Jerk Response Spectrum

Abstract

Jerk, also known as jolt, is the rate of change of acceleration. It is a vector quantity and its scalar magnitude is also the third derivative of position of a body or joint of a structure. Its dimension is [length/time 3 ]. Excessive "jerky motion" may result in an uncomfortable stay in buildings, bridges, and ride on trains or trams and it should be designed so as to reduce the influence of the jerk. This paper presents formulas and graphs for jerk response spectrum which should be taken into account among the other response spectra (displacement, velocity, acceleration) in structural engineering design for dynamic loading including earthquake impact.

Full text

39
ГОДИШНИК НА УНИВЕРСИТЕТА ПО АРХИТЕКТУРА, СТРОИТЕЛСТВО И ГЕОДЕЗИЯ
СОФИЯ
Том
51
2018
Брой
2
Volume
Issue
ANNUAL OF THE UNIVERSITY OF ARCHITECTURE, CIVIL ENGINEERING AND GEODESY
SOFIA
Получена: 03.11.2017 г.
Приета: 16.11.2017 г.
JERK RESPONSE SPECTRUM
A. Taushanov
1
Keywords: jerk, jolt, earthquake, response spectrum, pseudojerk
ABSTRACT
Jerk, also known as jolt, is the rate of change of acceleration. It is a vector quantity and
its scalar magnitude is also the third derivative of position of a body or joint of a structure. Its
dimension is [length/time3]. Excessive "jerky motion" may result in an uncomfortable stay in
buildings, bridges, and ride on trains or trams and it should be designed so as to reduce the
influence of the jerk. This paper presents formulas and graphs for jerk response spectrum
which should be taken into account among the other response spectra (displacement, velocity,
acceleration) in structural engineering design for dynamic loading including earthquake
impact.
1. Introduction
Acceleration response spectrum is widely used in earthquake engineering. Acceleration
is the rate of change of velocity of an object or degree of freedom in a structure with respect to
time. Response spectrum is a plot of the peak value of a response quantity (displacement,
velocity, acceleration, jerk or other) as a function of the natural vibration period of the system,
or a related parameter such as circular frequency. Usually it is presented with several curves for
different damping ratio values.
Elastic response spectrum is a function
 
eTS
defined as "elastic horizontal ground
acceleration response spectrum". At
0T
, the spectral acceleration given by this spectrum
equals to the design ground acceleration on type A ground multiplied by the soil factor S [6].
Sometimes this spectrum is called Elastic Design spectrum.
1
Alexander Taushanov, Assoc. Prof. Dr. Eng., Dept. “Structural Mechanics”, UACEG, 1 Smirnenski
Blvd., Sofia 1046, Bulgaria, e-mail: Taushanov@hotmail.com

40
Design response spectrum
 
dTS
includes the behaviour factor
q
according to [6] or
response reduction factor according to US standard. At
0T
, the spectral acceleration given
by this spectrum equals to the design ground acceleration on type A ground multiplied by the
soil factor S. Sometimes this spectrum is called Inelastic Design spectrum (with specified
ductility property).
Usually if it is not mentioned on the spectrum of which function it comes, it is
considered that it is a spectrum of acceleration, often referred with
a
S
in general, in [6] with
e
S
for elastic and
d
S
for design spectrum. The expression
D
S
is for spectrum of
displacement. The spectrum of jerk will be noted with
j
S
.
Jerk is a vector, and therefore it has direction and scalar magnitude, whose SI units are
m/s3 (metres per second cubed, or m·s−3). Jerk, also known as jolt, is the rate of change of
acceleration. It is a vector quantity and its scalar magnitude is also the third derivative of
position of a body or a joint of a structure. Excessive "jerky motion" may result in an
uncomfortable stay in buildings, bridges, and ride on trains or trams and it should be designed
so as to reduce the influence of jerk. There is a need for strong ground motion data and
additional studies to clarify the effects of the various factors on the intensity and shapes of
response spectrum curves. In developing response spectra for design application, one should
place emphasis on strong ground motions recorded in the region of the site where the spectra
are to be applied [3].
The name of term “jerk” is the most common and therefore preferred for rate of change
of acceleration. It is also recognised in international standards, for example in [12]. Some
authors have used other names as jolt, surge, lurch, terza or TDoA (time derivative of
acceleration). Jerk response spectrum is a plot of the peak or steady-state response of a jerk
function of a series of oscillators.
Figure 1. Horizontal component of the recordings of the 1978 Tabas earthquake in Iran [4]
t
Acc.
[g]
Jerk
[g/s]

41
2. Application in Common Areas
2.1. Jerk in Physics
In the physics of electromagnetism, the Abraham–Lorentz force is the recoil force on an
accelerating charged particle caused by the particle emitting electromagnetic radiation. It is
also called a radiation reaction force. Mathematically, that force is proportional to the jerk (the
derivative of acceleration, or the third derivative of displacement).
2.2. Jerk in Geophysics
In geophysics, a geomagnetic jerk or secular geomagnetic variation impulse is a
relatively sudden change in the second derivative of the Earth's magnetic field with respect to
time [5]. These events were noted by Courtillot and others in 1978. The clearest ones, observed
all over the world, happened in 1969, 1978, 1991, and 1999. It is believed that this
phenomenon originates from the interior of the Earth, but their precise cause is still a matter of
research. Some theories claim that geomagnetic jerk function is connected to strong
earthquakes [7].
2.3. Jerk in Harmonic Vibration Theory
In [14] and [15] the third order differential equations of motion of a SDOF system in
case of free and forced vibrations including the influence of third derivation of motion are
considered and solved. Since jerk’s influence enters in the equation of motion, to derive a
solution one needs not only the initial position and velocity of the DOF, but also its initial
acceleration. However, this obvious problem can be solved with a different approach. In this
Figure 2. Elastic horizontal jerk response spectrum Sj according to (8), in log scale
T
j
S
TD .
TC .

42
case, instead of the initial acceleration, additional condition is specified. This interpretation
restores the coherence of the physical interpretation of the theory.
2.4. Application in Motion of Mechanisms
Jerk is important when evaluating the destructive effect of motion on a mechanism or
the discomfort caused to passengers in a vehicle. The movement of delicate instruments needs
to be kept within specified limits of jerk as well as acceleration to avoid damage. When
designing a train, the engineers will typically be required to keep the jerk less than 2 metres per
second cubed for passengers comfort. In the aerospace industry they even have such a thing as
jerkmeter – an instrument for measuring jerk [8].
2.5. Application in Geometric Design of Roads and Tracks
The principles of geometric design of roads and tracks apply to the jerk oriented
orthogonally to the path of motion. Any change in curvature of the path implies non-zero jerk
arising from purely geometric reasons. To avoid the unbounded (centripetal) jerk when moving
from a straight path into a curve or vice versa, track transition curves are constructed, which
limit the jerk by gradually increasing the curvature, to the value that belongs to the radius of
the circle and the travelling speed. The theoretical optimum is achieved by the Euler spiral,
which is commonly referred as clothoid. This is a curve whose curvature changes linearly with
its curve length. The curvature of a circular curve is equal to the reciprocal of the radius. Using
an Euler spirals in road and railroad engineering guarantees minimal constant jerk whose
maximum value is 0.50 m/s3 in some design codes for convenience purposes. A value of about
0.35 m/s3 is recommended as a limit in railway design.
T
S
2.5S
TA=0
TBTB*
TC
TD
Figure 3. Elastic horizontal ground acceleration response spectra
C
const T
T
T
2
CD
T
const T
T
.const
C
Classical, according to EC8
Obtained, according to
the rule of constant jerk
spectrum for low periods
Se/ag
,1
2
j plat
g
a


S
1

43
In motion control the focus is on straight linear motion, where the need is to move a
system from one steady position to another (point-to-point motion). So, effectively the jerk
resulting from tangential acceleration is under control. Prominent applications are elevators in
people transportation, and the support of tools in machining. It is reported [11] that most
passengers rate a vertical jerk of 2.00 m/s3 in a lift ride as acceptable, 6.00 m/s3 as intolerable.
For a hospital environment 0.70 m/s3 is suggested. In any case, limiting jerk is considered
essential for riding convenience.
For passenger comfort, a train in operation will typically be required to keep jerk below
2.00 m/s3. In the aerospace industry, a type of instrument called jerk-meter is used for
measuring jerk [13].
2.6. Measurements for Seismic Analysis
Important characteristics of jerk are evaluated based on records from the 1999 Chi-Chi
earthquake and one of its aftershocks. It is found out that the maximum of jerk at a free-field
station was over 312 m/s3, and the effective duration, between the first and the last time,
20 m/s3, was almost one minute [10].
Although jerk has potential significance for earthquake engineering, the time-frequency
characteristics of jerk are not exhaustive until now, not to mention the study on the jerk
spectrum [10].
Jerk quantity can be used for analysing a time history record of earthquake. Fig. 1
shows the acceleration and jerk histories of the 1978 Tabas (Iran) earthquake. Spurious spikes
are obvious in the acceleration record at 10.8 and 16 sec. The derivative of the acceleration
trace (to produce the jerk) will convert a spike into a double sided pulse, making it easier to
identify spikes. By doing this, spikes at 12.3, 26 and 33.2 sec are also identified in [4]. From
the bottom panel of the same figure anyone can judge that the maximum jerk is about 120 m/s3
(at 16 sec.).
T
Sj
Figure 4. Elastic horizontal jerk response spectrum
2
C
const T
T
T
3
CD
T
const T
T
const
T
according to Se from EC8
applied constant jerk for
low periods
TA=0
TBTB*
TC
TD
,j plat
S

44
In [16] the principles and specifications (including the sensibility to noise) of a new
sensor for measuring the first order derivative of acceleration are presented. Experimental
results include the calibration result of the jerk sensor by using a low frequency vibration table
and the earthquake wave contrast experiment.
3. Jerk Spectrum for Earthquake Analysis
3.1. According to Recurrent Formula
In earthquake engineering it is usually necessary to calculate only the so-called pseudo
velocity spectral response
 
,
pv S
defined by:
       
0


          




t
pv g
S u t t d
max
, sin exp ,
(1)
instead of velocity spectral response
 
,
vS
, whose expression is:
       
0


          




t
vg
S u t t d
max
, cos exp .
(2)
The other basic desired spectra can be obtained by the relations [3]:
T
2.94
7.36
TB* =0.077; TB=0.10
TC=0.50
Figure 5. Modified elastic horizontal ground acceleration response spectrum
T
– region with smaller ordinates
Se

45
   
   
1
    

     
d pv
pa pv
SS
SS
, , ,
, , .
(3)
For damping values over the range 0 ÷ 0.20, we may use the approximate relation:
   
     
a pv
SS,,
, (4)
which is exact for
0%
. By using the Kelvin model, the maximum force developed in the
mass is:
   
     
s d pa
f k S m S
,max , , ,
(5)
where
k
is the spring stiffness,
m
is the mass of the system. The exact expression of jerk
function
 
jt
in [10] is given by:
   
 
 
 
 
3
32
2
2
2
0
2
43
1
14
1

      




        




   




D
t
Dg
D
t
j t u t d
t
sin
exp .
cos
(6)
For practice purposes in case of low and middle damping we may use shorter
expression of the (pseudo) jerk spectral response
 
,
jS
. The recurrence formula is:
   
   
3
     
     
ja
jd
SS
SS
,,
,,
(8)
or:
3
22




j a d
S S S
TT
. (9)
With the last formula we may analyse the shape of response spectra functions in the
design codes. On Fig. 2 a Jerk Response Spectrum obtained from Horizontal Elastic Response
Spectrum according to [6] for ground type C, agR/g=0.25, =5% is plotted. A logarithmic scale
for better view is used and it is obvious that for the lowest values of natural period (less than
0.03 sec.) the jerk is greater than 1000 m/s3 and tends to infinity. Experimental results show
that the maximum value of jerk during strong earthquakes is about 600 m/s3 [10]. An upper
bound value of jerk (in the design codes) may be applied, so that the Horizontal Elastic Jerk
Response Spectrum has a plateau with a value of Sj,plat. Almost every (National) design code
has a similar shape of Elastic Design spectrum: with zones with constant velocity (between TC
and TD) and constant acceleration (between TB and TC). Hence it follows that the zone between
TA =0 and TB should be with a constant jerk, see Fig. 4.

46
By assuming an upper bound of the jerk (Sj,plat), the jerk spectrum has a shape shown on
Fig. 3. Rightmost point plateau corresponds to a period of TB*. Corresponding function of the
acceleration response spectrum (continuous line) with zero ordinate for TA =0 is placed under
the classical elastic horizontal ground acceleration response spectrum curve with zero ordinate
of S for TA =0 s., see Fig. 2. The last fact is contrary to the common practice [3] to normalize
the intensity of these design spectra to the peak acceleration level:
   
0


pa g
TS T u t max
lim ,
. (10)
The difference is in disregarding the jerk in the dynamic analysis. The proposed theory
leads to zero value of the right-hand side of the equation (10). The slope of the line of the
acceleration spectrum is Sj,plat/2. The value of TB* can be easily found by the rule of three:
2a plat
Bj plat
S
TS
,
*
,
, (11)
where
25
a plat g
S a S
,.
(12)
is according to [6]. The jerk plateau of Sj,plat can be defined also by keeping the value of
TB* = TB . The last idea can be expressed by:
2a plat
j plat B
S
ST
,
,
. (13)
3.2. Modified Relation Between Acceleration and Jerk
As shown above, there is a contradiction between the requirement of normalization of
intensity of the design spectra to the peak acceleration level (10) and the recurrence formula
(8), (9). In order to keep the shape of the Elastic Design spectrum function Se (T) from [6] and
not to specify an infinite value of corresponding jerk for low periods we need different relation
between Acceleration and Jerk spectra. The proposed (modified) formula for the plateau value
of the Elastic Design Jerk spectrum is:
   
 
0
22 5 1

  


  
j plat B e B e
j plat g
B
S S T S
S a S
T
,
,.
(14)
for
B
0 T T
. For
B
T >T
the expression of jerk spectrum is according to the recurrent
relationship:
2

ja
SS
T.
(15)

47
4. Example
Build a jerk response spectrum curve corresponding to ground type C [6] (spectrum
type 1 according Bulgarian National Annex with soil factor
1.20S
), importance factor
1
I

,
0.25
gR
ag
, damping ratio
5%
, for three variants with the following assumptions:
– fixed upper bound value of the jerk and recurrent relationship;
– to keep the period
*
BB
TT
as rightmost point plateau and recurrent relationship;
– to keep the period
*
BB
TT
as rightmost point plateau and modified relationship.
Compare with variants based on recommended spectra types 1 and 2 in [6].
Solution: Assuming an upper bound of the jerk
3
,600 m/s
J plat
S
results in
acceleration plateau
,7.3575
a plat S
(11). With formula (10) the value of
*0.077 sec.
BT
In
that case the acceleration response spectrum function has an additional kink, see Fig. 5.
Assuming an upper bound of the jerk
3
,320 m/s
j plat
S
as mentioned in [10] leads to
*0.144 sec.
BT
, which graphics is similar to Fig. 3.
For the purpose of preserving the value of the period
*0.1 sec.
BB
TT
, the upper
bound of the jerk is
3
,462.3 m/s
j plat
S
, and the whole corresponding Elastic Design
spectrum should be as on the Fig. 6.
T
7.36
TA=0
TB=TB*=0.1
TC
TD
Figure 6. Modified elastic horizontal ground acceleration response spectrum in
case of constant jerk for T≤TB
T
obtained according to rule
of constant jerk spectrum
for low periods
Se
,73.6
2
j plat 

S
1

48
Table 1. Plateau Values of Elastic Design Acceleration and Jerk Spectra
Code (Annex),
type spectrum
(source)
EC8 (BG Annex),
type 1
(far field)
EC8 (Recomm.),
type 1
(far field)
EC8 (Recomm.),
type 2
(near field)
S
(soil factor)
1.20
1.15
1.50
B
T
0.10
s.
0.20
s.
0.10
s.
,a plat
S
7.36
2
m/s
7.05
2
m/s
9.20
2
m/s
,j plat
S
277.4
3
m/s
132.9
3
m/s
346.7
3
m/s
The last analysis with the modified relationship (14) is based on three variants including
far field and near field source [6]. In Bulgarian Annex no type 2 is specified. Using the
proposed formula (14), the source data and the plateau values of the Elastic Design Jerk
spectrum are presented in Tab. 1. These results are similar to the maximum of jerk at the free-
field station records from the 1999 Chi-Chi earthquake (
3
312 m/s
, M7.6).
5. Conclusion
Assuming that the four quantities of motion (displacement, velocity, acceleration and
jerk) are reliant by their spectra on recurrent relationships leads to decreasing the elastic
horizontal ground acceleration response spectrum curve in the low-period region. This will
reduce the influence of the higher natural frequencies in a modal dynamic analysis.
T
7.36
TB
TC
TD
Figure 7. Acceleration and Jerk response spectra according to modified relation
between them (14), with example values, based on BG Annex
T
Se
44.145
1
2.94
g
aS

,
277.4
j plat
S

Sj

49
Using an approach regarding a recurrence formula of jerk spectrum proportional to
acceleration spectrum gives results in contradiction with the common practice of normalizing
the intensity of these design spectra to the peak acceleration level. Additional disadvantage
may appear depending on the way of assuming the connection between second and third
derivative of motion: assuming a value of jerk spectrum plateau leads to additional point in the
acceleration spectrum function. Applying a modified Elastic Design spectrum (with shape as
on Fig. 5) will decrease the influence of higher modes in dynamic analysis. It is necessary to
carry out a more thorough analysis how to include the influence of jerk on the global dynamic
behavior of a structure.
Using an approach with the proposed formula (14) of jerk spectrum proportional to
acceleration gives a good results comparing with ground (earthquake) motion recorded data.
As seen from Tab. 1, near field case in design codes produces (2.6 times-) greater values
of jerk spectra than far field (type 1), which is a good confirmation, because the influence of
the jerk diminishes with the distance.
It is necessary to carry out a more thorough analysis including establishing the influence
of the change of position of the left end of the Elastic Design acceleration spectrum plateau
onto the dynamic behavior of a structure. The results also show two times greater maximum
jerk if the point
B
T
is moved from 0.2 to 0.1 sec. The last fact is expected, because the slope of
the acceleration spectrum curve becomes two times steeper in that region.
Further analysis should reveal the impact of jerk response spectrum function on the
MPMR (Modal Participating Mass Ratio) of the higher modes. However, in any case, it is clear
that for near field source of ground motion more eigenmodes should be taken into account.
Acknowledgement
The scientific work was performed in the department of “Structural Mechanics” of the
University of Architecture, Civil Engineering and Geodesy, Sofia.
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СПЕКТЪР НА РЕАГИРАНЕ ЗА ДЖЪРКОВЕТЕ
A. Таушанов
1
Ключови думи: джърк, земетръс, спектър на реагиране, псевдо-джърк
РЕЗЮМЕ
Джърк (jerk) се нарича степента на изменение на ускорението. Това е векторна
физична величина и нейната скаларна големина е също така третата производна по
време на положението на тяло или възел от конструкция. Единицата за джърк е
[дължина/време3] и може да се представи като втора производна на скоростта по време.
Прекалено „джърково“ движение може да доведе до некомфортен престой в сгради, на
мостове, при движение в превозни средства и при проектиране трябва да се редуцира
влиянието на джърка. В тази статия са представени формули и графики за спектър на
реагиране за джърковете, които трябва да бъдат взимани под внимание заедно с оста-
налите спектри на реагиране (преместване, скорост и ускорение) при проектиране на
строителни конструкции на динамични въздействия, включително земетръс.
1
Александър Таушанов, доц. д-р инж., кат. „Строителна Механика”, УАСГ, бул. „Хр. Смирнен-
ски“ № 1, 1046 София, e-mail: Taushanov@hotmail.com