**1. Introduction**

Motorcycle technology advancements have been evident throughout the years, following or even exceeding the trends of automotive technologies. Accordingly, electric motorcycles are becoming a reality and they already have an important market share (over 30% in 2019) [1]. Several technology breakthroughs have been introduced up to now, providing numerous production vehicles and stimulating a huge e ffort towards their research and development by motorcycle manufacturers and startups. Electric motors, batteries and powertrains, are the key factors mainly researched by the industry. But electrification of powertrains also enables the redesign of several critical motorcycle parts.

One critical structural part of all motorcycles is the front suspension system that connects the chassis with the front wheel. Several alternative concepts of front suspension can be found, such as girder forks, leading link and hub center steering [2], introducing various advantages in motorcycle handling. The majority of the production motorcycles globally install telescopic forks, as front suspension systems, even though this entails specific disadvantages [3]. As the choice of the suspension system has major effects on the performance and handling of the motorcycle, alternative front suspension systems have been researched and have also been installed on production motorcycles [2,4]. Another issue of motorcycle design is the use of two or three wheels and specifically the setup of two wheels front and one rear. These kind of motorcycles can be considered as trikes or motorcycles, depending on the way they tilt as well as the lateral distance between the front wheels. Trikes are a di fferent vehicle category, due to di fferent handling and cornering. It is clear that up to now, two-wheel motorcycles dominate

the market. However, in the last years di fferent three-wheel concept vehicles have been introduced (Honda Neowing, Tokyo, Japan, Kawasaki J, Tokyo, Japan), production models are already on the road (Yamaha NIKEN, Shizuoka, Japan), and motorcycle magazine reviews have highlighted the advantages of two front wheels for handling, braking and safety feeling of the riders [5]. Linked to the above facts, extensive research on three-wheel motorcycles and their front suspension systems design, is needed and should be carefully evaluated. On this track, our research team have been focusing on the design and development of a new three-wheel electric motorcycle [6], with an innovative two-wheel front system arrangemen<sup>t</sup> using a single-sided swingarm for each front wheel and hub center steering. This motorcycle was developed to meet the challenges of future electric vehicles' and the development of the two-wheel front system arrangemen<sup>t</sup> was customized for installation on the specific vehicle. It must be noted, that the replacement of a gasoline engine by an electric powertrain in the same space, provides additional freedom in chassis design of future electric motorcycles, which makes it easier to install such an alternative front system. The work presented here is mainly focused on the structural analysis of the swingarm, which is a critical component of the front system.

Research on three-wheel motorcycles reported in the literature is limited. Related studies are mainly focused on dynamic modelling and stability analysis of these vehicles [7,8]. Simple and advanced dynamic models of three-wheel motorcycles are developed and specific modes of operation (capsize, weave and wobble) are recognized and evaluated compared to two-wheel vehicles through simulation. Other researchers are focused on the e ffects of passing over low friction coe fficient surfaces, where the riders of three-wheel vehicles have indicated that one noteworthy characteristic is the vehicle's stability when cornering, over a low friction surface, like slippery or wet road. As indicated, under these conditions the decrease in lateral force on one front tire, passing over the low friction road surface, is compensated by the other front tire [9].

As mentioned, swingarm structural behavior is the main topic covered in this research. Based on related literature there are several publications focused on swingarm structural analysis, but they all explore single-sided [10–14] or double-sided [15–17] rear swingarms. Double-sided swingarms hold the rear wheel by both sides of its axle, while single-sided ones are a type of swingarm which lies along only one side of the rear wheel and hold it in one side, allowing it to be mounted like a car wheel. In all cases, a static analysis is considered, presenting stress and displacements based on di fferent loading conditions. Focusing on the single-sided swingarms, in [10] the loads applied correspond to maximum traction and lateral bending forces, comparing an aluminum and Carbon Fiber Reinforced Plastic (CFRP) version. A multi objective optimization is performed for the redesign of the carbon fiber version targeting maximum sti ffness and minimum weight. Bedeschi [14] presented a similar analysis, towards weight optimization, considering CFRP material. Smith [11,13] also used a static analysis based on torsional and vertical loads, trying to achieve higher torsional sti ffness and minimum weight. Considering double-sided swingarms, a relative procedure is followed where structural analysis is based on braking and cornering [15], torsional loads [16], or extreme loading such as performing a wheelie [17]. Regarding material use, the majority is focused on aluminum alloys such as 7075-T6 [15], CFRP [18] and comparisons of aluminum and steel versions [16,17]. Based on the aforementioned literature, other valuable information was also gathered (Table 1), including: a) weight of swingarms developed, b) vehicle motor power and c) safety factor. It is evident that rear swingarm's weight depends on the motor power used on the motorcycle, as well as if it is single or double-sided. In Table 1, double- or single-sided swingarms are defined as (D) or (S), respectively. For motor power less than 40 KW, only double-sided can be found and their weight is 2.6–2.7 kg for aluminum and CFRP versions, respectively. On powertrains ranging between 130 and 185 KW, single-sided swingarms weight is 4.1–4.2 kg for CFRP and 5 kg for aluminum, while double-sided swingarms are 5.3–6.85 kg for di fferent aluminum versions. The safety factor targeted by most researchers, even though data are limited, has values close to *N* = 2.


**Table 1.** Rear swingarm specifications in the literature.

Braking and cornering forces are accounted for in most of the literature for rear swingarms. The magnitude of these forces is related to tire-ground friction and a simplification called "friction circle or ellipse" can be used to understand the maximum forces available for braking and cornering [2]. It assumes that the maximum tire friction force possible in any direction is a constant, which means that a tire can support a specific maximum force either for cornering, or braking, or any combination of steering and braking forces that result to the same maximum resultant force available, which is a simplification of reality. Other research findings exist, where more accurate tire models for combined braking and steering force components can be found, such as the Pajecka Magic formula [19] and BNP-MNC [20]. But according to the above and considering that, in our case, we are exploring the use of a front swingarm, the maximum loading scenario can be defined under emergency braking conditions in a longitudinal driving direction.

Basic theory of braking forces calculation can be found in various publications, including related books [21–23], as well as research focused on evaluation of the effects of braking loads applied on single-sided swingarms [12]. It must be noted though, that when brake forces are calculated, motorcycle tires and especially tire coefficient of friction are important as detailed in Section 2.3. Another important factor, directly linked to braking conditions, is the effect of deceleration. Braking deceleration of motorcycles is differentiated according to vehicle brake design and driver's ability to avoid skidding during braking [21]. Experimental brake tests have been conducted by different research teams measuring deceleration values, revealing the effect of the driver but also the differences that occur by using front, rear or both brakes simultaneously [24–26]. This part is exclusively presented in the following sections of this work, highlighting the importance of deceleration on braking calculations, in order to adequately define the loads applied on a swingarm.

To the best of our knowledge, there is no other analysis of a front single-sided swingarm in the relevant literature, as the majority of research is related to rear swingarms. Comparing the loads applied on a front or rear swingarm, several differences exist, such as: (a) the effect of torque during acceleration, which is not considered on a front swingarm, (b) motorcycle weight distribution loads, as also (c) higher braking forces applied on the front wheels. It can be easily understood, that weight and stiffness of rear swingarms cannot be directly compared to those of a front swingarm. Hence, the main contribution of this work is to present valuable results and insights, regarding stress and displacements calculated for the front single-sided swingarm design proposed. To cope with the forces applied during vehicle operation and at the same time to be lightweight, different designs are evaluated under different loading scenarios. Since braking conditions correspond to a major factor of forces applied, this is the worst-case loading scenario considered. For the evaluation procedure, a dedicated CAE (ANSA, Beta CAE Systems, Thessaloniki, Greece) software is used for a series of finite element analysis simulations. Based on preliminary results obtained, the redesign of the swingarm is

discussed, targeting higher structural e ffectiveness and lower weight. For this purpose, a topology optimization procedure is carried out to assist and shape the new design. Finally, an evaluation of di fferent swingarm versions under specific loading conditions is discussed, considering simulation results of stresses and displacements, which depend on design modifications.

#### **2. Calculation of Forces on a Motorcycle**

For the definition of the loadcases used in the simulations, the weight distribution and brake forces acting on the front swingarm must be calculated. In this section, a brief analysis of the relevant theory and experimental results found in the literature is presented.

## *2.1. Weight Distribution*

Assuming a simple motorcycle model, the basic loads acting from weight distribution and braking can be determined as shown in Figure 1, where the motorcycle and the rider are modeled as a single rigid body. Supposing a flat road and ignoring the contribution from rolling resistance and aerodynamic forces, the loads on the vehicle's wheels are then calculated as shown below [21].

**Figure 1.** Simple motorcycle model of acting forces during braking.

In Figure 1, *a* represents the distance of front wheel from Center of Gravity (CoG), *b* is the distance of rear wheel from CoG and *h* measures the height of the CoG. Applying the sum of forces with respect to axes X and Z and sum of torques at the center of gravity, we obtain the following equations:

$$S\_f + S\_{\mathcal{I}} = md \tag{1}$$

$$N\_f + N\_r = mg \tag{2}$$

$$(N\_r \, b - N\_f \, a + \left(\mathcal{S}\_r + \mathcal{S}\_f\right)b = 0\tag{3}$$

where, *Sf* is the front braking force, *Sr* is the rear braking force, *Nf* is the reaction from the ground to the front wheel, *Nr* the reaction from the ground to the rear wheel, *d* represents the deceleration, *m* the total mass and *m* × *d* the inertial force from the deceleration.

Rewriting the Equations (1)–(3) with respect to *Nf* and *Nr*, results in:

$$N\_f = mg\frac{b}{a+b} + md\frac{h}{a+b} \tag{4}$$

$$N\_r = mg\frac{a}{a+b} - md\frac{h}{a+b} \tag{5}$$

Using the above equations, the weight distribution resulting in the loads acting on the front and rear wheels can be obtained.

#### *2.2. Braking Forces and Tire Friction*

As found in Equation (1), the total braking force (*Stot*) is:

$$S\_{tot} = S\_f + S\_r = md \tag{6}$$

A condition that should be avoided to maintain driver safety during braking is tire skidding, resulting in a loss of tire grip with the road [21]. Tire traction limit *D* is based on tire characteristics and is related to the ratio μ, which is the friction force (or otherwise the normalized braking force), effected by the braking force *S* and the vertical load *N*. In order to maintain road grip during braking, this ratio should not exceed a maximum available value of *D*, for the front (*Df*) and rear (*Dr*) tire traction, respectively:

$$
\mu\_f = \frac{S\_f}{N\_f} < D\_f \tag{7}
$$

$$
\mu\_r = \frac{S\_r}{N\_r} < D\_r \tag{8}
$$

As the total brake force (*Stot*) increases, tire skidding may occur either in the front or in the rear tire depending on braking force distribution [21]. In order to examine this phenomenon, the ratio of braking balance (ρ) is introduced, linking the braking force on the rear wheel and the total braking force:

$$\rho = \frac{S\_r}{S\_f + S\_r} = \frac{S\_r}{S\_{tot}} \tag{9}$$

By using Equations (1)–(3) and (7)–(9), normalized braking force μ may be expressed as a function of the deceleration *d* and the braking balance ρ, as follows [21]:

$$
\mu\_f = (1 - \rho) \frac{(a+b)d}{ag + hd} \tag{10}
$$

$$
\mu\_r = \rho \frac{(a+b)d}{a\lg - bd} \tag{11}
$$

When ρ = *0* then μ*r* = 0 meaning only the front brakes are applied, while if ρ = *1* just the rear brakes are applied.

In order to calculate the brake ratio at the verge of skidding, the actual deceleration at this point must be known [22]. Assuming the tires (front and rear) have the same coefficient of friction, the total braking force at the point of skidding is calculated by Equation (2):

$$F\_{\rm skid} = \left(N\_f + N\_r\right)\mu = mg\mu\tag{12}$$

By equating (6) with (12), the expression for maximum deceleration is obtained:

$$F\_{\text{skid}} = S\_{\text{tot}} = \succ d = -\mu \text{g} \tag{13}$$

The negative sign is an indication that the vehicle is in a state of deceleration.

In general, tire friction data are linked to road conditions (dry, wet, ice), as also on different road material types (such as asphalt, gravel etc.) [7,9,27]. Nevertheless, motorcycle tires are different than car tires and have different shapes of contact patch on the road [27,28]. Experimental results focused on motorcycle tire friction coefficient (μ) calculation, indicate values close to μ = 0.8 [22]. Experimental and theoretical results for car tires in contact with a road made from tarmac in dry conditions can be found in [23,27], presenting a range of values 0.7 < μ < 1.0. Research on motorcycle tires can also be found [22,27], indicating that for a dry road μ = 0.8. Considering all the above information, for our calculations the reference coefficient of friction used for the front and rear tires is μ*f* = μ*r* = 0.8. Therefore, based on Equation (13) and assuming μ = 0.8 as a reference value, the maximum theoretical deceleration so that no tire skidding occurs is calculated at *d* = 0.8 g.

#### *2.3. Urgent Braking in Real Conditions*

The way motorcycle drivers use brakes in emergency situations a ffects motorcycle response and di fferentiates deceleration magnitude. For this purpose, several experimental tests have been conducted by di fferent researchers, using professional or ordinary drivers, di fferent sport or sport-touring motorcycle models and di fferent braking conditions (using front, rear or both brakes) [24,29–36]. Relevant experiments on low-power motorcycles (100–150 cc) were conducted in [26], as well as in wet road conditions where decelerations are lower [25]. These tests reveal the range of decelerations achieved in reality (minimum–maximum) and can be compared to theory. A summary of these results is presented in Table 2 and at the last row the mean value for all minimum and all maximum decelerations is calculated. In this calculation, we have not accounted for the tests conducted in [26] due to the low power of motorcycles used in the tests. Also all values included are without the use of an anti-lock braking system (ABS).


**Table 2.** Braking deceleration rates in g reported in the literature (dry and wet road).

Combined braking (front and rear brake use) is considered as the best way to brake in an emergency situation and this is represented in the results, where the maximum deceleration achieved is 0.96 g. Lower values are found for rear brake use (0.46 g), considered as the worst choice. Finally, using only front brakes results in maximum decelerations up to 0.89 g. It can be seen that the range of decelerations vary from test to test. For this reason, a mean value of all maximum and all minimum decelerations is calculated in the last row of Table 2 and is considered as a more adequate magnitude of reference. Based on that, when only front brakes are applied, the mean value of maximum deceleration is *d* = 0.75 g.

Comparing this experimental reference value (0.75 g) with the theoretical value of deceleration (*d* = 0.8 g) calculated in the previous Section 2.2, we observe that theoretical calculations are 5.8% higher than the experimental. Due to the small di fference, we choose the worst case (theoretical deceleration value *d* = 0.8 g), as the reference value used for the maximum braking forces applied in calculations hereafter.

#### **3. Testbed Vehicle Specifications and Loading Scenarios**

The proposed swingarm design is installed on the front suspension system of a three-wheel electric motorcycle (Daedalus). Using two wheels at front, two identical single-sided swingarms are used to hold the wheels. The top and side view of this assembly is shown in Figure 2a,b, respectively. In order to calculate the forces acting on the testbed vehicle, the technical specifications and corresponding dimensions of CoG related to Figure 1 are presented in Table 3.

**Figure 2.** Front swingarm assembly: (**a**) top view; (**b**) side view.



#### *3.1. Front Swingarm Design and Materials*

The initial design of the single-sided swingarm is presented in Figure 3, while the assembly of related components that are included in the simulation, is shown in Figure 4. As shown, at the back side the swingarm is connected to the chassis through an axle inserted in the spacers, while on the front it is connected to the wheel through the swingarm axle. The front axle is attached on the swingarm with two bearings and its other side is connected to the wheel using a hub center steering system.

**Figure 4.** Front swingarm and related components assembly.

#### *3.2. Weight Distribution Forces—First Loading Scenario*

As explained in Section 2.1, the weight distribution forces on the wheels are determined by the position of the center of gravity (CoG) and the mass of the motorcycle including the driver, in our case 340 kg. By subtracting the unsprung mass of the wheels (21 kg), the total mass acting on the CoG is *m* = 319 kg. Weight distribution on the front and rear wheels is calculated using Equations (4) and (5), assuming deceleration *d* = 0, resulting in *Nf* = 1572.9 N and *Nr* = 1555.9 N. Force *Nf* is distributed equally on the front two wheels, resulting in *Nfleft* = *Nfright* = 786.45 N. These forces correspond to the first loading scenario considered in the simulations.

#### *3.3. Braking Forces—Second Loading Scenario*

Using Equation (4) force *Nf* can be calculated for different deceleration values (*d*). In addition, using Equation (10), the coefficient of friction (μ*f*) can be obtained, related to different braking distribution factors (ρ) and decelerations. The brake force acting on the front wheels (*Sf*) is then calculated based on *Nf* and μ*f*. The same procedure is used to calculate the rear braking force *Sr*. The worst-case scenario of forces applied is considered when only the front brake is used (ρ = 0). Assuming a friction factor μ*f* = 0.8 and a maximum deceleration of d = 0.8g, results in *Sfleft* = *Sfright* = *Sf*/2 = 1251.75 N on each front wheel. The brake moment (*Mbr*) generated by *Sf*, considering the front tire radius (*R* = 0.32 m), is *Mbr* = *Sf* × *R* = 400.5 Nm. The vertical load is also found equal to *Nfleft* = *Nfright* = 1210 N. These forces correspond to the second loading scenario considered in the simulations. Using Equation (13), the maximum deceleration that can be achieved in order to avoid tire skidding would be *d* = 0.8 g, assuming μ = 0.8.

### **4. Modelling and Simulation**

Finite elements modelling requires specific steps, depending on the pre-processor used, in order to prepare a CAD part for Finite Elements Analysis (FEA) simulation. This includes modelling of forces, constraints, connections of parts in the assembly, material specification and mesh generation. All these steps are detailed in the following paragraphs. A dedicated CAE software (ANSA, Beta CAE Systems, Thessaloniki, Greece) is used for this purpose [37], providing adequate results.

#### *4.1. Geometry Mesh and Materials*

The ANSA pre-processor is used for the development of the geometry mesh and volume elements are used for higher results accuracy (Figure 5). The mesh of the swingarm has 97,259 volume elements, from which 87,475 are Tetras and 9784 are Pyramids elements. The total mesh of all the components included in the swingarm assembly has 116,818 volume elements (Figure 6).

**Figure 5.** Mesh of the initial swingarm design.

**Figure 6.** Mesh of components assembly for the swingarm design.

The modelling of material properties used for the preliminary simulations can be found in Table 4, where the required properties used by the pre-processor are: (a) Elastic modulus, (b) Poisson ratio, (c) Shear modulus, (d) Density and (e) Yield strength. The materials initially considered are: (a) aluminum 7075-T6 for the swingarm and the suspension link, (b) stainless steel AISI-304 for the swingarm axle and the bearings, (c) steel AISI-130 for the suspension, all modelled as isotropic materials. Suspension has been simplified to a steel round beam (shown with a line in Figure 6) in order to reduce mesh elements number and is assumed to be a rigid component.


## *4.2. Connections and Contacts*

The connection of different components in the assembly is defined using bolts, related to suspension and suspension link, or contact functions for spacers and bearings. Figure 7a presents bolts connections placement and Figure 7b depicts the contact connections as modelled on the assembly.

**Figure 7.** Connections modeled: (**a**) with bolts; (**b**) with contacts.

### *4.3. Forces and Constraints*

Constraints are modelled as shown in Figure 8a, placed as fixtures on the rear part of the swingarm connected to the chassis. Suspension movement is also fixed using a constraint on the suspension link. The forces are applied considering the calculated values of the vertical force *Nf* (first loading scenario) and a clockwise moment *Mbr* (used with *Nf* for second loading scenario) placed on the end of the swingarm axle (Figure 8b).

**Figure 8.** Finite element analysis model: (**a**) constraints; (**b**) loads.

The assumption that suspension is rigid and that no tire modelling is considered, is a simplification used to reduce computational time under the static analysis presented here. Stress and displacements calculated in this way are expected to be higher than in the case when the suspension was operating normally and tires could also compensate part of the loads. These assumptions help us on evaluating different swingarm designs, which is the main target here, but in order to obtain more precise results and achieve higher weight reduction, these modelling procedures should be applied.

## *4.4. Topology Optimization*

A general optimization problem is identified firstly as a problem of finding the optimal topology and then as a problem of finding the optimal shape or finding the optimal cross sections [38]. Topology optimization is a mathematical process that aims to find the optimal distribution of the material of a construction, while satisfying its support conditions and loads. It is implemented by combining finite elements for analysis and mathematical programming techniques for solving [38,39]. ANSA's SOL200 was used to conduct topology optimization simulations in the present work [37]. The first important parameter defined is the design area, corresponding to the area of the part where the problem of topology optimization will be solved, while areas not included in the solution are called non-design areas [38]. In our case, the design area is the swingarm excluding the rear brackets connected to the chassis. The second parameter is the definition of the objective function and constraints. The objective function for this application corresponds to the minimization of the weighted deformation energy (min compliance). Using this objective function, the residual mass percentage and the static loading scenario are taken as constraints. The first restriction refers to the percentage of mass removed from the original mass of the swingarm.
