1

HOMEWORK 1

1. Derive equation of motion of SDOF using energy method

2. Find amplitude A and tanΦ for given x0, v0

3. Find natural frequency of cantilever, l=400mm, Φ=5mm, E=2e11Pa, m=2.7kg. Confirm with SW Simulation

4. Work with exercises in chapter 19 – blue book

2

RING

Ring.SLDASM

Energy method

3

ωn, x0, v0 fully define free undamped vibration

4

MODULE 02

DAMPED VIBRATION

Inman (3rd edition) section 1.3

5

SINGLE DEGREE OF FREEDOM SYSTEM WITHOUT DAMPING

massstiffness

0mx kx &&

This is the equation of motion of a single degree of freedom system with no damping. It states

that inertial forces are equal and opposite to stiffness forces.

Notice that mass and stiffness are completely separated.

6

Viscous damper, damping force is proportional to velocity

SINGLE DEGREE OF FREEDOM SYSTEM WITH DAMPING

Notice that mass, damping and stiffness are completely separated.

7

Equation of motion

We assume solution in the form of:

mx

8

2

t

t

t

x ae

x a e

x a e

Critical damping

Damping ratio (modal damping)

n

k

m

Characteristic equation

Determinant of characteristic equation

9

1

underdamped

1

Equation is linear, therefore the sum of solutions is also solution

10Two forms of solution of free under- damped vibration

Damped natural frequency

1

underdamped

or, using Euler’s relations:

11Inman p. 24

1

underdamped

A and Ф are calculated from initial conditions

ωd- natural frequency of damped oscillations

12

1

overdamped

This is a non-oscillatory motion

13

1

critically

Damping is critical when:

14

FREE VIBRATIONS WITH VISCOUS DAMPING

Depending on the sign of ζ we have three cases

ζ < 1

system is under damped, motion is oscillatory with an exponential decay in amplitude

ζ = 1

system is critically damped, at most one overshot of system resting position is possible

ζ > 1

system is over damped, motion is exponentially decaying

15Damped vibration amplitude decay.xls

ntbottoma Ae

nttopa Ae

16SDOF damped.SLDASM

m =10kg

k = 1000N/mc = 20Ns/m

17

Linear damper 20Ns/m Modal damping 10%

18

0.62, 56.48

1.23, 30.30

10 0.62

10 1.23

6.1

100010

10

56.481.86

30.30

1.86

6.1 ln1.86 0.62

0.10

n

x

x

Ae

Ae

e

t1 = 0.62s a1 = 56.48

t2 =1.23s a2 = 30.30

19swing arm.SLDASM

20

m =0.56kg

Assume that beam mass is negligible

kL=2000N/m cL=10Ns/m

Is this system under damped, critically damped or over damped?

L2=0.1m

L1=0.2m

21System is underdamped, modal damping is 7.4%

22

Modal damping 7.4%Linear damper 10Ns/m

23

ζ = 5%

ζ = 10%

ζ = 200%

ζ = 100%

ζ = 50%

24

SYSTEM Damping ratio (% of critical damping)

METALS 0. 1%

CONTINUOUS METAL STRUCTURES 2% - 4%

METAL STRUCTURES WITH JOINTS 3% - 7%

ALUMINUM/STEEL TRANSMISSION LINES ~0.4%

SMALL DIAMETER PIPING SYSTEMS 1% - 2%

LARGE DIAMETER PIPING SYSTEMS 2% - 3%

SHOCK ABSORBERS 30%

RUBBER ~5%

LARGE BUILDINGS DURING EARTHQUAKE 1%-5%

PRE-STRESSED CONCRETE STRUCTURES 2% - 5%

REINFORCED CONCRETE STRUCTURES 4%-7%

TIMBER 5% - 12%

TYPICAL VALUES OF DAMPING RATIO

25

Structure vibrating in a given mode can be considered as the Single Degree of Freedom (SDOF)

system. Structure can be considered a series of SDOF. For linear systems the response can be

found in terms of the behavior in each mode and these summed for the total response. This is the

Modal Superposition Method used in linear dynamics analyses.

A linear multi-DOF system can be viewed as a combination of many single DOF systems, as can

be seen from the equations of motion written in modal, rather than physical, coordinates.  The

dynamic response at any given time is thus a linear combination of all the modes.  There are two

factors which determine how much each mode contributes to the response: the frequency content

of the forcing function and the spatial shape of the forcing function.  Frequency content close to the

frequency of a mode will increase the contribution of that mode.  However, a spatial shape which is

nearly orthogonal to the mode shape will reduce the contribution of that mode.

MODAL SUPERPOSITION METHOD

26

MODAL SUPERPOSITION METHOD

The response of a system to excitation can be found by summing up the response of multiple SODFs.

Each SDOF represents the system vibrating in a mode of vibration deemed important for the vibration response.

27

TIME AND FREQUENCY

RESPONSE ANALYSIS

28

TIME RESPONSE ANALYSIS

In Time Response analysis the applied load is an explicit function of time, mass and damping

properties are both taken into consideration and the vibration equation appears in its full form:

Where:

[ M ] mass matrix

[ C ] damping matrix

[ K ] stiffness matrix

[ F ] vector of nodal loads

[ d ] unknown vector of nodal displacements

Dynamic time response analysis is used to model events of short duration. A typical example

would be analysis of vibrations of a structure due to an impact load or acceleration applied to the

base (called base excitation). Results of Time Response analysis will capture both the response

during the time when load is applied as well as free vibration after load has been removed.

29

TIME RESPONSE ANALYSIS

Force excitation or base excitation is a function

of time. Solution is performed in time domain i.e.

data of interest (displacement, stresses) are

computed as functions of time.

0

1

2

3

4

5

0 2 4 6 8 10

time time

F(t)

Examples of load time history in Time Response analysis.

F(t)

0

2

4

6

8

10

12

0 2 4 6 8 10

F(t)

30

FREQUENCY RESPONSE ANALYSIS

Force excitation or base excitation is not directly a function of

time, rather it is a function of the excitation frequency. Solution

is performed in frequency domain i.e. data of interest

(displacement, stresses) are computed as functions of

frequency.

Example of load time history that can be used in Frequency Response analysis.

time

Force amplitude does not have to be constant

31

Different ways to illustrate a unit impulse

timetime

force force

Impulse load

An impulse applied to a SDOF is the same as applying the initial conditions of zero displacement

and initial velocity v0 = F Δt /m (Δt here is duration of “square” impulse, Δt is very short)

Δt

TIME RESPONSE DUE TO IMPULSE LOAD

32

To model impulse load we set:

x 0 = 0

v 0 = F Δt /m

TIME RESPONSE DUE TO IMPULSE LOAD

0ˆ

0

d d d

v F t FA

m m

33

Analytical solution of unit impulse excitation problem.

Response of a system due to an impulse at t = 0

Inman p. 195

F̂ is force impulseWhere

TIME RESPONSE DUE TO IMPULSE LOAD

34

TIME RESPONSE DUE TO IMPULSE LOAD

Impulse load response.xls

35

Impulse 4.77Ns

Impulse load definition

A force is considered to be an impulse if its duration Δt is very short compared with the period T=1/f

Here Δt = 0.0075, T=0.03

TIME RESPONSE DUE TO IMPULSE LOAD

36

cr

Nsc = 2 km = 4000

m

The critical damping for SDOF is:

To define damping as 5% of critical damping we can either enter it as 200 in the Spring-Damper Connector window or as 0.05 in the Global Damping window.

Damping definition. Damping can be defined explicitly (left) or as a fraction of critical damping

(right).

The entries in both windows define the same damping.

TIME RESPONSE DUE TO IMPULSE LOAD

37Results of Dynamic Time analysis; displacement time history

Location for Time History graph

TIME RESPONSE DUE TO IMPULSE LOAD

38SDOF damped

In class exercise to demonstrate:

Equivalence of explicit damping and modal damping (study 01, study 02)

Equivalence of short impulse load to initial velocity (study 03, study 04)

39

01

Explicit damping 80Ns/m

02

Equivalent modal damping 0.02

Results are identical

40

04

Initial velocity 0.477m/s

03

Impulse 4.77Ns

Results are identical except for the very beginning (can’t be seen in these graphs)

41

HOMEWORK 2

1.

SDOF m=10kg, k=1000N/m is critically damped. Find a combination n of x0 and v0 that will make the SDOF cross the resting position. Prepare plot in Excel

2.

SDOF m=10kg, k=1000N/m performs damped vibration. After 10s the displacement amplitude is 5% of the original amplitude. Find linear damping

3.

Swing arm problem

4.

SDOF m=10kg, k=1000N/m is at rest. At t=0 it is subjected an impulse 10Ns. What is the amplitude of displacement, velocity and acceleration?