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Analysis of Natural Frequency of a Four-Wheeler Passenger Car by Combined Rectilinear and Angular Modes an Analytical Approach
Automotive vehicles are the first medium of transportation within the world and should consider the highest levels of safety and comfort for the passengers. Thehealthandsafetyofthedriversandpassengerswithin an automotive structure are often judged by finding out the dynamic interaction between the occupants and moving automotive. Dynamic investigations are quite common in numerous transportation industries like railways,naval,aviationandautomobilesectors,wherever moving structure and its occupants are exposed to unwanted vibration. The essential objective of research and its findings is to judge and reduce the amount of vibration by designing the system or sub system in the extreme operative conditions. Unwanted vibration within theautomotivestructuresisproducedbytheuncontrolled and random excitation which may cause the resonance and huge displacement on the automotive car seat and occupant human bodies. Dynamic simulation analysis of the car seat and human body is known as a very complicated and non linear phenomenon and hence, an entireanalysisofallthenon linearparametersisrequired tocarryoutaneffectivesimulationtodeterminethelevel of vibration within human body and car seat [3]. The vehicle suspension forms very important system for an automobile. This will often help to support the engine, vehiclebodyandpassengersandatthesametimeabsorbs the shocks arising due to roughness of the road. The normal arrangement consists of supporting the chassis throughspringsanddampers,bytheaxle.Theengineand
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Prof. S. V. Tawade1 Assistant Professor, Dept. of Mechanical Engineering, Navsahyadri Engineering College, Naigaon, Pune, India *** Comfortability of the vehicle is most important for the passengers. Automobile and aerospace industries are spending millions of rupees on research work to reduce the vibrations in vehicles. The present research work shows the results of two modes of natural frequencies with respect to variation in mass as well as stiffness of front and rear suspension system of four wheeler passenger cars. The results shows that the increase in mass as well as stiffness values of suspension system increases the two modes of natural frequencies of vibration of four wheeler passenger cars. Every mechanical system has its own natural frequency, also called as fundamental frequency. When the systems are put in practical applications, many external forces will act over the system and due to which the systems will vibrate with forced frequencies. If the fundamental frequency and the forced frequency matches then resonance condition will occur leading to partial or full failure of the systems. So, avoiding the resonance as well as reducing the vibration in four wheeler passenger cars is a challenging task for engineers to have a vehicle with more comfortability for the passengers.
Key Words: Natural frequency, Mode shapes, Passenger Car, Stiffness, Mass
1.INTRODUCTION
Every mechanical body or system has its own natural frequency of vibration or it is also known as the fundamental frequency of vibration. Once the mechanical systems are put into practical applications different kinds offorceswillactoverthesystemcausingforcedvibrations inthesystem.Thenaturalfrequencyofthesystemshould not become equal to forced frequency. If the natural frequency of the system becomes equals to forcing frequency, then resonance will occur leading to complete or partial system failure. Since at resonance condition the system vibrates with very high or infinite amplitude. So, analysis of natural frequency of any mechanical system plays a very important role. In our present work determination of natural frequency of vibration of a four wheeler passenger care is done by combined rectilinear and angular modes by analytical method. A four wheeler passenger car with 2000kg mass is considered for the analysis. Equation of motion are derived and finally the two modes of natural frequencies are found out. Also, the variation of two modes of frequencies with varying stiffnessoffrontandrearsuspensionsystemareanalyzed.
Abstract
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Also, the effect of varying mass of vehicle on the natural frequencyofvibrationofthefour wheelerpassengercaris analyzed. Up gradation of comfortability of the vehicle is the most significant for the passengers. This will be done bythereductionofvibrationwithinthevehicles.Vibration analysis finds its application in automotive industries, aerospace industries and many engineering fields. The automobile and aeronautical industries intensively work on the vibration analysis procedures and also on type of materials to achieve the objective of comfort of the passengers and also the durability and their life [1]. The essential ride model of a passenger car has suspension pointsateachofthefourwheels.Thewheelsoraxleshave the mass. The vehicle designer tries to keep the body bounce resonant frequency as less as possible however this can be obtained as a result of constraints on the stiffnessofthesuspensionimposedbytheavailablerange of suspension travel. Sport cars normally have higher natural frequencies and more damping due to considerations about their performance in handling [2].
1
Thedifferentialequationofmotionforthesystemare mx+K₁(x l₁θ)+K₂(x+l₂θ)=0 lθ K₁(x l₁θ)l₁+K₂(x+l₂θ)l₂=0 m +(K₁+K₂)x (K₁l₁ K₂l₂)θ=0 l (K₁l₁ K₂l₂)x+(K₁l₁2 +K₂l₂2)θ=0
Let at any instant the vehicle is displaced through a rectilinear distance x and angular displacement θ as showninfig1.SinceθissmallthereforespringsK1 andK2 arecompressedthrougha distance(x l1 θ)and(x+l2 θ) respectively. The free body diagram of the system is showninfigure2
Fig 3: FreeBodyDiagram
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alsothebodyofthevehiclearefittedtothechassisrigidly. Hence, the chassis along with the body of the vehicle, engine and the passengers may be considered to be one unitonly.Thespringsanddamperswhichconnecttheaxle and chassis have important role in absorbing shocks and keepingchassisaffectedtoaminimumlevel[4].Vibration evoked by the irregularities on the road surface and imbalanceofthetire/wheel istransmitted tothevehicle body through the suspension, a vibration transmission system.Enginevibrationisadditionallytransmittedtothe vehicle body through the suspension system via the drive shaft [5]. When talking about the suspension system, spring rates and damping levels are the parameters that areusuallyused.Eachelasticobjectormaterialwillhavea certain speed of oscillation which will occur naturally when there are zero outside forces or damping applied. This natural vibration happens solely at a precise frequency, referred as the natural frequency. Natural frequency is that frequency at which a system tends to oscillate in the absence of any driving or damping force. The motion pattern of a system oscillating at its own natural frequency is called the normal mode vibration if eachpartsofthesystemmovesinusoidallywiththatsame frequency.Iftheoscillatingsystemisdrivenbyanexternal forceatthefrequencyatwhichtheamplitudeofitsmotion islargestandalmostcloseorequaltoanaturalfrequency of the system, this frequency is called as the resonant frequency.Ifaspringthatissubjecttoavibratorymotion and is close to its natural frequency the spring will begin to surge. This case is extremely undesirable since the life ofthespringcanbereducedasexcessiveinternalstresses developed. The operative characteristics of the spring are also seriously affected. For many springs subject to low frequency vibrations surging is not a problem. However, forhighfrequencyvibratingapplicationsitisnecessaryto ensure, in the design stage, that the spring natural frequencyisfifteentotwentyormoretimesthemaximum operating vibration frequency of the spring [6]. The pneumatic tyre perceives the road unevenness and roughnessandtransmitsthemtothesuspensionelements during the movement of a vehicle. The generated vibrations will enter into the passenger compartment through the connections between the suspension and the vehicle body. Properly choosing of elastic and damping characteristics of the suspension elements (shock absorber, spring, rubber mounts), and modal parameters of the metal elements (rods, shells, plates) can improve the car comfort in terms of noise and vibrations. This reducesthedriverfatigue.Dependingonthevehicleclass, thevibrationandnoiselevelsinitscabinaredifferent.The driver feels these vibrations in the seat, the floor panels andthesteeringwheel.[7].
Fig1: Four WheelerPassengerCar
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2. COMBINED RECTILINEAR AND ANGULAR MODES OF VIBRATION OF A FOUR-WHEELER PASSENGER CAR
In actual practice, sometimes, the body is subjected to combinedrectilinearandangularmotions.Forexample,in case of vehicle, when brakes are applied on moving vehicle, two motionsof vehicleoccursimultaneously. One motion is rectilinear(x) and other motion is angular (θ). This type of combined motion in the system is due to the fact that C.G. of vehicle and centre of rotation do not coincide.ConsideracarofmassmandmomentofinertiaI supported on two suspension systems with stiffness K1 andK2 asshowninfigure1
Fig.2: Equilibriumanddisplacedposition
Case 1 when K₁ l₁ = K₂ l₂ when K₁ l₁ = K₂ l₂ the equation the equations of motion become,m +(K₁+K₂)x=0&l +(K₁l₁2 +K₂l₂2)θ=0 From these equations it is seen that rectilinear and angular motions can exist independently. Such equations are called as uncoupled differential equations. The two naturalfrequenciesofsystemare √ & √ Case 2 When K₁ l₁ ≠ K₂ l₂ TheSolutionforxandθundersteadystateconditionareX =Xsinωt,θ=Φsinωt,Where,XandΦareamplitudeof rectilinearandangularmotionrespectively. = Xωtsinωt & = Φω2sinωt mXω2sinωt+(K₁+K₂)Xsinωt (K₁l₁ K₂l₂)Φsinωt=0 mXω2 +(K₁+K₂)X (K₁l₁ K₂l₂)Φ=0 (K₁+K₂ mω2)X=(K₁l₁ K₂l₂)
The equations have both the term x and θ hence they are called the coupled differential equation. These equations indicate that system has rotary as well as translatory motion.
Φ lΦω2sinωt (K₁l₁ K₂l₂)Xsinωt+(K₁l₁2 +K₂l₂2)Φsin ωt=0 lω2Φ (K₁l₁ K₂l₂)X+(K₁l₁2 +K₂l₂2)Φ=0 (K₁l₁2 +K₂l₂2 lω2)Φ=(K₁l₁ K₂l₂)X (K₁+K₂ mω2)(K₁l₁2 +K₂l₂2 lω2)=(K₁l₁ K₂l₂) 2 K₁ 2 l₁ 2 +K₁K₂l2 2 K₁lω2+K₁K₂l₁ 2+K₂2 l₁2 K₂Iω2 K₁ l₁ 2 mω2 K₂l2 2 mω2 +Imω4 =K₁l₁ 2 2K₁K₂l₁l2 +K₂2 l22 Imω4 (K₁I+K₂I+K₁l12 m+K₂l22m)ω2 +K₁K₂l₁2 +K₁ K₂l22 +2K₁K₂l₁l2=0 Imω4 (K₁I+K₂I+K₁l12m+K₂l22m)ω2+K₁K₂(l1 +l1)2 =0 ( ) ( ) ω4 Bω2+C=0 B C== ω2= √ ω2n1 = √ ] ω2n2= √ ] 3. NATURAL FREQUENCIES RESULTS BY ANALYTICAL METHOD Table1: Thenaturalfrequencyresultsofvibrationswith constantmassandvaryingstiffness K1 K2 1L 2L K M I ωn1 ωn2 450 500 2 2 1.2 0100 0144 60.972 71.625 500 550 2 2 1.2 0100 0144 31.052 11.661 550 600 2 2 1.2 0100 0144 81.098 81.742 600 650 2 2 1.2 0100 0144 31.143 71.820 650 700 2 2 1.2 0100 0144 21.186 51.895 700 750 2 2 1.2 0100 0144 61.227 41.967 750 800 2 2 1.2 0100 0144 71.267 82.036 800 850 2 2 1.2 0100 0144 51.306 92.103 850 900 2 2 1.2 0100 0144 21.344 92.168 900 950 2 2 1.2 0100 0144 91.380 02.232 950 0100 2 2 1.2 0100 0144 61.416 42.293 0100 0105 2 2 1.2 0100 0144 51.451 22.353 0105 0110 2 2 1.2 0100 0144 51.485 52.411
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International Research Journal of Engineering and Technology (IRJET) e ISSN: 2395 0056 Volume: 09 Issue: 03 | Mar 2022 www.irjet.net p ISSN: 2395 0072 © 2022, IRJET | Impact Factor value: 7.529 | ISO 9001:2008 Certified Journal | Page322 0110 0115 2 2 1.2 0100 0144 81.518 52.468 0115 0120 2 2 1.2 0100 0144 41.551 12.524 0120 0125 2 2 1.2 0100 0144 31.583 52.578 0125 0130 2 2 1.2 0100 0144 51.614 92.631 0130 0135 2 2 1.2 0100 0144 21.645 12.684 0135 0140 2 2 1.2 0100 0144 31.675 42.735 0140 0145 2 2 1.2 0100 0144 91.704 72.785 0145 0150 2 2 1.2 0100 0144 01.734 12.835 0150 0155 2 2 1.2 0100 0144 61.762 72.883 0155 0160 2 2 1.2 0100 0144 71.790 52.931 0160 0165 2 2 1.2 0100 0144 41.818 52.978 0165 0170 2 2 1.2 0100 0144 71.845 73.024 0170 0175 2 2 1.2 0100 0144 61.872 33.070 0175 0180 2 2 1.2 0100 0144 11.899 23.115 0180 0185 2 2 1.2 0100 0144 21.925 53.159 0185 0190 2 2 1.2 0100 0144 01.951 13.203 0190 0195 2 2 1.2 0100 0144 51.976 23.246 0195 0200 2 2 1.2 0100 0144 62.001 73.288 0200 0205 2 2 1.2 0100 0144 52.026 73.330 Fig:4 StiffnessK1 VsFirstNaturalFrequency Fig:5 StiffnessK1 VsSecondNaturalFrequency Fig:6 StiffnessK2 VsFirstNaturalFrequency Fig:7 StiffnessK2 VsSecondNaturalFrequencies Thetable1showstheresultsoftwonaturalfrequenciesof vibration or fundamental frequenciesofvibration of four wheeler passenger cars calculated by means of using the two natural frequency equations (ωn1 & ωn2 equations) derived from the equation of motion of four wheeler passenger cars. The total mass of the passenger car considered here is 1000kg. L1 and L2 have same value as 2m. Front and rear suspension system stiffness K1 & K2 varied from 450 N/mm to 2000N/mm and 500N/mm to 2050N/mmrespectively. 0.0000 1.0000 2.0000 3.0000 ω n1 K1 Stiffness K1 Vs First Natural Frequency 4.00002.00000.0000 450 600 750 900 1050 1200 1350 1500 1650 1800 1950 ω n2 K1 Stiffness K1 Vs Second Natural Frequency 0.0000 0.5000 1.0000 1.5000 2.0000 2.5000 ω n1 K2 Stiffness K2 Vs First Natural Frequency 0.0000 2.0000 4.0000 ω n2 K2 Stiffness K2 Vs Second Natural Frequency
550 600 2 2 21. 0110 4158 81.033 81.683 600 650 2 2 21. 0115 6165 71.052 41.719 650 700 2 2 21. 0120 8172 81.069 41.751 700 750 2 2 21. 0125 0180 41.085 21.780 750 800 2 2 21. 0130 2187 51.099 41.806 800 850 2 2 21. 0135 4194 51.112 21.830 850 900 2 2 21. 0140 6201 41.124 11.852 900 950 2 2 21. 0145 8208 41.135
950 0100 2 2 21. 0150
ThevariationoffrontsuspensionsystemstiffnessK1 with ωn1 &ωn2 are shown in figure4 & figure 5 respectively. As the stiffness of front suspension system K1 of the four wheelerpassengercarincreasesthetwomodesof natural frequenciesωn1 &ωn2 alsoincreases.Asthestiffnessvalue K1 offrontsuspensionsystem increasedfrom450N/mm2 to 2000 N/mm2 the first natural frequency ωn1 value increased from 0.9726 rad/sec to 2.0265 rad/sec. As the stiffness value K1 of front suspension system increased from 450 N/mm2 to 2000N/mm2 the second natural frequency value varied from 1.6257 rad/ sec to 3.3307 Therad/sec.variation of rear suspension system stiffness K2 with ωn1 &ωn2 are shown in figure6 & figure7 respectively. As the stiffness of rear suspension system K2 of the four wheelerpassengercarincreasesthetwomodesofnatural frequenciesωn1 &ωn2 alsoincreases.Asthestiffnessvalue K2 of rear suspension system increased from 500 N/mm2 to 2050 N/mm2 the first natural frequency ωn1 value increased from 0.9726 rad/sec to 2.0265 rad/sec. As the stiffness value K2 of rear suspension system increased from 500 N/mm2 to 2050 N/mm2 the second natural frequency value varied from 1.6257 rad/ sec to 3.3307 rad/sec. frequencyresults with varyingstiffnessandmassvalues 81.643 11.872 0216 61.145 61.890 0155 2223 11.155 81.907 1.172 1.938 0 0 2 0 6 1 5 0115 0120 2 2 21. 0170 8244 81.179 31.952 0120 0125 2 2 21. 0175 0252 11.187 21.965 0125 0130 2 2 21. 0180 2259 81.193 31.977 0130 0135 2 2 21. 0185 4266 21.200 71.988 0135 0140 2 2 21. 0190 6273 31.206 51.999 0140 0145 2 2 21. 0195 8280 01.212 62.009 0145 0150 2 2 21. 0200 0288 41.217 12.019 0150 0155 2 2 21. 0205 2295 51.222 12.028 0155 0160 2 2 21. 0210 4302 31.227 72.036 0160 0165 2 2 21. 0215 6309 91.231 82.044 0165 0170 2 2 21. 0220 8316 31.236 52.052 0170 0175 2 2 21. 0225 0324 51.240 92.059 0175 0180 2 2 21. 0230 2331 51.244 92.066 0180 0185 2 2 21. 0235 4338 31.248 62.073 0185 0190 2 2 21. 0240 6345 91.251 92.079 0190 0195 2 2 21. 0245 8352 41.255 02.086 0195 0200 2 2 21. 0250 0360 71.258 92.091 0200 0205 2 2 21. 0255 2367 91.261 42.097 8 MassmVsFirstNaturalFrequency 0.0000 0.3000 0.6000 0.9000 1.2000 1.5000 1000 1150 1300 1450 1600 1750 1900 2050 2200 2350 2500
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K1 K2 1L 2L K M I ωn1 ωn2 450 500 2 2 21. 0100 0144 60.972 71.625 500 550 2 2 21. 0105 2151 81.012
0100 0105 2 2 21.
0105 0110 2 2 21. 0160 4230 91.163 71.923 110 115 2 2 1. 165 237
Fig:
Table2: Thenatural
ω n1 Mass of Vehicle (m) Mass Vs First Natural Frequency
ofvibrations
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[1] Neelappagowda Jagali, Chandru B, Dr.Maruthi B, Dr.Suresh P, Evaluation of Natural Frequency of Car Door With and Without Damper Using Experimental Method and Validate Using Numerical Method, International Journal of Advanced and Innovative Research(2278 7844)/36/Volume5Issue6,2016
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ω n2 Mass
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Fig:9 MassmVsSecondNaturalFrequency
Table2showstheresultsoftwonaturalfrequenciesωn1& ωn2 of the four wheeler passenger car with respect to variation of front and rear suspension system as well the variation in mass. The stiffness K1 value varied from 450 N/mm2 to 2000 N/mm2 and the stiffness value k2 varied from500N/mm2 to2050N/mm2andthemassofthefour wheeler passenger car increased from 1000kg to 2550kg.
The results of mass against two modes of natural frequencies ωn1 & ωn2 are plotted in figure8 and figure9 respectively. The results show that as the mass of the vehicle increases, the two modes of natural frequencies alsoincrease. 4. CONCLUSION: The present research work shows the results of two modesoffundamentalfrequenciesωn1&ωn2ofvibration of a four wheeler passenger car. The research work is donebyusinganalyticalmethod.Theequationsofmotion for the four wheeler passenger car in practical condition are devolved. The equations of motion are solved for calculating the two modes of fundamental frequencies ωn1 & ωn2. In first case the mass of vehicle is kept constant and the mas considered was 1000kg. The stiffness values of front suspension systems varied from 450 N/mm2 to 2000 N/mm2 and stiffness values of rear suspension systems varied 500 N/mm2 to 2050 N/mm2 . The first fundamental frequency values achieved are 0.9726 rad/sec to 2.0265 rad/sec and the second fundamental frequency values achieved are 1.6257 rad/ secto3.3307rad/seccorrespondingtothestiffnessvalues of front and rear system. In second case the mass of vehicleisvariedfrom1000kgto2550kgcorrespondingto frontsuspensionsystemstiffnessvaluesfrom450N/mm2 to 2000 N/mm2 and rear suspension system stiffness values from 500 N/mm2 to 2050 N/mm2. The results show that as the front and rear suspension system stiffness values increases with constant mass of vehicle the fundamental frequencies also increase linearly. Also, as the mass of vehicle along with front and rear suspension systems stiffness values increases the two the two modes of fundamental frequencies also increase linearly.
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©
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[5] Hideto Nishinaka, Noritaka Matsuoka, Noise, Vibration, and Harshness Analysis Technique Using a Full Vehicle Model, Sei Technical Review, Number 88, April2019,page122t0126.
Mass Vs Second FrequencyNatural
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