Acta Mechanica 117, 165-179 (1996)
ACTA MECHANICA
9 Springer-Verlag 1996
Rocking instability of a pulled suitcase with two wheels
R. H. Plaut, Blacksburg, Virginia
(Received November 11, 1994; revised March 1, 1995)
Summary. Rocking motions of a two-wheeled suitcase are considered. The suitcase is pulled on a horizontal
ground and may rock back and forth, first with one wheel in contact with the ground, then the other, and so
on. When a wheel impacts the ground, some energy is lost. It is assumed that the puller's walking motion
induces a periodic force or moment on the handle of the suitcase. In addition, the puller may apply an
additional restoring moment in an attempt to suppress the rocking motion. Under certain conditions, the
motion may grow until the suitcase overturns. The effects of the excitation frequency and the coefficient of the
restoring moment on the critical excitation amplitude are examined for the special case in which yaw and
pitch motions are neglected and the suitcase is pulled in a straight line. Due to the nonlinearities of the
problem, the results exhibit some irregular behavior.
1 Introduction
Based on personal experiences of the a u t h o r and others, a suitcase with two wheels m a y become
unstable as it is pulled. Due to forces at the handle or disturbances from the ground, the suitcase
m a y start to rock sideways a b o u t one wheel (with the other wheel lifting off the ground), then
switch and rock about the other wheel, and so on. Despite the application of a restoring m o m e n t
at the handle, the rocking amplitude m a y increase, and the suitcase may fall onto one of its sides
(i.e. . . . overturn" or ,,topple"), to the annoyance of the puller. This dynamic behavior and
instability are investigated here.
A related p r o b l e m that has been studied extensively is the p l a n a r rocking of a rigid block on
a moving foundation. It is motivated by the effects of earthquakes on flee-standing equipment,
furniture, tombstones, storage tanks, and other structures [1]-[3]. Another related problem is
rollover of vehicles [4] - [6]; however, this instability usually occurs without rocking, and m a y be
described as ,,immediate overturning." Lift-off of some of the wheels of an articulated vehicle
during a lane-change maneuver was described in [7].
A number of assumptions are m a d e in the present study. The surface on which the suitcase is
being pulled is assumed to be rigid and horizontal. The suitcase (with its contents) and its Wheels
are treated as rigid bodies, and the wheels are assumed to roll without slipping. The thickness of
the wheels is neglected. W h e n a wheel has lifted off the ground and then hits the ground again, the
loss of energy at impact is modeled using a coefficient of restitution. At least one wheel is in
contact with the g r o u n d at all times until overturning occurs (i.e., no ,,bouncing"' is considered).
In the examples, the puller is assumed to walk in a straight line and to impart a periodic m o m e n t
or force to the handle of the suitcase. W h e n the suitcase rocks, the puller sometimes applies an
additional m o m e n t in an attempt to restore the suitcase to a vertical state with both wheels on the
ground.
166
R . H . Plaut
The problem will be formulated in the following Section. In Section 3, small motions about
a steady motion will be discussed. A special case will be examined in Section 4, in which yaw
motion is neglected and the pitch angle is assumed to remain constant. Numerical results for this
special case will be presented, followed by concluding remarks in Section 5.
2 Formulation
The suitcase is depicted in Fig. 1. The author's suitcase which motivated this study has the
following dimensions:
H = 48 cm, D = 70 cm, g~ = 8 cm, g2 = 14 cm, B = 18.4 cm, Rw = 1.9 cm,
ho = 0.6 cm, do = 4.1 cm, da = 12 cm, h3 = 34 cm, h4 = 5 cm.
(1)
Its handle has a T shape and is located eccentrically on the front.
It is assumed in the formulation in this Section that the right wheel (as viewed from the back)
is in contact with the ground and the left wheel is uplifted. The coordinate system X, Y,, Z, with
origin O, is fixed in space, with the ground being the X Y plane (see Fig. 2). The wheel is thin and
circular, and its contact point has position vector rxy = xi + yj where the unit vectors i,j, k are
parallel to X, Y, Z, respectively. The angle between the contact tangent and the positive X axis is
denoted ~b, positive as shown, which represents yaw. The xl, yt, z~ axes have their origin at the
contact point, and their unit vectors il, ji, k~ are obtained from i, j, k by rotation about the
vertical axis through the angle ~b, so that
{it [cos sin i]{/t
Jl
=
kl
-sinq5
cosq5
j
0
0
k
[4
.
(2)
,I
D
z5
ot
p x5
h3
Rw ~
1
-'~.',
(
Jr
Lr-
'ao"
dl
N- d34,,I
=-I
(a) Side view
, gl.._,_ g2 ,,
z5
H
l
I
'5 d
D- . . . . .
(b) Front view
'5
I,
(c) Back view
Fig. 1. G e o m e t r y of suitcase
Rocking instability of a pulled suitcase with two wheels
167
z
~
j ~ ' x ,
~,, o)
(a)
Y4
x3
Zl
Y2 le~ z2
r
.•
x2
(b)
(c)
Fig. 2. C o o r d i n a t e s y s t e m s
The rocking angle (i.e., the angle of inclination of the wheel from the vertical) is denoted 0 and
is positive as shown in Fig. 2. The x2, Y2, z2 axes are obtained by rotation about the xl axis by the
angle 0, so that z2 lies in the plane of the wheel and passes through its center. The unit vectors are
given by
{'t [ '~ ~
J2
=
k2
0
0
cos 0
--sin 0
sin 0
COS 0 l
jt
kl
9
(3)
In Figs. 1 and 2, the point J lies on the line connecting the two wheels and is in the plane that
contains the center of mass and the principal axes xs, z5 of the suitcase. The axes x3, Ya, za have
their origin at J and are parallel to x2, Y2, z2, so that/3 = &,J3 =J2, k3 = k2, and the vector
connecting their origins is r2a = Rwk2 + b,j2 where Rw is the radius of the wheel and b, is the
distance from the wheel to J.
The pitch angle is denoted r and is positive as shown in Fig. 2 c. Rotation of the x3, Y3, z3 axes
about Y3 by the angle r gives x4, Y4, z4, so that
f} I c~
j4
k4
=
0
--
sin r
1
0
cos r J
j3
k3
9
(4)
Finally, axes xs, ys, z5 have their origin at the center of mass C of the suitcase (without its wheels)
and are its principal directions (see Fig. 1). They are assumed to be parallel to x4, Y4, z4, and the
origins are connected by the vector r45 = hlk4 + dli4.
It follows that the position vector from O to the center of mass C is given by
rc = xi + yj + R w k 2 + brj 2 + h~k4 + dti4.
(5)
The expression for r, in terms of the unit vectors i, j, k is given in the Appendix. Then the
translational kinetic energy of the suitcase (without wheels), with mass ms, is
1
T~,,, = ~- m~c'~c.
(6)
168
R.H. Plaut
With the use of Eq. (A1), Ts,tr can be expressed in terms of x(t), y(t), O(t), O(t), •(t), and their first
derivatives.
The rotational kinetic energy of the suitcase is given by [8]
1
%,ro~ = ~- ms' ts'm~
(7)
where Is is the inertia dyadic
Is = I~isi~ + Iy.isj~ + I~ksk5
(8)
and ms is the angular velocity vector
ms = Oil - ~ja + (bkl,
(9.1)
i.e.,
ms = 0il - ~ cos Oj~ + (~ - (J sin 0) k~.
(9.2)
In Eq. (8), Ix, Iy, I~ are the principal mass moments of inertia with respect to the axes xs, Ys,
z5 through C, respectively. If Eq. (8) is put in terms of il, jl, kl, Eqs. (7)-(9) lead to
1
1
1
T~.~ot = ~ I,(O cos ~ + 4 sin ~ cos 0) 2 --~ ~ / y ( 4 sin 0 - //))2 _~ 2- I~(0 sin ~ -- q~ cos ~0 cos 0) 2 .
(10)
The kinetic energy of the right wheel involves its rolling angle ~, shown in Fig. 1. The position
vector from O to the center of the wheel is
rw = xi + yj + R~k2,
(11.1)
i.e.,
rw = (x + Rw sin 0 sin q~) i + (y - R~ sin 0 cos 4 ) j + (R~ cos O)k,
(11.2)
and the translational kinetic energy is
1
Yw,tr .= ~ mwi*w "tkw
(12)
where m,~ is the mass of the wheel and
/3w"J;w = 22 _~. ~72 d- Rw202 -}- Rw2~ 2 sin 2 0 + 2R~2(~ cos ~b sin 0 + 0 cos 0 sin 4)
(13)
+ 2Rw~((~ sin 0 sin 4~ - 0 cos 0 cos q~).
If A~ and Cw are the mass moments of inertia of the right wheel about a diameter and about
the axle, respectively, then the inertia dyadic is
I w = Awili 1 + Cwj2j2 + Awk2k2,
(14)
and the angular velocity vector is
(151
Rocking instability of a pulled suitcase with two wheels
169
This leads to the rotational kinetic energy
1
T,~,rot = ~1 Aw(02 + (~2 cos 2 0) "}- 7 Cw(~r + q$ sin 0) 2
(16)
which is needed since a, will be involved in the rolling constraint.
The kinetic energy of the uplifted left wheel is neglected. Thus, the total kinetic energy T
is taken to be the sum of the quantities in Eqs. (6), (10), (12), and (16). The total potential energy
V of the suitcase and two wheels, due to gravity, is given by
V= msgrc" k + mwg(2Rwk 2 + Bj3)" k,
(17.1)
i.e.,
V= msg(Rw cos 0 + b, sin 0 + hi cos ~Ocos 0 + dl sin ff cos 0) + mwg(2Rw cos 0 + B sin 0).
(17.2)
The equations defining the constraint that the wheel rolls without slipping are [9]
- Rw~r cos q~ = 0,
(18.1)
R~c~, sin ~b = 0.
(18.2)
3) -
Lagrange's equations are used to obtain the equations of motion. The rolling constraint
equations are included with Lagrange multipliers 21 and 2z. In terms of the Lagrangian
L = T - - V, the equations are [t0]
dt ~
- ~xx + 21 = Q~,
(19.1)
dt
- ~yy + 22 = Qy,
(19.2)
dt
~(~ - Qo,
(19.3)
d(O/)
dt ~
~L
O0 = Oo,
(19.4)
d5 ~
d
( OL~
- ~
-
aL
= oo'
-
2~Rw cos q~ -- 22R~ sin q~ = Q,,
(19.5)
(19.6)
where the Q's are generalized forces associated with actions at the handle of the suitcase.
Equations (19.1) and (19.2) can be solved for 21 and 22. The velocities 2, ;9 can be obtained
from Eqs. (18.1) and (18.2), and Y, ~ can be found by differentiation. With the use of these
relations, Eqs. (19.3)-(19.6) can be written in terms of ~b, O, ~, ~,, and their first two derivatives.
The second derivatives appear linearly, so that the equations can be set up for standard
techniques of numerical integration. If the generalized forces are specified, and initial conditions
on qS, 0, ~9, c~, and their first derivatives are chosen, the ensuing motion can be computed
numerically.
170
R.H. Plaut
Equations (19) are only valid if the left wheel is off the ground, i.e., if 0 > 0. A similar set of
equations can be derived for the case 0 < 0, when the suitcase rolls on its left wheel and the right
wheel is uplifted. At the moment of transition from rocking about one wheel to rocking about the
other, an impact occurs as the uplifted wheel hits the ground [10]. The behavior of the related
problem of impact of a rocking block has been discussed extensively, and often the loss of energy
is modeled analytically by a coefficient of restitution [2], [11].
3 Small motions
For the case of small motions about a steady motion, an approximate set of linear equations can
be obtained. The problem still is nonlinear if the suitcase rocks from one wheel to the other.
Assume that the suitcase is being pulled at constant velocity v parallel to the X axis, with both
wheels rolling on the ground, and that the pitch has the constant value ~o. The corresponding
constant angular velocity of the wheels is denoted ~o, and the coordinates 0, ~b, and y are zero. For
small motions about this steady motion, with the left wheel uplifted, O(t) > 0 and the quantities
O(t), ~(t), qJa(O, ~a(t), 2a(t), and 3)(t) are assumed to be small, where
~(t) = v + ~d(t),
~,(0 = ~o + ~d(t),
0(t) = 0 o + ~,d(t).
(20)
With the use of Eqs. (18) and (20), one can obtain the linear relations
~o = v/Rw,
xa(t) = Rw&a(t),
p(t) = v~)(t),
(21)
Ji(t) = Rwdie(t),
y(t) = vd)(t).
After eliminating x, y, 21, and 22 from Eqs. (19) and neglecting nonlinear terms, the resulting
linear equations have the form
(22)
M~(t) + D(l(t) + Kq(t) = f
where
(23)
q = [O~Oeae] T.
The elements of the symmetric matrix M are listed in the Appendix. The matrix D has only
three nonzero elements, which are proportional to the velocity v. They are
d12 = - ( m f l c
+ m~Rw) v - (C~/Rw) v,
d21 ---- (Cw/Rw)
v,
(24)
d22 =
mfl~v
where H~ is the height of the center of mass C the suitcase above the ground, i.e.,
H~ = R~ + H~w,
H~w = hi cos ~o + dl sin ~o,
(25)
and d~ is the distance from the contact points of the wheels to C parallel to the X axis, i.e.,
d~ = dl cos 0o - ht sin Oo,
in the steady motion. The velocity v only appears in the matrix D in Eq. (22).
(26)
Rocking instabifity of a pulled suitcase with two wheels
171
The matrix K in Eq. (22) also has only three nonzero elements, given by
ka 1 = - msgH~ - 2mwgR~,
k33 = - m~gH~,
k42
=
--
R~Q r.
(27)
The components of the column vector f are
f l = Qo - m s g b , - mwgB,
f z = Qo,
(28)
f3 = Q~, - msgdc,
f4 = Oa + R~Q~.
The second and third terms in fa represent the moments, about the contact point, of the weight of
the suitcase and the uplifted wheel, respectively, that resist rocking. One cannot simply p u t f = 0
to investigate small rocking vibrations, due to the restoring moments and the constraint on the
sign of 0 (i.e., 0 > 0 for Eq (22) to be valid).
4 Rocking without pitch or yaw
4.1 Equation o f motion
As a special case, it is assumed that the suitcase travels in a constant direction and rocks but does
not have yaw or pitch motions. Therefore q~(t)= 0, ~ ( t ) = ~9o, y ( t ) = 0, and the varying
generalized coordinates are x(t), O(t), and a,(t).
For rocking about the right wheel, it follows from Eqs. (6), (10), (12), (13), (16), and (A.1) that
t;c = :~i - (b~ sin 0 + Hc cos 0) Oj + (b, cos 0 -- Hc sin 0) Ok,
(29.1)
1
Ts,tr = ~ ms(X 2 + br202 + Hc202),
(29.2)
1
1
"2
T~,rot = ~ I~02 cos 2 ~o + ~ IzO sin 2 ~o,
1
T~,t, = ~ mw(x 2 + Rw202),
1
1
Tw.rot = ~- AwO 2 q- ~ Cwar 2.
(29.3)
(29.4)
(29.5)
The equations of motion (19.1), (19.4), and (19.6) and constraint (18.1) lead to the relations
~, = 21Rw,
(30.1)
{ms + m w + [Cw/(Rw2)]} 5~ = Q,,,
(30.2)
I0" + Mb cos 0 -- Mh sin 0 = Qo,
(30.3)
where
I = m~(Hc2 + b, 2) + mwR,, 2 + Aw + Ix cos 2 ~o + Iz sin 2 ~o,
(31.1)
Mb = msgbr + m~gB,
(31.2)
Mh = msgHc + 2mwgRw.
(31.3)
172
R.H. Plaut
It is assumed now that the two wheels are identical and bz = hr. When the right wheel is
uplifted and the suitcase rolls on the left wheel (i.e., 0 < 0), the sign of the second term in Eq. (30.3)
changes. Therefore one can write the governing equation for rocking motion in the form
(32)
IO" q- SMb cos 0 - Mh sin 0 = Qo
where
S= +1
if
0>0,
S=-I
if
0 < 0.
(33)
Equation (32) has the same form as the governing equation for planar rocking of a rigid block
[2]. If the block is rectangular and homogeneous with mass too, base bo, and height ho, then the
quantities I, Mb, and Mh in Eq. (32) correspond to 4mo(ho 2 + bo2)/3 (the mass moment of inertia
about either bottom corner), mogbo/2, and mogho/2, respectively. As in many of the studies of the
rocking block, a coefficient of restitution e (0 < e < 1) is utilized here to model the loss of energy
that occurs instantaneously when one wheel lands on the ground (i.e., 0 changes sign). The
angular velocities 0- just before impact and 0 + just after impact are assumed to satisfy the
relation
0 + = e0-.
(34)
The generalized force Qo in Eq. (32) represents a moment about the contact point of the wheel
with the ground, due to actions on the handle of the suitcase. In most of the examples to be
discussed, it is assumed that the puller applies a periodic moment, dependent on the speed of
walking, and also a restoring moment to try to reduce rocking motion. The restoring moment in
this initial study is assumed to be simply proportional to the rocking angle 0. Hence Qo is
assumed to have the form
Qo = qo sin (oat + II) - koO,
qo >- O,
(35)
ko >_ O.
It is convenient to introduce the nondimensional quantities
z = t]/Mh/I,
~ = Mb/Mh,
A = qo/Mh, f2 = o9 I l l ' h ,
fl = ko/Mh.
(36)
Then Eqs. (32) and (35) lead to the nondimensional equation of motion
d20
dz z + S? cos 0 - sin 0 = A sin (f2z + 17) - flO.
If 0 o = 0 - 3 5 , the suitcase data in Eq.(1) lead to I = 3 . 8 4 k g m
Mh = 81.3 kg m2/se&, Hc = 36.5 cm, and dc = 24.6 cm.
(37)
2, Mb = 2 0 . 2 k g m Z / s e c 2,
4.2 Small motions
If the rocking angle 0 is assumed to be small, Eq. (37) can be approximated by the equation
dZO
dr2 + ( / ? - l ) 0 = - S T + A s i n ( O r + q )
(38)
which is linear between impacts and has a simple analytical solution. However, the times at which
impacts occur usually must be computed numerically. One exception is the case A = 0, where
periodic forcing is not included [12]. For example, assume that A = 0 and that the suitcase is
Rocking instability of a pulled suitcase with two wheels
173
given an initial impulse leading to initial conditions 0(0) = 0, 0(0) = Wo. Let the impact times be
denoted z = % with Zo = 0, and let 0 = Ok and S = Sk for Zk- 1 < Z < Zk. (If W0 > 0, then 01 > 0
and Sk = (--1)k+1; if W0 < 0, Sk = (--1)k.) Also, let Wk = O+(Tk), SO that Ok has the initial
conditions Ok("(,k - 1) = O, Ok("(.k - 1) = W k - 1 "
If fl < 1, the solution for this example is
Wk- 1 9
Ok(V) = ~ - - - smh [2(z - Zk- 1)] +
1
Zk
{1 -- cosh [2(z -- Zk- t)]},
[?Sk + 2 W k - 1 ]
)CWk- l J
(39.1)
(39.2)
Zk- 1 + ~- In L?Sk
(39.3)
Wk = eOk(Zk)
where 2 = l//1 -- ft. If fl > 1 and tr = l//fl - 1, the solution is
7S~ {cos [cr(z -- Zk- 1)] -- 1},
0k(V) = Wko.
-~ sin [~r(z -- zk- ~)] + -~1
Zk = Zk- 1 + -- COS- 1
(40.1)
[-7 2 - aEw2-1]
_ _
(40,2)
along with Eq. (39.3). These solutions assume that the initial velocity Wo is not large enough to
cause immediate overturning (without rocking). The motions decay if the coefficient of
restitution e is less than unity.
4.3 Numerical results
In this Section, the nonlinear equation of motion (37) is integrated numerically for twenty cycles
of excitation (i.e., from z = 0 till z = 40 n/f~). Impact times are computed to high accuracy by
repeated reduction of the time step when 0 approaches zero. The numerical procedure was
validated with the use of the analytical solutions in Eqs. (39) and (40). If 0 reaches +_n/2, the
suitcase is said to have overturned, and the lowest positive value of A for which overturning
occurs is called the critical excitation amplitude Act.
In the results, ~ = 0.248, which corresponds to the suitcase described in Eq. (1) if the steady
pitch angle is 00 = 0.35. The coefficient of restitution is taken to be e = 0.913, corresponding to
purely inelastic impacts [3]. The excitation phase q is chosen so that the excitation A sin t/at z = 0
is equal to the restoring m o m e n t about either wheel when the suitcase is vertical, which is given
by 7 in nondimensional terms (in the examples to be treated, A > 7).
The remaining nondimensional parameters are the excitation amplitude A, excitation
frequency f2, restoring force coefficient r, and initial conditions 0(0) and 0(0). Unless otherwise
stated, 0(0) and 0(0) will be equal to zero. The average step frequency of a person walking is
approximately 2 Hz [13]. This corresponds to a sideways frequency of 1 Hz [14] and the
nondimensional value f2 = 1.37 based on the data in Eq. (1). In the examples, the range
1 < O < 2 will be considered.
Tables 1 and 2 present results for various combinations of A and f2 when fl = 0 and fl = 0.5,
respectively. The symbol ,,N" indicates that the suitcase does not overturn, while a dash
represents overturning. In Table 1, for t2 --- 1.0, 1.1, ..., 2.0, respectively, the critical values Ac, are
0.250, 0.259, 0.258, 0.259, 0.259, 0.258, 0.258, 0.258, 0.259, 0.252, and 0.250, and in Table 2 the
174
R . H . Plaut
Table 1. Combinations of A and Q for non-overturning (N) and overturning ( - ) when fi = 0
A
Q
1.0
0.280
0.279
0.278
0.277
0.276
0.275
0.274
0.273
0.272
0.271
0.270
0.269
0.268
0.267
0.266
0.265
0.264
0.263
0.262
0.261
0.260
0.259
0.258
0.257
0.256
0.255
0.254
0.253
0.252
0.251
0.250
0.249
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N
N
N
N
N
N
-N
1.1
1.2
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1.3
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N
N
N
N
N
N
N
N
N
N
1.4
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N
N
N
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N
N
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N
1.5
1.6
1.7
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N
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N
N
N
N
N
N
N
N
N
N
1.8
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N
N
N
N
N
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N
N
N
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N
N
N
N
N
N
N
N
N
2.0
-
-
N
N
-N
N
N
N
N
N
N
N
N
N
N
N
N
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N
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N
N
N
N
N
N
N
N
N
1.9
N
N
N
N
N
N
N
N
N
N
c o r r e s p o n d i n g values o f Act are e i t h e r 0.259 or 0.260 e x c e p t at ~2 = 2.0, w h e r e Act = 0.266. F o r
a fixed e x c i t a t i o n f r e q u e n c y , if A is g r e a t e r t h a n Act, the suitcase d o e s n o t n e c e s s a r i l y o v e r t u r n .
T h e p r e s e n c e of the r e s t o r i n g m o m e n t at the h a n d l e a d d s m a n y m o r e n o n - o v e r t u r n i n g
c o m b i n a t i o n s , especially a b o v e the critical e x c i t a t i o n a m p l i t u d e . As for the case o f b a s e e x c i t a t i o n
o f a rigid b l o c k [3], the b e h a v i o r h e r e is erratic.
F i g u r e 3 s h o w s t h e v a r i a t i o n o f Ac, as a f u n c t i o n of f2 for h i g h e r values o f fl, i.e., fl = 1.0, 1.5,
2.0, a n d 2.5. Values w e r e c o m p u t e d at g2 = 1.00, 1.05 . . . . . 2.00, a n d c o n n e c t e d w i t h s t r a i g h t
s e g m e n t s . A n i n c r e a s e in t h e r e s t o r i n g force coefficient ]3 d o e s n o t a l w a y s lead t o a n i n c r e a s e in the
critical e x c i t a t i o n a m p l i t u d e . E a c h o f t h e f o u r curves is a s s o c i a t e d w i t h the l o w e s t value o f Act for
s o m e r a n g e o f f2.
T h e effect o f i n c r e a s i n g t h e r e s t o r i n g force coefficient/3 d e p e n d s o n t h e e x c i t a t i o n f r e q u e n c y ~ .
T h e case of~2 = 1.37, c o r r e s p o n d i n g t o t h e a v e r a g e w a l k i n g speed, is i l l u s t r a t e d in Fig. 4. R e s u l t s
f o r / 3 = 0, 0.1, 0.2 . . . . . 5.0 are c o n n e c t e d w i t h s t r a i g h t s e g m e n t s . As ~2 is i n c r e a s e d , the value o f
Act is a r o u n d 0.259 till ~2 -- 0.7, t h e n i n c r e a s e s a n d h a s a m a x i m u m o f 1.016 at g2 = 1.2, t h e n
d e c r e a s e s to a m i n i m u m of 0.269 at f2 = 2.4, a n d finally i n c r e a s e s as t h e r e s t o r i n g force coefficient
i n c r e a s e s further.
Rocking instability of a pulled suitcase with two wheels
175
Table 2. Combinations of A and f2 for non-overturning (N) and overturning ( - ) when/~ = 0.5
A
O
1.0
0.280
0.279
0.278
0.277
0.276
0.275
0.274
0.273
0.272
0.271
0.270
0.269
0.268
0.267
0.266
0.265
0.264
0.263
0.262
0.261
0.260
0.259
0.258
0.257
0.256
0.255
0.254
0.253
0.252
0.251
0.250
0.249
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
-N
N
N
N
N
N
N
N
N
N
1.1
.
.
.
.
.
1.2
.
.
.
.
1.3
.
.
.
.
.
.
.
.
.
.
.
N
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
N
N
N
N
N
N
N
N
N
N
1.8
1.9
2.0
N
N
N
N
.
.
N
N
N
N
N
N
N
.
N
N
N
N
N
N
N
--N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
.
.
N
.
N
N
.
.
.
.
.
1.7
.
.
.
N
---
.
.
.
.
.
.
.
.
.
.
.
.
.
.
N
N
N
N
N
N
N
N
N
N
.
N
N
N
N
N
N
N
N
N
N
.
.
.
N
N
N
N
.
.
.
.
N
.
.
.
.
.
.
.
.
1.6
N
N
.
.
.
1.5
.
.
.
.
.
.
.
.
.
.
.
.
.
.
N
N
N
N
N
N
N
N
N
N
N
1.4
.
-
-
.
.
.
.
.
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
.
.
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
--
.
.
.
N
N
N
N
N
N
N
N
N
N
.
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
T h e effect o f a n initial i m p u l s e d u e t o a d i s t u r b a n c e is d e p i c t e d in Fig. 5. T h e r o c k i n g a n g l e is
a s s u m e d t o h a v e a n initial p o s i t i v e a n g u l a r velocity in t h e s a m e d i r e c t i o n as t h e p e r i o d i c m o m e n t ,
l e a d i n g t o initial r o c k i n g a b o u t t h e r i g h t wheel. I n Fig. 5 t h e e x c i t a t i o n f r e q u e n c y is O = 1.37 a n d
t h e r e s t o r i n g force coefficient is fl = 1. T h e critical e x c i t a t i o n a m p l i t u d e for o v e r t u r n i n g d e c r e a s e s
m o n o t o n i c a l l y as t h e initial a n g u l a r v e l o c i t y increases, w h i c h is n o t surprising.
It h a s b e e n a s s u m e d in this S e c t i o n t h a t t h e puller exerts a p e r i o d i c m o m e n t a b o u t
the contact point of the suitcase with the ground. N o w suppose that a periodic horizontal
force is a p p l i e d , r a t h e r t h a n a m o m e n t . L e t F d e n o t e t h e a m p l i t u d e o f t h e force a n d ,
in a vertical p r o j e c t i o n o f the b a c k view o f the s u i t c a s e at t h e c o n s t a n t p i t c h angle
with no
r o c k i n g , let R1 a n d
R2 b e t h e d i s t a n c e s f r o m the h a n d l e t o t h e r i g h t a n d
left w h e e l c o n t a c t p o i n t s , respectively, a n d let ~1 a n d 42 be t h e angles o f t h e s e d i a g o n a l s
w i t h t h e vertical. T h e p e r i o d i c p a r t o f
Qo in Eq. (35) is r e p l a c e d b y
FR1 cos (41 - 0) sin (cot + t/)
if
0 > 0,
Qo = [FR2 cos (42 -~- 0) sin (cot + t/)
if
0 < 0.
I
(41)
176
R. H. P l a u t
2.0
I"$=1.5
"'~'"
1.5
Acr
"
/
/"
,
1.0 :
1.2
1.4
1.6
1.8
2.0
Fig. 3. Variation of critical excitation amplitude with excitation frequency
The resulting equation of motion is similar to that of a rigid block with harmonic, horizontal,
base excitation [2].
If the handle is not located symmetrically with respect to the two wheels, as in Eq. (1), Qois
discontinuous when the suitcase rocks from one wheel to the other (i.e., when 0 changes sign).
Based on the data in Eq. (1) and ~o = 0.35, one obtains R1 = 68.6 cm, R 2 = 69.3 cm, 41 = 0.088,
and ~z = 0.174. With/~ = 0, the critical excitation amplitude was computed for this case and also
for the case in which the handle was located in the symmetric position, halfway between the two
wheels. The critical values were the same to three digits, indicating that the eccentricity of the
handle in this example does not have a significant effect on the overturning of the suitcase.
5 Concluding remarks
Rocking motions of a two-wheeled suitcase have been examined. The general equations of
motion were formulated. It was assumed that the ground is rigid and horizontal, the suitcase is
a rigid body, one wheel is in contact with the ground during rocking motion and it rolls without
slipping, and loss of energy only occurs during impact of the uplifted wheel with the ground.
Numerical results were presented for the special case of motion in a straight line with no variation
of yaw or pitch. In this case the problem can be reduced to the study of a single equation in the
3.0
2.5
2.0
Acr ~.5
1.0
0.5
O
. . . .
0
=
. . . .
i
1
. . . .
i
. . . .
J
. . . .
i
2
. . . .
I
3
i
,
,
i
I
i
,
,
,
I
4
,
,
~
~
I
i
~
i
i
5
8
Fig. 4. Variation of critical excitation a m p l i t u d e with restoring force coefficient
177
Rocking instability of a pulled suitcase with two wheels
1.00
0.95
0.90
Acr
0.85
o.8o
'.
0.75
0
. . . . . . .
i ....
0.02
j ....
, ....
0.04
, ....
i . . . . . . . . . . . . . . . . . .
0.06
0.08
0.10
Fig. 5. Variation of critical excitation amplitude with initial rocking angular velocity
rocking angle. It was assumed that the puller exerts a periodic moment or force on the handle of
the suitcase, and possibly also a restoring moment proportional to the rocking angle.
The problem involves nonlinearities due to large motions, switching of contact from one
wheel to the other, and loss of energy at impact (modeled with the use of a coefficient of
restitution). These nonlinearities lead to some irregular behavior. In Tables 1 and 2 it is seen that
the suitcase may overturn at one value of the excitation amplitude but not overturn at a higher
value (with the same excitation frequency). The variation of the critical excitation amplitude
Ac, with the excitation frequency ~ is not monotonic. Figures 3 and 4 demonstrate that an
increase in the restoring moment does not necessarily increase the stability against overturning.
In the related rocking block problem, the critical amplitude of foundation acceleration is
approximately constant for small excitation frequencies Q and then tends to increase as
s increases [3]. Here it is seen in Fig. 3 that Ac, also tends to increase as Q is increased above the
average walking speed o f ~ = 1.37. According to experience, however, suitcases tend to be more
susceptible to rocking and overturning at higher walking speeds, for which the excitation
frequency Q at the handle is increased. This apparent discrepancy may be partially explained by
a corresponding increase in excitation amplitude A as the puller walks faster. The puller's (Q, A)
curve, for example, may be lower than the (Q, A+r)curve in Fig. 3 at low walking speeds, but then
may intersect it at some value of ~2 and thus cause overturning.
Many extensions of this study could be considered. Numerical results for the full set of
equations of motion should be obtained. Among the effects that could be examined are the
following: (a) variations in the suitcase dimensions; (b) pulling of the suitcase on a curved path,
such as turning a corner; (c) persistent disturbances due to a rough ground, including the case of
regular impulses caused by a tiled floor with grooves; (d) a ground that is not horizontal; (e)
a time delay in the restoring moment, which could be destabilizing (e.g., [15]); (f) a nonlinear
restoring moment; and (g) sliding or bouncing when the uplifted wheel impacts the ground (e.g.,
[16]). The possibility of quasi-periodic motions, chaotic motions, and fractal behavior, which
occur for the rocking block under horizontal base oscillation (e.g., [2], [17]), also could be
investigated.
Acknowledgements
The author wishes to acknowledge S. Suherman for carrying out the numerical computations for the tables
and figures, and S. L. Hendricks and C. M. McCourt for helpful discussions.
178
R.H. Plaut
Appendix
ff Eq. (5) is put in terms of the unit vectors i, j, k which are fixed in space, the result is
(A.1)
rc = r l i + r z j + r 3 k
where
ri = x + (dl cos ~ - hi sin ~b) cos q~ - br cos 0 sin ~b + (Rw + hi cos ~ + dl sin ~) sin 0 sin ~b,
(A.2)
r2 = y + (dl cos ~ - hi sin ~) sin q5 + b, cos 0 cos ~b - (Rw + hlcos ~ + dl sin Ip) sin 0 cos qS,
(h.3)
ra = (Rw + ht cos O + dl sin ~) cos 0 + br sin 0.
(A.4)
The upper triangular elements of the symmetric matrix M in Eq. (22) are as follows:
roll = A w + m w R w 2 + m~(Hc z + br z) + Ix cos 2 ~o + Iz sin 2 Oo,
m l 2 = (Ix - Iz) sin ~0 cos ~0 - msHcdc,
mi4 = O,
m13 = rn~brdc,
m22 : A w q- ms(de 2 q- br 2) q- Ix sin 2 ~0 + Iy cos 2 ~Po,
m23 = m~b~Hcw,
m2~ = - m s b , R w ,
m33 = Iy + msdc 2 + rnsH~2 ,
rrt44 = Cw q- (ms q- row)Rw 2 9
m34. = - m ~ R w H c w ,
(A.5)
References
[1] Ishiyama, Y.: Review and discussion on overturning of bodies by earthquake motions. Building
Research Institute, Ministry of Construction, Japan, Research Paper No. 85 (1980).
[2] Augusti, G., Sinopoli, A.: Modelling the dynamics of large block structures. Meccanica 27, 195- 211
(1992).
[3] Plaut, R. H., Suherman, S.: Fractal boundary for overturning of rigid blocks under base excitation. In:
Nonlinear dynamics: the Richard Rand 50th birthday volume (Guran, A., ed.). Singapore: World
Scientific 1995.
[4] Verma, M. K., Gillespie, T. D.: Roll dynamics of commercial vehicles. Vehicle Sys. Dyn. 9, 1 - 1 7 (1980).
[5] Nalecz, A. G.: Influence of vehicle and roadway factors on the dynamics of tripped rollover. Int.
J. Vehicle Design 10, 321--343 (1989).
[6] Rakheja, S., Ranganathan, R., Sankar, S.: Field testing and validation of directional dynamics model of
a tank truck. Int. J. Vehicle Design 13, 251-275 (1992).
[7] Mallikarjunarao, C., Segel, L.: A study of the directional and roll dynamics of multiple-articulated
vehicles. In: The dynamics of vehicles on roads and on tracks (Wickens, A. H., ed.). Lisse: Swets
& Zeitlinger 1982.
[8] Meirovitch, L.: Methods of analytical dynamics. New York: McGraw-Hill 1970.
[9] Rosenberg, R. M.: Analytical dynamics of discrete systems. New York: Plenum Press 1977.
Rocking instability of a pulled suitcase with two wheels
179
[10] Neimark, Ju. I., Fufaev, N. A.: Dynamics of nonholonomic systems. Providence, Rhode Island:
American Mathematical Society 1972.
[11] Smith, C. E., Liu, E-P.: Coefficients of restitution. J. Appl. Mech. 59, 963-969 (1992).
[12] Perry, J.: Note on the rocking of a column. Trans. Seismological Soc. Japan. 3, 103-106 (1881).
[13] Murray, T. M.: Building floor vibrations. Eng. J. AISC 28, 102-109 (1991).
[14] Fujino, Y., Pacheco, B. M., Nakamura, S.-I., Warnitchai, P.: Synchronization of human walking
observed during lateral vibration of a congested pedestrian bridge. Earthquake Eng. Struc. Dyn. 22,
741--758 (1993).
[15] Bhatt, S. J., Hsu, C. S.: Stability charts for second-order dynamical systems with time lag. J. Appl. Mech.
33, 119-124 (1966).
[16] Shenton, H. W. III, Jones, N. P.: Base excitation of rigid bodies, I: formulation. J. Eng. Mech. 117,
2286-2306 (1991).
[17] Yim, S. C. S., Lin, H.: Nonlinear impact and chaotic response of slender rocking objects. J. Eng. Mech.
117, 2079-2100 (1991).
Author's address: Prof. R. H. Plaut, The Charles E. Via, Jr. Department of Civil Engineering, Virginia
Polytechnic Institute and State University, Blacksburg, Virginia 24061-0105, U.S.A.