos)()(sin)(

)( 110

1111

1

1

ttttm

dttptttutt

tuttu

t

(2.43)

It can be proved that the impulse response in damped system is

)( )(sin)(

)( 11

)(01

1

1

ttttem

dttpttu D

tt

D

t

(2.44)

If 1)(1

0

t

dttp , Eqs.(2.43) and (2.44) are called unit impulse response functions which are shown

respectively for undamped and damped systems as

)( )(sin1

)( 111 ttttm

tth (2.45)

)( )(sin1

)( 11

)(

11 tttte

mtth D

tt

D

(2.46)

t1<<T

t

p(t)

t1

11

2.4 Response to Arbitrary Loading*

General arbitrary loading, Duhamel integral (Time-domain)

For an arbitrary loading as shown in Fig.2.15. it can be considered as a combination of numerous

(infinite in fact) successive unit impulses (loading). At time t= , the loading acting during the interval

of time d represents a very short-duration impulse p( )d on the system, which leads a displacement

response increment effected from that time t- 0 .

Hence the response increment due to p( )d can be

calculated based on either Eq.(2.43) or Eq.(2.45)

for undamped system

)( )(sin)(

)( ttm

dptdu (2.47)

For linearly elastic system, the total response can

be obtained by summing all these differential

response increment developed during the load

history, that is a Duhamel integral

)0( )(sin)(1

)(0

tdtpm

tut

(2.48)

It is seen that Eq.(2.48) suggests the loading was

initiated at t=0 and the system was at rest at that

time. If the initial conditions are not zeros, the

additional free vibration response must be added to

this solution.

Fig.2.15

)0( )(sin)(1

sin)0(

cos)0()(0

tdtpm

tu

tutut

(2.49)

Should the nonzero initial conditions be produced by loading p(t) for t<0, the response can also be

found through Eq.(2.48) by changing the lower limit of the integral (from zero to minus infinity).

Similarly, the under critically damped response can be derived

t

D

t

D

dttepm

tu0

)( )(sin)(1

)( (2.50)

Arbitrary periodic loading, Fourier series expansion

Any periodic loading can be express as a series of harmonic loading via Fourier expansion

11

0 sincos)(n

nn

n

nn tbtaatp (2.51)

in which T

nn

2 and T is the period of the loading p(t).

T

dttpT

a0

0 )(1

sin)(2

and cos)(2

00

T

nn

T

nn dtttpT

bdtttpT

a (2.52)

Undamped steady-state response (superposition) ( /nn )

120 )sincos(

1

11)(

n

nnnn

n

tbtaak

tu (2.53)

Damped steady-state response (superposition)

1222

22

0)2()1(

)sin2)1(cos2)1(1)(

n nn

nnnnnnnnnn tabtbaa

ktu (2.54)

d

p( )

t

p(t)

t- 0

Response increment

t-

du(t)

12

References

[1] R. W. Clough and J Penzien. Dynamics of Structures. McGraw-Hill Inc., 2nd

edition, 1993,

Singapore.

[2] M. Paz. Structural Dynamics--theory and computation. Van Nostrand Reinhold Company Inc., 2nd

edition, 1985, New York.

[3] R. R. Craig. Structural Dynamics--an introduction to computer methods. John Wiley & Sons, 1981,

Canada.

[4] G. Vertes. Structural Dynamics. Developments in civil engineering; V.11. Elsevier, 1985,

Amsterdam.

[5] Wu Z. J., Han F. and Wu H. J. Elasticity, University Press of Beijing Institute of Technology, 2010,

ISBN 978-7-5640-3267-8. Beijing.