Abstract
Researchers worldwide have developed various semi-active control devices for seismic protection of structures. Most of these devices are electromechanical in nature and thus require a power source for their operation. In this paper, a newly developed rotation-based mechanical adaptive passive device is presented. These unique devices are able to mechanically change stiffness, either by adding positive or negative stiffness, by using different types of rotational elements. The devices are compact due to their use of rotational elements, facilitating their implementation in structures. The conceptual development of these devices is presented herein along with analytical models and numerical simulation results that demonstrate their potential for providing seismic protection. In addition, an extension of the stiffness modulation concept is introduced wherein damping is modulated.
1. Introduction
One of the most common approaches used to protect existing structures against earthquakes has been to add a seismic protection system that modifies stiffness, strength, or energy dissipation capacity. In general, seismic protection systems incorporate passive, active, hybrid or semi-active control systems (Soong and Constantinou, 1994; Spencer and Nagarajaiah, 2003). In the case of active, hybrid or semi-active systems, computer control systems are used to control the real-time behavior of electromechanical devices based on the measured response of the structure during earthquakes. However, these systems may not be reliable during an earthquake in the event of potential power failure or control system malfunction (Battaini, 1999). In addition, the implementation of active and semi-active systems is more expensive and incurs higher maintenance costs in comparison to passive systems. In contrast, passive systems are regarded as being highly robust, but lack the adaptability of active or semi-active systems. Passive systems are therefore generally regarded as not being as effective as active or semi-active systems.
In the field of mechanical engineering, researchers have worked on another type of control device in the form of an adaptive passive device. Although such devices are passive, they are able to change their properties in real time. The device described in this paper is a mechanical adaptive passive device that can change the behavior of the structure using mechanical means while the earthquake is occurring, responding in an optimal fashion to the unique aspects of the earthquake, while retaining the robust characteristics of fully passive systems.
Using rotational elements, many adaptive passive seismic protection devices can be configured such that each has unique capabilities. In this paper, two configurations are presented: one that is used to modify stiffness and the other to modify damping. Although such devices can be used to protect a variety of structural systems, the focus of this paper is on their implementation in bridges. Bridges are often vital links (lifelines) in transportation systems and thus any damage they might sustain during an earthquake can have major implications on the post-earthquake recovery of a community. Researchers have developed many different types of seismic protection devices for bridges. The most common passive devices used in bridges are elastomeric bearings, sliding bearings, viscous dampers and viscoelastic dampers. Generally, these devices are effective in reducing the displacement of the bridge deck but may not reduce the acceleration of the bridge deck or shear forces within the piers. In addition, many active and semi-active devices have been tested and implemented in bridges (both field and lab implementations). For example, Kawashima and Unjoh (1994) implemented a viscous damper with variable damping in a bridge model. The damping coefficient of the damper could be changed by controlling a servo-valve based on the amount of the bridge deck deformation with respect to the piers.
In recent efforts to advance the field of seismic protection as applied to bridges, Attary et al. (2012, 2013, 2015a, 2015b, 2015c) used a mechanical device, in a bridge model, that could produce negative stiffness behavior in order to reduce earthquake forces in structures through softening behavior. This was the first time such a device has been used in seismic testing of a bridge model. Negative stiffness behavior was produced by using a pre-compressed spring within a geometrical arrangement of steel braces and levers. Although the device was passive, it was considered to have adaptive characteristics in that the displacement at which negative stiffness behavior initiates could be controlled via mechanical means. The negative stiffness device (NSD) was tested experimentally within a quarter-scale bridge model on a shake table at the University at Buffalo; a range of earthquake ground motions was considered together with a variety of bridge model configurations (Attary et al., 2015a, 2015b). The results from the shake table testing showed that the device was able to reduce significantly the acceleration of the deck and, correspondingly, the shear forces within the piers. The large deformations that would typically accompany a system with softening behavior were controlled through energy dissipation that occurred within the device and its support system via friction at various sliding interfaces. By engaging the NSD at a certain displacement and using its negative stiffness to create apparent ‘weakening’ or ‘softening’ behavior, the combined behavior of the primary structure (which has positive stiffness) and the NSD (which has negative stiffness) exhibits a virtual yield point at a small force level. The framing of the primary structural system does not actually yield; rather, the response of the combined system appears to have a yield point. Thus, the softening response is of a nondamaging nature (Pasala et al., 2013; Sarlis et al., 2013; Attary et al., 2015c).
The effects of the negative stiffness and virtual yield point on the dynamic response of structures can be explained using the capacity spectrum method and acceleration–displacement response spectrums (ADRS) (Attary et al., 2015c) in which the capacity diagram of the structure is overlaid with the ground motion demand diagram to determine the performance point (intersection of the two diagrams). Figure 1 shows a comparison of the effects of hardening (e.g., adding lateral bracing) versus adding a negative stiffness device to a linear structure. As can be seen, the effect of hardening is to reduce the displacements of the structure while increasing the accelerations, and therefore forces, in the structure. In contrast, by employing an NSD a virtual yield point is created and thus accelerations of the structure are reduced while displacements are increased. However, dampers can be added in parallel with the negative stiffness device, to shrink the ground motion demand curve and thereby limit the displacements. The end result is a substantial reduction in force demands on the structure while limiting the displacements (see Figure 1).
Use of capacity spectrum method to compare the effects of hardening versus adding a negative stiffness device (green line = base structure; yellow line = hardening of base structure; blue line = adding negative stiffness device to base structure: red markers = performance points; black markers = virtual yield points).
As can be seen in Figure 1, another major benefit of adding a device with negative stiffness behavior is that the effective stiffness (secant stiffness), and thus the natural frequency, of the combined system continuously changes, thereby avoiding resonance. Such nonresonant behavior was observed in experimental tests performed by Attary et al. (2015a, 2015b). As an example, one of the common and successful methods of protecting bridges during earthquakes is to isolate the deck from the piers using flexible deck support bearings (e.g., elastomeric bearings). However, there have been studies on seismically-isolated bridges which indicate that these passive systems may have reduced effectiveness (and sometimes can degrade performance) in cases where the ground motion exhibits strong velocity pulses (Nagarajaiah et al., 1993). For such near-fault ground motion conditions, the velocity pulse can amplify the dynamic response of the isolated bridge when the pulse period is close to the fundamental period (Shen et al., 2004). In this case, an NSD can be added to eliminate possible resonance of the system.
For an optimal design of an NSD, the device should be able to generate sufficient negative stiffness to cancel out the positive stiffness of the structure at the virtual yield point (thus resulting in a system with zero tangent stiffness at the yield point (see Figure 1). In addition, since the structural framing system could yield under very strong earthquakes, the optimum device should exhibit a hardening region at large deformations so that the combined system would remain stable. From this perspective, it is also apparent that negative tangent stiffness should always be avoided for the combined system.
For the NSDs that were tested previously by the present authors (Attary et al., 2015a, 2015b), the functional dependence on the structure response (displacement in this case) could not be readily changed in real time. In these rotation-based adaptive passive (RBMAP) devices, the response of the structure is used to provide mechanical feedback to the device, resulting in modification of stiffness and/or damping and thus offering the ability to shape the rich range of force–displacement behaviors of the devices.
It should be noted that the adaptive passive devices that are described in this paper are devices that are able to produce a variety of nonlinear hysteretic behaviors which can be designed in such a way as to produce a specific desired behavior depending on the specific application (either for seismic resistance of structures or vibration control of general dynamic systems). This is in contrast to nonlinear passive devices, such as nonlinear viscous dampers, which can produce a single type of hysteretic behavior.
In this paper, the general characteristics and abilities of RBMAP devices are introduced and two examples of such devices are discussed in detail. Analytical solutions and experimental proof-of-concepts tests are presented to illustrate their force–displacement behavior. One of the devices was designed for application to a highway bridge model and its effectiveness for seismic protection is evaluated by means of numerical simulations.
2. Overview of RBMAP device
As mentioned above, a new rotation-based mechanical adaptive passive device has been developed by the authors. The device provides adaptive passive stiffness and can also be designed to provide adaptive passive damping. The devices are mainly mechanical control devices that are able to produce forces based on displacement, velocity and/or acceleration of the structure, without using sensors or any other electronic devices, thus resulting in a robust and reliable system with near-optimal performance. The devices convert the translational motion of a structure to rotational motion wherein the rotational elements of the device are designed in such a manner that they provide stiffness and/or damping modulation. They can also be used to mechanically control other seismic protection devices (mainly to replace control systems within a semi-active seismic protection system). A typical configuration of the developed devices consists of three gears (see Figure 2); the primary gear, which is attached to the structure through a pinned-pinned connecting arm, and two secondary gears, which are designed to control the motion of the primary gear and thus to influence the response of the structure.
Possible RBMAP configuration: plan view (left) and 3D view (right).
Translational and/or rotational damping devices can also be added to different parts of the assembly to dissipate energy according to a chosen control strategy. For example, Figure 2 shows a linear viscous translational damper attached to the main gear wherein the damping coefficient of the damper can be controlled based on the rotation of the main gear. Field implementation of such devices in bridge structures might include encapsulation within a steel enclosure (box) that could be attached to the top of the bridge piers and with the connecting arm attached to the bottom of the deck (see Figure 3).
Method of implementing RBMAP devices within a bridge.
3. Modulation of stiffness using the RBMAP device
To demonstrate some of the capabilities of an RBMAP device, a particular type of device is examined wherein negative stiffness is produced through the use of gears and torsional springs. The device, RBMAP-S (the suffix ‘S’ indicates that the device is capable of changing stiffness), consists of three gears (see Figure 4): the primary gear, which is attached to the structure through a pinned-pinned hook-shaped connecting arm, and two secondary gears, each restrained by a pre-torqued torsional spring. The secondary gears are held in place using parking pawls that also employ pre-torqued springs. Both the primary and secondary gears have gear teeth over only a portion of their circumference (i.e., partial teeth).
Undeformed (top) and deformed (bottom) shape of RBMAP-S (plan view).
The connecting arm transfers the earthquake-induced motion of the structure to the device. This translational motion (point A moving to point A' in Figure 4) is transferred to the primary (center) gear which can rotate freely back and forth until a certain angle is exceeded (the angle associated with point B moving to point B'). When the earthquake-induced displacement response of the structure becomes large enough, the last tooth of the primary gear (located originally at point C in Figure 4) comes into contact with the first tooth of one of the pre-torqued secondary gears (located originally at point D in Figure 4), unlocking the secondary gear from the parking pawl and releasing the pre-torque in the spring and thus rotating the primary gear in the same direction in which it was initially rotating (rotating primary gear such that point C moves to point C' due to rotation of the secondary gear where point D moves to point D'), thereby creating negative stiffness behavior and a virtual yield point at that displacement.
By preventing immediate engagement of the primary gear teeth with the secondary gear teeth, an initial dead-zone is created whereby no force develops in the device. This dead-zone ensures stability of the structure/device system (i.e., it delays the development of negative stiffness). When the motion of the structure changes direction, the primary gear first returns to its original position, rotating the secondary gear that was providing negative stiffness back to its original position such that it is engaged by the parking pawl. Next, the structure continues to displace in the opposite direction, eventually engaging the other secondary gear and producing the same negative stiffness behavior. The general behavior of this type of device (softening/hardening/other) is a function of the displacement response of the structure to which it is attached, the functional dependence being dictated by the degree to which partial gear teeth are used along the circumference of the three gears (i.e., engaging and disengaging the two secondary gears at different stages).
As noted above, this device can be used to create negative stiffness behavior (by using suitably pre-torqued torsional springs) which, when applied to an existing structure, results in virtual softening behavior and thus generally reduces earthquake force demands. If the negative stiffness of the device starts affecting the structure after a certain structural deformation is exceeded, a virtual yield point is created within the combined system, resulting in a system that behaves nonlinearly but without inelastic response. As mentioned previously, the device is capable of creating such a delay through the use of a rotational gap between the partial teeth along the circumference of the primary and secondary gears. In general, the configuration of gear teeth (the pattern of discontinuity of teeth around the circumference) and the use or nonuse of pre-torqued torsional springs can be used to create softening or hardening at multiple stages of displacement and with different functions defining the softening or hardening at specified displacements. The shape of the individual gears can also be used to modify the functional form of the force–displacement relation of the device (e.g., in mechanical engineering field, it is not uncommon to use elliptically-shaped gears, eccentric circular gears, and logarithmic spiral gears).
3.1. Derivation of force–displacement relation
For the device with circular gears, the force–displacement relation can be derived by examining the geometry of the device in both the nondeformed and deformed positions (see Figure 5). The derivation begins by determining an expression for the translational displacement, u, of the connecting arm (where it attaches to the structure) as a function of the main gear rotation, Parameters that define geometry of device for undeformed (top) and deformed (bottom) position of RBMAP.
Based on the geometry of the device in its deformed position (see Figure 5), the following relation can be obtained
3.2. Proof-of-concept testing
A small-scale prototype (about 2 ft × 1 ft, 610 mm × 305 mm, in plan) of the device was constructed and tested (see Figure 6) as a proof-of-concept. The device was designed to provide negative stiffness after a specified deformation was exceeded ( RBMAP-S prototype. Values of RBMAP-S parameters.
The device was tested experimentally by putting it in parallel with a strong translational spring having positive stiffness. One end of the spring was attached to the connecting arm of the device by means of a pinned connection and the other end was attached to a fixed support. Prescribed forces were applied incrementally to the combined system and the deformation of the device was measured at each load increment. The same test was performed for the strong translational spring alone and the results of the two tests were subtracted to obtain the force–displacement relation of the device. The results from the experimental testing (see Figure 7) showed that the device was capable of generating the desired negative stiffness behavior (i.e., an initial gap where no force develops followed by negative force for increasing displacement and therefore negative stiffness). In Figure 7, an analytical prediction of the device behavior is also provided in which the analytical relations given by Equations (3) and (12) were used and with the device parameter values being those shown in Table 1. The resulting analytical force–displacement relation compares well with the experimental data (see Figure 7), indicating that the analytical model can be used to reliably predict the behavior of the device.
Measured data from experimental testing and prediction from analytical model.
3.3. Effects of RBMAP-S on the behavior of a highway bridge model
Attary et al. (2015a) performed seismic testing of a quarter-scale highway bridge with various seismic protection devices – including negative stiffness devices. The negative stiffness devices used in the testing used a large pre-compressed translational spring. To illustrate the effects of the new RBMAP-S device on the behavior of structures, the devices were employed in the same bridge model via numerical simulations. The model consisted of a 158 kN deck (the deck is represented by a steel framework supporting concrete blocks) supported on four low damping elastomeric bearings having a total effective stiffness of approximately 12 kN/cm and two steel piers which can be braced or unbraced to mimic various bridge configurations (see Figure 8). The two NSDs were placed within support frame systems located under the bridge deck (yellow components in Figure 8a).
Bridge structure used for numerical simulations: (a) model on shake table and (b) finite element model.
The numerical simulations were performed for the case of the bridge with braced piers and under the assumption that four RBMAP-S devices were installed at the isolation level (two devices located at each pier and installed between the piers and the deck; the enclosure of the device would be mounted to the top of the piers and the connecting arm would be attached to the underside of the bridge deck – see Figure 3 for the general installation concept). It should be noted that since the effect of the device on the dynamic behavior of the structure is related to the relative displacement of the deck and the piers the devices would generally work best if they are placed in parallel with flexible isolation bearings (e.g., elastomeric bearings).
The parameter values used to define the behavior of the RBMAP-S device were the same as those used in the proof-of-concept testing (see Table 1) except that the stiffness of the torsional springs was increased to 0.7 kip-in/rad (0.078 kN-m/rad) (correspondingly, the pre-torque of the secondary gears was increased). The resulting force–displacement relation of the RBMAP-S devices is shown in Figure 9. It should be noted that the behavior that is shown of the device was based on the assumptions of linear behavior of all of the elements and no friction at sliding surfaces. In reality, there would be some friction force but it is expected to be minor relative to the total force developed by the RB-MAP device (based on prior experience with testing of negative stiffness devices (Attary et al., 2015c). Thus in this study, which was focused on conceptual development of the device and preliminary assessment of performance, the effects of friction in the RBMAP-S device were not considered.
Force–displacement behavior of RBMAP-S for use in numerical simulations of bridge model.
With four RBMAP-S devices and the stiffness of the four bearings (for the isolated bridge with braced piers (IB case), the bridge is essentially a single degree-of-freedom system with its effective stiffness being equal to the total stiffness of the four bearings), the combined lateral force–displacement behavior of the bridge/RBMAP-S system (IB + RBMAP case) is as shown in Figure 10. The force–displacement behavior of the combined system shown in Figure 10 has two major problems. One is that at large deformations the device develops high negative stiffness such that the combined system no longer has positive tangent stiffness and thus the structure becomes unstable. Because the behavior of the combined system is elastic, if the bridge with the RBMAP-S devices were subjected to an earthquake motion and the deformations of the isolation system were more than about 60 mm, it is possible that the structure would become trapped in this large deformation region (negative tangent stiffness region) and not be able to return to the positive stiffness region and thus, at the end of the ground motion, the combined system would have a permanent deformation of about 70 mm.
Force–displacement behavior of bridge model with four RBMAP-S devices.
The second problem for the combined system behavior is associated with the initial development of negative stiffness. As shown in Figure 9, the device engagement is so sudden that, at the time that the secondary gear comes into contact with the main gear, the negative stiffness of the device is very strong and thus creates another region of deformation where the structure is trapped (like the one described previously). It should be noted that since the behavior is nonlinear elastic and the combined behavior has positive stiffness after that point it is possible to use such a device, but the structure must be checked after the ground motion ends to determine if it has any permanent deformation, in which case it could be pushed back manually to its original position. Also, it should be noted that the RBMAP devices have a limited deformation capacity as dictated by the value of y0 (after 90 degrees of rotation of the main gear, the system becomes locked and thus develops very large positive stiffness).
3.4. Modified RBMAP-S design for application to the bridge model
The main advantage of RBMAP devices is that there are many options available for altering the behavior of the device, thus allowing the designer to achieve the desired behavior. As an example, the parameters of the RBMAP-S device that was described above for application to a highway bridge model could be modified to overcome the problems that were noted. To resolve the issue of sudden engagement of the secondary gear, a linear translational spring can be connected to the circumference of the secondary gear as illustrated in Figure 11.
Connection of an additional spring to the RBMAP-S device and deformation of the spring at four stages of motion (from left to right, secondary gear rotation of 0, 45, 90 and 180 degrees).
This additional spring resists the initial moment caused by the secondary gear (its stiffness would be selected such that the moment it applies to the gear would balance the moment due to the pre-torqued torsional spring attached to the secondary gear). As the secondary gear rotates, it compresses the additional spring but, since the spring will also rotate, the orientation of its force changes and the moment it applies to the gear will gradually decrease. When the secondary gear has rotated by 45 degrees, the line of action of the spring force will pass through the center of the secondary gear and thus the moment becomes zero. In addition, at this stage the additional spring reaches its maximum compression and thus, when the secondary gear rotates beyond 45 degrees, the effect of the additional spring would be reversed and the moment it applies to the gear will therefore be reversed. Because the compressed additional spring will elongate after this point, it will create a force in the same direction of the motion, causing negative stiffness. At 90 degrees of rotation of the secondary gear, the additional spring reaches its original length and thus is no longer compressed. Overall, the purpose of this additional spring is to create a delay in developing a portion of the negative stiffness of the device by adding positive stiffness which reduces the amount of the sudden change in force at the beginning of engagement of the secondary gear. Furthermore, as noted above, for larger rotations the additional spring provides negative stiffness that is additive to the inherent negative stiffness of the device. The effect of adding the additional spring on the device behavior can be determined analytically. Using the geometry shown in Figure 12, the initial length of the additional spring is Rs. Using the coordinate system shown in the figure, if the secondary gear rotates by Geometry of secondary gear and additional spring when gear is rotated.
As shown in Figure 11, the connection of the additional spring to the secondary gear is through a telescopic tube which is completely compressed at zero rotation of the secondary gear and remains compressed until the secondary gear rotation reaches 90 degrees. The reason for using this type of connection is that, if the spring were connected to the secondary gear with a simple pinned connection, when the rotation of the secondary gear exceeds 90 degrees the additional spring would produce positive stiffness which would reduce the negative stiffness of the device in this region, thus decreasing the effectiveness of the device. By using the telescopic connection, after the 90 degrees of rotation (when the additional spring is at its original length) the telescopic tubes stretch without applying force to the secondary gear, thus resulting in disengagement of the additional spring beyond 90 degrees of rotation of the secondary gear.
Using Equation (13), the RBMAP-S described previously for application to the bridge model was modified by selecting a value of 0.7 kips/in (1.225 kN/cm) for the stiffness of the additional spring. In addition, the stiffness of the rotational spring that is attached to the secondary gear was increased to ks = 1 kip-in/rad (0.197 kN-m/rad) to produce additional negative stiffness behavior (the stiffness values were selected in such a way as to cancel out the positive stiffness of the elastomeric bearings with the initial negative stiffness of the device). It should be noted that since the radius of the main and secondary gears are the same in this example and, due to the geometry, the main gear cannot rotate by more than 90 degrees, the secondary gear will not rotate by more than 90 degrees and thus there is no need for a telescopic connection. Because the radius of the main and secondary gears are the same in this example, the telescopic connections were not needed; but if their radii were chosen to be different from each other and the secondary gears needed to rotate by more than 90 degrees, this type of connection would be necessary. For example, if the radius of the main gear was 6 in (152.4 mm) and the secondary gears were 3 in (76.2 mm), the secondary gear would rotate twice as much as the main gear with a maximum rotation of 180 degrees. Another benefit of the additional springs is that the parking pawls with their pre-torqued torsional springs are no longer needed, resulting in simpler and more reliable devices. Note that although the modified device provides improvements over the original device, the original device has been discussed here at length because the basic mechanics are the same – that is, Equations (1) to (12) – and, furthermore, experimental data that validate the basic mechanical behavior were available for comparison with the analytical model.
To overcome the large negative stiffness that occurs in the region of large deformations, another torsional spring can be attached to the secondary gear in which its connection to the secondary gear is made through a slotted arc-shaped hole (see Figure 13). This spring would add positive stiffness to the combined system at a specific deformation and rotation. For the modified RBMAP-S, this hole was designed to engage the positive stiffness after π/5 radians of rotation of the main gear.
Method of connecting additional torsional spring to cancel out high negative stiffness of the device for large deformations: undeformed shape (left) and deformed shape (right) of the secondary gear.
Different types of rotational springs can be used in RBMAP devices (e.g., constant or variable force springs). For application of the modified RBMAP-S to the bridge model, a rotational spring that develops moment proportional to the third power of the rotation was used (because the designer would have flexibility in selecting the moment–rotation relationship, the particular relationship used here was arbitrary). The stiffness of the spring in this case is ks = 30 kip-in/rad3 (3.39 kN-m/rad3). In an effort to reduce further the forces in the structure caused by earthquakes, the value of Force–displacement behavior of modified RBMAP-S (used in numerical simulations of the bridge model).
As can be seen, the modified device design results in a smooth transition to negative stiffness with an approximately constant negative stiffness over a wide range of deformations. In addition, the modified device exhibits a smooth hardening region at large deformations, which helps to avoid damage to the structure due to excessive deformations.
Using four of the modified RBMAP-S devices in parallel with the elastomeric bearings in the isolation system of the bridge model with braced piers, the behavior of the combined system is as shown in Figure 15.
Force–displacement behavior of bridge model with the modified RBMAP-S devices.
As can be seen, the modified device results in a smooth transition to a zero tangent stiffness region, which promotes large force reductions in the bridge model and, at large deformations, provides a smooth transition to a positive tangent stiffness region, thus avoiding damage to the bridge through possible excessive deformations. It should be noted that, if necessary, the rotational springs can be placed on the top and bottom of the gears to increase the number of spring attached to a gear. Based on the modification described here, the final modified device would be of the form shown in Figure 16.
Sketches of the complete modified RBMAP-S devices, 3D view (top): plan view from top (bottom left), plan view from bottom (bottom right).
3.5. Numerical simulation of seismic response of bridge with modified RBMAP-S devices
The effectiveness of the RBMAP-S device for seismic protection was evaluated by means of numerical simulations of the aforementioned bridge model. The simulations were performed using SAP2000 software whereby the RBMAP-S device was modeled using a multi-linear link element that made use of the force–displacement behavior of the device. For implementation of the modified device in the bridge model, the low-damping/low-stiffness elastomeric bearings in combination with the negative stiffness of the RBMAP-S devices produces softening behavior that could result in an undesirable increase in the deformations at the isolation level. To limit such deformations, a small linear viscous damper with damping coefficient of 0.05 kips-s/in (8.76 kN-s/m) was connected to the main gear of the RBMAP-S device with a pin–pin connecting rod similar to that shown in Figure 2. These dampers were included in the numerical model by means of viscous damper link elements in parallel with the RBMAP devices. Note that the amount of damping that is needed is relatively small because the RBMAP-S devices produce a softening behavior which reduces the effective stiffness of the structure. This reduction in stiffness, combined with a small increase in damping, results in a significant increase in the effective viscous damping ratio of the structure. In addition, since this paper focuses on development and preliminary assessment of the performance of these types of devices, the relatively minor influence of friction on the hysteretic behavior of the devices was not considered. Thus the idealized model of the device used here does not exhibit any hysteresis. Because the device is a mechanical system, in reality it would of course have some level of damping associated with friction which would reduce the required amount of additional viscous damping.
The numerical model of the bridge with RBMAP-S devices and the viscous damping device was analyzed for various earthquake ground motions using nonlinear time-history analysis. Due to space limitations, only selected results for the Kobe earthquake (KJM-000 ground motion) are presented here. A comparison of the base shear (shear force in bridge piers) for the case of the isolated bridge (IB) with unbraced piers relative to that of the same bridge, except with four modified RBMAP-S devices installed (IB + RBMAP), is shown in Figure 17. As can be seen, these devices reduce the peak base shear by about 60%. In addition, Figure 18 shows that the RBMAP-S devices have a similar effect on the peak deformation of the isolation system. The reduction in base shear is related to the negative stiffness provided by the RBMAP-S devices while the reduction in deformation is attributed to the energy dissipation provided by the viscous dampers together with the particular characteristics of the ground motion used in the simulations.
Base shear of bridge model for the case of IB and IB + Modified RBMAP devices for Kobe KJM-000 ground motion. Shear deformation of isolation system for the case of IB and IB + Modified RBMAP devices for Kobe KJM-000 ground motion.
4. Modulation of damping using the RBMAP device
Using the same general approach described above for producing adaptive stiffness, the proposed device can also be modified in a variety of ways in order to produce adaptive damping. One simple approach is to attach a rotational viscous damper to the main gear. Because a functional relationship exists between displacement of the structure and rotation of the main gear (see Equation (2)), their derivatives can also be related to each other and thus the damping force of the rotational damper would be mechanically dependent on the response of the structure. Another method is to add rotational dampers to the secondary gears and then control the secondary gears in a manner similar to that described for the RBMAP-S device (as an example, by having teeth on a portion only of the gear circumferences). A third method is to use the rotational elements to control a viscous damper located outside of the device. As discussed above, one of the first semi-active viscous dampers for implementation in a bridge model for seismic testing was developed by Kawashima and Unjoh (1994), in which a viscous damper was controlled by a servo-valve and its properties were controlled based on the relative deformation of the bridge deck with respect to piers (large damping at small and high deformations, low damping at intermediate deformations). The philosophy behind the control scheme was that at low deformations the damping coefficient should be high so that the damping device behaves like a snubber (loads that result in such low deformations include passing traffic, wind loading and thermal expansion). For earthquake loading, the damping coefficient reduces linearly as the deformation increases and, upon reaching some deformation threshold, the damping coefficient remains constant at a relatively low value in order to allow deformations to increase, thereby dissipating energy and reducing accelerations. Finally, upon reaching some higher deformation threshold, the damping coefficient is increased linearly to limit the maximum deformation between the bridge deck and piers (i.e., the damping device again acts as a snubber).
As mentioned previously, RBMAP devices can be used to control passive devices and replace semi-active control devices. For example, the semi-active control device developed by Kawashima and Unjoh (1994), could be replaced with the RBMAP-D device shown in Figure 19 (the suffix ‘D’ indicates that device is capable of changing damping).
Schematic of RBMAP-D showing viscous damper and the two parallel bypass pipes.
This device has two gears with teeth around their complete circumference. As with other RBMAP devices, the motion of the structure is transferred to the device using a crank mechanism. Rotation of the main gear is transferred to a secondary gear whose radius is one-half that of the main gear (and thus the secondary gear rotation is twice that of the main gear). The device is used to control a translational viscous damper which is attached to the structure in the same manner as a passive viscous damping device would be attached. To change damping of the device, a mechanical control valve is used to modulate the fluid flow through an external bypass pipe. The mechanical control valve is a plug valve which has a plug (rotating piece) that covers 30 degrees of the inlet port of the valve. For the control scheme developed by Kawashima and Unjoh (1994) – see Figure 20 – the damping coefficient was reduced because the damper displaced up to 1/6 of its maximum stroke. Beyond that, the damping coefficient was held constant for a wide range of deformations and then increased as the deformation became large. Using the crank mechanism, the main gear would be able to rotate by ± 90 degrees, meaning that the secondary gear would be able to rotate by ± 180 degrees. The secondary gear is connected to the plug valve so that the plug that covers 30 degrees of the inlet port would also be able to rotate ± 180 degrees. Because the plug is covering 30 degrees of the inlet port and it starts rotating from a closed position, it takes 30 degrees of rotation (1/6 of the 180 degrees of maximum rotation) to open the inlet port completely. After that, as the gears are rotating the valve is open for 120 degrees of rotation of the secondary gear (2/3 of the maximum rotation) and then the plug would gradually close the inlet port in 30 degrees of rotation (1/6 of the maximum rotation). Using this device, behavior similar to that devised by Kawashima and Unjoh (1994) can be achieved with the slight difference that, for the RBMAP-D device, the increase in damping coefficient at larger deformations is delayed by an amount corresponding to 1/6 of the maximum rotation (see Figure 20). As a result, the increase in damping coefficient occurs over a shorter amount of deformation (in 1/6 of the maximum rotation instead of 1/3).
RBMAP-D behavior versus the control scheme proposed by Kawashima and Unjoh (1994).
Using other types of mechanical valves, different behavior can be achieved for these types of devices. As an example, valves that work with velocity or acceleration can also be used to control the flow of the fluid in the bypass pipe, thus enabling the device to change its properties based on velocity, acceleration or displacement-based mechanical feedback from the structure. This concept can be used to improve the RBMAP-D device that was described above. For instance, as can be seen in Figure 19, it is possible to use two bypass pipes on the device that are parallel to each other and on one of which there is a plug valve, as discussed previously; and on the other there is an inertia control valve, which would be attached to the main gear. The inertia control valve is controlled by the acceleration of the motion; it would normally be closed, but it has an acceleration limit and, in the event of ground motion, if the acceleration of the structure is higher than the limiting value, the valve would mechanically open to protect the structure against possible sudden increases in acceleration.
5. Conclusions
A new concept for seismic protection devices is described in this paper. The devices are adaptive passive and are able to modify stiffness and/or damping of a structure mechanically using mechanical feedback from the structure to which they are attached. In general the devices operate by transforming the linear motion of a structure into rotation using a crank mechanism attached to rotational elements that are designed to control the behavior of the structure during seismic loading events. Two examples of this class of devices are presented here: one is a device that is able to change the stiffness of a structure (RBMAP-S); the other is able to modify the damping provided by a viscous damper (RBMAP-D).
The RBMAP-S device adds negative stiffness to a structure after a particular deformation is reached, resulting in linear behavior followed by a virtual yield point and subsequent softening behavior. The main effect of the device when applied to highway bridges is to reduce the acceleration of the deck and thus the shear forces in the piers. A small prototype of the device was tested to verify the concept at a small scale. In addition, numerical simulations were conducted to evaluate the effectiveness of the device when installed within a bridge model. The simulations showed that the device can substantially reduce shear forces in the bridge piers during earthquakes.
A second device configuration was explored (RBMAP-D) in which adaptive passive damping was used. The device was designed to mimic the behavior of a semi-active control system that had been used to apply variable damping to highway bridges. Theoretically, the device produced similar behavior and thus would be expected to be capable of producing simultaneous reductions in bridge deck accelerations and displacements.
The main advantage of RBMAP devices over other seismic protection devices is the wide range of possible adaptive passive behavior that can be achieved using different rotational elements. In addition, there are many parameters that can be adjusted to achieve the desired behavior. In this paper, examples included the use of torsional and translational springs (which could be pre-torqued), torsional viscous dampers, plug valves and inertia control valves. The effects of adjusting some of the parameter values has also been discussed. In these examples, devices with only two secondary gears were considered. The number of secondary gears can be increased as necessary to change the behavior of the device at different stages of operation. Other possibilities for controlling the device behavior include the use of gears with noncircular shapes (e.g., oval or square-shaped gears). More advanced rotational elements, similar to those found in vehicle transmissions, could be used to adjust the behavior of the device. In summary, it is argued that the RBMAP devices represent a new concept in seismic protection devices in which passive mechanical means are used to control the response of a structure. Such a concept potentially opens up a range of new opportunities in the field of earthquake and structural engineering, taking advantage of advances in the field of mechanical engineering field.
Footnotes
Acknowledgements
Thanks are due to Mr Jon LaPointe, Academic Support Technician for the Department of Civil and Environmental Engineering at Rensselaer Polytechnic Institute, who assisted with construction and testing of the prototype RBMAP device.
Conflict of interest
The authors declare no conflict of interest
Funding
This work was supported by the National Science Foundation (NSF) under grant number CMMI-0830391. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of NSF.