Wire rope isolators possess both of the above properties. ... 1) Cases for electronic equipment, .... case of changing spring stiffness k, from Figure 13, it is easy.
中華民國力學學會第三十屆全國力學會議 彰化縣大葉大學機械與自動化工程學系 95 年 12 月 15-16 日 The 30thNational Conference on Theoretical and Applied Mechanics, December 15-16, 2006, DYU, Changhwa, Taiwan, R.O.C.
An Investigation on Mathematical Models of Wire Rope Isolators Feng Tyan Computational Dynamics and Control Lab Dept. of Aerospace Engineering Tamkang University Tamshui, Taipei County, Taiwan 25147, R. O. C.
Bulletin No. CCA/94
Shun-Hsu Tu and Jeffrey Wu Mechanical and Systems Research Laboratories Industrial Technology Research Institute, Chu Tung, Hsin Chu, Taiwan 310, R. O. C.
CCA Series Circular Arch Isolators
ITRI Project: 3000108944
Abstract— Vibration isolation systems using helical wire rope isolators are known to be highly effective in controlling both shock and vibration and have been used in numerous space and military applications. For damping response, the forcedisplacement relationship of a wire rope isolator is highly nonlinear and history-dependent. For satisfactory analysis of such behavior, it is important to be able to characterize and to model the phenomenon of hysteresis accurately. Both Tinker and Bouc-Wen models that had been proposed for response studies of wire rope isolators are examined in this paper. For helical type wire-rope isolator Bouc-Wen’s model, compared with Tinker’s model, predicts much more intimate behavior as observed in experiment datum. keywords: wire rope, Tinker model, Bouc-Wen model, hysteresis loop.
I. I NTRODUCTION Wire rope isolators are simple devices made of stainless steel stranded cable and retaining bars. They provide a high performance level of shock and vibration isolation. They are not affected by temperature and are environmentally stable, hence have been used in numerous space and military applications. The structure of typical wire rope isolators can be described by the followings [1]: 1) comprised of stainless steel stranded cable, 2) threaded through aluminum alloy retaining bars, 3) crimped and mounted for effective vibration isolation. Figures 1 and 2 illustrate some typical commercially available types of wire rope isolators.
Fig. 2.
A New Concept in Shock & Vibration We use a helical wire rope isolator as an example to further Control illustrate its configuration. As we have mentioned previously, a wire rope, or cable, typically consists of several strands of wire wrapped around a metallic or fibrous core, by winding the cable in the form of a helix and fastening it with clamps to maintain the shape, helical isolators are constructed, see Figure 3. These helical devices can then be arranged as desired to form a vibration isolation system. An example system configuration is shown Figure 4.
Fig. 3.
Enidine Wire Rope Isolators
Fig. 1.
Typical Wire Rope Isolators
Wire Rope Isolators
Circular Arch Isolator
Fig. 4.
Helical Wire Rope detail configuration
An electrical cabinet using wire rope isolation system
The main principle by which wire rope systems are designed and by which they work are inherent interfacial damping, or sliding friction, and adjacent wires move relative to each other, friction causes part of the kinematic energy of the wires to be converted into heat, thereby dissipating vibrational energy. Quantitatively, the damping characteristics of helical isolators depend on the following factors [1]: 1) the cable diameter Rc , Rh , 2) number of strands, 3) number of loops per isolator, 4) lay angle of the strands α, 5) the overall dimensions of the isolator, See Figure 5([2]), for the structure of a typical wire rope and definitions of the variables used above.
support the load and a damping element to dissipate the input energy. Wire rope isolators possess both of the above properties. In addition, owing to the capability of working under severe environment, wire Ropes are suitable for a broad range of applications. They are ideal for applications such as 1) 2) 3) 4) 5)
Cases for electronic equipment, Mobile electronic equipment [3] (See e.g. Figure 7), Chemical machines, Medical equipment, Pump, Generator and Compressor Isolation, Transportation of sensitive equipment, or shipping containers and skids to eliminate damage caused by shock and vibration to equipment during transport, see Figure 8.
lang lay rope. If the in a fashion similar to ey are right lay rope the rope to the left are inations of these wire ireRope Users Manual ontains more detailed and construction. Most ed; this means that the helical form they will process eliminates the azardously, when they s cut.
Volume 3 Number 15
Fig. 7. Wire Rope Application
WIRES
Eliminating Damage to Automobile Door Panels
wires have a circular n they are stretched or along the center of a f the wire and used to at are related to wire al wire of a strand also
By Toshihiko Yanada Figure 1.C Structural elements in a typical wire rope.
s in a wire rope as aight wire in a straight C or IWRC rope. The eometrically into two s. The outer wires in a a single helical form nly once around a straight into a rope, the central of the other wires have re wound twice, once around the rope axis. ative to the central wire d. This relationship is ble helical wire.
TERS
specifying the helical resenting them in the parentheses. xis of the rope around form a rope or around form the center strand e axis is defined to be s. ntroidal axis of the helelically wound to form f a helical strand. The to be the direction that
Wire rope isolator used in a PDA and hard disk drive
Fig. 5.
Structural elements in a typical wire rope
A major automotive manufacturer in Japan was looking for a solution to a problem on its cargo retainers. In the automotive industry, many manufacturers are trying to reduce shock and vibration as a requirement of ISO 14000. In addition, the trend toward computerization and faster throughput can create increasing shock and vibration challenges for manufacturing equipment.
Since wire rope isolators are made of stainless steel stranded cable, they have significant advantages over conventional rubber or elastomeric isolators for the severe operational environments of mechanical systems. In summary, they ◦ assembly process, During the perform well through a wider range of temperatures (−400 F the company moves automobile ◦ 8. Wirea rope isolator in shipping door panels to 700 F), and are less susceptible to wear and deterioration The manufacturer is also prototype system. used door panels from one building to Fig.manufactured interested in applying Wire Rope They mounted six WR6-800 next. The doors were loaded than elastomers, and are able to provide isolationthe Isolators to the controller of its Isolators to the base of the cart. on ain cargoall retainer (cart), which welding robots. The Wire Rope Isolators were was propelled by an Automatic directions (see Figure 6). mounted horizontally, one on Guided Vehicle (AGV). The cart Solid opportunities exist in The major feature of the nonlinear restoring force of wire carries 16 door panels to the next each corner and two in the center industrial material handling of the cart. assembly area, where the doors rope isolator is its hysteretic characteristic (See applications (SIC 3537: AGVs, Figure 9), are mounted on the car frames. The test results exceeded their Industrial Trucks, Tractors, When the panels arrive, many i.e., it depends not only on the instantaneous Trailers & Stackers) to reduce deformation expectations. The maximum have sustained damage caused shock and vibration inputs vertical shock was reduced from by shock and vibration fromalso the but on the past history of deformation. In this paper, which can cause damage to 24G to 2.5G. The customer road and the AGV. Often, nearly expensive components. Enidine’s appreciated themodelling simultaneous the mathematical of the dynamic behavior of a half of the panels were being experience with semi-conductor shock and vibration attenuation damaged during transport. C andinvestigated. medical carts are other capabilities of the Wire Rope wire rope vibration isolator is Two kinds of good examples of how the Isolator, as well as its Enidine contacted the simultaneous shock and vibration maintenance-free long service automotive manufacturer to mathematical modelandwill be considered, namely, Figure 2. Comparison of typical wire rope lays.
introduce Wire Rope Isolators. The company selected the Wire Rope Isolators on a trial basis and
life attributes. They decided to
retrofit their existing carts with 1) Tinker’s model, Wire Rope Isolators from Enidine. 2) Bouc-Wen’s model.
capabilities of Wire Rope Isolators can be used effectively.
Enidine Incorporated 7 Centre Drive, Orchard Park, New York 14127 Phone: 1-716-662-1900 Fax: 1-716-662-1909 Hot Line: 1-800-852-8508 www.enidine.com
II. T INKER ’ S M ODEL CGA36AEB
Fig. 6.
Mounting Configurations
A good isolator system has two components, a spring to
Several nonlinear one-dimensional mathematical models proposed by Tinker will be investigated in an effort to model the frequency response and damping characteristics of the wire rope isolator model shown in Figure 10 [1]. The Tinker’s model is described by a second-order differential equation and can be divided into the following categories:
-
751
5) nth -Power Velocity Damping + Variable Coulomb Friction + Modifications - Hysteresis Added with Modification m¨ x + r k[x − xs ± ε2 (x − xs )3 ] + b + cn |(x˙ − x˙ s )n | + f [1 − a(x − xs )2 ] sgn(x˙ − x˙ s ) = 0, (2.1)
where b: the distance each hysteresis backbone curve lies Figure 7. Experimental hysteresis loops with different frequencies. above the origin, Fig. 9. Hysteresis curves from test r: stiffness parameter, to account for the reality A total of 20 groups of recorded displacement and restoring force signals with that dynamic stiffness curves do not necessarily different frequencies and amplitudes are used for identification, i.e., M = 20. The pass through the centers of corresponding hystereorder number of harmonics truncated is taken as N = 5. The parameter identification is performed by taking the weight factor wj equal to 1, A−1/2 and A−1 sis curves. j j respectively, where Aj is the amplitude of the jth displacement signal xj (t). Four sets Figure 11 demonstrates the contributions to the model of initial guesses, from viscous and Coulomb damping forces. Linear Systems Display Similar Hysteresis (0) T (0) T {y } = {0·1 0·1 0·1 0·1 1·0} ,
{y(0)} = {5·0 5·0 5·0 5·0 1·0}T,
{y } = {1·0 1·0 1·0 1·0 1·0} ,
{y(0)} = {10·0 10·0 10·0 10·0 1·0}T
are tried to carry out the iterative solution using formula 1 and to check the uniqueness of convergency. The convergency takes place for all these initial guesses. With a specific weight factor, the iteration starting from different initial guesses approaches the same set of converged values. Table 5 lists the identification results. The parameter estimation is also performed using formula 2 and identical values are obtained. Fig. 10. Test for Tinker’s Modelthe Bouc–Wen model By using the identified values of Setup the model parameters, generates the theoretical hysteresis loops as shown in Figure 9. It is seen that the Bouc–Wen model with the identified parameters provides a good representation of the measured hysteresis loops. Though the values of the model parameters
•
Spring
•
150
Viscous damper
150
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-5 -50
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Spring, viscous & Coulomb dampers
150
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0
0 -10
m¨ x + c[x˙ − x˙ s ] + k[x − xs ± ε2 (x − xs )3 ] = 0. Fig. 11.
-5
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1) Nonlinear Stiffness and Simple Linear Damping - No Hysteresis
5
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Spring & Coulomb damper
c 0
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APPLIED TRUCTURAL
Linear spring with 38 viscous and Coulomb damperSDYNAMICS
where c: is the viscous damping coefficient, Example 2.1: The following typical values of data for k: is the stiffness, Tinker’s model ( 2.1) are taken from [1]. The hysteresis 2 ε : nonlinear spring constant, x: is the absolute displacement of the center of curves for different parameters are illustrated in Figures 12Figure 8. Measured hysteresis loops with different amplitudes after low pass filtering: (a) hysteresis 13. In Figure 12, the damping coefficients are changed to mass to the loops at f = 5 Hz; (b) hysteresis loopsshaker, at f = 10 Hz. see its effects. As we can tell from these figures, in the xs : is the absolute displacement of the shaker. 2) Nonlinear Stiffness and Coulomb Friction - No Hys- steady state, when c increases, the “size” of the hysteresis loop shrinks, and the loop becomes rounder. While for the teresis case of changing spring stiffness k, from Figure 13, it is easy m¨ x + f sgn(x˙ − x˙ s ) + k[x − xs ± ε2 (x − xs )3 ] = 0. to identify that the slope of the hysteresis loop increases with the increasing of k. where f is the Coulomb friction force. 3) nth -Power Velocity Damping- Hysteresis Added m¨ x+c|(x− ˙ x˙ s )n |sgn(x− ˙ x˙ s )+k[x−xs ±ε2 (x−xs )3 ] = 0,
TABLE I PARAMETERS USED FOR E XAMPLE 2.1 parameters
value
mass m = 0.225817 (lb) where spring stiffness k = 168 c is the velocity damping coefficient dampting ratio ζ = 0, 0.25, 0.5, 0.75 nonlinear spring constant ε2 = 1043 and bias b = 0 (lb) n = 0 - Coulomb damping, Coulomb friction f = 0.2678 (lb) damping exponent n=2 n = 1 - Viscous damping, frequency ω = 60 · 2π (rad/sec) n ≥ 2 - hysteresis. th 4) n -Power Velocity Damping + Variable Coulomb Friction III. B OUC -W EN M ODEL m¨ x + k[x − xs ± ε2 (x − xs )3 ] We adopt a single-degree-of-freedom model (Figure 14) + c|(x˙ − x˙ s )n | + f [1 − a(x − xs )2 ] sgn(x˙ − x˙ s ) = 0.that has nonlinear stiffness and various damping mecha-
is depicted in Figure 1. Its main components are the lumped mass m and the nonlinear restoring force element r (x, x˙ ). A photograph of the completely assembled electro-mechanical apparatus is shown in Figures 2 and 3. The labels indicate the major subcomponents of the apparatus. The system mass consists of a rigid block that is constrained to slide uniaxially on ball bearings. A computer-controlled electro-mechanical device is used to apply, through a stepper motor, a time-varying normal force to a bar which slides r=0.95, k =168, 2 =1043, b =0, c =0.0467, f =0.2678, a =12453, n
8
2
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1.5
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Force (lb)
Force (lb)
r=0.95, k =168, 2 =1043, b =0, c =0.0022, f =0.2678, a =12453, n
0 −2 −4
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Relative Displacement (in) r=0.95, k =168, 2 =1043, b =0, c =0.1, f =0.2678, a =12453, n =
0
0.005
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Force (lb)
1 0.5
−0.5 −1 −1.5
Figure 1. Idealized mathematical model of a generic nonlinear SDOF system.
Fig. 14.
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x 10
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x 10
Fig. 12. Hysteresis Curves for ω = 60 · 2π(rad/sec), u0 = 0.0125in and various c
2
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A standard Bouc-Wen model describing the hysteresis restoring force can be found in, e.g. [4, 5], it is expressed as Fr (t) = kx(t) + cx(t) ˙ + dx3 (t) + z(t), (3.3)
0
−1
−0.005
A. Standard Bouc-Wen Model
−0.5
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0.015
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Relative Displacement (in) Relative Displacement (in) r=0.95, k =201.87, 2 =1043, b =0, c =0.0467, f =0.2678, a =12453, r=0.95, k =301.87, 2 =1043, b =0, c =0.0467, f =0.2678, a =12453, 2
2.5 2
1.5
1.5 1
Force (lb)
Force (lb)
1 0.5 0 −0.5
0.5 0
where x(t) is the absolute displacement of isolator, z(t) is the hysteretic auxiliary variable. The hysteretic auxiliary variable is governed by the following differential equation
−0.5
−1
−1
−1.5 −2 −0.01
nature. A new parameter is proposed to describe the character between linear and dry friction damping. Owing to this new parameter, the order of the differential equation becomes three instead of two. The result of parameter identification shows this model can express hysteretic character effectively.
r=0.95, k =168, 2 =1043, b =0, c =0.0467, f =0.2678, a =12453, n
2
Force (lb)
Force (lb)
r=0.95, k =101.87, 2 =1043, b =0, c =0.0467, f =0.2678, a =12453,
−2 −0.015
SDOF Math Model
−0.5
−2 −2.5 −8
−0.005
Relative Displacement (in) r=0.95, k =168, 2 =1043, b =0, c =0.5, f =0.2678, a =12453, n =
1.5
Force (lb)
0 −0.5
z(t) ˙ = x(t) ˙ {α − |z(t)|n [γ + β · sgn(x)sgn(z)]} ˙ ,
−1.5 −0.005
0
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−2 −6
Relative Displacement (in)
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−2
0
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Relative Displacement (in)
6 x 10
Fig. 13. Hysteresis Curves for ω = 60 · 2π(rad/sec), u0 = 0.0125in and various k
nisms, including linear viscous damping, constant Coulomb friction, nth-power velocity damping, and combined nthpower and variable Coulomb friction damping, m¨ x(t) + Fr (t) = F (t),
(3.1)
where Fr (t) is the restoring force Fr (t) , kx(t) + cx(t) ˙ + r[x(t), x(t)], ˙
(3.4)
8 −3
(3.2)
and m: the mass of the isolator, x(t): the absolute displacement of isolator, F (t): external excitation, r[x(t), x(t)]: ˙ hysteretic restoring force. Elastic element of cable wire assembly exhibit hysteretic nonlinearity. It is found, however, the restoring force can be expressed as the sum of “nonlinear non-hysteretic” force and “pure hysteretic” force. In the actual system it is difficult to describe the damping force exactly due to its complicated
where α, β, γ and n are the “loop parameters” to be calibrated from experimental tests which control the shape and the magnitude of the loop [5]. The hysteretic variable z(t) has a finite ultimate value zm which corresponds to the displacement xm , that is n1 α . (3.5) zm = β+γ Two cases can be considered here: 1) x(t) ˙ > 0 (loading), a) β + γ > 0: softening stiffness, b) β + γ = 0: quasi-linear stiffness, c) β + γ < 0: hardening stiffness. 2) x(t) ˙ < 0 (unloading), a) β − γ > 0: softening stiffness, b) β − γ = 0: quasi-linear stiffness, c) β − γ < 0: hardening stiffness. Figure 15 illustrates the effects of β and γ in Bouc-Wen model. Example 3.1: Consider the following parameters shown in Table II for the stand Bouc-Wen model. The hysteresis curves are shown in Figure 16. From these curves we know that larger damping coefficient c widens the hysteresis loop.
requirements in commercial, industrial, and defense industries, including MIL-STD-810, MIL-STD-167, MIL-S-901, MIL-E-5400, STANAG-042, BV43-44 and DEF-STND 0755. 670
Load Axis Mounting Orientations
S. Dobson ef al.
small deformations. However, when the deformation of the isolator reaches a certain level, the stiffening of hysteresis For the best in vibration isolation capabilities, choose loops occurs (see Figure 18). This feature is referred to as Enidine’s Compact Wire Rope Isolators. Smaller than soft-hardening hysteresis. COMPRESSION 45˚COMPRESSION/ROLL traditional wire ropes, these unique isolators provide a = as (al cost-effective, (cl a .a9and vibration (b) a *aI simultaneous shock t3 =a5 ma9 B = 0.1 attenuation where4 package space is at a premium. Compact Wire Rope Isolators
Enidine Compact Wire Rope Isolators feature an easy, single-point installation, which allows them to be installed in virtually any application. Their small size also permits FIXED ROLL FIXED SHEAR the isolation of individual system components, making (for Wire Rope Isolators) them ideal for use in sensitive equipment and electronics. Fig. 17. The shear model and roll mode of helical isolator Just as with our standard Enidine Wire Rope Isolators, Enidine Compact Wire Rope Isolators feature a patented, all-metal design and components that ensure maximum reliability, regardless of temperature or substrate requirement, and that can help meet MILSPECS similar to those of our Wire Rope Isolator series. i⫹j (1) h(2) hij ij = (⫺1)
(31)
and a one-stage least-squares estimate scheme consequently arises as
If your application is outside the standard Compact Wire Rope Isolator product range, please consult the r standard Wire Rope Isolator portions of this catalog. Ifua standard solution is still not available, Enidine Min. y(H ) = 冘 冋g ⫺ 冘 冘 h 冉 冊 冉 冊 册 u r engineers can design an isolator to suit your specifications. (1)
冘 冋 冘冘 m
Fig. 1. Examples of hysteresis generated by the Bout-Wen
m
n
i=1
j=1
(1) k
k for u˙k>0
n
⫹ (fromWen g ⫺ [9]). (⫺1) model (2) k
k for u˙k
