Suspension device for low-frequency structures

A suspension device is provided for simulating the free-free boundary conditions of space for a low frequency structure. A support cable is connected at one end to the test structure and is vertically guided by a guiding ring. The other end of the cable is connected to a cam having an outer circumference which supports the cable. A drive axle passes through the cam center of rotation and is rotatably journalled in a suitable manner to a rigid structure. Two torsion springs are provided about the drive axle. One end of each spring is connected to a respective face of the cam and the other end is connected to the fixed support. The cam is shaped and the torsion springs selected such that Wr.sub.(t) =T.sub.s(t), wherein W is the weight of the test structure; r.sub.(t) is the instantaneous moment arm defined as the perpendicular distance from the rotational center of the cam to the cable at time t, and T.sub.s(t) is the total spring torque exerted by the two springs on the cam at time t. The test structure is accordingly vertically suspended by the cable and the instantaneous moment arm compensates for any increased spring torque arising from a vertical displacement of the test structure to simulate space conditions.

BACKGROUND OF THE INVENTION 
1. Field of the Invention 
The present invention relates generally to ground-based dynamic testing of 
low-frequency structures and more particularly to a suspension device for 
testing such structures. 
2. Description of the Related Art 
Space structures in general experience free-free boundary conditions that 
are not readily replicable in ground based dynamic testing. Numerous 
devices have been proposed to conduct dynamic tests while supporting the 
weight of the structure without introducing any constraining forces which 
can impose boundary conditions not found in space. For example, long 
cables have been used to suspend the structures from a high ceiling. 
Dynamics testing is conducted in the horizontal plane to reduce any 
gravitational effects on the structural dynamics. However, this suspension 
gives rise to an undesired pendulum effect as the induced horizontal 
motion combines with the vertical cable support. In addition, this 
proposal has the additional drawback of large space requirements for the 
high ceilings and the introduction of undesired low frequency vibrations 
to the structures during movement due to elasticity of the long cables. 
Various other designs are based on the principle of reducing or eliminating 
friction in the horizontal plane. The most common design involves air pads 
which act as hydrostatic air bearings to suspend the structure. Once again 
the testing is performed in the horizontal plane. These air bags are 
incorporated into the structure and may have a mass sufficient to distort 
the actual dynamic characteristics of the structure. In addition, this 
proposal requires a large air table as well as air pumps, filters and 
other pneumatic equipment to reduce the friction between the air pads and 
the air table. 
A pneumatic electric device is also used comprising an external air tank 
under pressure for driving a piston supporting the test structure. Since 
the pneumatic system incurs a positive spring stiffness, a linear D.C. 
motor is incorporated to introduce a negative spring stiffness to the 
pneumatic system to create a perspective of a low stiffness toward the 
structure. This requires a very complex control system with closed loop 
feedback to ensure proper operation and stiffness compensation. 
Finally, various spring configurations have been proposed which attempt to 
introduce a near zero stiffness of the suspension system. These 
configurations only result in a very small domain of operation, i.e., a 
very small stroke, through which the test structure can move without any 
constraining force. 
OBJECTS OF THE INVENTION 
It is accordingly an object of the present invention to provide a 
suspension device for low frequency structures for ground based dynamic 
testing which approximates the free-free boundary conditions of space. 
It is another object of the present invention to accomplish the foregoing 
object in a simple manner. 
It is a further object of the present invention to accomplish the foregoing 
objects with minimal space requirements. 
It is yet another object of the present invention to achieve the foregoing 
objects while ensuring an adequate domain of operation. 
Additional objects and advantages of the present invention are apparent 
from the specification and drawings which follow. 
SUMMARY OF THE INVENTION 
The foregoing and additional objects are obtained by a suspension device 
for low frequency structures according to the present invention. A support 
cable is connected at one end to the test structure and is vertically 
guided by a guiding ring. The other end of the cable is connected to a cam 
having an outer circumference which supports the cable. A drive axle 
passes through the cam center of rotation and is rotatably journalled in a 
suitable manner to a rigid structure. Two torsion springs are provided 
about the drive axle. One end of each spring is connected to a respective 
face of the cam or to the drive axle and the other end is connected to the 
fixed support. The cam is shaped and the torsion springs selected such 
that Wr.sub.(t) =T.sub.s(t), wherein W is the weight of the test 
structure; r.sub.(t) is the instantaneous moment arm defined as the 
perpendicular distance from the rotational center of the cam to the cable 
at time t, and T.sub.s(t) is the total spring torque exerted by the two 
springs on the cam at time t. The test structure is accordingly vertically 
suspended by the cable and the instantaneous moment arm compensates for 
any increased spring torque arising from a vertical displacement of the 
test structure to simulate space conditions.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
Referring to FIGS. 1(a), 1(b), and 2, a suspension device 10 according to 
the present invention is shown. A thin cable 12 is wrapped around the 
circumference of a non-circular disc or cam 14 and is connected at one end 
thereto. This circumference may be grooved to facilitate any movement as 
discussed below. The cable extends from the circumference of the cam 14 
through an immovably fixed, vertical orienting ring 16 and then extends 
vertically downward to suspend a schematically represented test structure 
18 connected to the other end of the cable. 
Turning now to FIG. 2, cam 14 is supported by a drive axle 20 fixed through 
the cam rotational center and rotatably journalled at both ends in 
immovably fixed supports 22a and 22b. Collars 24a and 24b are located 
about axle 20 at the outside faces of the supports. Torsional springs 26a 
and 26b extend around axle 20 and are respectively connected at one end to 
supports 22a and 22b. Each torsional spring is connected at its other end 
to respective axial hubs 28a and 28b fixed to opposite faces of cam 14 or 
to drive axle 20. To prevent cable 12 and the attached structure 18 from 
driving cam 14, the torsional springs 26a and 26b are selected such that 
the combined torque T.sub.s1 of the two torsional springs equals the load 
imposed by the test structure 18 on the cam 14, i.e., such that: 
EQU Wr.sub.1 =T.sub.s1 (1) 
wherein W is the weight of the test structure 18 and r.sub.1 is the 
instantaneous moment shown in FIG. 1(a) for position 1. 
Referring once again to FIGS. 1(a) and (b), test article 18 may be 
displaced vertically downward a distance l from position 1 to position 2. 
To maintain test structure 18 in equilibrium at position 2, the new 
instantaneous moment arm r.sub.2 needs to be larger than r.sub.1 to 
compensate for the increased torsional spring torque T.sub.s2 in the new 
equilibrium equation wherein the load W and resultant cable tension is the 
same, i.e., such that: 
EQU Wr.sub.2 =T.sub.s2. (2) 
This difference in moment arms (r.sub.2 -r.sub.1) allows the profile of cam 
14 to be determined such that a continuous change in the moment arm will 
result to compensate for any increased torsional spring torque arising 
from a vertical displacement of the structure 18. In general terms the cam 
profile is generated by integrating the relationship Wr.sub.(t) 
=T.sub.s(t) over time for rotating the cam wherein W is the weight of test 
structure 18, r.sub.t is the moment arm at time t, and T.sub.s(t) is the 
torque generated in cam 14 by torsional springs 26a and 26b at time t. The 
equilibrium equation of equation (1) can be further written as 
EQU k.sub.s (.theta..sub.s0 +.theta..sub.1)=Wr.sub.1 (3) 
where .theta..sub.s0 is the angle due to the preload in the torsional 
springs S, and .theta..sub.1 is the rotational displacement of cam 14 
caused by the test structure 18. Note that this equation provides an 
explicit relationship between the angle of rotation .theta..sub.1 of the 
noncircular cam 14 and the moment arm r.sub.1. 
In the new equilibrium position 2: 
EQU k.sub.s (.theta..sub.s0 +.theta..sub.2)=Wr.sub.2 (4) 
where .theta..sub.2 is new rotational displacement of cam 14, as 
illustrated in FIG. 1(b). Note that the moment arms r.sub.1, r.sub.2 are 
not the radial distances to the points of tangency of the cable 12 at the 
cam profile, but rather are the perpendicular distances from the cam 
rotational center O to the cable 12. Since the moment arm r.sub.2 is 
different from r.sub.1, it is possible to determine the profile of the 
noncircular cam 14 such that a continuous change in the moment arm is 
obtained for any given position of the test structure 18, in such a way 
that when displaced from one position of static equilibrium to another 
position, the test structure will remain in static equilibrium at this new 
position. This constant static equilibrium results in a weightless 
situation which simulates a space environment. 
The static characteristic of the suspension system is thus governed by 
equations (1)-(4). Compared with prior suspension systems discussed 
hereinabove, where complicated electrical devices or the huge facilities 
are needed, this band drive suspension system is a rather simple 
mechanical system. Obviously, the circumferential profile of noncircular 
cam 14 plays a very crucial role in this suspension system. The profile 
coordinates of the noncircular cam 14 are derived by using envelope theory 
in conjunction with the equilibrium equations given by equations (1)-(4). 
Envelope theory will be applied to generate the coordinates of the cam 
profile given in FIG. 3. Using kinematic inversion, as the cam 14 rotates 
an observer fixed on the cam would view the sequential positions of the 
cable 12, as a sequence of straight trajectories 
EQU P.sub.0 T.sub.0,P.sub.1 T.sub.1,P.sub.2 T.sub.2, . . . , P.sub.n T.sub.n 
as shown in FIG. 3. The swinging point P.sub.i (i=1,2, . . . ,n) is 
observed to lie on a circular path CP with a radius r.sub.a, which is the 
distance from the rotational center O of cam 14 to the ring 16. A 
combination of these straight trajectories when taken infinitesimally 
apart gives the envelope which forms the cam profile. Assuming that the 
initial swinging point P.sub.o is tangent both to the cam profile and to a 
base circle BC sharing origin O and having a radius r.sub.b, the angle 
.phi..sub.o, which denotes the starting rotational position of the cable 
12, is given by 
##EQU1## 
Suppose that the cable is viewed by the observer at center O, while the 
cam rotates through an angle .theta.. Then the thin cable will subtend an 
angle of .phi..sub.o +.phi. with the vertical, at the ring. The increment 
angle .phi. is the rotational displacement of the string trajectory PT 
from its initial orientation. Therefore there exists a relationship 
between cam rotational angle .theta. and the string angular displacement 
.phi.. This relationship will be derived using the equilibrium equations 
(2) and (4). A general equation of the family of lines forming the 
envelope is governed by a straight line which is: 
EQU y=mx+b (6) 
where the slope of the swinging string at the disk angular position .theta. 
is given by 
EQU m=tan(.phi.+.phi..sub.o -.theta.) (7) 
and y-intercept of the cable PT, based on the cartesian system in FIG. 3, 
is 
EQU b=r.sub.a cos .theta. tan (.phi.+.phi..sub.o -.theta.)+r.sub.a sin 
.theta..(8) 
This general equation of the cable PT in equation (7) gives a one-parameter 
family of strings as a function of the cam angle of rotation .theta.. From 
the theory of envelopes, an envelope of the family of the straight lines 
is governed by an equation: 
EQU F(x,y,.theta.)=y-mx-b=y- tan (.phi.+.phi..sub.o -.theta.)[x+r.sub.a cos 
.theta.]-r.sub.a sin .theta.=0. (9) 
Equation (9) is continuous and is a continuously differentiable function in 
the coordinates x and y as well as in the variable .theta.. 
Differentiating the equation (9) with respect to the cam angle .theta. 
provides 
##EQU2## 
where .beta. equals .phi.+.phi..sub.o -.theta.. The coordinates of cam 
profile at a given angle .theta. may be obtained by solving equations (9) 
and (10), i.e.: 
EQU x=-r.sub.a [A sin .beta.+cos .theta.] (11) 
where 
##EQU3## 
Substituting equation (11) into equation (9) provides 
EQU y=r.sub.a [-A sin .beta.+sin .theta.]. (13) 
Initially, the angles .theta. and .phi. equal zero so that the starting 
coordinate of the noncircular cam becomes: 
##EQU4## 
which coincides with the point at which the starting cable PT.sub.o is 
tangent to the base circle in FIG. 3. 
The rate of change of the cable orientation with respect to cam rotation, 
.differential..phi./.differential..theta., can be determined by 
investigating the relationship between the angles .phi. and .theta.. Based 
on equation (2) and illustrated in FIG. 2, the equation of the initial 
equilibrium is governed by: 
EQU Wr.sub.a sin .phi..sub.o =k.sub.s .theta..sub.o (16) 
For the incremental angles of .theta. and .phi., from the initial 
orientation angles .theta..sub.o and .phi..sub.o, the new equilibrium 
state becomes: 
EQU Wr.sub.a sin (.phi..sub.o +.phi.)=k.sub.s (.theta..sub.o +.theta.)(17) 
Subtracting equation (16) from equation (17) provides: 
EQU Wr.sub.a [ sin (.phi..sub.o +.phi.)-sin .phi..sub.o ]=k.sub.s .theta.(18) 
which can be rewritten as 
##EQU5## 
Differentiating equation (19) with respect to the anlge .theta. yields 
##EQU6## 
Then, the profile of the noncircular cam is determined by substituting 
.phi. and .differential..phi./.differential..phi. from equations (19) and 
(20) into the equations for the cam coordinates given by equations (11) 
and (13). Note that the profile of the cam must be convex. 
Several parameters are needed to generate the profile of the noncircular 
cam, and they include r.sub.a, r.sub.b, k.sub.s, and W. It can readily be 
shown that each cam profile can be specified according to a parameter 
which is the ratio of the weight of the test article to the stiffness of 
the torsional spring, i.e., W/k.sub.s. This means that if testing is to be 
conducted for another test article twice its original weight, the 
torsional spring stiffness must be increased by the same factor so that 
the same cam can again be used. More generally, the new springs are 
selected such that the ratio W/k.sub.s is unchanged for the new weight 18. 
Such a design, therefore, permits tremendous flexibility since different 
loads can be used on this device, without the need to fabricate a new cam 
every time a new test article with a different mass is used. 
A particular cam profile is shown in FIG. 4. Thus, any vertical 
displacement of the structure 18 from an original static equilibrium 
position results in a new static equilibrium position. This is exactly the 
motion experienced in space, i.e., that any object displaced from one 
position of static equilibrium to another position will remain in static 
equilibrium at the other position. 
This cam profiling also allows simulation of an article subjected to an 
impulse in space which will translate at a constant velocity. When the 
test structure 18 is imposed with an initial velocity V.sub.o, the test 
structure will continue to travel over a considerable range at a constant 
velocity V.sub.o because the tension in the cable is equal to the weight 
of the test article, resulting in no net driving force on the article 
during its entire range of motion as is the case in space. 
The torsional springs 26a and 26b have been described as linear. However, 
non-linear rate torsional springs may also be used so long as the cam 
profile generated by the required moment arm reflects this new non-linear 
spring rate. If this non-linear spring rate is a zero rate, i.e., a 
constant load torsional spring, the profile of the cam would be circular. 
Accordingly, the present invention simulates the conditions of space in a 
simple, inexpensive manner without the need for complicated drivers or 
large space requirements. The range of motion of the article permitted by 
the present invention is relatively large, allowing for an increased 
collection of experimental data. Also, the inertia of the suspension 
device is small, thereby avoiding any appreciable modification of the 
dynamic characteristics of the test article. 
Many modifications, improvements and substitutions will become apparent to 
one skilled in the art without departing from the spirit and scope of the 
present invention as described in the specification and defined in the 
following claims.