17
Free
Single-DOF
Oscillator
17–1
Lecture 17: FREE SINGLE-DOF OSCILLATOR
TABLE OF CONTENTS
Page
§17.1
§17.2
§17.3
§17.4
Introduction
. . . . . . . . . . . . . . . . . . . . . 17–3
Notation
. . . . . . . . . . . . . . . . . . . . . .
17–3
Free Vibrations of Undamped SDOF Oscillator
. . . . . . . . 17–3
§17.3.1
Equation of Motion Of Undamped Oscillator
. . . . . .
17–4
§17.3.2
Undamped Response in Terms of Complex Exponentials . . . 17–4
§17.3.3
Undamped Response in Terms of Trigonometric Functions . .
17–5
§17.3.4
Effect of Initial Conditions . . . . . . . . . . . . . 17–6
§17.3.5
Energy Conservation Property
. . . . . . . . . . .
17–7
Free Vibrations of Viscous-Damped SDOF Oscillator
. . . . . . 17–7
§17.4.1
Equation of Motion of Damped Oscillator
. . . . . . .
17–7
§17.4.2
Damped Response in Terms of Trigonometric Functions . . . 17–8
§17.4.3
Underdamped Case: ξ < 1
. . . . . . . . . . . .
17–9
§17.4.4
Critically Damped Case: ξ = 1
. . . . . . . . . . . 17–10
§17.4.5
Overdamped Case: ξ > 1 . . . . . . . . . . . . . 17–10
§17.4.6
Energy Dissipation
. . . . . . . . . . . . . . . 17–11
17–2
§17.3
FREE VIBRATIONS OF UNDAMPED SDOF OSCILLATOR
§17.1. Introduction
This lecture begins Part VI of the course. This Part provides a quick introduction to structural
dynamics and in particular free and forced vibrations. Nowadays the term dynamics has acquired
several interpretations, especially in the social sciences. It is used here in the traditional sense:
“The study of the relationship between motion and the forces affecting motion”
This is in fact meaning (1a) given in the American Heritage Dictionary of the English Language.
The corresponding adjectives are dynamic1 and its equivalent dynamical.2
But even the traditional definition is far too broad in two respects. First, Mechanics embodies
a wide range of scales that span from cosmological through atomic and sub-atomic. Second, the
study can focus on three aspects: theoretical, applied and computational. Our focus is restricted to a
particular flavor: Classical Mechanics, which is the subset of Mechanics that obeys Newton’s laws.
This restriction allows the use of continuum (field) models as well as certain “lumped” idealizations
(for example, point masses) that can be derived directly from such laws.
An important application area of Classical Mechanics is to structures, a subject naturally called
Structural Dynamics. This is the application considered here. More advanced treatments, which
fall under the purview of Solid and Continuum Mechanics, are studied in dedicated courses at the
senior and graduate level.
§17.2. Notation
As just observed, Structural Dynamics studies the motion of structures under time-dependent forces.
The time will be always denoted by t. Derivatives with respect to t will be abbreviated by
superposed dots. For example, if u(t) is a scalar motion, the associated velocity and acceleration
are compactly written
du(t)
d 2 u(t)
u̇ ≡
.
(17.1)
,
ü ≡
dt
dt 2
Sometimes we will denote velocity by v and acceleration by a where appropriate. Note that partial
differentiation with respect to time is not needed: the ordinary differential symbol d suffices.
§17.3. Free Vibrations of Undamped SDOF Oscillator
We start up this study with the simplest dynamic system that obeys the laws of Classical Dynamics:
the unforced spring-mass oscillator undergoing free vibrations.
A weight of mass m > 0 hangs from an extensional spring of stiffness k ≥ 0 under gravity
acceleration g directed along the spring direction. The weight will be treated as a point mass. The
weight force is W = m g. We will assume small displacements throughout. The static deflection
of the mass is
δs = W/k = m g/k.
(17.2)
This configuration is called a Single Degree of Freedom oscillator, or SDOF oscillator for short.
See Figure 17.1(a,b).
1
Etymology: French dynamique, from Greek dunamikos: powerful, from dunamis, power, from dunashai, to be able.
2
The variant dynamical tends to be often used in a more abstract sense. For example, MathWorld defines dynamical
system as “a means of describing how a state evolves into another over the course of time.”
17–3
Lecture 17: FREE SINGLE-DOF OSCILLATOR
Apply now an initial displacement u 0 and initial velocity v0 to the mass at t = 0, so that
u(0) = u 0 ,
u̇(0) = v0 .
(17.3)
These are called the initial conditions or ICs. Then release the mass and do not apply any force
for t > 0. The oscillator is set in motion, and a free vibrations response ensues. Let u(t) be the
deviation from the static equilibrium position so that the total motion is δs + u(t), as shown in
Figure 17.1(c). The function u(t) is called the dynamic response. The only DOF is u = u(t). We
next study the equation of motion (EOM) and its solution.
§17.3.1. Equation of Motion Of Undamped Oscillator
Remove the spring and replace it by the force Fs = k(δs + u(t)), as drawn in Figure 17.1(d). The
other two forces acting on the mass are its weight W = m g, which is independent of time, and the
inertia force m ü(t), which acts in the opposite direction to the acceleration ü = ü(t) (because it
resists it). The resulting picture is called a Dynamic Free Body Diagram, or DFBD.
Equilibrium of forces in the x direction requires m ü = W − k(δs + u) = m g − kδs − ku. On
canceling m g − kδs = 0 we get
m ü + k u = 0.
(17.4)
This is the equation of motion (EOM) of the free, undamped, single DOF, linear oscillator. It is
a second order, linear ODE in u(t). Notice that the static deflection δs has disappeared from this
EOM; in fact the mass will oscillate about that position.
To convert to canonical form, divide (17.4) through by m and introduce ωn2 = k/m:
ü +
ωn2
u = 0,
in which
ωn = +
k
.
m
(17.5)
This parameter ωn is called the undamped circular natural frequency, or natural frequency for short.
Its units are radians per second (rad/s).
Key features of the equation of motion (17.5) are:
1.
EOM is linear: superposition applies
2.
EOM has constant coefficients: solutions are elementary circular functions
3.
EOM is second-order in time: two constants of integration ⇒ two initial conditions required
§17.3.2. Undamped Response in Terms of Complex Exponentials
Let u(t) = C eλt , where C = 0 and λ are generally complex. Since ü = λ2 C eλt , substitution into
(17.5) requires
(17.6)
(λ2 + ωn2 ) C eλt = 0
But since C = 0 (else the solution is null) and the exponential never vanishes, the term in parenthesis
must be zero. This gives the characteristic equation
λ2 + ωn2 = 0,
⇒
17–4
λ1,2 = ±i ωn ,
(17.7)
§17.3
FREE VIBRATIONS OF UNDAMPED SDOF OSCILLATOR
;;; ;; ;;
(a)
(b)
k
(c)
g
k
(d)
g
k
..
FI = m u
δs
δs = W/k
u=u(t)
W
x
Fs = k (δs+u)
W
W
..
FI = m u
Figure 17.1. Undamped, unforced spring-mass SDOF oscillator undergoing free vibrations.
√
in which i = −1 denotes the imaginary unit. Because both eiωn t and e−iωn t satisfy the linear ODE
(17.4), so does any linear combination of them. We thus arrive at the general solution in terms of
complex exponentials:
u(t) = C1 eiωn t + C2 e−iωn t ,
(17.8)
in which C1 and C2 are generally complex numbers.
§17.3.3. Undamped Response in Terms of Trigonometric Functions
We know that the motion u(t) is supposed to be real. Therefore it is convenient to recast (17.8) as
a real expression. Using Euler’s relation for the complex exponential
e±iθ = cos θ ± i sin θ,
(17.9)
u(t) = (C1 + C2 ) cos ωn t + i(C1 − C2 ) sin ωn t.
(17.10)
the solution (17.8) becomes
To simplify the linkage of this expression to the initial conditions, introduce A1 = C1 + C2 and
A2 = i(C1 − C2 ). Then we compactly express the response in terms of trigonometric functions as
u(t) = A1 cos ωn t + A2 sin ωn t.
(17.11)
Here A1 and A2 are real constants that can be directly determined from the initial conditions (17.3):
u(0) = u 0 = A1 ,
u̇(0) = v0 = A2 ωn ,
(17.12)
v0
sin ωn t.
ωn
(17.13)
Replacing into (17.11) yields
u(t) = u 0 cos ωn t +
17–5
Lecture 17: FREE SINGLE-DOF OSCILLATOR
(a)
(b)
2
1.5
2
u(t)
1.5
u0 = 1, v 0 = 0
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
−1.5
−1.5
−2
0
2
4
6
Time t
8
10
12
−2
u(t)
0
u0 = 0, v0 = 1
4
2
6
Time t
8
10
12
Figure 17.2. Response of undamped spring-mass oscillator with k = m = 1: (a) unit initial
displacement and zero initial velocity; (b) zero initial displacement and unit initial velocity.
This is the free vibration response of an undamped SDOF system expressed in terms of trigonometric
functions and initial conditions. Although (17.13) holds only for free vibrations, this solution will
be later reused as the homogeneous portion of the response of a forced SDOF oscillator. Another
common version of (17.13) is the phased response (also called phase-shifted response) form
u(t) = U cos(ωn t − α).
(17.14)
Here the amplitude U and the phase angle α are linked to the initial conditions (17.3) by
U=
u 20
+
v0
ωn
2
,
tan α =
v0
.
ωn u 0
(17.15)
§17.3.4. Effect of Initial Conditions
To study the effect of initial conditions, consider first the case where the mass is displaced from its
static equilibrium position by u 0 and released. If so v0 = 0, and (17.13) reduces to
u(t) = u 0 cos ωn t.
(17.16)
This is plotted in Figure 17.2(a). The resulting response is a simple harmonic motion with amplitude
u 0 , undamped natural frequency f n and undamped natural period Tn , given by
fn =
ωn
,
2π
Tn =
1
2π
=
,
fn
ωn
(17.17)
Frequency f n is expressed in cycles per second or Hertz, abbreviated to Hz (1 Hz = cycle/s). The
period is given in seconds per cycle, or simply seconds (s). It s easily verified that in this case the
phased response (17.14) coalesces with (17.16) since U = u 0 and α = 0.
17–6
§17.4
FREE VIBRATIONS OF VISCOUS-DAMPED SDOF OSCILLATOR
Figure 17.2(b) shows a response plot when u 0 = 0 and v0 is nonzero. In this case the phased
response (17.14) gives a similar harmonic curve but shifted in time. If both u 0 and v0 are nonzero,
the response has amplitude U related to the initial conditions by the first of (17.15). The maxima
and minima occur at t = α/ωn + 12 nTn , n = 1, 2, . . . .
§17.3.5. Energy Conservation Property
(Not covered in class.) It is instructive to consider the energy conservation (also called energy balance and
energy invariance in the literature) property exhibited by the response of this oscillator. At any time t during
the motion the total energy is
H = T + V.
(17.18)
Here T = T (t) is the kinetic energy due to the moving mass and V = V (t) the potential energy stored in the
spring. For a general motion u(t), their differentials are dT = FI du = m ü du and d V = Fs du = k u du,
respectively, in which the time dependence of forces and displacements is omitted for brevity. Integrating in
time from t = 0 to a final time t f gives the energy change
tf
FI du = m
T =
0
tf
V =
ü du,
0
tf
Fs du = k
0
tf
u du,
(17.19)
0
But the integrands ü du and u du are the differentials of 12 u̇ 2 and 12 u 2 , respectively, for any time-differentiable
u(t). Thus
t f
T =
1
2
m u̇ 2 0 =
V =
1
2
k u 2 0 =
t f
1
2
1
2
m u̇ 2f − 12 mv02 = T f − T0 ,
k u 2f − 12 k u 20 = U f − U0 .
(17.20)
in which u 0 = u(0) and v0 = u̇(0). Since t f is arbitrary, it is convenient to rename t f → t by supressing all
f subscripts; whence (17.20) can be presented in a more concise form:
T = T0 + 12 m u̇ 2 ,
V = V0 + 12 k u 2 .
(17.21)
These expressions hold for arbitrary time responses u(t) and u̇(t). If the specific response (17.13) is inserted
and the energies added we get
H = T + V = T0 + U0 =
1
2
m v02 + 12 k u 20 .
(17.22)
Consequently the total energy of the undamped unforced oscillator is conserved for all t ≥ 0. Although both
T and U vary in time (for example, T vanishes when the velocity is zero) any change in one is compensated
by an equal and opposite change in the other. This is a consequence of the absence of damping.
§17.4. Free Vibrations of Viscous-Damped SDOF Oscillator
The response plots of Figure 17.2 convey the idea that, once an undamped SDOF oscillator is set
in motion, that motion will continue for all times t > 0 without any change in amplitude. In reality
all mechanical systems exhibit some damping, which dissipates energy and causes the motion
eventually to die out. In the present section we consider the case of a SDOF oscillator under the
simplest damping model: viscous damping, which is linearly proportional to the velocity. It is
modeled by a dashpot element of viscosity c ≥ 0, as illustrated in Figure 17.3(a).
17–7
Lecture 17: FREE SINGLE-DOF OSCILLATOR
;; ;;
(a)
k
;;
(b)
c
k
c
(c)
g
(d)
c g
k
Fs = k (δs +u)
δs
δs = W/k
x
.
Fd = cu
..
mu
u=u(t)
W
W
..
W mu
Figure 17.3. Damped spring-dashpot-mass oscillator undergoing free vibrations: (a) initial unloaded
position; (b) static equilibrium position; (c) dynamic equilibrium position; (d) Dynamic Free Body Diagram
(DFBD).
§17.4.1. Equation of Motion of Damped Oscillator
From the Dynamic Free Body Diagram (DFBD) shown in Figure ?(d) we obtain the physical form
of the EOM:
m ü + c u̇ + k u = 0.
(17.23)
Divide this equation through by m and denote
k
k
2
ωn = , ωn = +
, c = 2ξ ωn m,
m
m
ξ=
c
c
= √
.
2ωn m
2 km
(17.24)
Here ωn is the undamped
√ circular natural frequency introduced in the previous section, while
ξ = 12 c/(ωn m) = 12 c/ k m is the viscous damping factor, also called damping ratio and damping
coefficient. We thus convert (17.23) to the canonical form of the EOM:
ü + 2ξ ωn u̇ + ωn2 u = 0.
(17.25)
This is a linear, second-order ODE with constant coefficients. The only difference with respect to
the undamped case is the presence of the velocity term 2ξ ωn u̇.
§17.4.2. Damped Response in Terms of Trigonometric Functions
As usual in solving ODEs with the canonical form (17.25), assume a complex exponential solution
u(t) = A eλt .
(17.26)
where A and λ are generally complex values to be determined. Inserting into (17.25), we obtain
the characteristic equation
(17.27)
λ2 + 2ξ ωn λ + ωn2 = 0.
17–8
§17.4
FREE VIBRATIONS OF VISCOUS-DAMPED SDOF OSCILLATOR
which is quadratic in λ. Its roots are given by
λ1,2 = −ξ ωn ± ωn
ξ 2 − 1.
(17.28)
Note that for the undamped case: ξ = 0, the roots reduce to ±iωn , as found previously. The
magnitude of the damping factor ξ compared to unity can be used to distinguish three cases:
ξ <1
Underdamped case. Damping is called subcritical. The roots λ1,2 in (17.28) are complex
conjugate. The motion is oscillatory with decreasing amplitude. This is the most common
case in typical structures, and thus the most practically important one.
ξ >1
Overdamped case. Damping is called overcritical. The roots λ1,2 in (17.28) are negative
real and distict. The motion is non-oscillatory. Its amplitude decays monotonically except
possibly for one zero crossing.
ξ =1
Critically damped case. The roots λ1,2 are negative real and coalesce. The motion
is non-oscillatory. Its amplitude decays monotonically except possibly for one zero
crossing. Since the response exhibits the most rapid decay toward the static position,
critical damping is of interest in the design of devices such as instruments and suspensions.
We now proceed to study those three cases in more detail. Emphasis is placed on the underdamped
case, which is the most important one for structural dynamics.
§17.4.3. Underdamped Case: ξ < 1
For convenience the roots (17.28) of the characterist equation may be expressed as follows
λ1,2 = −ξ ωn ± iωd ,
in which ωd denotes the damped circular natural frequency given by
ωd = ω n 1 − ξ 2 .
(17.29)
(17.30)
Like ωn , this is expressed in radians per second. The corresponding damped period is
Td =
2π
.
ωd
(17.31)
With the help of this symbols and Euler’s formula the general solution can be expressed as
u(t) = e−ξ ωn t (A1 cos ωd t + A2 sin ωd t) .
(17.32)
Again using the initial conditions u(0) = u 0 and u̇(0) = v0 , we obtain A1 = u 0 and A2 =
(v0 + ξ ωn u 0 )/ωd , which substituted into (17.32) gives the response
u(t) = e
−ξ ωn t
v0 + ξ ωn u 0
sin ωd t .
u 0 cos ωd t +
ωd
(17.33)
This equation may in turn be rewritten in the phased form
u(t) = U e−ξ ωn t cos(ωd t − α)
17–9
(17.34)
Lecture 17: FREE SINGLE-DOF OSCILLATOR
1.5
1
0.5
(a)
1.5
u(t)
u0 = 1, v 0 = 0
1
ξ=0
u(t)
u0 = 0, v0 = 1
0.5
ξ=1
0
ξ=0.5
ξ=0.2
ξ=0.5
ξ=1
0
−0.5
−0.5
−1
−1.5
(b)
ξ=0.2
ξ=0
−1
0
2
4
6
8
10
12
−1.5
0
2
4
Time t
6
Time t
8
10
12
Figure 17.4. Response of underdamped (and critically damped) spring-dashpot-mass oscillator with k = m = 1,
for various values of the damping factor ξ that range from 0 to 1: (a) unit initial displacement and zero initial velocity;
(b) zero initial displacement and unit initial velocity. Note that the critically damped oscillator does not move in (b).
in which
U=
u 20 +
v 0 + ξ ωn u 0
ωd
2
,
tan α =
v0 + ξ ωn u 0
.
ωd u 0
(17.35)
This kind of response is illustrated in Figure 17.4.
§17.4.4. Critically Damped Case: ξ = 1
In this case the characteristic roots coalesce: λ1 = λ2 = −ωn . The second order ODE theory says
that the solution is u(t) = (A1 + A2 t)e−ωn t . When the initial conditions are taken into account we
get
u(t) = [u 0 + (v0 + ωn u 0 ) t] e−ωn t .
(17.36)
Plots of critically damped system responses are illustrated in Figures 17.4 and 17.5 by taking ξ = 1
(red curve response).
§17.4.5. Overdamped Case: ξ > 1
If ξ > 1 the characteristic equation has two distict negative real roots. For convenience define
(17.37)
ω∗ = ωn ξ 2 − 1.
Then the solution may be written in terms of the hyperbolic sine and cosine as
u(t) = e−ξ ωn t (A1 cosh ω∗ t + A2 sinh ω∗ t),
(17.38)
in which A1 and A2 depend on initial conditions. Introducing these we arrive at the response
v0 + ξ ωn u 0
−ξ ωn t
∗
∗
u 0 cosh ω t +
u(t) = e
sinh ω t .
(17.39)
ω∗
For a pictorial effect of damping factor on the response of an overdamped system (including the
transition critically damped case ξ = 1) see Figure 17.5.
17–10
§17.4 FREE VIBRATIONS OF VISCOUS-DAMPED SDOF OSCILLATOR
(a)
1.4
0.5
u(t)
u0 = 1, v 0 = 0
1.2
0.4
1
0.8
ξ=2
0.2
2
ξ=5
0.1
ξ=1
0
ξ=2
0.2
ξ=5
0.4
u0 = 0, v0 = 1
ξ=1
0.3
ξ=10
0.6
(b)
u(t)
ξ=10
4
6
Time t
1.5
1
8
10
0
12
(c)
u(t)
2
4
6
Time t
8
10
12
u 0 = 1, v0 = −5
ξ=10
0.5
ξ=5
0
ξ=2
−0.5
ξ=1
−1
0
4
2
6
Time t
8
10
12
Figure 17.5. Response of overdamped and critically spring-dashpot-mass oscillator with k = m = 1, for various
values of the damping factor ξ that range from 1 to 10: (a) unit initial displacement and zero initial velocity; (b)
unit initial displacement and unit initial velocity; (c) unit initial displacement and high negative initial velocity.
§17.4.6. Energy Dissipation
(Not covered in class.) The main physical effect of damping is conversion of mechanical energy (the sum of
kinetic and potential energy) into heat. This process is known as energy dissipation. The dissipated energy at
time t ≥ 0 is denoted by D = D(t). For convenience we set D(0) = 0 so D = D. Proceeding along the
lines of §17.3.5 we express D as
tf
Fd du = c
D=
0
tf
u̇ du = c
0
tf
u̇ dt = 2ξ ωn m
2
0
tf
(17.40)
u̇ 2 dt.
0
The integrand u̇ du is not an exact differential for arbitrary u̇(t), so it is necessary to insert specific response
functions from the start. We restrict attention to the underdamped case. On inserting (17.33) into the last of
(17.40), Mathematica gives
C1 + e2ξ ωn t f (−1 + ξ 2 ) C2 − ξ C3 cos(2ωd t f ) + ξ
D=m
2 e2ξ ωn t f (ξ 2 − 1)
1 − ξ 2 C4 sin(2ωd t f )
,
(17.41)
in which C1 = v02 + 2 u 0 v0 ξ ωn + u 20 ωn2 , C2 = v02 + u 20 ωn2 , C3 = v02 ξ + 2 u 0 v0 ωn + u 20 ξ ωn2 , and C4 =
−v02 + u 20 ωn2 . This exact expression can be further manipulated to furnish more practical results. First, it is
17–11
Lecture 17: FREE SINGLE-DOF OSCILLATOR
Energy decay ratio
1
Linearized
0.8
Exact
0.6
0.4
0.2
0
0.2
0.4
0.6
ξ
1
0.8
Figure 17.6. Energy decay ratio (17.44) plotted for damping
ratio ξ ∈ [0, 1], and its linearization 2π ξ at ξ = 0.
evaluated at t f = 2π/ωd to give the dissipation work over one cycle:3
Dc =
1
2
m 1 − exp −4πξ/ 1 − ξ 2
(u 20 ωn2 + v02 ).
(17.42)
This expression can be expanded in Taylor series about ξ = 0 to yield
Dc = 2π ξ m (u 20 ωn2 + v02 ) + O(ξ 2 ) = 2π ξ k (u 20 +
v02
) + O(ξ 2 ).
ωn2
(17.43)
which is convenient if the damping is light: ξ << 1. At t = 0 the total energy is, as per (17.22), H0 =
1
(k u 20 + m v02 ) = 12 k (u 20 + v02 /ωn2 ) = 12 m (u 20 ωn2 + v02 ). The energy decay ratio Uc /H0 gives a measure of
2
the energy dissipated per cycle:
4πξ
Dc
= 1 − exp − H0
1 − ξ2
= 2πξ + O(ξ 2 ).
(17.44)
This ratio is independent of IC. The exact formula and its light damping linearization 2πξ are plotted in Figure
17.6. Plainly the linearization is way off if ξ exceeds ≈ 5%.
The exact expressions in (17.41) and (17.44) are new.
3
Most publications integrate over the undamped period 2π/ωn . The result is actually more complicated than the exact
expression (17.42). But the first term of the Taylor series (17.43) is identical.
17–12