ISSN: [Alkhatib* et al., 5(10): October, 2016] Impact Factor: 4.116

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1 [Akhatib* et a., 5(10): October, 2016] Impact Factor: IC Vaue: 3.00 IJESRT INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY VIBRATION ANALYSIS, SHAPE AND DESIGN OF MOTORCYCLE MOUNTING SYSTEM SUBJECTED TO SHAKING FORCES Fadi Akhatib*, Anoop K. Dhingra * Department of Mechanica Engineering Austraian Coege of Kuwait Bock 5 - A Aqsa Mosque Street, Kuwait DOI: /zenodo ABSTRACT This artice presents a study of a mounting system that is used to isoate vibrations in motorcyces. The isoating system consists of the powertrain assemby, swing-arm assemby and the mounts that are used for isoation. In addition to mounting the powertrain in pace, engine mounts are used to minimize the transmitted forces from the engine to the frame. The governing equations of motion are being derived for a V-Twin engine which is generay used in motorcyce appications. The oads that need to be minimized are the oads due to the shaking force. These forces are generated in part as a resut of the rotating unbaance due to eccentric masses and due to the acceeration of the moving masses that the reciprocating engine consists of. An optimization probem is first set up to determine the mount stiffness rates in a three principa directions. Then the effect of the varying engine speed on mount characteristics is studied. Shape optimization is aso performed to find the geometrica mount shape due to the stiffness vaues in the principa direction of the mount. KEYWORDS: Motorcyce mounts design; Engine mounts; Vibration isoation; Shape optimization. INTRODUCTION This paper discusses the V-Twin engine configuration that is commony used in motorcyce appications. There are two sources of vibration that affect the performance of a motorcyce engine mount system; the first one is due to the shaking forces which are generated due to the engine imbaance in the moving parts inside the engine. This force is transmitted to the frame through the mounting system. The second force is due to the road oads which are caused by the irreguarities in the road profie. These forces are transmitted to the frame thorough the tire patch. The road oad coud be periodic or non-periodic whereas the shaking oad is periodic. This paper focuses on ony on the shaking oads. This work focuses on designing the most suitabe mounting system that provides isoation against forces transmitted from the powertrain to the frame. It is known that force and motion isoation are the major probems that engineers encounter when designing an engine mount. Motorcyce engines contain reciprocating parts that produce shaking forces due to the movement of various parts of the engine. The main objective herein is to minimize these shaking forces. This objective is achieved by supporting the powertrain by using a resiient support or an isoator. The argest umped mass that the vehice carries is the powertrain, which is attached to the frame using rubber mounts. These engine mounts are generay reinforced bonded-rubber bushings (figures 1 and 2). Internationa Journa of Engineering Sciences & Research Technoogy [698]

2 [Akhatib* et a., 5(10): October, 2016] Impact Factor: IC Vaue: 3.00 Figure 1. Front engine mount A substantia amount of research has been done in the area engine mount design for automotive and aircraft appications. However, the mounting system in motorcyce appications has its own unique chaenge. The motorcyce rear suspension system is couped to the powertrain through the shaft which to the compexity of the probem. On top of this, the motorcyce mounting system has to withstand the road oads as we. The mounting system that is used in these cases must ensure ow vibration transmission from/into the engine. There are a ot of factors to consider when ooking at the source of vibration, which coud be interna or externa or both. Figure 2. Rear engine mount Spiekermann, et a. (1985), discussed the issue of minimizing the forces that are transmitted through the mounting system. He expored the forces that are generated from the rotationa imbaance and reciprocating masses. Ford (1985), presented a design procedure for the front whee drive engine ide isoation in which a six degree of freedom umped system is used to represent the engine mount. Swanson at a. (1993), studied the mounting system of aircraft engines by minimizing the oads transmitted to the frame with an addition of constraint due to the defection present from the static weight of the engine. Ashrafiuon (1993), expored the same behavior taking into consideration the fexibiity of the frame. Sui (2003) emphasized on the roe of mounts in achieving better vehice handing characteristics and rider comfort as we as a resuting vibration caused by engine firing force and other sources. Liu (2003) presents a method used in the optimization design of engine mounts. The constraint probem is soved using some of the known parameters such as engine center of gravity, mount stiffness rates and mount ocation and/or orientation. Courteie and Mortier (2005) present a new technique to find an optimized and robust soution for the mounting system. Muti objective agorithm (Pareto optimization) is used as a base to the muti objective robust optimization probem. The use of this technique enhances the vehice isoation characteristics. This paper presents a new method to design the mounting system in a motorcyce. It focuses on the interna shaking forces which are created due to the engine imbaance as the source of vibration which wi be minimized. The shaking force is defined as the sum of the inertia and static forces that are transmitted to the frame through the mounting system as suggested by Kau, (2006). The mounting system consists of a pair of mount that support Internationa Journa of Engineering Sciences & Research Technoogy [699]

3 [Akhatib* et a., 5(10): October, 2016] Impact Factor: IC Vaue: 3.00 the front of the powertrain and another pair supports the rear end. The optimization probem is formuated such that the mount stiffness, ocation and orientation are optimized. Before deveoping the expressions for shaking forces in a V-Twin engine, an anaysis wi be performed to deveop expressions for shaking forces in a singe cyinder engine. The paper is organized as foows. Section 2 discusses the dynamic anaysis in which the engine mount is characterized and the six DOF mode is set up. Section 3 discusses the shaking forces and deveops expression for the shaking forces for a singe cyinder engine first, then expanding these expressions to accommodate the V- Twin engine configuration. The optimization probem formuation is presented in section 4. The concept of shape optimization is presented in section 5. Section 6 and section 7 presents the numerica resuts and concusions. DYNAMIC ANALYSIS Figure 3 shows a schematic diagram of the powertrain aong with a pair of mount used. Figure 4 shows a schematic diagram of the mount that wi be used in this study. The chaenge that is faced in this stage is identifying the mount properties which are represented in stiffness in a three principa directions, its ocation and orientation. The mode represented in Fig. 4 is a simpe Voigt mode that consists of a rigid body that resembes the powertrain which is connecting to the frame using the mounting system. The stiffness k and the damper c represent a singe D.O.F system with an equation of motion shown in Eq. (1). Mx + Cx + Kx = Fe iωt (1) In the Eq. (1), F denotes the input force vector that can be caused be either the shaking force or the road oad or both. M, C and K represent the mass, damping and stiffness matrices respectivey whereas x represents the dispacement vector. Figure 3. Schematic diagram of the engine Internationa Journa of Engineering Sciences & Research Technoogy [700]

4 [Akhatib* et a., 5(10): October, 2016] Impact Factor: IC Vaue: 3.00 Figure 4: Schematic diagram of the mount mode. The terms of the inertia matrix M of the powertrain are with respect of the goba coordinate system. In order to account for different orientations of the mounts; the stiffness and damping which are represented in the oca coordinate system of the mount must be expressed in the goba coordinate system which is achieved by using a transformation matrix. The mass matrix of the powertrain is represented in Eq. (2). m e m e 0 M p.t = 0 0 m e 0 m e z e m e y e m e z e 0 m e x e [ m e y e m e x e 0 0 m e z e m e y e m e z e 0 m e x e m e y e m e x e 0 I xxe I xye I xze I xye I yye I yze I xze I yze I zze ] (2) In Eq. (2), M p.t is the mass of the powertrain assemby, (x e, y e, z e) is the ocation of the center of gravity of the powertrain with respect to the origin of the coordinate system and I xxe, I yye, I zze, are the inertia of the powertrain with respect to the origin of the coordinate system. The stiffness and damping matrices of an individua mount expressed about its own coordinate system is given by Eqs. (3) and (4). k xi 0 0 k i = [ 0 k yi 0 ] (3) 0 0 k zi c xi 0 0 c i = [ 0 c yi 0 ] (4) 0 0 c zi A transformation matrix (A) is used in order to transfer both, the stiffness and damping matrices to the goba coordinate system as foows: k i = A i T k i A i and c i = A i T c i T A i where c i and k i are the individua mount damping and stiffness matrices expressed in the goba coordinate system. The matrix A i is a transformation matrix which is a combination of the three different rotations θ 1, θ 2 and θ 3 about x, y and z axes with respect to the goba coordinate system. Cθ 2i Cθ 3i Cθ 1i Sθ 3i + Sθ 1i Sθ 2i Cθ 3i Sθ 1i Sθ 3i + Cθ 1i Sθ 2i Cθ 3i A i = Cθ 2i Sθ 3i Cθ 1i Cθ 3i + Sθ 1i Sθ 2i Sθ 3i Sθ 1i Sθ 3i + Cθ 1i Sθ 2i Cθ 3i (5) [ Sθ 2i Sθ 1i Cθ 2i Cθ 1i Cθ 2i ] In Eq. (5) Cθ i = cos(θ i ) and Sθ i = sin(θ i ). The transformed damping and stiffness matrices are shown in Eq. (6) Internationa Journa of Engineering Sciences & Research Technoogy [701]

5 [Akhatib* et a., 5(10): October, 2016] Impact Factor: IC Vaue: 3.00 C e = [ C 11 C 12 ], K e = [ K 11 K 12 ] (6) C 21 C 22 K 21 K 22 K 11 = k i, K 12 = k i r i, K 21 = K 12 K 22 = r ik i r i (7) C 11 = c i, C 12 = c i r i, C 21 = C 12 C 22 = r ic i r i (8) K e and C e represents the overa damping and stiffness matrices of the powertrain assemby shown in Eq. (6). r i represents the skew-symmetric matrix that corresponds to an individua mount position (r xi, r yi, r zi ) and it s given by: 0 r zi r yi r i = [ r zi 0 r xi ] (9) r yi r xi 0 In this paper, a six degree of freedom (DOF) mode is used to describe the powertrain as a rigid body. In the case, the powertrain and the exhaust system which are fixed to the frame via four mount system. The frame is assumed to be infinitey rigid. The overa equation of motion (EOM) of the six DOF is given as foows: M e X e + C e X e + K e X e = F e e jωt (10) In Eq. (10), M e, C e and K e are 6 x 6 mass, damping and stiffness matrices respectivey. F e denotes the force vector at frequency ω due to the engine shaking force caused by engine imbaance described in which wi be discussed in the next section. The generaized mass matrix is defined with respect to the center of gravity of the powertrain. X e is a 6 x 1 vector that contains three transationa dispacements x, y and z and three rotationa dispacements α, and of the powertrain. To account for different orientations for each mount, the stiffness and damping matrices are assembed in their oca coordinate systems and then transformed to the goba coordinate system using the Eq. (3) to Eq. (9). SHAKING FORCES In this section, deveopment of the shaking forces in a singe cyinder engine is done first, and then a the expression are expanded to accommodate the V-Twin configuration. Shaking Forces in a Singe Cyinder Engine Figure 5 shows a schematic diagram of a singe cyinder sider crank mechanism. The standard sider crank mechanism is the basic buiding bock of virtuay a interna combustion engines. Presented next is the position, veocity, acceeration and the forces anaysis of the sider-crank mechanism. Let the crank radius be r and the connecting rod ength be. The crank ange is θ and the ange that the connecting rod makes with the x axis is, the crank rotates at a constant speed ω then: q = r sin θ = sin φ (11) θ = ωt (12) sin φ = r sin ωt (13) s = r cos ωt and u = cos φ (14) The distance x that is measured from the pivot point O to the sider at point B is given as foows: x = s + u = r cos ωt + cos φ (15) cos φ = 1 sin 2 φ = 1 ( r sin ωt) 2 (16) x = r cosωt + 1 ( r sinωt) 2 (17) The expression given by Eq. (17) gives the position of the piston aong the x axis as a function of crank ange θ. If a derivative of Eq. (17) is taken once with respect to time, the veocity of the piston wi be determined as shown beow: Internationa Journa of Engineering Sciences & Research Technoogy [702]

6 [Akhatib* et a., 5(10): October, 2016] Impact Factor: IC Vaue: 3.00 x = rω sin ωt + r 2 sin 2ωt (18) [ 1 ( r sin ωt)2 ] Figure 5. Sider Crank Mechanism If a derivative of the piston veocity is taken once with respect to time, the piston acceeration is obtained as shown beow: x = rω 2 {cos ωt r[2 (1 2cos 2 ωt) r 2 sin 4 ωt] [ 2 (r sinωt) 2 ] 3 } (19) 2 In the veocity expression shown in Eq. (18) and the acceeration expression shown in Eq. (19), a steady state soution is considered where it is assumed that the crank speed ω is constant. Using the binomia theorem, an approximate expression for the position, veocity and acceeration of the piston can be written as foows: x r2 4 + r (cos ωt + r cos 2ωt) 4 x rω (sin ωt + r sin 2ωt) (20) 2 x rω 2 (cos ωt + r cos 2ωt) The inertia force F i is the sum of the inertia forces at points A and B on the sider crank mechanism. F i = m A a A + m B a B (21) Internationa Journa of Engineering Sciences & Research Technoogy [703]

7 [Akhatib* et a., 5(10): October, 2016] Impact Factor: IC Vaue: 3.00 In Eq. (21), the acceeration term a B is the acceeration of the piston which is given in Eq. (20). The acceeration term a A coud be found by taking the second derivative of the position vector at point A with respect to time. The position vector that describes the ocation of point A is given as foows: R A = r cos ωt i + r sin ωt j (22) Differentiate the position vector given in Eq. (22) twice with respect to time and an expression for the acceeration at point A is achieved as foows: a A = [ rα sin ωt rω 2 cos ωt] i + [rα cos ωt rω 2 sin ωt] j (23) In Eq. (22), i and j are unit vectors defined aong the x and y axis. The inertia force aong the x and y axis are given as foows: F ix = (m A + m B ) rω 2 r cosωt m 2 ω 2 B cos2ωt (m A + m B ) rα sinωt m B sin2ωt 2 F iy = m A rα cosωt m A rω 2 sinωt (24) In Eq. (24), m A and m B are the equivaent rotating and reciprocating masses respectivey. The shaking force is F s = F i. It is fuy described taking into account the equivaent baancing masses as shown beow: F sx = (m A + m B m cb ) rω 2 r 2 ω 2 cosωt + m B cos2ωt +(m A + m B m cb ) rα sinωt + m B sin2ωt 2 F sy = (m A m cb ) rω 2 sinωt (m A m cb ) rα cosωt (25) In Eq. (25), F sx and F sy denote the net shaking forces in the x and y directions respectivey and m cb is the equivaent mass. These shaking forces resut from a singe cyinder. Shaking Forces in a V-Twin Engine Presented next is the deveopment of shaking force expressions for a V-twin engine shown in Fig. 6. The shaking force anaysis that was done on a singe cyinder engine is generaized to accommodate the V-twin engine and the shaking forces wi be computed aong the goba X-Y coordinate system. Figure 6. V-twin Engine Configuration. Internationa Journa of Engineering Sciences & Research Technoogy [704]

8 [Akhatib* et a., 5(10): October, 2016] Impact Factor: IC Vaue: 3.00 In order to determine the shaking forces for the V-Twin engine, the shaking force expression for singe cyinder engine given in Eq. (25) are used. The forces in each bank wi be computed separatey. Then by combining the corresponding terms of the shaking forces in each bank, the tota shaking forces and moments can be computed in the goba X-Y coordinate system for the V-Twin engine. The shaking force in the eft cyinder (bank) (F s) eft is given as foows: (F s ) eft = {(m A + m B )rω 2 cos(θ β) m cb1 r 1 ω 2 r 2 ω 2 cos(θ β) + m B cos2(θ β) r + (m A + m B )rα sin(θ β) m cb1 r 1 α sin(θ β) + m 2 α B sin2(θ β)} + {m A rω 2 sin(θ β) m cb1 r 1 ω 2 sin(θ β) m A rα cos(θ β) + m cb1 r 1 α cos(θ β)} m (26) (F s ) right = {(m A + m B )rω 2 cos(θ + β) m cb2 r 2 ω 2 r 2 ω 2 cos(θ + β) + m B cos2(θ + β) + (m A + m B )rα sin(θ + β) m cb2 sin(θ + β) + m B sin2(θ + β)} r 2 + {m A rω 2 sin(θ + β) m cb2 r 2 ω 2 sin(θ + β) m A rα cos(θ + β) + m cb2 cos(θ + β)} n (27) In Eq. (26) and Eq. (27), r and n are the unit vectors aong the x and y axis of the oca coordinate system for the right cyinder. and m are the unit vectors aong the x and y oca coordinate system for the eft cyinder. m cb1 and m cb2 are the equivaent masses at distances r 1 and r 2 for the eft and right banks respectivey. Combining the shaking forces for the right and eft cyinders in their corresponding oca coordinate system and transferring them into the goba coordinate system X-Y to come up with the overa shaking forces of the V-twin engine yieds: F sx = sinβ {2(m A + m B )rω 2 sinθsinβ 2(m A + m B )rα cosθsinβ r 2 ω 2 + 2m B sin2θsin2β m B cos2θsin2β} + ω 2 sinβ {m cb2 r 2 cos(θ + β) m cb1 r 1 cos(θ β)} + αsinβ {m cb2 r 2 sin(θ + β) m cb1 r 1 sin(θ β)} + cosβ {2m A rω 2 sinθcosβ 2m A rα cosθcosβ} + αcosβ {m cb1 r 1 cos(θ β) + m cb2 r 2 cos(θ + β)} ω 2 cosβ {m cb1 r 1 sin(θ β) + m cb2 r 2 sin(θ + β) } (28) F sy = cosβ {2(m A + m B )rω 2 cosθcosβ + 2(m A + m B )rα sinθcosβ r 2 ω 2 + 2m B cos2θcos2β + m B sin2θcos2β} ω 2 cosβ {m cb1 r 1 cos(θ β) + m cb2 r 2 cos(θ + β)} αcosβ {m cb1 r 1 sin(θ β) + m cb2 r 2 sin(θ + β)} + sinβ {2m A rω 2 cosθsinβ + 2m A rα sinθsinβ} + αsinβ {m cb2 r 2 cos(θ + β) m cb1 r 1 cos(θ β)} + ω 2 sinβ {m cb1 r 1 sin(θ β) m cb2 r 2 sin(θ + β) } (29) The shaking forces shown in Eq. (28) and Eq. (29) can be empoyed to find the shaking moments by mutipying each term by the moment arm. The moments exist within each bank and their vectors wi be orthogona to the cyinder panes. For the right bank, a moment unit vector n is defined which is perpendicuar to the unit vector r. Simiary, a moment unit vector m is defined which is perpendicuar to the unit vector for the eft bank as shown in Fig. 2. The shaking moment in the eft cyinder (bank) (M s) eft is given as foows: (M s ) eft = {(m A + m B )rω 2 cos(θ β) m cb1 r 1 ω 2 r 2 ω 2 cos(θ β) + m B cos2(θ β) + (m A + m B )rα sin(θ β) m cb1 r 1 α sin(θ β) + m B sin2(θ β)} z m (30) 2 The shaking moment in the right cyinder (bank) (M s) right is given as foows: 2 Internationa Journa of Engineering Sciences & Research Technoogy [705]

9 [Akhatib* et a., 5(10): October, 2016] Impact Factor: IC Vaue: 3.00 (M s ) right = {(m A + m B )rω 2 cos(θ + β) m cb2 r 2 ω 2 r 2 ω 2 cos(θ + β) + m B cos2(θ + β) + (m A + m B )rα sin(θ + β) m cb2 sin(θ + β) + m B sin2(θ + β)} z n (31) 2 In Eq. (30) and Eq. (31), z is the moment arm. Combining the shaking moments for the right and eft cyinders that have been shown Eq. (30) and Eq. (31) in their corresponding oca coordinate system and transferring them into the goba coordinate system X-Y to come up with the overa shaking moments for the V-twin engine yieds: r 2 ω 2 M sx = cosβ {2(m A + m B )rω 2 cosθcosβ + 2m B cos2θcos2β + 2(m A + m B )rα sinθcosβ + m B sin2θcos2β}. z ω 2 cosβ {m cb1 r 1 cos(θ β) + m cb2 r 2 cos(θ + β)}. z αcosβ {m cb1 r 1 sin(θ β) + m cb2 r 2 sin(θ + β)}. z (32) M sy = sinβ { 2(m A + m B )rω 2 r 2 ω 2 sinθsinβ 2m B sin2θsin2β + 2(m A + m B )rα cosθsinβ + m B cos2θsin2β}. z + ω 2 sinβ {m cb1 r 1 cos(θ β) m cb2 r 2 cos(θ + β)}. z + αsinβ {m cb1 r 1 sin(θ β) m cb2 r 2 sin(θ + β)}. z (33) The shaking torque of one cyinder is cacuated using the inertia force acting on the piston F i14 mutipied by the distance x from the piston at point B to the origin of the coordinate system at point O as shown in Fig. 5. The free body diagram of the piston showing a the acting forces are shown beow in Fig. 7. F i34 F i34 B m B x F i14 m B x F i14 Figure 7. Free Body Diagram of the Piston T s = (F i14 x)k = m B x tan x (34) In Eq. (34), x is the piston acceeration represented in Eq. (5.10). Substitute the piston acceeration expressed in Eq. (5.10) into Eq. (20), the shaking torque wi be expressed as foows: T s = m B [ rω 2 (cosωt + r cos2ωt) rα (sinωt + r r2 sin2ωt)] tan r [cosωt + r cos2ωt] (35) 4 tan r sinωt (1 + r2 2 2 sin2 ωt) (36) Internationa Journa of Engineering Sciences & Research Technoogy [706]

10 [Akhatib* et a., 5(10): October, 2016] Impact Factor: IC Vaue: 3.00 In Eq. (35), x is described in Eq. (20) and k is unit vector acting aong the z axis which is perpendicuar to the pane of the sider crank mechanism shown in Fig. 5. Assume that the anguar acceeration α is zero and approximating tan as shown in Eq. (36) we get an expression for the shaking torque for both the eft and the right banks. The shaking torque in the eft bank (T s) eft is as foows: (T s ) eft = 1 2 m Br 2 ω 2 [ r sin(θ β) sin2(θ β) 2 3r 2 sin3(θ β)] k (37) The shaking torque in the right bank (T s) right is as foows: (T s ) right = 1 2 m Br 2 ω 2 [ r sin(θ + β) sin2(θ + β) 2 3r 2 sin3(θ + β)] k (38) The combined shaking torque T s due to shaking torque from both eft and right banks is the agebraic sum of both (T s) eft and (T s) right shown beow: T s = 1 2 m Br 2 ω 2 [ r 3r sinθcosβ 2sin2θcos2β sin3θcos3β] k (39) The fina shaking force vector is a 6x1 vector shown beow: F s = [F sx F sy F sz M sx M sy T s ] T (40) OPTIMIZATION PROBLEM The optimization probem for the mode represented in Fig. 4, is perused using the Sequentia Quadratic Programming (SQP) technique described in Rao (2009). The objective function is formuated such that the transmitted forces are minimized taking into consideration the constraints imposed on the engine dispacement due to the static and dynamic oads. The objective function that is used in this work is the weighted sum of the transmitted force through the individua mount. The transmitted forces through the mounts are due to the shaking forces generated inside the engine. The force transmitted to the frame through the individua mount is given as foows: f i = [ k i k i r i ] [X ti X ri ] (41) X ti and X ri represents the transationa and rotationa dispacement at the center of gravity of the powertrain as resut of the shaking oad. k i is the oca stiffness matrix for the individua mount and r i is the skew symmetric from the position vector of the individua mount represented in Eq. (9). The objective f w function is assembed by summing the Eucidean norm of the individua force transmitted through each mount. f w = λ j f i j i In Eq. (42), λ i is the weighting parameter that corresponds to different oading conditions. The maximum engine defection that is aowed creates the constraints on the optimization probem. The static defection, X st is cacuated at the origin of the coordinate system is cacuated as: X st = K 1 F st (43) where F st is the static oad acting on the system. The engine mounts optimization probem therefore, can be stated as foows: Min f w (k i, r i, θ i ) (44) subject to g j (k i, r i, θ i ) 0 j = 1,, N In Eq. (44), the mount stiffness, ocation and orientation (k i, r i, θ i ) are the design variabes that are subjected to a tota of N number of constraints g j. The constraints that are used in the above probem consist of constraints on the engine mount stiffness, constraints on the mount ocation based on the avaiabe space, constraints on the mount orientation that is dictated by symmetry and finay a constraint on the defection of the center of gravity of the powertrain due to the static weight of the powertrain. The objective function f w is defined in Eq. (23). Both f w and g j are functions of the design variabes (k i, r i, θ i ). (42) Internationa Journa of Engineering Sciences & Research Technoogy [707]

11 [Akhatib* et a., 5(10): October, 2016] Impact Factor: IC Vaue: 3.00 SHAPE OPTIMIZATION The geometric dimensions of the isomeric mount shown in Fig. 8 are determined via a parametric study. These optimum vaues for the dimensions are chosen such that a compete deception description of the mount is achieved. The mount fina shape is determined by minimizing the difference between the mount stiffness vaues obtained from the dynamic anaysis performed in the previous section and the mount stiffness vaues based on its geometry which can be found from the finite eement anaysis. The objective function that is empoyed herein is described in Eq. 45 must satisfy aongside with the bound on the design variabes the condition described in Eq. 46, where x i is the i th design variabe and n is the number of the design variabes mentioned in Kim (1997). ψ = wt(1)(k x k des x ) 2 + wt(2)(k y k des y ) 2 + wt(3)(k z k des z ) 2 (45) min max x i x i x i for i = 1,, n (46) In Eq. (45), wt(i) is the weighting function that corresponds to the stiffness in the i th direction. The superscript des indicates the desired stiffness that is obtained from the dynamic anaysis of the mounting system, meanwhie the design parameters seected wi determine the stiffness vaues for the geometry that is obtained from the noninear finite eement anaysis. The process of determining the design variabes is expensive and time consuming, therefore in order to reduce the number of function evauations, the east effective stiffness coud be dropped from the objective function ψ. Figure 8. Schematic of a rubber mount Figure 8, shows the actua geometry of an engine mount that is used in cars aong with its defining parameters. This mount is a bush type that is made of rubber. There are a tota of six parameters that dictates the shape of the mount in which four are used as the design variabes namey t s, t z, t r and θ. The other two parameters (r i and r o ) are constants. These design variabes affects the mount stiffness directy. The weighting function that is used in the objective function coud be used to take into account the importance of the stiffness in a particuar direction. The dynamic anaysis is done for a motor cyce powertrain in which is supported by four isomeric mounts. The connection between the powertrain and the swing-arm are taken into consideration generating a tweve DOF system. The exciting force is imited to the interna shaking force at 4000 rpm. In this work, the stiffness vaues are obtained using a noninear finite eement anaysis. The geometry shown in Fig. 8 is used to generate a mesh for the anaysis. The optimization is carried out using ANSYS. Soid 186 is the eement that has been used for this purpose. Appropriate boundary conditions has been appied to the mode which is assumed to exhibit sma defections, for this reason the Mooney Rivin mode is sufficient to describe the fuy incompressibe hypereastic materia behavior of rubber as presented by Kim (1997) and Rivin, (1992). The Mooney Rivin mode of the strain energy is expressed as: U = C 10 (I 1 3) + C 01 (I 2 3) (47) Internationa Journa of Engineering Sciences & Research Technoogy [708]

12 [Akhatib* et a., 5(10): October, 2016] Impact Factor: IC Vaue: 3.00 I 1 and I 2 are the first and second strain invariants. The coefficients C 10 and C 01 are determined from the uniaxia tension test. The rubber that is used in this work is carbon back fied natura rubber. The vaues of the coefficients are: C 10 = and C 01 = A the design variabes must satisfy the design range which coud be considered as inequaity constraints that dictates the ower and upper bound of these variabes. Each one of these ranges that specify the upper and ower imit of the design variabes are considered as inequaity constraints and are incorporated in the finite eement optimizer. The static defection that is due to the static weight of the engine is measured aong the axis of gravity. δ = F g k (48) F g represents the static weight of the engine due to gravity and k represents the stiffness in the gravity direction. NUMERICAL EXAMPLE In this exampe, the force transmitted through the engine mounts due to the shaking force at 4000 rpm is used to formuate the objective function shown in Eq. (42). The mounting system used the exampe herein is based on four identica mounts with a circuar cross section as shown in Fig. 9. The design vector contains the individua mount stiffness, orientation and position. Lower and upper bound for the design variabes are isted in Tabe 1. Defection constraints on the powertrain are considered due to the static and dynamic oads. The static constraints which are paced on the defection of the powertrain are as foows: U st U max (49) In Eq. (49), U st represents the static defection of the powertrain at its C.G. due to the static oad and U max represents the maximum aowabe defection due to the static oad. For this exampeu max = [0.025 " 0.05 " " 0.5 o 0.5 o 0.5 o ]. An additiona constraint is paced on the maximum dispacement at the mount ocation in the y direction of 0.05 in to prevent premature snubbing. The shaking force at 4000 rpm and the static oad vector due to the engine weight are given by Eq. (50) and Eq. (51) respectivey. F s = [ ] T (50) F st = [ ] T (51) Figure 9. Schematic diagram showing the Mount System Layout Figure 9, shows the mounting system used in this exampe which consists of four identica circuar cross section eastomeric mounts in which, the radia and axia stiffness vaues fuy defines the stiffness characteristics of each Internationa Journa of Engineering Sciences & Research Technoogy [709]

13 [Akhatib* et a., 5(10): October, 2016] Impact Factor: IC Vaue: 3.00 mount and both of these stiffness vaues are used as design variabes. A oss factor of 0.3 and a dynamic-to-static stiffness coefficient of 1.2 have been used. In order to reduce the tota number of design variabes, symmetric constraints are imposed. This is done by pacing two mounts on each side of the x-y pane resuting six position variabes instead of tweve and four orientation variabes instead of tweve. The radia and axia stiffness vaues are identica for a four mounts resuting a tota of tweve design variabe for the mounting system. The mass of the powertrain is 0.5 b-s 2 /in and inertia vaues of the powertrain are given in Tabe 2. The optimization probem is done using the SQP technique that empoys an optimization routine to minimize the objective function. The design variabes resuting from the optimization process are shown in Tabe 3. Once the operation is over, the design vector that corresponds to the optimum vaue of the objective function is known. The second part of the probem starts by setting the objective function described in Eq. (45) to minimize the difference between the desired stiffness vaues obtained from the first optimization done through the dynamic anaysis and the stiffness vaues obtained from the geometric shape of the mount. The mount is connected to the frame via meta stee pates on both sides. These pates are bonded to the mount and the connection is at the mount attachment hoes. Since the stiffness of the stee pates is higher than the mount stiffness, the constraints are moved from the pate hoes directy into the mount surface as shown in Fig. 10. The boundary conditions are appied by constraining the dispacement of the surface of the mount in a directions. The resuts of the shape optimization process are shown in Tabe 4. The shape optimization takes into account the range of the design variabes that acts ike ower and upper bounds. These bounds are shown in Eq. (52) and the starting shape of the mount and the optimized shapes are shown in Figs. 11 and 12. The resuting force pots in the x and y direction for engine speed of 4000 rpm is shown in Fig. 13 and the resuting torque pot for the engine speed of 4000 rpm is shown in Fig t r t s 1.5 (52) 0.5 t z 1.77 π 18 θ π 6 Figure10. Isometric View Showing the Boundary Conditions Internationa Journa of Engineering Sciences & Research Technoogy [710]

14 [Akhatib* et a., 5(10): October, 2016] Impact Factor: IC Vaue: 3.00 Figure 11: (a) Isometric View of the Initia Geometry, (b) Front View of the Initia Geometry Figure 12: (a) Isometric View of the Optimized Geometry, (b) Front View of the Optimized Geometry Shaking Force corresponding to 4000 rpm Fore-aft (x) Vertica (y) 4000 Shaking force (b) Crank ange (deg) Figure 13. Shaking Force in the x and y Directions (4000 rpm) Internationa Journa of Engineering Sciences & Research Technoogy [711]

15 [Akhatib* et a., 5(10): October, 2016] Impact Factor: IC Vaue: Shaking torque corresponding to 4000 rpm Torque (b-ft) Crank ange (deg) Figure 14. Shaking Torque (4000 rpm) Tabe 1: Bounds for Design Variabes Min. Max. Mount Stiffness (x,y) b/in Mount Stiffness (z) Orientation Anges deg Tabe 2: Inertia Tensor of Powertrain Assemby. x y z I x I y I z Tabe 3: MatLab Optimization Resuts. Load Transmitted Mount Stiffness (b/in) (b) x y z Initia Guess Optimized Design Design Variabes Tabe 4: Parameter Optimization Resuts Initia Optimized Target Stiffness θ (rad) t r (in) t s (in) t z (in) Stiffness (kb/in) kx Internationa Journa of Engineering Sciences & Research Technoogy [712]

16 [Akhatib* et a., 5(10): October, 2016] Impact Factor: IC Vaue: 3.00 ky kz Obj. Function ψ CONCLUSION The design of a shear (bush) type engine mount has been obtained using the geometrica shape optimization using the parameterization technique. The method was done through utiizing a noninear finite eement anaysis. Part of the design was done using SQP method provided by a MatLab buit in function in order to find the target stiffness vaues. It shoud be noted that mutipe starting guess have been used in order to determine the optimum vaue of the objective function in the optimization probem. This is due to the fact that the engine mount optimization probem is noninear and coud provide oca minima as a soution to the optimization probem. As it can be noticed from the resuts above that the optimum shape of the mount is acceptabe and can be used as the fina shape. It is worth mentioning that this approach is appicabe for any type of engine mounts. The stiffness vaues that are obtained from the shape optimization are sighty different than those vaues obtained from the dynamic anaysis; however, the shape obtained from the parametric optimization is acceptabe and can be used in rea design situations. A more accurate representation of the mount connection to the frame is yet to be considered in the future work. REFERENCES [1] ANSYS 12.0, Hep Documentation, 2009, ANSYS Inc. [2] Ashrafiuon, H., Design optimization of aircraft engine-mount systems. ASME J. Vib. Acoust., 1993, 115, [3] Courteie, E. and Mortier, F., 2005, Muti-Objective Robust Design Optimization of an Engine Mounting System, SAE, paper no [4] Ford, M. D., 1985, An Anaysis and Appication of a Decouped Engine Mount System for Ide Isoation, SAE, paper no , pp [5] Kau, S., 2006, Modeing Techniques for Vibration Isoation in Motorcyces, PhD Thesis, University of Wisconsin, Miwaukee. [6] Kim, J. J. and Kim, H. Y., 1997, Shape Design of an Engine Mount by a Method of Parameter Optimization, Computers and Structures, Vo. 65, No.5, pp [7] Liu, C. Q., 2003, A Computerized Optimization Method of Engine Mounting System, SAE, paper no [8] MATLAB User Guide, Version 7.10, Reease 2010, Math Works. [9] Norton, R. L., 2011, Design of Machinery: An Introduction to the Synthesis and Anaysis of Mechanisms and Machines, 5th edition, McGraw Hi. [10] Rao, S. S., 2009, Engineering Optimization Theory and Practice, 4th edition, John Wiey & Sons, Chapter 7. [11] Rivin, R. S., 1992, The Easticity of Rubber, Rubber Chem. Techno. 65, G51-G66. [12] Snyman, J. A. and Heyns, P.S. and Vermenuen, P. J., 1995, Vibration Isoation of a Mounted Engine Through Optimization, Mechanica and Machine Theory, Vo. 30, No.1, pp [13] Spiekermann, C. E., Radciff, C. J. and Goodman, E. D., 1985, Optima Design and Simuation of Vibrationa Isoation Systems, Journa of Mechanisms, Transmission and Automation in Design, Vo. 107, pp [14] Sui, S. J., 2003, Powertrain Mounting design Principes to Achieve Optimum Vibration isoation with dimension Toos, Society of Automotive Engineers, paper no [15] Swanson, S. R., 1985, Large Deformation Finite Eement Cacuations for Sighty Compressibe Hypereastic Materias, Computers and Structures, Vo. 21, pp [16] Tao, J. S. and Liu, G. R and Lam, K. Y., 2000, Design Optimization of Marine Engine-Mount System, Journa of Sound and Vibration, Vo. 253, No.3, pp Internationa Journa of Engineering Sciences & Research Technoogy [713]