Habiba Lahdhiri
email: habiba.lahdhiri@ensea.fr

Jordane Lorandel
email: jordane.lorandel@ensea.fr

Emmanuelle Bourdel
email: emmanuelle.bourdel@ensea.fr

Threshold-based routing algorithm for RF-NoC OFDMA architecture

Keywords: RF, NoC, OFDMA, Routing, Reconfigurable

Network-on-chip (NoC) is currently the chosen paradigm for interconnection in multicore architectures. On the other hand, the development of these architectures with thousands of cores leads to several issues regarding latency and power consumption, especially with conventional wired interconnects. Moreover, they do not support specific features like broadcast and multicast. Radio-Frequency NoC based on OFDMA seems an appropriate solution to overcome such issues, providing low latency for long communications and a significant communication bandwidth with a high spectral efficiency. It also permits a dynamic allocation of the available bandwidth among the different elements of the NoC, according to the communication needs. The routing algorithm has a significant impact on the overall performance of a RF-NoC, by diffusing the traffic in a suitable way between the wired and the wireless paths in order to improve the throughput and reduce the latency. In this paper, a threshold-based routing algorithm is proposed and evaluated for a 32×32 RF-NoC OFDMA architecture with 32 concurrent wireless channels. We demonstrate the importance to finely tune the threshold parameter to achieve a certain level of performance. Simulation results show a significant improvement in terms of latency and throughput.

I. INTRODUCTION

The development of new architectures with thousands of cores on the same chip, communicating together, reveals new challenges. In particular, communications between the cores using conventional wires create a bottleneck, leading to poor performance and suffering from a high latency. This is particularly true when long communications across the chip occurs. Several interconnect technologies have been proposed since the last decades, permitting to overcome the main issues of classical wired NoC.

Firstly, optical technology is a promising alternative to conventional NoC, providing shortcuts for destination and source that are faraway from each other. The use of optical interconnect allows to achieve high data transmission rate and a very low latency. A critical limitation is the need for an external light source. 3D-NoC is an attractive approach to solve interconnect scaling. This architecture fulfils the demands for future multicore architectures, exploiting the short vertical links between adjacent layers that significantly improves the network performance [START_REF] Ye | Analysis of power consumption on switch fabrics in network routers[END_REF]. A Multi-layer NoC architecture reduces the distance between routers in a considerable way. In 3D-NoC energy reduction can be achieved up to 29%, in the average packet latency up to 2% and in the area occupied by the routers of the NoC up to 18% [START_REF] Ye | Analysis of power consumption on switch fabrics in network routers[END_REF]. However, the advantages of this technology cannot overlook thermal issues when the number of layers increases. Some proposed architectures benefit from both 3D and optical technologies [START_REF] Ye | 3D optical networks-on-chip (NoC) for multiprocessor systems-on-chip (MPSoC)[END_REF]. The upper layer contains the optical data transmission network, which is responsible for optical payload packets transmission. The bottom layer contains an electronic control network used to route control packets and to configure the optical network. Power analysis based on synthesis results shows that the optical NoC can have a 70% power saving for 2048B packets.

RF interconnects can also be used for long-range communications. RF interconnects add a set of shortcuts to the network that significantly reduces the latency. Moreover, due to the reconfigurability of RF interconnections, it is possible to modify the bandwidth allocation and adapt it to the demand. RF waves can be transmitted through a wave-guide or an antenna. Solutions using antennas have greater flexibility, but they also increase consumption in comparison to wave-guides, and are less robust against interferences. Another solution of interconnect is the combination of optical and wireless Network OWN [START_REF] Sikder | OWN: Optical and Wireless Network-on-Chip for Kilo-core Architectures[END_REF]. The communications within the same cluster (64 cores) are realised with one-hop photonic interconnect while communications between clusters use one-hop wireless interconnect. Such architecture enhances scalability and simulation results show that the energy/bit for OWN is 30.36% lower than wireless and 13.99% higher than photonic architecture. Several studies [START_REF] Carloni | Networks-on-chip in emerging interconnect paradigms: Advantages and challenges[END_REF], [START_REF] Wang | Wireless network-on-chip: a survey[END_REF] have made preliminary comparisons among these different interconnect technologies. Table I gives an overview of the main advantages and limitations of the interconnect technologies mentioned previously.

To fully exploit an interconnect technology as optical or RF, new efficient communication mechanisms have to be proposed. In particular, all these interconnect technologies require a multiple access technique to efficiently share the spectrum resources between the numerous elements wishing to communicate. In addition, resource allocation has to be fairly allocated among the elements of the network [START_REF] Luo | Channel Allocation protocol for reconfigurable Optical Network-on-Chip[END_REF].

A lot of multiple access techniques exists in the literature, such as Frequency Division Multiple Access (FDMA), Time-Division Multiple Access (TDMA), Wavelengh Division Multiple Access (WDMA). Multi-carriers modulation tech- Multi-hop paths replaced by single hop links [START_REF] Wang | Wireless network-on-chip: a survey[END_REF], [START_REF] Hamieh | Sizing of the physical layer of RF inra-chip communications[END_REF] Challenges thermal issue because of high-power density Integration of on-chip photonic components mixed-signal physical design problem integration of onchip antennas. [START_REF] Chang | Power reduction of CMP communication networks via RF-interconnects[END_REF] niques such as Orthogonal Frequency Division Multiplexing (OFDM), achieve a high spectral efficiency while transmitting data among multiple orthogonal sub-carriers. OFDM provides many advantages such as robustness against inter-carrier interference, high spectral efficiency, simple equalisation in the frequency domain at the receiver side, etc. However, OFDM is very sensitive to synchronisation issues. Orthogonal Frequency Division Multiple Access (OFDMA) is based on OFDM allowing multiple users to share a common bandwidth easily. From a design point of view, OFDMA needs fewer hardware resources for the RF transceiver as compared to FDMA, which necessitates as many local oscillators as the number of elements connected to the interconnect. OFDMA is thus a more scalable technique. Moreover, sub-carriers allocation can be directly done from the base-band and dynamically changed while the application is running, allowing the frequency bandwidth used by each core or group of cores to be adapted according to their needs [START_REF] Unlu | An OFDMA based RF interconnect for massive multicore processors[END_REF]. For the aforementioned reasons, OFDMA was chosen as the most well-suited technique in our study.

A key element for NoC is the routing algorithm. To keep increasing NoC performance implies the development of new routing algorithm that can fully exploit the interconnect capabilities. In this paper, a routing algorithm for OFDMAbased RF-Noc is proposed. It allows to distribute the traffic communication by comparing the distance between source and destination to a user-defined threshold. From our knowledge, the algorithm has already been proposed in [START_REF] Hu | Design and Analysis of a Meshbased Wireless Network-on-Chip[END_REF] but was never applied to an OFDMA RF-NoC, combining RF-waveguide and wired NoC. Moreover, the choice of threshold was not deeply investigated. A deep study is thus provided to deeply investigate the choice of threshold depending on the target latency and throughput. This paper is organised as follows. First, Section II details the new interconnect technologies used in multicore architectures and their associated topologies and routing algorithms. Then, the RF-NoC architecture based on OFDMA is presented in Section III and the proposed routing algorithm is given in Section IV. Simulation results are presented in Section V and finally conclusion and future works are described in Section VI.

II. BACKGROUND AND RELATED WORKS

A. Topology

The topology defines the physical layout and the connections between nodes in the network. The topology impacts network cost and performance. It influences the latency by determining the number of hops a packet must perform from the source to its destination.

For conventional NoCs built around a wired interconnect between nodes, topologies can be divided into 3 categories: direct, indirect and irregular. Direct topologies refer to meshes or tori. Indirect topologies refer to butterflies networks. Finally, irregular topologies refer to multiprocessor system-on-chip (MPSoC) with a large range of heterogeneous intellectual property (IP) blocks.

To benefit from new interconnect technologies, novel topologies have been proposed. For example, WiNoC use the wireless interconnect between nodes to reduce the network latency, especially for these long-distance nodes [START_REF] Ditomaso | A-WiNoC: Adaptive Wireless Network-on-Chip Architecture for Chip Multiprocessors[END_REF].

In order to provide scalability and flexibility, the wireless links are used to replace the wired interconnect. In a purely wireless NoC, all the links between two nodes should be wireless. As the number of nodes is growing, wireless channels cannot handle all the links of simultaneous communications. Thus, purely wireless NoC topologies scale poorly. To overcome this issue a multi-channel WiNoC was proposed [START_REF] Zhao | Design of multi-channel wireless NoC to improve on-chip communication capacity![END_REF]. The network is composed of RF nodes working as a wireless router with ultra-wide-band (UWB) transceivers and antenna. The wired links in the conventional 2D mesh are replaced by high-bandwidth RF links.

Another topology, called hybrid, is based on wired links for short-distance communications and wireless/optical links for long-distance communications. Hybrid topology is constructed by replacing some of the routers in the 2D mesh-based topology with wireless routers, generally one wireless router per cluster (sub-net) which have both wired and wireless links [START_REF] Wang | A Wireless Network-on-Chip Design for Multicore Platforms[END_REF]. An example of hybrid topology is the multiple tiers hybrid topology proposed to deal with scalability problems [START_REF] Lee | A scalable micro wireless interconnect structure for CMPs[END_REF]. The particularity of this topology is the recursive wireless interconnection structure named WCube which is also a hybrid topology with a multi-tier structure, including the wireless backbone and wired edges. The wired edge (CMesh), which is a 2D mesh with 4-way concentration (concentration in on-chip networks is to share the router among different (processors) nodes). Another example is the Small-world-based topology which divides the whole network into multiple small groups (sub-nets) with different heterogeneous topologies. It can be any of the conventional NoC topologies, such as mesh, tree, ring etc. The upper level of the network contains hubs connected by both wired and wireless links [START_REF] Wang | Wireless network-on-chip: a survey[END_REF].

Irregular NoC topologies are usually used for MPSoCs with heterogeneous IP blocks. In [START_REF] Wu | Low-cost deadlock-free design of minimaltable rerouted xy-routing for irregular wireless nocs[END_REF] a low cost and efficient distributed minimal table-based routing for an irregular WiNoC topology is presented. With these particular topologies, the routing algorithm is very important to fully exploit the capacity of the additional interconnect, conditioning NoC performance.

B. Routing Algorithm

The routing algorithm defines the way to route packets from source to destination. It must distribute the traffic in an appropriate manner, avoiding deadlocks and significant delays. An efficient routing algorithm allows thus the improvement of the network throughput and the reduction of the latency. In conventional wired NoCs, routing algorithms can be classified into two categories: deterministic and adaptive. routing algorithms route packets from point A to another point B along a fixed path and offer the best choice for uniform or regular traffic patterns. Adaptive routing algorithms use information about the network traffic and channel conditions to avoid congested network regions. An adaptive approach is preferable when dealing with irregular traffic or in networks with unreliable nodes and links.

For RF-NoC architecture, the routing algorithm is specific to the topology. As an example, Location-based algorithm (LBR) has been proposed in for McWiNoC [START_REF] Zhao | Design of multi-channel wireless NoC to improve on-chip communication capacity![END_REF]. It is based on the knowledge of the router node and all its neighbours to make the routing decision. When routing a packet towards the destination, it will choose to send the packet to its neighbour which is geographically closest to the destination to the destination. In order to optimise the LBR algorithm, the authors divided the chip area into four quadrants from the perspective of a node. When a router receives a packet, by checking the position of destination D, only one quadrant containing node D will be chosen. Then, only the neighbours of the selected quadrant will be calculated for the shortest path.

In [START_REF] Lee | A scalable micro wireless interconnect structure for CMPs[END_REF], a hybrid WiNoC is divided into 3 tiers. For each tier, a routing algorithm must be chosen first. Then, further routing strategies are required to decide if and how packets are transmitted between the different layers. The authors devise new two-tier wormhole based routing algorithms that are deadlock free and ensure a minimum latency route on a 1000core on-chip interconnect network, showing a reduction of the observed latency going from 20% up to 45%.

In Small-world-based routing [START_REF] Wang | Wireless network-on-chip: a survey[END_REF], the routing within the cluster can be any of the traditional NoC routing algorithms according to the topology of each sub-net, for example XY routing algorithm. For the inter cluster transmission, the packet will be routed through the hubs. Therefore, whether using the wireless links or just taking the wired link along the ring is very important to the network performance.

Adaptive routing is based on the same concept as the previous algorithms by using a traditional NoC routing algorithm within the cluster (wired sub-net). However, for communications between different clusters an adaptive routing algorithm is needed. For example, in [START_REF] Wang | A Wireless Network-on-Chip Design for Multicore Platforms[END_REF] the path selection is a function of travelling distance and buffer utilisation status to dynamically avoid the congestion.

Finally, for irregular WiNoC topologies, traditional routing algorithms may not work in some situations. Therefore Wu et al. [START_REF] Wu | A Low-Cost Deadlock-Free Design of Minimal-Table Rerouted XY-Routing for Irregular Wireless NoCs[END_REF] proposed a distributed minimal table based routing scheme by utilising the geographic distance information. The proposed routing can be also applied to the conventional irregular mesh topology connected by a wired network.

RF-NoC OFDMA topology combines both wired mesh topology and radio hubs for RF interconnections. For this specific architecture a dedicated routing algorithm is required which will be presented instead.

As the number of RF interfaces in the same chip is growing, an efficient medium access control (MAC) strategy is required for every RF-NoC architecture to efficiently share the RF medium. The MAC is a major factor of performance in WiNocs since it forbids simultaneous accesses to the same channel.

Related works on MAC for WNoCs can be classified into two types. Contention free [START_REF] Matolak | Wireless networks-on-chips: architecture, wireless channel, and devices[END_REF] approaches are based on the use of a coordinated access protocol. The most common example is the token-ring, allowing only the RF interface holding the token to access the medium and transmit data. This group of MAC strategies has the advantages of delivering a high throughput and avoiding collisions, but does not work well under variable workloads since bandwidth is statically allocated. An other limitation is the overhead related to the token circulation that increases with the number of RF interfaces.

The second type of MAC strategy is called Contention based. This group is specified by the random access that means all nodes can try to transmit on the shared medium at any time instant, offering flexibility and reducing the latency. However, the risk of collision is increased, impacting negatively the throughput of the network [START_REF] Abadal | Medium Access Control in Wireless Network-on-Chip: A Context Analysis[END_REF].

For our study, we selected a hybrid topology for the RF-NoC that uses the RF link for long range communication. To share the common bandwidth, Orthogonal Frequency Division Multiple Access (OFDMA) was chosen. Finally, for the routing algorithm, we adapted the algorithm proposed in [START_REF] Wang | A Wireless Network-on-Chip Design for Multicore Platforms[END_REF] for the RF-NoC OFDMA architecture detailed in the next section.

III. RF-NOC ARCHITECTURE WITH OFDMA

The studied RF-NoC architecture, illustrated in Figure 1, contains two levels of hierarchy: clusters and cores. At each hierarchical level corresponds a specific interconnect, respectively RF interconnect and wired NoC. Each cluster contains 32 routers (1 per core), which are interconnected with a wired NoC while the communications between clusters are realised through the RF wave-guide. Note that communications between clusters are also possible through the wired interconnect. Finally, the global RF-NoC architecture integrates 1024 cores (and routers). Each cluster features an RF-NoC interface consisting in one OFDMA transmitter and one receiver, called radio hub [START_REF] Drillet | Flexible Radio Interface for NoC RF-Interconnect[END_REF]. The RF-NoC interface is connected on one side to the four central routers as illustrated in Figure 1, and on the other side to the RF wave-guide. Accessing the is thus possible using Orthogonal Frequency-Division Multiple Access (OFDMA) that achieves a high spectral efficiency by dividing the bandwidth into several orthogonal narrow sub-channels. The use of OFDMA allows simultaneous communications between several radio radio using different frequency channels.

In our configuration, a bandwidth B of 20 GHz is divided into 1024 sub-carriers. A frequency spacing of 19.53 MHz between each sub-carrier is thus obtained. The OFDMA symbol duration of T s is computed as follows and is equal to 51.2 ns:

T s = (1/B) * N sc (1)
with N sc representing the number of sub-carriers. The data rate R can be changed to transmit more information per OFDMA symbol, by modifying the symbol modulation:

R = (M * N sc )/T s ( 2
)
where M is the modulation order, representing the number of bits per symbol (e.g. 2 for QPSK, 4 for 16-QAM, etc.).

Each cluster can transmit data through its RF interface using a group 32 contiguous sub-carriers, but can receive the entire bandwidth, making possible multicast and broadcast communications between clusters. Throughout the choices of MAC technique and topology, such RF-NoC architecture ensures scalability while being physically compatible with existing CMOS technology. However, to efficiently exploit the RF-NoC architecture, a main issue is the definition of an appropriate routing algorithm.

Cluster RF NoC Interface

IV. THRESHOLD-BASED ROUTING ALGORITHM

The routing algorithm decides which path a packet will take from the source to its destination. It is responsible for diffusing the traffic on the network in a suitable way to improve the throughput of the network and reduce the latency.

In fact, the algorithm is based on the calculation of the distances (number of hops) from the packets source node S to its destination node D using the wireless link and the wired NoC, as shown in eq. 3 and eq. 4. If the wireless distance is greater than the wired distance then the packets are transmitted using the wireless link, leading to a reduction of the average delay of the network. A naive solution could be to take the minimum distance but this could rapidly create a bottleneck at the radio hub. Therefore, an adjusting threshold γ is defined to control the utilisation of the RF link to avoid network congestion.

D wired = |(x s -x d ) + (y s -y d )| (3)
D wireless = |(x s -x cs )+(y s -y cs )|+|(x d -x cd )+(y d -y cd )| (4)
The threshold mainly depends on the network size and in our study, an optimum value of γ has been determined and explained later. To illustrate the routing algorithm, we present three cases. Firstly, if the source S and the destination D belong to the same cluster, there is no need to use wireless links. The packet is transmitted through the wired links according to a XY routing. Secondly, if the S and D are not from the same cluster but they are just close enough, satisfying D wired + γ < D wireless , then packets are routed using the wired NoC. Thirdly, only for long distance communications, where D wired + γ > D wireless , the routing decision is to take the wireless path by reaching the closest router attached to the radio hub in the cluster of S using XY routing algorithm. The packet is then transmitted using the wireless link and is delivered to the closest router from the destination that is attached to the radio hub. Finally, if needed, a XY routing is used to route the packet towards the destination router.

To avoid deadlock situations, virtual channels (VCs) are used. Two different sets of VCs are deployed in each router and each radio hub. The first set of VCs is used for communications going from the source to CS using the wired path. The second set of VCs is dedicated to packets going from the source to the destination using wired path or packets going from CD to the destination. As shown in [START_REF] Hu | Design and Analysis of a Meshbased Wireless Network-on-Chip[END_REF], VCs ensure the deadlock free property because they separate the allocation of physical channels from buffers, so that the physical channels are given to another packet when a packet is blocked.

The Threshold-Based Routing (TBR) algorithm is summarised in algorithm 1.

V. SIMULATION RESULTS

To evaluate the performance of the routing algorithm for our 1024-core RF-NoC OFDMA architecture, the Noxim simulator was used [START_REF] Catania | Noxim: An open, extensible and cycle-accurate network on chip simulator[END_REF]. It is an existing SystemC-based NoC simulator, allowing to perform cycle-accurate simulations. It supports the simulation of both conventional wired and wireless NoC architectures and several topologies, while including different routing algorithms, e.g. XY, Odd-Even, West-first, North-last, and many traffic distributions. By default, a tokenring technique is used to specify how to access the radio channels, and only one radio hub can transmit information on the wireless link at a time. However, Noxim was extended to support OFDMA and concurrent accesses to wireless channels. The TBR routing algorithm described in Section IV was also integrated into the simulator.

Table III indicates the simulation parameters used to evaluate the architecture and the performance of the TBR routing algorithm. We first evaluate the average latency of the architecture, corresponding to the time spent by a packet to cross the network from the source to the destination node. Figure 2 gives the average delay versus the threshold value for TBR and conventional XY routing algorithm, respectively illustrated with the blue and red bars. The simulations are done with a constant value of Packet Injection Rate (PIR) of 5E-3 packet/cycle/node. It can be observed that the average delay is very important for TBR when the threshold is equal to zero, giving an average use of the RF link about 96%, as indicated in Table III. When the threshold increases, we notice that the latency gets reduced in a significant way until it reaches an optimal value for a threshold of 20. When γ is 20, the latency is reduced by 93% as compared to the conventional XY routing. Referring to the RF-NoC architecture, this particular threshold value corresponds to a communications with a distance of at least two clusters, indicating that the RF link has to be used for long communications only. However, once the threshold is greater than the optimal value, the average delay increases again, but it is still lower than for XY routing, highlighting the benefit of integrating RF interconnect in addition to a wired NoC. Finally, the choice of the threshold value has a direct impact on the network latency while depending of the RF-NoC size. For our 1024-core RF-NoC OFDMA architecture, the optimal threshold value is γ = 20. This value will be used as reference in the rest of the paper. Figure 3 gives the average delay against PIR for TBR, XY with RF path and XY without RF. By comparing the red and black curves is demonstrated the importance of providing an appropriate routing algorithm to efficiently exploit the RF interconnect. As illustrated, the red curve representing a wired XY routing suffers from a slightly higher latency as compared when RF communications are activated. TBR achieves a lower average latency when the PIR is getting higher than 4E-3. A reduction of 95% of the latency is denoted as compared to wired XY routing algorithm at PIR = 6E-3. We can note that there is no significant gain in terms of latency to use RF communications at low PIR.

Then, the performance of the TBR algorithm is compared to existing routing algorithms, i.e. North Last, Odd-Even and West-First, under a random traffic. Figure 4 gives the evolution of the network latency for several values of PIR. Recall that the threshold value was fixed at 20 hops for TBR algorithm. As it can be noticed, TBR provides the lowest latency at any PIR value in the range [5E-4,1E-2], but the gap is remarkable from a PIR of 4E-3. The average wireless utilisation for PIR = 6E-3 is equal to 35%. The optimal value of threshold has prevented congestion of radio hubs and delayed network saturation that's way TBR provides better performance. The IP throughput was also evaluated for different values of PIR for the routing algorithms. As illustrated by Figure 5, threshold-based routing algorithm makes the network able to support a more important IP throughput, for PIR values greater than 5E-3. At PIR of 75E-4, an improvement of the IP throughput of 28% is achieved for TBR in comparison to XY routing with wireless links. For lower PIR values, the same IP throughput is achieved for the 3 cases studied.

The Performance of the TBR algorithm was also evaluated for another traffic pattern, Transpose 1, which provides more unbalance to the network, showing more differences between routing algorithms [START_REF] Wu | A survey of routing algorithm for mesh Network-on-Chip[END_REF]. As illustrated by Figures 6 and7 TBR achieves better performance in terms of latency and throughput. The first observation is the similarity of the two curves wired XY and XY with the RF link, which shows that such is our architecture (only the four routers in the center of the cluster that are attached to the radio hub) and with this kind of traffic the RF link does not bring any gains in terms of latency and throughput. However, a reduction of 70% of the average latency is obtained for TBR at a PIR of 6E-3 while the throughput is increased of around 27%. Throughout these results, TBR routing algorithm provides an efficient way of distributing the traffic on both wired and RF links, reducing the network latency and increasing the IP throughput for different traffic patterns. However, the choice of threshold is a critical step conditioning a certain level of performance. Finally, RF communications supporting by our 1014-core RF-NoC OFDMA architecture should only be used for long communication with the right threshold value to achieve better performance.

VI. CONCLUSION AND FUTURE WORK

In this paper, a threshold-based routing algorithm is evaluated for a 1024-core RF-NoC OFDMA architecture. We demonstrated the benefit of using RF-NoC for long communications and the impact of the threshold value on critical metrics such as latency and throughput, for different traffic patterns. The routing algorithm TBR is a preliminary step towards the development of a completely distributed cognitive RF-NoC based on OFDMA. Future work will focus on the development of a spectrum sensing technique into the RF-NoC architecture, allowing to determinate spectrum access opportunities and dynamically allocate spectrum resources ondemand. The proposed architecture will also be evaluated using real traffic patterns to better investigate the behaviour of the network in these conditions.

Fig. 1 .

 1 Fig. 1. RF-NoC architecture

Algorithm 1

 1 Proposed routing algorithm Inputs: S(x s ,y s ), D(x d ,y d ) and traffic information. Output: A decision for routing the packet. S(x s ,y s ): Source router coordinates. D(x d ,y d ): Destination router coordinates. CS(x cs ,y cs ): Closest attached router from S node coordinates. CD(x cd ,y cd ): Closest attached router from D coordinates. D wired : Wired distance between S and D using wired path. D wireless : Wireless distance between S and D using wireless. γ: Routing threshold. Procedure: if S, D ∈ same cluster then XY routing from S to D else if D wireless < D + γ then XY routing from S to CS Wireless routing from CS to CD XY routing from CD to D else XY routing from S to D end if

Fig. 2 .

 2 Fig. 2. Average delay vs threshold for TBR and XY routing algorithms

Fig. 3 .

 3 Fig. 3. Average delay vs PIR under random traffic

Fig. 5 .

 5 Fig. 5. Average IP throughput vs PIR under random traffic

Fig. 6 .Fig. 7 .

 67 Fig. 6. Average delay obtained for Transpose 1 traffic

TABLE III RF

 III UTILISATION AND DELAY FOR TBR ALGORITHM

	Threshold	Average delay	Average	RF
	γ	(cycles)	utilisation (%)
	0	1558	96%
	5	795	87%
	10	651	71%
	15	236	56%
	20	52	41%
	25	54	28%
	30	74	18%
	35	122	13%
	40	190	10%
	45	229	9%
	50	268	8%

ACKNOWLEDGMENT

The authors would like to thank M. Palesi and S. Monteleone for their support during the extension of Noxim.

This work is supported by a public grant overseen by the French National Research Agency (ANR) as part of the Investissements d'Avenir program (reference: ANR-16-IDEX-0008).


B Hervé

J-J Sinou

H Mahé

L Jézéquel

ANALYSIS OF SQUEAL NOISE AND MODE COUPLING INSTABILITIES INCLUDING DAMPING AND GYROSCOPIC EFFECTS

This paper deals with an audible disturbance known as automotive clutch squeal noise from the viewpoint of friction-induced mode coupling instability. Firstly, an auto-coupling model is presented showing a non-conservative circulatory effect originating from friction forces. Secondly, the stability of an equilibrium is investigated by determining the eigenvalues of the system linearized equations. The effects of the circulatory and gyroscopic actions are examined analytically and numerically to determine their influence on the stability region. Separate and combined effects are analysed with and without structural damping and important information is obtained on the role of each parameter and their interactions regarding overall stability. Not only is structural damping shown to be of primary importance, as reported in many previous works, this article also highlights a particular relationship with gyroscopic effects. A method of optimizing both the stability range and its robustness with respect to uncertainty on system parameters is discussed after which practical design recommendations are given.

INTRODUCTION

In cars with manual transmission, different unforced vibrations may be observed during the sliding phase of clutch engagement and are classified according to their frequency range. Such friction-induced self-generated vibrations can be explained by four general and independent mechanisms, namely stick-slip, speed dependent friction force, sprag-slip and mode coupling [START_REF] Ibrahim | Friction-induced vibration, chatter, squeal and chaos: Part I -Mechanics of contact and friction[END_REF][START_REF] Ibrahim | Friction-induced vibration, chatter, squeal and chaos: Part II -Dynamics and modelling[END_REF][START_REF] Spurr | A theory of brake squeal, Proceedings of the Automobile Division[END_REF][START_REF] Kinkaid | Automotive disc brake squeal[END_REF][5][START_REF] Earles | Instabilities arising from the frictional interaction of a pin-disc system resulting in noise generation[END_REF][START_REF] North | A mechanism of disc break squeal[END_REF][START_REF] Millner | An analysis of disc break squel[END_REF][START_REF] Ouyang | Friction-induced parametric resonances in discs: effect of a negative friction velocity relationship[END_REF][START_REF] Ouyang | Friction-induced vibration of an elastic slider on a vibrating disc[END_REF][START_REF] Ouyang | Unstable travelling waves in the friction-induced vibration of discs[END_REF][START_REF] Chambrette | Stability of a beam rubbed against a rotating disc[END_REF][START_REF] Sinou | Analysis of friction and instability by the center manifold theory for a non-linear sprag-slip model[END_REF][START_REF] Sinou | Friction induced vibration for an aircraft brake system. part I : experimental approach and stability analysis[END_REF][START_REF] Lignon | Stability analysis and mu-synthesis control of brake systems[END_REF][START_REF] Oden | Models and computational methods for dynamic friction phenomena[END_REF][START_REF] Earles | Disque brake squeal noise generation: Predicting its dependency on system parameters including damping[END_REF]. The former two rely on tribological properties whereas the latter two are due to geometrical conditions. Interested readers may refer to papers [START_REF] Ibrahim | Friction-induced vibration, chatter, squeal and chaos: Part I -Mechanics of contact and friction[END_REF][START_REF] Ibrahim | Friction-induced vibration, chatter, squeal and chaos: Part II -Dynamics and modelling[END_REF][START_REF] Spurr | A theory of brake squeal, Proceedings of the Automobile Division[END_REF][START_REF] Kinkaid | Automotive disc brake squeal[END_REF] for an overview of the four general mechanisms. In the automotive industry, low frequency phenomena such as judder , felt as jolts of the vehicle when starting, can often be attributed to tribological properties. Other works on the subject generally show good agreement between analytical predictions and experimental observations [START_REF] Chevallier | Etude des Vibrations de Broutement Provoquées par le Frottement Sec, Application aux Systèmes d'Embrayage[END_REF]. However, squeal noise is a relatively high frequency phenomenon (up to several kHz), and may arise even when the friction coefficient is almost constant versus sliding speed. It cannot be related to stick-slip behaviour because of the speed range of the vibrations measured. In this case the assumption of a purely tribological origin appears unrealistic. Consequently, sprag-slip and to a greater extent mode coupling instabilities due to the intrinsic structure of the system are more likely to be responsible for this phenomenon. We decided to focus on mode coupling for the following study because of the absence of a static sprag condition. Firstly, a phenomenological model is presented. Then, important information is obtained concerning the influence of each parameter on the stability domain and practical design recommendations are discussed. As reported in many previous works, structural damping is of primary importance to the stability of mode couplings [START_REF] Sinou | Mode coupling instability in friction-induced vibrations and its dependency on system parameters including damping[END_REF][START_REF] Hoffmann | Effects of damping on mode-coupling instability in friction induced oscillations[END_REF][START_REF] Fritz | Effects of damping on brake squeal coalescence patterns -application on a finite element model[END_REF][START_REF] Shin | Nonlinear analysis of friction induced vibrations of a two degree of freedom model for disc brake squeal noise[END_REF][START_REF] Shin | Analysis of disc brake noise using a two-degree-of freedom model[END_REF]. The present study investigates in detail its influence on the effects of the combined circulatory and gyroscopic actions.

ANALYTICAL MODEL

The phenomenological model proposed is depicted in Fig. 1. It includes a homogeneous disc (A) rotating about the Oz axis, free to swing along the Ox and Oy axes. Oxyz is the principal inertial frame of (A) and is therefore a turning frame. Two sets of flexible elements in each swinging direction characterized by linearized stiffnesses k θ and k φ and damping d θ and d φ exert restoring torques on (A). The disc (A) is assumed to be in permanent contact with a second disc (B), also rotating about the Oz axis. For the sake of simplicity, this contact is represented by four flexible elements equally distributed on radius R from axis Oz, in planes Oxz and Oyz and characterized by equal stiffness k c and, for the sake of completeness, damping d c . The distance between the swinging axis and the contact surface is denoted h. For the sake of simplicity, friction coefficient µ is assumed to be constant and the rotation speeds to be constant and in the same sense. The sign of the sliding speed is assumed not to change, even locally due to the fact that the observed amplitude of the vibrations remains low. Lastly, the equilibrium position is assumed to correspond to the parallelism of the two discs as illustrated in Fig. 1. Disc (A) represents a friction disc and disc (B) the engine flywheel. They will be named the driven and driving discs respectively. The springs depicted between the two discs in the model represent the flexibility of the contact area, while h is related to the thickness of the driven disc. This simple model is sufficient to obtain mode coupling, show different phenomena by varying stiffness and damping and propose several analytical treatments. Writing the equations of motion linearized in the vicinity of the equilibrium for the driven disc gives 0

= + + KX X D X M , (1) where
[ ] T φ θ = X
. The mass, damping and stiffness matrices are given by

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = φ θ J J 0 0 M , (2) ( ) ( ) ( ) ( )
⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + - Ω + - + - Ω - - + - + = c c c c d R d sign Rhd J J J sign Rhd J J J d R d 2 2 2 . 2 . 2 2 φ ϖ φ θ ϖ φ θ θ ϖ µ ϖ ϖ µ ϖ D , (3) ( )
( ) ( ) ( ) ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - + + - Ω - Ω - - + + = 2 2 2 2 2 . 2 . 2 2 ϖ ϖ µ ϖ µ ϖ θ ϖ φ φ ϖ θ J J k R k sign Rhk sign Rhk J J k R k c c c c K . ( 4
)
The upper dot denotes a derivative with respect to time t. The specific (static and uncoupled) characteristics of each DOF appear in the diagonal parts: inertia, damping and stiffness. Moreover, the rotation of the disc (A) is responsible for (diagonal) centrifugal stiffening terms and a(n) (antisymmetric skew-diagonal) gyroscopic action in the stiffness and damping matrixes respectively. Finally, the friction forces exert on each DOF a torque dependant on the position (through k c ) and speed (d c ) on the other DOF due to lever arm h, resulting in antisymmetric skew-diagonal terms, i.e. a circulatory action in the stiffness matrix and an additional gyroscopic action.

In order to generalize the associated analysis, the following non-dimensionalization is proposed, assuming J J J = = φ θ for the sake of simplicity

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ - + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ - + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ 0 0 1 ' ' 2 2 ' ' ' ' 2 1 2 2 1 2 1 x x x x x x α ϕ ϕ αβξ ρ ρ ξ . ( 5
)
The non-dimentionalization factors are given by ( )

J J J k R k c 2 2 0 2 ϖ ω ϖ θ - + + = , (6)
J d R d c 2 0 2 2 1 + = θ ω ξ , (7) ( ) ( )
J J J sign Rhd c ϖ ϖ µ ω ρ ϖ - + - Ω - = 2 . 2 1 0 , (8)
( )

J sign Rhk c ϖ µ ω ϕ - Ω - = . 2 1 2 0 , (9)
( )

J J J k R k c 2 2 0 2 1 ϖ ω α ϖ ϕ - + + = , ( 10
) J d R d c 2 0 2 2 1 + = ϕ αξω β . ( 11
)
The upper dash denotes a derivative with respect to the new time scale t 0 ω τ = .

Thus the system is parameterized by five values, namely the ratio of the natural pulsations α which is assumed to be higher than or equal to 1, the damping factor of the reference DOF ξ, the ratio of the damping factors β, the circulatory action φ and the gyroscopic action ρ.

STABILITY ANALYSIS

In order to investigate the stability and the influence of each parameter, an eigenvalue analysis is performed. Considering the linearized governing equation, the non-resonant solutions can be written in the complex exponential form given by ( ) ( )

n n n i e a 0 0 ∑ + = X X τ τ λ τ . ( 12
)
Consequently the following generalized eigenvalue equation is derived from Eq. 5.

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ - + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ - + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ 0 0 1 2 2 1 0 0 1 20 10 2 2 x x α ϕ ϕ αβξ ρ ρ ξ λ λ . ( 13
)
Eigenvalues and associated eigenvectors are obtained by solving this equation. The eigenvectors represent the mode shape. The imaginary part of the associated eigenvalues is the pulsation and the real part is the exponential growth rate of the amplitude of the modes.

According to the Hartman-Grobman theorem, when the real parts of the eigenvalues of the linearized system are negative, the equilibrium studied is stable and when at least one is positive, the equilibrium then becomes unstable independently of the non-linear behaviour.

RESULTS WITHOUT STRUCTURAL DAMPING

In this section, the effects of the circulatory and gyroscopic actions on the eigenelements and the stability of the equilibrium are investigated by assuming no structural damping.

Purely circulatory system

The corresponding equation is given by

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ - + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ 0 0 1 1 0 0 1 20 10 2 2 x x α ϕ ϕ λ . ( 14
)
The eigenvalues and eigenvectors are derived. We obtain

2 2 2 2 2 2 1 2 1 ϕ α α λ - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - ± + - = (15)
and

( ) ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + - = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = 1 2 4 .. 1 20 10 n n x x λ ϕ . ( 16
)
Four eigenvalues and only two linearly independent modal vectors are obtained in this circumstance, related to the eigenelements of the sole stiffness matrix, as suggested by Eq. 14.

From Eq. 15, a critical value can be derived for the amplitude of φ

2 1 2 - = α ϕ . ( 17
)
Below this limit, the eigenvalues are purely imaginary thus, according to Eq. 16, the eigenvectors have a real ratio between their two components, thus the modal trajectories are non-turning trajectories. As φ increases, the two pulsations approach each other and the direction of the two modes tends to be the same until the modes merge at the coalescence point. As soon as the limit value is exceeded the squared eigenvalues become conjugate complex, thus the eigenvalues display two opposite pairs of conjugate values. As a consequence the system has a unique pulsation and the equilibrium is unstable due to the eigenvalues with a positive real part, leading to flutter. This is the basic destabilizing effect due to the circulatory action. Moreover, the modal trajectories become turning ones and turn in opposite senses since the eigenvectors are conjugate and, as shown in Eq. 16, the modal trajectory associated with the unstable eigenvalues turns in the direct sense when the circulatory action is positive and viceversa.

Direct and reverse senses are relative to oriented variable space x 1 , x 2 and is different from the usual sense taken into account when working on rotating machinery, which is relative to rotation.

As an example, a negative circulatory action is obtained in Eq. 4 when ϖ > Ω

. In this situation, for a positive value of θ the friction forces induce a torque that tends to decrease φ; for a negative value of φ, it tends to decrease θ. This is the basic antisymmetric coupling mechanism of the system. Thus, the associated motion appears to be turning in reverse sense, in accordance with the previous conclusion and the physical intuition according to which the rotation of the unstable modal trajectory is oriented by the circulatory action. A typical evolution of the eigenvalues is depicted in Fig. 2 and the associated modal trajectories are in the centre line of Fig. 3, corresponding to a null gyroscopic action.

Purely gyroscopic system

The following equation of the purely gyroscopic undamped system is considered

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ - + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ 0 0 0 0 1 0 0 1 0 0 1 0 0 2 2 ϕ θ α ρ ρ λ λ . ( 18
)
The characteristic equation and its roots are then given by ( )

0 1 2 2 2 2 4 = + + + + α λ ρ α λ (19) and 2 2 2 2 2 2 2 2 1 2 1 α ρ α ρ α λ - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + ± + + - = . ( 20
)
The term in the square root is positive, thus the square root is real; moreover this root has a smaller absolute value than the preceding term. Thus the eigenvalues remain purely imaginary and, as the gyroscopic action increases, the highest pulsation increases to infinite while the lowest pulsation decreases to zero, producing the well-known Campbell effect. In this system, four modal vectors can be written in the form

( ) ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + - = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = 1 2 4 .. 1 20 10 n n n x x λ ρλ . ( 21
)
This result shows the gyroscopic splitting of the modes. The four vectors are complex, and the modal trajectory associated with each squared eigenvalue rotates in the opposite senses. The sign of the gyroscopic action does not influence the pulsations, but determines the sense of rotation of the associated modal trajectory. The modal trajectory turning in the direct sense is associated with the highest pulsation when the gyroscopic action is positive and vice-versa.

It can be concluded that the gyroscopic action alone changes the pulsations and transforms the modal trajectories into turning trajectories but does not destabilize the equilibrium. The modal trajectories are in the central column of Fig. 3, corresponding to a null circulatory action.

Combination of circulatory and gyroscopic actions

Finally, in presence of both gyroscopic and circulatory actions, the modal equation becomes

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ - + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ - + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ 0 0 1 0 0 1 0 0 1 20 10 2 2 x x α ϕ ϕ ρ ρ λ λ . ( 22
)
The associated characteristic equation is given by

( )( ) ( ) 0 1 2 2 2 2 = + + + + ϕ ρλ α λ λ . ( 23
)
The equation cannot be balanced with either an assumed purely imaginary or a real value. Therefore the eigenvalues always have both non-zero imaginary and real parts and are conjugated by pair since the characteristic equation is real. Moreover, since the third order term of the characteristic polynomial is null, the two pairs of eigenvalues have opposite real parts. As a consequence, the equilibrium of the undamped system subjected to a combined circulatory and gyroscopic action is systematically unstable and leads to flutter. Moreover, by using Eq. 23 it is possible to check that the proportion factor between the two modal vectors expressed in Eq. 24 is not real, except in the case of the purely circulatory system below the coalescence point.

( ) ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + - + = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = 1 2 4 .. 1 20 10 n n n x x λ ϕ ρλ . ( 24
)
Thus, because of the continuity of the roots of a polynomial with respect to its coefficients, the sense of rotation of the modal trajectory associated with each individual eigenvalue remains the same when traversing the parameter space. However in the case of the purely circulatory system below the coalescence point, it has been shown that the modal trajectories do not turn. Therefore, the sense of rotation of the unstable and stable modal trajectories remains determined by the sign of the circulatory action, and the relationship between the corresponding pulsations remains determined by the sign of the gyroscopic action. Fig. 4 shows the pulsations and the associated exponential growth rate in the gyroscopic/circulatory plane. Bold lines are also plotted to indicate the locus of the intersection and junctions. Fig. 3 shows the associated modal trajectories drawn at a few points of the plane. These figures provide a complete overview of the qualitative behaviour of the undamped system regarding the stability of the equilibrium. Indeed, the results given in all the previous discussions do not depend qualitatively on the ratio of the natural pulsations or the domain of study, assuming it includes the coalescence point.

The region of potential stability (limiting case of the Hartman-Grobman theorem) can be observed as corresponding either to a purely gyroscopic or a purely circulatory action (within a limited range), therefore the robustness of the stability is null (its sensitivity is infinite) with respect to the vanishing action. From this point, the fact that neither the gyroscopic nor the circulatory action could be neglected a priori at the same time as damping in a stability investigation should be strongly underlined.

RESULTS WITH STRUCTURAL DAMPING

In the presence of strictly positive structural damping the characteristic polynomial shows a strictly negative third order coefficient, which indicates that the sum of the real parts of the (paired conjugate) eigenvalues is negative. As a consequence only one pulsation at the same time can be associated with an unstable nature. The stable and unstable domains are separated by frontiers due to the continuity of the eigenvalues. Assuming one pair of eigenvalues has a null real part and identifying the coefficients of the characteristic polynomial, the following equation of the frontiers is derived

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 1 4 1 1 2 2 2 2 2 2 2 2 = + + + + + + + + + - + + ϕ α αβ ξ αβξ ρ α β α αξ ϕρ αβ ξ β α αξ ϕρ .
(25) This is a second order polynomial of φ and a third order polynomial of ρ. When the gyroscopic action tends to infinite, the circulatory actions on the frontier of stability tend to the two following equations respectively,

0 = ϕ , (26)
( )

αβ ρξ ϕ + = 1
(27) describing two asymptotic lines for the frontier.

Moreover, it can be shown that the frontiers remain outside the asymptotes when no DOF is overdamped. The asymptotes can therefore be considered as stability frontiers by default. In order to clarify the global influence of the damping matrix comparatively to the basic destabilizing effect of the circulatory action, we propose writing the equation of motion in the eigenbase of the stiffness matrix. Assuming a non-null circulatory action different from the coalescence value, the eigenvalues and the associated eigenvectors of the stiffness matrix are given by

⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎨ ⎧ - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - + + = - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - - + = 2 2 2 2 2 2 2 2 2 1 2 1 2 1 2 1 2 1 ϕ α α ϕ α α e e (28)
and

[ ] ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ - - = 1 2 1 2 e e ϕ ϕ α 2 1 ψ ψ . ( 29
)
In the new base the equation of motion can be re-written as in Eq. 30 with the new matrix expressed in Eq. 31-33.

0 ΚΨ ΓΨ' ∆Ψ' ' Ψ' = + + + , ( 30
) ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = 2 1 0 0 e e Κ , (31) ( ) ( )
⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ - + - ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ - + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ - ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - = - 1 0 0 1 1 1 1 0 0 1 0 0 1 1 2 1 2 2 2 2 1 2 2 2 e αβ ξ ϕ ϕρ αβ ξ α ϕ α ∆ , (32) ( )
⎥ ⎦ ⎤ ⎢ ⎣ ⎡ - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - = - 0 1 1 0 1 2 1 2 1 2 2 1 2 2 2 ξϕ αβ ρ α ϕ α Γ . ( 33
)
In this expression the new stiffness matrix is intentionally diagonal and all the coupling terms have been combined in the damping matrix, split into a diagonal damping element ∆ and a purely gyroscopic element Γ. It should be noted that the "total" gyroscopic action obtained not only includes the "external" gyroscopic action but also a contribution from the circulatory action through the defect of proportionality of the structural damping. While the circulatory action is below the coalescence point, the base vectors are real and all the terms are real. As a consequence the system can be investigated as a purely gyroscopic real system. When the circulatory action is beyond the coalescence point, the base vectors are complex as are the terms. In this case, the system cannot be investigated as a purely gyroscopic real system. Nevertheless some characteristic behaviour remains unchanged. Fig. 5 shows a characteristic plot of the frontiers in the (original) gyroscopic/circulatory action plane with and without the total gyroscopic action. Below the coalescence point, the Eq. 30 is real. Without considering Γ the diagonal system shows unstable equilibrium only when one of the diagonal damping terms of ∆ becomes negative. Beyond the coalescence point, the criterion becomes slightly more complicated since the system becomes complex. Whatever the case, the frontiers clearly appear to be different all over the gyroscopic/circulatory action plane with and without Γ. This illustration shows that the intuitive idea according to which the stability of equilibrium would only be related to the terms of ∆ is false. Indeed, the only coinciding point corresponds to the cancellations of Γ. Thus it appears that the terms of both ∆ and Γ both influence stability. In order to further understand the role played by the different elements in stability, we provide results on the purely gyroscopic and purely circulatory damped systems in the following parts which include the examination of the complete system.

Purely gyroscopic system

Without circulatory action, the equation of motion fits the desired form naturally

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ - + 0 0 0 0 1 2 2 2 Ψ Ψ' ' Ψ' α αβξ ρ ρ ξ λ . ( 34
)
Thus Eq. 25 can be re-written giving

( ) ( ) ( )( )( ) 0 4 1 1 1 2 2 2 2 = + + + + - + αβξ ρ β α αβ α α β . ( 35
)
No real value of ρ can be obtained from this expression. What is more, since the sum of the real part of the eigenvalues is negative, the equilibrium is always asymptotically stable. In order to investigate the sense of rotation of the modal trajectories, a modal base can be written in the original base

( ) ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + + - = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = 1 2 2 4 .. 1 20 10 n n n n x x ξλ λ ρλ . ( 36
)
This expression and the characteristic polynomial permit checking that that the relationship between the pulsations and the sense of rotation is the same as in the undamped case.

The modal trajectories are illustrated in Fig. 6. The asymptotic stability as well as the association of the pulsations to the senses of rotation can be noted. This should be compared to the central column of Fig. 3 corresponding to the associated undamped system. According to Eq. 30-33, the behaviour of the complete system below the coalescence point can be directly linked to these results. Moreover, the latter allow explaining the influence of Γ on the stability frontiers illustrated in Fig. 5. Writing Eq. 30, which is assumed to be real, in the form of Eq. 34 with not necessarily positive diagonal damping terms, with ξ and β being either positive or negative, gives Eq. 35 for the frontiers. In this case, the existence of real (opposing) frontiers requires β to be negative. The stability at the origin is obtained only when both the diagonal damping terms are positive or null and vice-versa, which allows determining the stable and unstable sides of the frontiers when they exist. When β is null, the equilibrium is stable at the origin and asymptotically stable for any non-null value of ρ. These few observations are sufficient to show that the frontiers obtained below the coalescence point, without considering Γ in Eq 30, are stability frontiers by default, as illustrated in Fig. 5. The analysis of the system's behaviour beyond the coalescence point could also be performed in a comparable manner but would require taking into consideration complex structural damping and gyroscopic action.

Purely circulatory system

The stability analysis for the purely circulatory damped system is of particular interest when the gyroscopic action is slight. In this part, the stability in the case of the system described by

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ - + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ 0 0 1 2 0 0 2 1 0 0 1 20 10 2 2 x x α ϕ ϕ αβξ ξ λ λ (37)
is investigated. The following equation for the frontiers can then be derived from Eq. 25 .

( )

( ) ( )( ) [ ] β α αβ αξ α αβ αβ ϕ + + + - + = 1 4 1 1 2 2 2 2 2 . ( 38
)
Since the damping parameter ξ is strictly positive, the equilibrium is stable at the origin and two opposite real values of the circulatory action delimit the stability domain, in a comparable manner to the undamped case. However, it should be noted that the stability domain is not necessarily broader and even tends to the sole origin as the damping ratio β tends to zero. Therefore both the amount of damping and damping distribution have a strong influence on stability. Varying β alone not only changes the damping ratio between the two DOF but also modifies the amount of damping of the system. Even if a possible increase of stability with the total amount of damping seems favourable, it should be underlined that such unbalanced evolution may lead to under-utilization of its stabilizing capacity. In order to distinguish the respective effects of the amount and distribution of damping over the two DOF, the total damping expressed ( )

αβ ξ ξ + = 1 t (39) is considered, which leads to the new expression ( ) ( ) ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + + + - + = αβ β α αξ α αβ αβ ϕ 1 4 1 1 2 2 2 2 2 t ( 40
)
for the stability limits. Assuming the damping is low enough to preserve the order of the pulsations, it can be seen that the width of the stability domain increases with the ratio of the natural pulsations and with the amount of damping. Nevertheless, regarding damping distribution, its evolution is not monotonic and always shows one unique maximum at the optimum value given by Eq. 41. The existence of this optimal distribution allows concluding that damping may have a destabilizing effect. Moreover, at this optimum, the stability domain has null sensitivity, or infinite robustness with respect to the sole damping distribution.

( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 4 2 2 2 2 2 4 2 4 1 4 1 4 1 1 1 4 1 16 - + - - - + - + + + - = α ξ α ξ α α ξ α α ξ α α β t t t t opt . ( 41
)
As the damping amount tends to zero, the value defined in Eq. 41 tends to 1/α, which corresponds to a proportional distribution. Both criteria are compared in Fig. 7 in the damping amount/natural pulsations ratio plane. For a total damping amount up to 20%, the difference between the exact and approximated values is small. Therefore, the proportional distribution can be assimilated with the optimum for the low damping purely circulatory system. In order to explain this optimum, it should be noted that it corresponds to Γ being null in Eq. 30. Thus the system behaves qualitatively as being without any coupling action. Below the coalescence point and for low damping, the pulsations remain very close to the undamped case and the real part is common to the two eigenvalues, since the terms of ∆ are equal on each DOF, as indicated by Eq. 32. The eigenvalues are equal at the coalescence point, but beyond it the real parts become separated whereas the pulsations remain equal. The similarities to the undamped case are clearly illustrated in Fig. 8 which shows the evolution of the eigenvalues with respect to circulatory action for various damping distributions. The evolution of the real parts, shown in Fig. 8b, highlights the correspondence of the widest range of stability to a configuration close to the proportional damping distribution.

Outside the proportional damping case, the eigenvalues are observed to have a splitting effect that can be attributed to the additional gyroscopic term related to the damping distribution in Eq. 33. However, from the coalescence point it is still possible to observe a rapid separation of the real parts and a parallel trend for the pulsations. This indicates that the coalescence phenomenon still occurs and is only partially blurred by the additional gyroscopic action. In order to investigate the sense of rotation of the modal trajectories, the following modal base can be proposed

( ) ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + + - = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = 1 2 2 4 .. 1 20 10 n n n x x ξλ λ ϕ . ( 42
)
From this expression and the characteristic polynomial, it can be shown that non-turning modal trajectories exist only for the proportional distribution and below the coalescence point. Therefore, the optimised stability also corresponds to the existence of non-turning modal trajectories. Moreover, the unstable eigenvalue is continuously the eigenvalue having the highest real part at the origin, i.e. the lowest diagonal damping term in Eq. 32 for φ=0, as illustrated on Fig. 9. Nevertheless, the sense of rotation of the unstable modal trajectory remains related to the sign of the circulatory action, and the relationship between the pulsation distribution and the senses of rotation with the sign of the total gyroscopic action. These observations generalize the undamped case, considering the total gyroscopic action instead of the sole "external" gyroscopic action. This is illustrated in Fig. 10 which shows the modal trajectories versus the circulatory action and for various damping distributions.

Stability analysis of the complete system

Damping has two effects on the system. Firstly, the presence of positive damping globally decreases the real parts of the eigenvalues since their sum becomes negative, which is the stabilizing effect expected. Secondly, the damping distribution modifies the total gyroscopic action so that the topology of the surfaces may be strongly modified. This latter effect is illustrated in Figs. 11 and 12 which show the eigensurfaces for the proportionally damped and unbalanced damped systems respectively. Both of them have the same total damping amount and the same natural pulsation ratio, as for Fig. 4. It appears that (low) damping has a rather weak effect on pulsations, Figs. 4a, 11a and 12a, whereas the real part of the eigenvalues is strongly affected, Figs. 4b, 14b and 12b.

In comparison to the associated undamped system, the proportionally damped system shows a very slight deformation of the surfaces of the real parts, so that they remain topologically similar. The destabilizing separation of the real parts still occurs beyond the coalescence point for the purely circulatory system, and when both circulatory and gyroscopic actions are combined as in the undamped case. However, this destabilizing separation is partially counterbalanced by the generally lower amount of the real parts so that the domain of potential stability in the undamped case becomes a domain of asymptotic stability in the damped case and the separation of the real parts no longer leads to immediate destabilization. In the unbalanced case, the topology of the surfaces of the real parts is clearly different; they are separated at the origin and their junction curve is discontinuous. Moreover, for the pulsation surfaces, the junction curves no longer correspond to a null gyroscopic action. Thus the purely circulatory system shows a pseudo-coalescence phenomenon with neither perfect merging of the pulsations nor equality of the real parts as shown previously. These modifications also affect the stability frontiers; these are shown in Figs. 13 and14 for different values of the total damping amount and different distributions respectively. The evolution of the frontiers in the proportional configuration with respect to the damping amount is in continuity with the undamped case. The frontiers smoothly envelop the frontiers of the associated undamped system, highlighting the antagonistic effect of damping and destabilization due to the combination of the actions.

As the damping distribution evolves, it can be seen that the frontiers deform progressively, in such a way that the bump associated with the coalescence phenomenon does not stay on the circulatory action axis and is deviated towards a direction combining both actions. At the same time, the frontiers contract in the vicinity of the origin, up to complete collapse when only one DOF is damped. This is the cause of the previously observed restriction of the stability domain to the sole origin in the purely circulatory system. Finally, the additional gyroscopic term also affects the relationship between the pulsation distribution and the sense of rotation of the modal trajectories. In the undamped and proportionally damped cases, not-turning modal trajectories exist only for the purely circulatory system. In the presence of gyroscopic action, the modal trajectories are turning ones with a sense of rotation related to the pulsation distribution in accordance with the sign of the gyroscopic action. This is illustrated in Fig. 15 which shows the modal trajectories at several points of the circulatory/gyroscopic action plane and the locus of existence of nonturning modal trajectories for the proportionally damped system. However, for the unbalanced damped systems and for a circulatory action below the coalescence point, the effect of the additional gyroscopic term is in agreement with the intuitive idea that the sense of rotation of the modal trajectories, as well as the existence of non-turning trajectories, is not related only to the external gyroscopic action but to the total gyroscopic action. The modal trajectories are illustrated in Fig. 16. The locus of existence of non-turning modal trajectories appears to be the segment where the total gyroscopic action is null below the coalescence point. An interpretation using Caughey's condition also gives the same locus for the real modes. Moreover, it can be observed that this segment nearly connects with the top of the bump of the frontiers.

Optimization of low damping quasi-purely circulatory systems

Observation of the behaviour of the frontiers with respect to damping makes it possible to propose a stability optimisation strategy for the quasi-purely circulatory systems, as a generalization of the purely circulatory case. This also requires knowledge of the relationship between the gyroscopic and circulatory actions. Two major aspects have to be taken into consideration: firstly the width of the stability region, and secondly the robustness of the criterion with respect to the uncertainties on parameters, especially the gyroscopic actions. In a stability investigation, gyroscopic action may be considered from two different angles: either it is assumed to be "external" and constant, as the consequence of a given rotation speed, or it is assumed to be linked to the circulatory action, as the gyroscopic action induced by the friction forces in Eq. 3. The load exerted on the system therefore shows different trajectories in the gyroscopic/circulatory action plane that depend on this consideration. Assuming a low constant gyroscopic action, one way of optimizing the stability range with respect to the circulatory action is to set the damping distribution so that the bumps associated with the coalescence phenomenon are located at the desired value of the gyroscopic action. Generally, assuming a relationship between the two actions provides a similar optimization strategy. Nevertheless, these apply only for low gyroscopic actions since it appears that the influence of the damping distribution on the location of the bumps is restricted.

In the particular case of a linear relationship between circulatory and gyroscopic actions, for low damping systems it can be seen that the optimized system is close to that displaying nonturning modal trajectories in the stable area, similar to the undamped case. This particular property is a generalization of the criterion for systems having a null total gyroscopic action.

Regarding robustness, the effect of proportional damping is clearly positive compared to the undamped system, in both cases of purely circulatory and gyroscopic actions. However, when investigating a system combining the two actions, the criteria of the sole damping distribution for optimizing the stability range and the robustness may be antagonistic. An example of this is when the stability optimizations lead to placing most of the damping on only one DOF, the robustness of the system for a null action tends to zero with respect to a purely circulatory action. Therefore, optimizing both stability and robustness may require acting on both the damping amount and the damping distribution.

PRACTICAL DESIGN FOR AVOIDING SQUEAL NOISE

In the proposed model, Fig. 1 and Eq. 1-4, the circulatory action controlled by the friction coefficient is chosen as the control parameter. Without the gyroscopic action originating from the rotation, the curve of the parameterized load in the gyroscopic/circulatory action plane is a straight line crossing the origin. The rotation of the disc provides an additional constant gyroscopic action and centrifuge stiffening. The additional constant gyroscopic term then acts as an offset on the load line, whereas the centrifuge stiffening modifies both diagonal stiffness terms, and therefore the reference natural pulsation and natural pulsation ratio. However, in the pulsation range of squeal noises, these latter effects can be neglected. Both these gyroscopic actions and the friction forces responsible for the circulatory action in the proposed model are related to the rotations. Therefore, assuming a constant ratio of natural pulsations, it appears possible to apply certain optimization strategies to improve the stability domain and robustness in order to reduce the propensity of clutches to squeal. The most influential parameter allowing stabilization is the ratio of the natural pulsations. Indeed, by changing the inertia and stiffness associated to the coupled modes, the coalescence point may be shifted, thus stabilizing the equilibrium. The possible gyroscopic effects as well as their relationship with the friction forces then have to be evaluated in order to choose an optimized stabilization strategy.

CONCLUSION

A linearized two DOF mode-coupling model has been studied from the viewpoint of stability with respect to its parameters. Particular attention has been paid to damping distribution and its gyroscopic aspect, both analytically and through numerical illustrations. The results on the determination of stability frontiers are of particular interest in the case of quasi-purely circulatory mechanisms such as clutch squeal noise. Provided the damping is proportional it has been shown that the stability range of the quasipurely circulatory damped systems could be approached via their associated purely circulatory undamped systems. Indeed, in this configuration, the effect of damping is to add robustness to the stability domain rather than to change its range or shape. Therefore, neglecting slight gyroscopic actions is possible and the coalescence point remains close to Hopf's bifurcation point. Moreover, the modal trajectories remain the same. However, in the general case of an unspecified damping distribution, such an approach becomes irrelevant. Because the external gyroscopic action is likely to interact with the gyroscopic aspect of damping, it can no longer be neglected. Moreover, both the stability domain and its robustness become strongly related to the load curve in the circulatory/gyroscopic action plane. As a consequence, damping is a key parameter which requires highly detailed analysis when modelling the occurrence of instabilities. Design can be optimized from the viewpoint of stability and robustness by efficient control of the damping structure of the system relative to circulatory and gyroscopic actions. , with and without consideration for the total gyroscopic term in the eigenbase of the stiffness matrix; dashed lines denotes the coalescence points Fig. 9 Locus of the eigenvalues of a purely circulatory system in the complex plane with parametric variations of the amplitude of the circulatory action, for a natural pulsation ratio 5 . 1 = α , for a total damping amount of 10% and various damping ratio (β at 0, 1/α≈0.66 and 2)

= α -1 0 1 -2 0 2 -1 0 1 -2 0 2 -1 0 1 -2 0 2 -1 0 1 -2 0 2 -1 0 1 -2 0 2 -1 0 1 -2 0 2 -1 0 1 -2 0 2 -1 0 1 -2 0 2 -1 0 1 -2 0 2 -1 0 1 -2 0 2 -1 0 1 -2 0 2 -1 0 1 -2 0 2 -1 0 1 -2 0 2 -1 0 1 -2 0 2 -1 0 1 -2 0 2 -1 0 1 -2 0 2 -1 0 1 -2 0 2 -1 0 1 -2 0 2 -1 0 1 -2 0 2 -1 0 1 -2 0 2 -1 0 1 -2 0 2 -1 0 1 -2 0 2 -1 0 1 -2 0 2 -1 0 1 -2 0 2 -1 0 1 -2 0 2 -1 -0.5 0
-1 0 1 -2 0 2 x 1 x 2 -1 0 1 -2 0 2 x 1 x 2 -1 0 1 -2 0 2 x 1 x 2 -1 0 1 -2 0 2 x 1 x 2 -1 0 1 -2 0 2 x 1 x 2 -1 0 1 -2 0 2 x 1 x 2 -1 0 1 -2 0 2 x 1 x 2 -1 0 1 -2 0 2 x 1 x 2 -1 0 1 -2 0 2 x 1 x 2 -1 0 1 -2 0
(a)

-1 0 1 -2 0 2 x 1 x 2 -1 0 1 -2 0 2 x 1 x 2 -1 0 1 -2 0 2 x 1 x 2 -1 0 1 -2 0 2 x 1 x 2 -1 0 1 -2 0 2 x 1 x 2 -1 0 1 -2 0 2 x 1 x 2 -1 0 1 -2 0 2 x 1 x 2 -1 0 1 -2 0 2 x 1 x 2 -1 0 1 -2 0 2 x 1 x 2 -1 0 1 -2 0 2 x 1 x 2 -1 -0.5 0 0.5 1
Mode 1, highest pulsation mode Mode 2, lowest pulsation mode Gyroscopic action (ρ)

(b) -1 0 1 -2 0 2 x 1 x 2 -1 0 1 -2 0 2 x 1 x 2 -1 0 1 -2 0 2 x 1 x 2 -1 0 1 -2 0 2 x 1 x 2 -1 0 1 -2 0 2 x 1 x 2 -1 0 1 -2 0 2 x 1 x 2 -1 0 1 -2 0 2 x 1 x 2 -1 0 1 -2 0 2 x 1 x 2 -1 0 1 -2 0 2 x 1 x 2 -1 0 1 -2 0 2 x 1 x 2 -1 -0.5 0 0.5 1 Gyroscopic action (ρ) Mode 2, lowest pulsation mode Mode 1, highest pulsation mode (c) -1 0 1 -2 0 2 x 2 x 1 -1 0 1 -2 0 2 x 1 x 2 -1 0 1 -2 0 2 x 2 x 1 -1 0 1 -2 0 2 x 2 x 1 -1 0 1 -2 0 2 x 2 x 1 -1 0 1 -2 0 2 x 2 x 1 -1 0 1 -2 0 2 x 2 x 1 -1 0 1 -2 0 2 x 2 x 1 -1 0 1 -2 0 2 x 2 x 1 -1 0 1 -2 0

Fig. 1 Fig. 2

 12 Fig. 1 Model of mode-coupling between swinging modes of a rotating disc with friction

Fig. 3 Fig. 4 Fig. 5

 345 Fig.3Modal trajectories for one period of an undamped system in the circulatory/gyroscopic actions plane, for a natural pulsation ratio 5 . 1 = α ; for each graph the X-coordinate is the first variable and the Y-coordinate is the second variable

Fig. 6 Fig. 7 Fig. 8

 678 Fig. 6 Modal trajectories for one period of a damped purely gyroscopic system versus the amplitude of the gyroscopic action (ρ at -1, -0.5, 0, 0.5 and 1), for a natural pulsation ratio 5 . 1 = α

Mode 2 ,Fig. 11 Fig. 12 Fig. 13 Fig. 14 Fig. 15 Fig. 16

 2111213141516 Fig.10Modal trajectories for one period of a damped purely circulatory system versus the amplitude of the circulatory action (φ at -1, -0.5, 0, 0.5 and 1), for a natural pulsation ratio 5 . 1 = α