Mechanical resonance – theory and applications
Introduction
In nature, resonance occurs in various situations. In physics, resonance is the tendency of a system to
oscillate with greater amplitude at some frequencies than at others (http://en.wikipedia.org/wiki/Resonance)
Interaction of electromagnetic waves with matter, nuclear magnetic resonance (e.g.
http://en.wikipedia.org/wiki/Nuclear_magnetic_resonance),
http://en.wikipedia.org/wiki/Electrical_resonance)
resonance in electric circuits (see e.g.
or mechanical resonance represent typical examples of
various types of resonance.
Frequency, at which the amplitude of response has local maximums, is called the system's resonance
frequency. Frequencies of self-oscillations correspond to resonant frequencies.
In mechanical systems, self-oscillations occur only in some specific combinations of inertial, elastic
and viscose properties. Some mechanical systems do not oscillate. The same holds for resonance.
System´s behavior at resonant frequency (or close to it) dramatically differs from the system´s
behavior in other frequencies. Generally holds, response of system´s output on its input is distinctively
higher in the state of resonance.
Resonance principle is applied in many sensitive measurements (NMR, EPR, RMA and others). On
resonance principle is also based operation of antennas in electronics and functioning of many musical
instruments. On the other hand, resonance in mechanics is often connected with origin of dangerous
vibrations which is important namely in building engineering and industry.
Resonance measurements in mechanics (RMA, resonance mechanical analysis, see also Resonance meters
for viscoelasticity measurement-ppt.pptx
) are more sensitive and more precise alternative to direct method
(DMA, dynamic mechanical analysis, see also http://en.wikipedia.org/wiki/Dynamic_mechanical_analysis) of
measurement of viscoelasticity.
Theory
Mechanical resonance of elastic systems
Simple mechanical oscillator (Fig. 1)
System consists of spring (elastic element) and weight (inertial element). Analysis of resonance of this
system may be based on classical solution of differential equation of the system´s movement.
Nevertheless, application of Laplace transformation and Fourier transformation provide more synoptic
methodological approach ( Operational calculus in viscoelaticity.docx ) which is applicable also for more
complicated situations. Thus, the methodology will be described step by step.
spring
weight
závaží
self-oscillations
energy inserted into system
Fig. 1. Simple mechanical oscillator.
Force equilibrium:
F (t )  FE (t )  FI (t )
(1)
Energy is inserted into system using short impulse of force F. The force is the sum of the elastic force
(FE) and of the inertial force (FI).
Constitutive equation:
d 2 y (t )
,
F (t )  H y (t )  M
d t2
(2)
where H is the spring constant, y is the deformation of spring, M is the mass of weight.
Self-oscillations
Application of Laplace transform (see Operational calculus in viscoelaticity.docx) leads to equation (3).
F ( p)  H  p 2 M . y ( p) ,
(3)
where p is new variable, instead of t.
For the impulse A of the force holds eq. (4).
A  H  p 2 M . y ( p)
For simplicity, the variable p will not be further written, following formulas.
Consequently, for deformation it holds:
(4)
y A
Roots of the denominator of eq. (5) are 
1
M p2  H
(5)
y(t )  A sin t ,
(6)
H
.
M
For deformation in time domain it holds:
Where:

H
M
(7)
Elastic rod as mechanical oscillator (Fig. 2 and 3)
elastic rod
mass in
centrum of
gravity
external mass
F
F
a)
b)
Fig. 2. Elastic rod with external inertial weight (a) and single elastic rod (b) in tensile loading.
F
F
a)
b)
Fig. 3. Elastic rod with external inertial weight (a), single elastic rod (b) in bending loading.
For resonance of elastic rod, formulae (1) – (7) holds. But in reality, mass is acting in the centrum of
gravity of whole system. Consequently, the inertial force (eq. 2 and further) must be adequately
converted ( see Operational calculus in viscoelaticity.docx, page 5).
Resonance curves of elastic systems
Resonance curves may be derived from frequency characteristics of systems. According definition,
amplitudes of vibrations are maximal at resonance frequency.
Frequency characteristic may be obtained from by help of Fourier transformation (see Operational
calculus in viscoelaticity.docx ). If
Fourier transformation is applied on eq. (2), it is obtained:
F (i )  H   2 M . y(i ) .
(8)
y (i )
1

F (i ) H   2 M
(9)
Consequently it holds:
Equation (9) describes frequency characteristic, provided F(t) and y(t) are harmonic functions.
Amplitudes of vibrations are maximal if this holds:
H  2 M  0 .
(10)
Resonance frequency corresponds to eq. (7).
In resonance, the amplitude of vibration is theoretically infinite. In real bodies, the amplitude is limited
do to energy losses.
In practical situation, entire resonance curve may be important. Namely, how “sharp” is the resonance
curve. Simulation is in resonance elastická.xlsx. Example is on Fig. 4.
amplitude
amplituda
1
0,9
0,8
0,7
0,6
0,5
0,4
0,3
0,2
0,1
0
1
1,2
1,4
1,6
frequency
(Hz)
frekvence (Hz)
1,8
2
Fig. 4. Example of resonance curve of elastic system. Parameters: of: H= 100 N/m, M= 1kg.
Mechanical resonance of viscoelastic systems
Resonance of simple viscoelastic systems
Structure of simple viscoelastic system is on Fig. 5. System consists of elastic element (spring),
inertial element (weight) and dumping element (dash pot).
spring
weight
self oscillations
dash pot
energy inserted
Fig. 5. Scheme of simple viscoelastic system
Force equilibrium
F (t )  FE (t )  FI (t )  FD (t )
(11)
Energy is inserted into system using short impulse of force F. The force is the sum of the elastic force
(FE), the inertial force (FI) and the dumping force (FD). Thus it holds:
dy (t )
d 2 y(t )
,
F (t )  H y (t )  N
M
dt
d t2
(12)
where H is the spring constant (Hooke´s coefficient), y is the deformation, N is the Newton´s
coefficient, M is the mass.
Self-oscillations
Laplace transformation of (12) leads to eq. (13):
F ( p)  H  pN y( p)  p 2 M . y( p) ,
(13)
For the impulse A of the force it holds:
A  p 2 M . y ( p)  p N  H .
(14)
For deformation it holds:
y A
1
M p N p H
2
(15)
Resonance occurs only if this holds: 4MH  N 2  0 .
In this case, from theory of Laplace transformation for deformation it holds:
A. e  a t . sin  t .
(16)
Where:

4M .H  N 2
2M
(17)
and

N
2M
(18)
Resonance curves of simple viscoelastic systems
Resonance curves may be derived from frequency characteristics of systems. According definition,
amplitudes of vibrations are maximal at resonance frequency.
Frequency characteristic may be obtained from by help of Fourier transformation (see Operational
calculus in viscoelaticity.docx ). If
the Fourier transformation is applied on eq. (12), it is obtained:
F (i )  H  iN   2 M . y(i ) ,
(19)
y (i )
1

F (i ) H  iN   2 M
(20)
Equation (20) describes frequency characteristic, provided F(t) and y(t) are harmonic functions.
For ratio of amplitudes it holds:
y A (i )
1

FA (i ) H  iN   2 M
(21)
where yA is the amplitude of deformation and FA is the amplitude of force.
Consequently:
y A (i )

FA (i )
1
(22)
( H  M 2 ) 2  N 2 2
For maximum of vibrations it holds:
d ( ( H  M 2 ) 2  N 2 2 )
0
(23)
4M .H  N 2
2M
(24)
dt
Thus for resonance frequency it holds:
fR 
1
2
Resonance frequency corresponds to eq. (17).
For the amplitude of vibrations at resonance it holds:
y AR 
1
N
(25)
In practical situation, entire resonance curve may be important. Namely, how “sharp” is the resonance
curve. Simulation is in viscoelastic resonance.xlsx. Example is on Fig. 6.
deformace (mm)
deformation (mm)
200
180
160
140
120
100
80
60
40
20
0
0
10
20
30
40
50
60
70
frekvence
(Hz)
frequency
(Hz)
Fig. 6. Example of resonance curve of real viscoelastic system. Parameters: of: H= 100 N/m, N= 0,02
Ns/m , wooden rod 100x4x4 mm.
Resonance curves of real viscoelastic systems
General constitutive differential equation for linear systems is:
n
m
i 1
j 1
a0 y   ai y (i )  b0 F   F j x ( j )
(26)
where a and b are constant coefficients, i and j are degree of derivatives, x is the input quantity
(force), y is the output quantity (deformation).
In Laplace transformation it holds:
n
m
i 1
j 1
a0 y   ai y (i )  b0 F   b j F ( j )
(27)
For impulse A of force in Laplace transformation it holds:
n
a 0 y ( p)   ai y ( p ) ( i )  A
(28)
i 1
For deformation it holds?
y ( p) 
A
a0  a1 p  a 2 p 2  ....
(29)
Previous equation is possible to convert into form:
y ( p) 
A1
A2
B1
B2

 .. 
..
...
2
p  1 p   2
p  1   1 p
p  2   2 p2
(30)
Members with coefficient B are relevant for resonance:
y R ( p) 
B1
B2
..
…..
2
p  1   1 p
p  2   2 p2
(31)
In time domain it holds:
y R (t )   B j . e
  ji .t
 sin ( j .t   j )
(32)
j
Resonance occurs at local maxims of the function (32).
In complex mechanical system several resonance frequencies may be found.
Resonance of long rods and cables
Introduction
Analysis of long rods and cables resonance is important namely for building industry, and engineering.
Main practical problems are connected with calculation of resonance frequency, dumping of vibration,
elimination of self-oscillations etc. Application of classical approach fails, as long structures are
systems with distributed parameters. Consequently, models with lumped parameters (e.g. classical
rheological models) cannot be used. Satisfactory theory of mechanical behavior of mechanical systems
with distributed parameters currently does not exist.
Nevertheless, practical solutions of some problems on this field may be based on analysis of
mechanical wave propagation.
Principle of solution for long rod
As mechanical impedance of rod differs from mechanical impedance of endings of rod, mechanical
wave consists in interference of direct wave and wave reflecting from endings. Resulting are standing
waves. Distance between nodes and antinode is one quarter of wavelength. Measurement of
mechanical wave propagation (Fig. 7) may be based on this fact.
L
λ/4
antinode
node
Fig. 7. Principle of measurement of speed of mechanical wave propagation in rod.
Measurement of speed of wave propagation
1) Short impulse energy is inserting into rod. The rod oscillates on resonant frequency f.
2) Resonance frequency f is measured,
3) Speed v of wave propagation is calculated according formula
v 4 f L .
Example of application – calculation of maxim of vibration alongside rod (Fig. 8)
Positions of antinodes correspond to locations of maxims of deformation. Positions of nodes
correspond to locations of maxims of stress.
(33)
antinodes
λ/4
λ/2
λ/2
node
λ/4
node
Fig. 8. Calculation of maxims of vibrations and stresses.
More detailed analysis may also lead to calculation of dumping as well as to the way of vibration
elimination.