European Erasmus Mundus Master Course

Sustainable Constructions under Natural Hazards and Catastrophic Events

520121-1-2011-1-CZ-ERA MUNDUS-EMMC

2C09 Design for seismic and climate changes

Lecture 02: Dynamic response of single-degree-of-freedom systems I

Daniel Grecea, Politehnica University of Timisoara

10/03/2014

L2 – Dynamic response of single-degree-of-freedom systems I

European Erasmus Mundus Master Course

Sustainable Constructions under Natural Hazards and Catastrophic Events

L2.1 – Introduction to dynamics of structures. L2.2 – Solutions methods for the equation of motion. L2.3 – Free vibration analysis of SDOF systems.

2C09-L2 – Dynamic response of single-degree-of-freedom systems I

Dynamics of structures Dynamics of

structures determination of response of structures under the effect of dynamic loading

Dynamic load is one whose magnitude, direction, sense and point of application changes in time

t

p(t)

p(t)

p(t)

t

t

t

üg(t)

Single degree of freedom systems Simple structures:

– mass m – stiffness k

Objective: find out response of SDOF system under the effect of: – a dynamic load acting on the mass – a seismic motion of the base of the structure

The number of degree of freedom (DOF) necessary for dynamic analysis of a structure is the number of independent displacements necessary to define the displaced position of masses with respect to their initial position

k

m

Single degree of freedom systems (SDOF)

Single degree of freedom systems One-storey frame =

– mass component – stiffness component – damping component

Number of dynamic degrees of freedom = 1 Number of static degrees of freedom = ?

Force-displacement relationship

Force-displacement relationship

Linear elastic system: – elastic material – first order analysis

Inelastic system:

– plastic material – First-order or second-order analysis

Sf k u

,S Sf f u u

Damping force Damping: decreasing with time of amplitude of vibrations

of a system let to oscillate freely Cause: thermal effect of elastic cyclic deformations of the

material and internal friction

Damping Damping in real structures:

– friction in steel connections – opening and closing of microcracks in r.c. elements – friction between structural and non-structural elements

Mathematical description of these components

impossible

Modelling of damping in real structures equivalent viscous damping

Damping Relationship between damping force

and velocity: c - viscous damping coefficient

units: (Force x Time / Length) Determination of viscous damping:

– free vibration tests – forced vibration tests

Equivalent viscous damping modelling of the energy dissipated by the structure in the elastic range

Df c u

Equation of motion for an external force Newton’s second law of motion

D'Alambert principle

Stiffness, damping and mass components

Equation of motion: Newton’s 2nd law of motion Forces acting on mass m:

– external force p(t) – elastic (or inelastic) resisting force fS

– damping force fD External force p(t), displacement u(t), velocity and

acceleration are positive in the positive direction of the x axis

Newton’s second law of motion:

( )u t( )u t

S Dp f f mu

S Dmu f f p

( )mu cu ku p t

Equation of motion: D'Alambert principle Inertial force

– equal to the product between force and acceleration – acts in a direction opposite to acceleration

D'Alambert principle: a system is in equilibrium at each time instant if al forces acting on it (including the inertia force) are in equilibrium

I S Df f f p

If mu

S Dmu f f p

( )mu cu ku p t

Equation of motion: stiffness, damping and mass components

Under the external force p(t), the system state is described by – displacement u(t) – velocity – acceleration

System = combination of three pure components: – stiffness component – damping component – mass component

External force p(t) distributed to the three components

( )u t( )u t

If mu Df c u

Sf k u

I S Df f f p

SDOF systems: classical representation

Equation of motion: seismic excitation Dynamics of structures in the case of seismic motion

determination of structural response under the effect of seismic motion applied at the base of the structure

Ground displacement ug Total (or absolute) displacement of the mass ut

Relative displacement between mass and ground u ( ) ( ) ( )t

gu t u t u t

Equation of motion: seismic excitation D'Alambert principle of dynamic equilibrium Elastic forces relative displacement u

Damping forces relative displacement u

Inertia force total displacement ut

0I S Df f f

Df c u

Sf k u

tIf mu

0tmu cu ku

( ) ( ) ( )tgu t u t u t

gmu cu ku mu

Equation of motion: seismic excitation Equation of motion in the case of an external force

Equation of motion in the case of seismic excitation

Equation of motion for a system subjected to seismic

motion described by ground acceleration is identical to that of a system subjected to an external force

Effective seismic force

gmu cu ku mu

( )mu cu ku p t

gmu gu

( ) ( )eff gp t mu t

Problem formulation Fundamental problem in dynamics of structures:

determination of the response of a (SDOF) system under a dynamic excitation – a external force – ground acceleration applied to the base of the structure

"Response" any quantity that characterizes behaviour

of the structure – displacement – velocity – mass acceleration – forces and stresses in structural members

Determination of element forces Solution of the equation of motion of the SDOF system

displacement time history

Displacements forces in structural elements – Imposed displacements forces in structural elements – Equivalent static force: an external static force fS that produces

displacements u determined from dynamic analysis Forces in structural elements by static analysis of the structure subjected to equivalent seismic forces fS

( )u t

( ) ( )sf t ku t

Combination of static and dynamic response Linear elastic systems:

superposition of effects possible total response can be determined through the superposition of the results obtained from: – static analysis of the structure under

permanent and live loads, temperature effects, etc.

– dynamic response of the structure Inelastic systems: superposition of

effects NOT possible dynamic response must take account of deformations and forces existing in the structure before application of dynamic excitation

Solution of the equation of motion Equation of motion of a SDOF system

differential linear non-homogeneous equation of second order

In order to completely define the problem: – initial displacement – initial velocity

Solution methods:

– Classical solution – Duhamel integral – Numerical techniques

( ) ( ) ( ) ( )mu t cu t ku t p t

(0)u(0)u

Classical solution Complete solution u(t) of a linear non-homogeneous

differential equation of second order is composed of – complementary solution uc(t) and – particular solution up(t)

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

Second order equation 2 integration constants initial conditions

Classical solution useful in the case of – free vibrations – forces vibrations, when dynamic excitation is defined analytically

Classical solution: example Equation of motion of an undamped (c=0) SDOF system

excited by a step force p(t)=p0, t≥0:

Particular solution:

Complementary solution: where A and B are integration constants and

The complete solution

Initial conditions: for t=0 we have and

the eq. of motion

0mu ku p 0( )ppu tk

( ) cos sinc n nu t A t B t

n k m

0( ) cos sinn npu t A t B tk

(0) 0u (0) 0u

0 0pA Bk

0( ) (1 cos )npu t tk

Duhamel integral Basis: representation of the dynamic excitation as a

sequence of infinitesimal impulses Response of a system excited by the force p(t) at time t

sum of response of all impulses up to that time

Applicable only to "at rest" initial conditions

Useful when the force p(t) – is defined analytically – is simple enough to evaluate analytically the integral

0

1( ) ( )sin[ ( )]t

nn

u t p t dm

(0) 0u (0) 0u

Duhamel integral: example Equation of motion of an undamped (c=0) SDOF system,

excited by a ramp force p(t)=p0, t≥0:

Equation of motion

0mu ku p

0( ) (1 cos )npu t tk

0 (1 cos )np tk

00

0 0

cos ( )1( ) sin[ ( )]tt

nn

n n n

p tu t p t dm m

Undamped free vibrations General form of the equation of motion: Equation of motion in the case of undamped free

vibrations:

Vibrations the system disturbed from the static equilibrium position by – initial displacement – initial velocity

Classical solution where

( )mu cu ku p t

0mu ku

(0)u(0)u

(0)( ) (0)cos sinn nn

uu t u t t

n k m

Undamped free vibrations Simple harmonic motion (0)( ) (0)cos sinn n

n

uu t u t t

Undamped free vibrations Natural period of vibration - Tn - time needed for

an undamped SDOF system to perform a complete cycle of free vibrations

Natural circular frequency

Natural frequency of vibration fn represents the number of complete cycles performed by the system in one second

mass stiffness

"Natural" - depends only on the properties of the SDOF system

2n

n

T

1n

n

fT

2n

nf

n k m

n k m

Undamped free vibrations Alternative expressions for n, fn, Tn:

elastic deformation of a SDOF system under a static force equal to mg

Amplitude: magnitude of oscillations

1 22

stn n n

st st

g gf Tg

st mg k

22

0

00

n

uu u

Damped free vibrations General form of the eq. of motion: Equation of motion in the case of damped free vibrations:

Dividing eq. by m we obtain

with the notations:

Critical damping coefficient

Damping coefficient c - a measure of the energy

dissipated in a complete cycle - critical damping ratio: a non-dimensional measure of

damping, which depends on the stiffness and mass as well

( )mu cu ku p t

0mu cu ku 22 0n nu u u

n k m 2 n cr

c cm c

22 2cr nn

kc m km

Types of motion c=ccr or = 1 the system returns to the position of

equilibrium without oscillation c>ccr or > 1 the system returns to the position of

equilibrium without oscillation, but slower c<ccr or < 1 the system oscillates with respect to the

equilibrium position with progressively decreasing amplitudes

Types of motion ccr - the smallest value of the damping coefficient that

completely prevents oscillations

Most engineering structures - underdamped (c<ccr)

Few reasons to study: – critically damped systems (c=ccr) – overdamped systems (c>ccr)

Underdamped systems Solution of the eq. for systems with

c<ccr or < 1:

with the notation:

(0) (0)( ) (0)cos sinnt nD D

D

u uu t e u t t

0mu cu ku

21D n

The effect of damping in underdamped systems Envelope of damped vibrations

Lowering of the circular frequency from n to D Lengthening of he period of vibration form Tn to TD

nte

22 0 (0)

0 n

D

u uu

Attenuation of motion Ratio between displacement at time t and the one after a

period TD is independent of t:

Using and

( ) exp( ) n D

D

u t Tu t T

2

nn

T

21n

DTT

2

( ) 2exp( ) 1D

u tu t T

21

2exp1

i

i

uu

Attenuation of motion Natural logarithm of the ratio

is called logarithmic decrement and is denoted with :

For small values of the damping

Determination of logarithmic decrement based on peaks several cycles apart

21

2exp1

i

i

uu

21

2ln1

i

i

uu

21 1 2

31 1 2

1 2 3 4 1

j j

j j

uuu u u eu u u u u

1 11 ln 2jj u u

Free vibration tests Determination of damping in structures: free vibration

tests 1 1ln ln

2 2i i

i j i j

u usauj u j u

References / additional reading Anil Chopra, "Dynamics of Structures: Theory and

Applications to Earthquake Engineering", Prentice-Hall, Upper Saddle River, New Jersey, 2001.

Clough, R.W. şi Penzien, J. (2003). "Dynammics of structures", Third edition, Computers & Structures, Inc., Berkeley, USA

daniel.grecea@upt.ro

http://steel.fsv.cvut.cz/suscos