Large signal non-linear analysis and validation of the suspension of a transport tool
Araz Abbasi, Joep Tax, Pieter Nuij
NTS Systems Development, Eindhoven, the Netherlands
In the design of ultra-high precision positioning equipment the design effort is mainly focused on the process performance of the equipment under operating conditions. External disturbance sources like floor vibrations, temperature fluctuations, particle pollution, air pressure variations and acoustic excitation etc. have a significant influence on process quality. The magnitude of these error sources however often is relatively small and one may realistically assume linear behaviour of the system. This assumption significantly eases the complexity of modelling. A completely different set of operating conditions is encountered during transport of the system or it’s modules from the manufacturing site to the customer site. Drop heights of 5 to 10 cm, horizontal impacts of 3 to 5km/h, temperature fluctuations of 15?C, air pressure variations of 350mbar, 2G static acceleration, 35? rolling/pitching angles etc. are realistic magnitudes of external disturbances which must be accounted for. Due to these large magnitudes linear system behaviour often is a fiction. Therefore, pioneering in the field of non-linear system design is required to deal with the extreme magnitudes of external disturbances as encountered in for example transport tooling for highly sensitive (optics) modules.
One of the main external disturbances during transport is vibrational input from the floor, representing the mean of transportation, e.g. a truck or airplane. During loading and unloading incidental shock related events occur. The floor vibrations have to be isolated and shocks have to be absorbed. A vibration isolation system is needed to achieve this. The combination of the minimum required amount of vibration isolation, shock absorption properties, and other constraints, such as the maximum allowable deflection of the isolation elements results in contradictory system requirements.
The vibrations requirements are based on the Power Spectral Density (PSD) of the input floor vibration (related to transport method) and allowable machine vibration (based on character of the protected precision equipment). Most high-tech customers provide their own input and output PSD spectrum. For shock, the input specification consists of a certain maximal drop height or a certain constant velocity before hitting a hard wall. The shock requirement specifies the maximum allowable rest shock level, i.e. maximum absolute acceleration experienced by the equipment.
Two of the main elements used in transport tool suspension systems are wire rope and elastomer isolators. Wire ropes are great for absorbing shocks (due to their highly nonlinear behaviour) but they act as a rigid connection for small random vibration input. Elastomers are able to provide reasonable damping for those vibrations but they do not provide the same big amount of damping during large deflection shocks.
This paper proposes a series connection of a wire rope spring and an elastomer isolator resulting in a hybrid damper. Such a well-designed passive isolation system combines and optimally uses the properties of these separate components. First a mathematical model of each component is presented and then model parameters are identified by performing measurements on a SOCITEC BFL-12 elastomer mount and a HH8-38TCM wire rope spring. The behaviour of the new hybrid damper configuration is analysed and design considerations are discussed.
Wire rope isolators
Wire rope isolators show both compliant and frictional behaviour. The bending of the wire strands results in compliant behaviour. The rubbing between the strands results in friction forces. Both phenomena characterize the force-displacement relation. The damping properties are determined from the resulting so-called hysteretic behaviour. This also implies that the response of a wire rope isolator is highly non-linear.
Figure 1: Wire rope spring and Iwan model
Figure 2: Elastomer isolator and Maxwell model
A wire-rope can be represented by Bouc-Wen as well as Iwan models. Iwan model has the advantage of having a more physically based structure. The model parameters are directly related to the force-displacement characteristic. This results in a better understanding and a less complex identification process. In this study, a parallel-series Iwan model structure is used  which is shown in Figure 1. The series spring and Coulomb friction elements are so-called Jenkin elements. The force through the symmetric wire rope is expressed by:
The characteristic force through a single Jenkin element is expressed in terms of its time derivative . In this form, only one equation is sufficient to describe the behaviour of both the sping and friction element:
The Iwan model in this form cannot describe non-linear softening, hardening or asymmetry behaviour. A non-linear parallel stiffness instead of the linear term is introduced to allow for these effects. The resulting modified Iwan model will then be:
The non-linear spring force consists of a linear, a quadratic, and a cubic term. Hysteresis, softening, hardening, and asymmetry are all described by this modified Iwan model.
Viscoelastic isolators are in most cases made out of rubberlike polymer materials, called elastomers. Viscoelasticity encompasses both viscous and elastic behaviour. Elasticity ensures stiffness, whereas due to the viscosity energy is dissipated. Viscoelastic behaviour is frequency and temperature dependent. The viscoelastic behaviour can be described by a frequency dependent complex Young’s modulus, consisting of a storage modulus (real part) and a loss modulus (imaginary part). The magnitude of the complex modulus and the ratio of the imaginary and real part are measures for respectively stiffness and damping.
A viscoelastic elastomer isolator is considered to behave like a viscoelastic solid. A generalized Maxwell model for viscoelastic solids consists of Maxwell elements and a single spring in parallel, as shown in Figure 2. The complex stiffness of this model is given by:
The loss factor of this viscoelastic model, the measure of dissipation, is given by:
Measurement and validation
Wire rope model
Two types of quasi-static measurements are performed with the wire rope isolator: large input global behaviour and small input local behaviour measurements. The Iwan model consists of a set of global parameters and local parameters. The global parameters describe the characteristic of the (non-linear) parallel spring, whereas the local parameters describe the Coulomb friction force and linear stiffness of the Jenkin elements.
An optimization algorithm is used to fit the model on the measurement data and hence to identify the model parameters. The deflection vector is assumed to be known input, the force vector is output. Moreover, the objective function to be minimized for each set is defined as a relative root mean square (RMS) error of the force value.
Figure 3: Wire rope Iwan model fit for small deflections
Figure 4: Wire rope Iwan model fit for large deflections
The optimization for an Iwan model of order 2 is shown in Figure 3 for a small deflection set and Figure 4 for a large deflection set. The mean error over the measurement sets for small deflections is reasonable (7.70%). The most significant deviation is seen for increasing deflection, the force is slightly underestimated in this region. The mean error over the large deflection sets is 7.07 %.
Quasi-static measurements are used to determine and validate the static stiffness of the elastomer mount (Figure 5). Furthermore, possible amplitude dependent behaviour can be detected. The behaviour is observed to be almost linear without significant hysteretic effects.
Figure 5: Elastomer quasi-static measurement
Figure 6: Elastomer FRF measurement
For the dynamic measurements a dynamical shaker setup was used with sinusoidal and white noise excitation signals. Linearity of the elastomer is assessed by the sinusoidal input measurements. White noise inputs are used for the identification of the (linearized) model parameters.
Figure 6 shows the frequency response functions (FRF) from the base velocity to the isolated mass velocity for three different input amplitudes. The shift in eigen-frequency is negligible and consequently, a linear amplitude independent model describes the dynamic behaviour sufficiently good.
The quasi-static results and white noise shaker measurements are used to identify the generalized Maxwell model. In Figure 2 an overview of the model with parameters is shown. The (linear) static stiffness is easily derived from the quasi-static measurements. The white noise measurements are used to identify the parameters of the Maxwell elements. Using different top masses, the transmissibility FRF is determined from each of these measurements. These FRF’s are used to fit a second order Maxwell model of the elastomer used in the isolation system. In the optimization process the simulated transmissibility of the model is compared to the measured transmissibility. The result fits are shown in Figure 7. The coloured lines show the measured transmissibility as function of the varying top mass. The dotted black lines represent the fitted model. The eigen-frequencies and corresponding damping values are well described.
Figure 7: Fit transmissibility elastomer (magnitude)
Figure 8: Schematic overview hybrid isolation system
Hybrid damper concept
The concept of this hybrid composition is the series utilization of the (large response signal) shock absorbing properties of the wire rope isolator and the (small response signal) damping properties of the elastomer isolator (Figure 8). For vibrations with low amplitudes the wire rope spring remains in its stick regime and is stiff with low damping. The damping required for damping the resonance is generated by the (less stiff) elastomer. For high amplitude input and response (i.e. shock) the wire rope spring is in its slip regime and will absorb much energy due to the high friction between the strands.
The static stiffness ratio () of the used elastomer and wire rope elements turns out to be an important parameter for tuning the isolation characteristics. Furthermore, the performance of the hybrid damper is highly dependent on the input amplitude. The shock response of the hybrid damper is bounded by that of the elastomer isolation system and wire rope isolation system. For shock absorption a hybrid damper with a dominant wire rope component is beneficial (low ). Considering the overall performance of an isolation system for vibration isolation and shock absorption, the use of hybrid isolators with respect to single wire rope or elastomer isolators is beneficial, however, the specific component selection is much more complex as the system response lies somewhere in between the two individual responses. Figure 9 shows that the output PSD level of a hybrid isolation system under white noise excitation is between the PSD levels of pure wire rope and elastomer responses. Figure 10 shows the same effect for the rest shock level after experiencing a shock.
Figure 9: Output PSD comparison for different isolation system configurations
Figure 10: Rest shock level comparison for different isolation system configurations
Conclusion and future developments
The behaviour of wire rope isolators is highly amplitude dependent. For small deformations the behaviour is stiff and close to undamped. For increasing deformations the cable strands start to slide, resulting in a decreasing stiffness and increasing damping. Further, the behaviour is characterized by softening in compression and hardening in tension. The modified Iwan model describes the amplitude dependent hysteretic loop of the wire rope isolator accurately.
The characteristic behaviour of an elastomer isolator is frequency, temperature and geometry dependent. For low frequencies, the stiffness and damping are low, for high frequencies the stiffness is high and the damping is low. The intermediate frequency region is interesting: the stiffness is increasing and there is significant damping. The linear generalized Maxwell model is able to properly describe the elastomer behaviour both in the time and frequency domain.
The concept of the hybrid damper is introduced which aims at utilization of the (large deformation) shock absorbing capabilities of the wire rope isolator and the small signal (vibrational) damping properties of the elastomer isolator. For small deformations the wire rope is stiff and consequently the elastomer component dominates the overall stiffness and damping. For increasing input, the cable strands start to slide and the wire rope stiffness decreases, resulting in beneficial shock absorption properties. The use of a hybrid system is beneficial to optimize both the vibration isolation and shock absorption performance. However, the non-linear behaviour is dependent on the stiffness ratio () of its wire rope and elastomer components. In contrast to linear passive systems, an input amplitude dependent trade-off between vibration isolation and shock absorption performance can be made, to optimally suit the application requirements.
W. Iwan, "The distributed-element concept of hysteretic modelling and its application to transient response problems," International Journal of Non-linear Mechanics, vol. 4, pp. 45-57, 1969.
D. S. a. P. Hagedorn, "On the hysteresis of wire cables in Stockbridge dampers," Journal of Non-linear Mechanics, vol. 37, pp. 1453-1459, 2002.