Numerical and experimental investigation of vibration of a car chassis and its effect with road gradient on noise and response optimization with factorial design method

Abstract

This paper delves in to a numerical and experimental investigation of vibration of a car chassis and its effect with road gradient on noise and response optimization. Peak acceleration, peak amplitude as vibration response parameters in vertical motion, internal and pass by noise as a noise response parameters. Vibration parameters obtained numerically by developing a mathematical model of car and equation of motion. Newmark Beta method programmed in MATLAB software used to solve the equation of motion. Experimental investigation of the vibration were carried on a car consist of monocoque chassis, three cylinder engine, Macpherson front suspension and twist beam rear suspension. FFT analyzer and digital sound level meter were used for the data collection. Pass by noise data of vehicle were collected at two locations with gradient 1.15° and 11.31° on state highway (MH SH-44) near Konchi Maharashtra state, India. Comparison between numerical and experimental investigated values of vibration parameters shows minimum error of 0.6035% and average error of 2.81% which validates the car model. Factorial design method was used to optimize the index of response parameters which shows reduction in peak acceleration, peak amplitude, internal noise and pass by noise level by 65.61%, 17.70%, 7.13% and 19.41% respectively. The methodology proposed in this study provides reference to improve the performance of car suspension and passenger comfort.

Keywords

Vibration noise factorial design road gradient response optimization passenger comfort

Introduction

Vibration and noise level is the important issue in the entire vehicle design process. The vibration analysis of vehicle full car model with set of realistic values for suspension parameters the natural frequencies evaluated using matrix iteration method shows that the impulse disturbance is more severe than the pulse displacement disturbance.¹ The engine vibration and its isolations, the modeling of multi-body engine of different components of internal combustion engine and their mounting orientation affects the vibration level of vehicle.² Modal analysis method commonly used for the experimental vibration analysis to measure the vibration of commercial vehicle.³ The energy flow path based vibration reduction technique can be used to reduce the level of noise and vibration of various automotive systems.⁴ The influence of damping vibrations and panel thickness needs to be considered for the effective vibro-acoustic design, namely wave based Sub structuring for competent local redesign and modification of modal.⁵ The stress analysis and computational modal transient response to forecast the dynamic characteristics of the car chassis.⁶ The level of noise and vibration produced by the Automobile vehicles is directly proportional to the different speed of driving and surface of road.⁷ Vibration produced by the Automobile vehicles affected by the different surface roughness of roadway.^8,9 The vibration control algorithm to reduce the vibration for off road vehicle along with damper system shows the comparative results.¹⁰ The substructure power flow based analysis used for measuring and reducing the noise and vibration of automobile full vehicle model.¹¹ The vibrations measured in particular points in the comparative study of test of two spark ignition engines for the experimental vibration analysis.¹² The technique of optimizing the stiffness and damping of suspension system for non-linear quarter car model which is subjected to excitations from arbitrary road.¹³ Approach of vibration signature of the two degrees of freedom shock absorber system utilized in vibration analysis.¹⁴ The coupling effect analysis and mathematical model used for coupled vibration system of road vehicle with ride dynamics useful for vibration analysis.^15,16 The potential failure of the machine could be detected or even predicted in case of predictive maintenance implementation.¹⁷ The noise-vibration-harshness analysis method used to reduce the vehicle noise and vibration.¹⁸ In the literature the vibration and noise analysis not performed at shifting of gear on a full car model, lack of noise analysis with effect of change in road gradient. Hence, contributions of the present paper are significant to improve performance of suspension and comfort level.

Methods

Methodology in this paper can be summarized in to the following steps as shown in Figure 1.

Figure 1.

Methodology of experiment.

Numerical method for the vibration investigation

Mathematical modelling of car

Compact and lightweight structure makes a monocoque chassis perfect for small vehicle construction. The car components like exterior panel, seats, suspension, engine and gearbox all remain attached to the chassis or frame structure. This structure is considered safer than most other frames and suitable for constructing modern compact vehicles. Figure 2 shows a full car model with a sprung mass or chassis mass m_B supported by spring and dampers at four corners. The unsprung mass m_F and m_R are considered to have included the masses of tyres. The motion of the entire system is represented by seven coordinates x_F, θ_F, x_R, θ_R, x_B, θ_B and ϕ_B. x_F, θ_F represents vertical and roll motion at the centre of mass g₁ of the mass m_F. x_R, θ_R represents vertical and roll motion at the centre of mass g₂ of the mass m_R. x_B, θ_B, ϕ_B represents vertical, roll and pitch motion respectively at the centre of mass g₃ of the mass m_B. y₁, y₂, y₃, y₄ represents the vertical displacements due to the road roughness.^1,9

Figure 2.

Mathematical model of car.

Equation of motion of the car model

The equations of motion were obtained by giving positive displacement to each coordinates,

m_{F} \ddot{x} F + 2 (k_{1} + k_{2}) x_{F} - 2 k_{2} x_{B} - 2 k_{2} l_{1} + 2 (c_{1} + c_{2}) \dot{x} F - 2 c_{2} \dot{x} B - 2 c_{2} l_{1} \dot{ϕ} B = k_{1} (y_{1} + y_{2}) + c_{1} ({\dot{y}}_{1} + {\dot{y}}_{2})

(1)

I_{1} \ddot{θ} F + 2 (k_{1} + k_{2}) b^{2} θ_{F} - 2 k_{2 b}^{2} θ_{B} + 2 (c_{1} + c_{2}) b^{2} \dot{θ} F - 2 c_{2 b}^{2} \dot{θ} B = - k_{1} b (y_{1} - y_{2}) - c_{1} b ({\dot{y}}_{1} - {\dot{y}}_{2})

(2)

m_{R} \ddot{x} R + 2 (k_{1} + k_{2}) x_{R} - 2 k_{3} x_{B} - 2 k_{3} l_{2} ϕ B + 2 (c_{1} + c_{2}) \dot{x} R - 2 c_{3} \dot{x} B - 2 c_{3} l_{2} \dot{ϕ} B = k_{1} (y_{3} + y_{4}) + c_{1} ({\dot{y}}_{3} + {\dot{y}}_{4})

(3)

I_{2} \ddot{θ} R + 2 (k_{1} + k_{2}) b^{2} θ_{R} - 2 k_{3 b}^{2} θ_{B} + 2 (c_{1} + c_{2}) b^{2} \dot{θ} R - 2 c_{3 b}^{2} \dot{θ} B = - k_{1} b (y_{3} - y_{4}) - c_{1} b ({\dot{y}}_{3} - {\dot{y}}_{4})

(4)

\begin{array}{l} m_{B} \ddot{x} B - 2 k_{2} x_{F} - 2 k_{3} x_{R} + 2 (k_{2} + k_{3}) x_{B} + 2 (k_{2} l_{1} - k_{3} l_{2}) ϕ B - 2 c_{2} \dot{x} F - 2 c_{3} \dot{x} R + 2 (c_{2} + c_{3}) \dot{x} B \\ + 2 (c_{2} l_{1} - c_{3} l_{2}) \dot{ϕ} B = 0 \end{array}

(5)

I_{3} \ddot{θ} B - 2 k_{2 b}^{2} θ_{F} - 2 k_{3 b}^{2} θ_{R} + 2 (k_{2} + k_{3}) b^{2} θ_{B} - 2 c_{2 b}^{2} \dot{θ} F - 2 c_{3 b}^{2} \dot{θ} R + 2 (c_{2} + c_{3}) b^{2} \dot{θ} B = 0

(6)

\begin{array}{l} I_{4} \ddot{ϕ} B - 2 k_{2} l_{1} x_{F} + 2 k_{3} l_{2} x_{R} + 2 (k_{2} l_{1} - k_{3} l_{2}) x_{B} \\ + 2 (k_{2} {l_{1}}^{2} + k_{3} {l_{2}}^{2}) ϕ B - 2 c_{2} l_{1} \dot{x} F + 2 c_{3} l_{2} \dot{x} R + 2 (c_{2} l_{1} - c_{3} l_{2}) \dot{x} B + 2 (c_{2} {l_{1}}^{2} + c_{3} {l_{2}}^{2}) \dot{ϕ} B = 0 \end{array}

(7)

The seven variables split up into two groups as (x_F, x_R, x_B and ϕ_B) as one group for pitching motion and (θ_F, θ_R and θ_B) as second group for rolling motion.

Considering X₁ = x_F, X₂ = x_R, X₃ = x_B, X₄ = ϕ_B, X₅ = θ_F, X₆ = θ_R and X₇ = θ_B, the equations (1)–(7) may be represented in matrix form as,

M \ddot{X} + C \dot{X} + KX = F

(8)

Where F, K, C and M are the force, stiffness, damping and mass matrices of size (7 × 7). The equations (1)–(7) splits in to two groups as,

M^{(1)} {\ddot{X}}^{(1)} + C^{(1)} {\dot{X}}^{(1)} + K^{(1)} X^{(1)} = F^{(1)}

(9)

M^{(2)} {\ddot{X}}^{(2)} + C^{(2)} {\dot{X}}^{(2)} + K^{(2)} X^{(2)} = F^{(2)}

(10)

Where,

M^{(1)} = {[\begin{array}{l} \begin{array}{l} \begin{array}{l} m_{F} & 0 & 0 \end{array} & 0 & : \end{array} & \begin{array}{l} 0 & m_{R} & \begin{array}{l} 0 & 0 & \begin{array}{l} : & 0 & \begin{array}{l} 0 & m_{B} & \begin{array}{l} 0 & : & \begin{array}{l} 0 & 0 & \begin{array}{l} 0 & I_{4} \end{array} \end{array} \end{array} \end{array} \end{array} \end{array} \end{array} \end{array}]}^{T}

(11)

\begin{array}{l} C^{(1)} = [2 (c_{1} + c_{2}) 0 - 2 c_{2} - 2 c_{2} l_{1}; 0 2 (c_{1} + c_{3}) - 2 c_{3} 2 c_{3} l_{2}; - 2 c_{2} - 2 c_{3} 2 (c_{2} + c_{3}) 2 (c_{2} l_{1} + c_{3} l_{2}); \\ 2 c_{2} l_{1} 2 c_{3} l_{2} 2 (c_{2} l_{1} - c_{3} l_{2}) 2 {(c_{2} l_{1}^{2} + c_{3} l_{2}^{2})]}^{T} \end{array}

(12)

\begin{array}{l} K^{(1)} = [2 (k_{1} + k_{2}) 0 - 2 k_{2} - 2 k_{2} l_{1}; 0 2 (k_{1} + k_{3}) - 2 k_{3} 2 k_{3} l_{2}; - 2 k_{2} - 2 k_{3} 2 (k_{2} + k_{3}) 2 (k_{2} l_{1} + k_{3} l_{2}); \\ 2 k_{2} l_{1} 2 k_{3} l_{2} 2 (k_{2} l_{1} - k_{3} l_{2}) 2 {(k_{2} l_{1}^{2} + k_{3} l_{2}^{2})]}^{T} \end{array}

(13)

X^{(1)} = {[\begin{array}{l} x_{F} & x_{R} & \begin{array}{l} x_{B} & ϕ_{B} \end{array} \end{array}]}^{T}

(14)

F^{(1)} = {[k_{1} (y_{1} + y_{2}) + c_{1} ({\dot{y}}_{1} + {\dot{y}}_{2}) : k_{1} (y_{3} + y_{4}) + c_{1} ({\dot{y}}_{3} + {\dot{y}}_{4}) : 0 : 0]}^{T}

(15)

M^{(2)} = {[\begin{array}{l} I_{1} & 0 & \begin{array}{l} 0; & 0 & \begin{array}{l} I_{2} & 0; & \begin{array}{l} 0 & 0 & I_{3} \end{array} \end{array} \end{array} \end{array}]}^{T}

(16)

C^{(2)} = {[2 (c_{1} + c_{2}) b^{2} 0 - 2 c_{2} b^{2}; 02 (c_{1} + c_{3}) b^{2} - 2 c_{3} b^{2}; - 2 c_{2} b^{2} - 2 c_{3} b^{2} 2 (c_{2} + c_{3}) b^{2}]}^{T}

(17)

K^{(2)} = {[2 (k_{1} + k_{2}) b^{2} 0 - 2 k_{2} b^{2}; 02 (k_{1} + k_{3}) b^{2} - 2 k_{3} b^{2}; - 2 k_{2} b^{2} - 2 k_{3} b^{2} 2 (k_{2} + k_{3}) b^{2}]}^{T}

(18)

X^{(2)} = {[\begin{array}{l} θ_{F} & θ_{R} & θ_{B} \end{array}]}^{T}

(19)

F^{(2)} = {[k_{1} b (y_{1} + y_{2}) + c_{1} b ({\dot{y}}_{1} + {\dot{y}}_{2}) : k_{1} b (y_{3} + y_{4}) + c_{1} b ({\dot{y}}_{3} + {\dot{y}}_{4}) : 0]}^{T}

(20)

The equations (9) and (10) solved by Newmark Beta method programmed in MATLAB software to get the values of amplitude, acceleration and frequency of vibrations.

Newmark beta method

The finite difference approximations for the Newton Beta method are

x_{B + 1} \approx x_{B} + h {\dot{x}}_{B} + h^{2} [(\frac{1}{2} - β) \ddot{x_{B}} + β {\ddot{x}}_{B + 1}]

(21)

x_{B + 1} \approx x_{B} + h [(1 - γ) \ddot{x_{B}} + γ {\ddot{x}}_{B + 1}]

(22)

If β = 1/6 and γ = 1/2 Newton Beta method is identical to the linear acceleration model.

The Newton Beta method is conditionally stable if γ < 1/2. For γ = 1/2 the Newton Beta method is at least second order accurate, it is first order accurate for all other values of γ.

Formulating the finite difference relationships WP in terms of increments of displacement $({δ x}_{B} = x_{B + 1} - x_{B})$ , velocity $δ {\dot{x}}_{B} = {\dot{x}}_{B + 1} - {\dot{x}}_{B})$ , acceleration ( ${\ddot{x}}_{B} = {\ddot{x}}_{B + 1} - {\ddot{x}}_{B})$ and non linear restoring force $({δ F}_{B} = F (x_{B + 1}, {\dot{x}}_{B + 1}) - F (x_{B,} {\dot{x}}_{B}))$

δ {\ddot{x}}_{B} = \frac{1}{β h^{2}} {δ x}_{B} - \frac{1}{β h} {\dot{x}}_{B} - \frac{1}{2 β} {\ddot{x}}_{B}

(23)

δ {\dot{x}}_{B} = \frac{γ}{β h^{2}} {δ x}_{B} - \frac{γ}{β} {\dot{x}}_{B} + h (1 - \frac{γ}{2 β}) {\ddot{x}}_{B}

(24)

Now satisfying incremental equilibrium over the time step h

M δ {\ddot{x}}_{B} + C δ {\dot{x}}_{B} + K {δ x}_{B} + {δ F}_{B} = f_{B + 1}^{u} - f_{B}^{u} = {δ f}_{B}^{u}

(25)

Regrouping the terms and solving for the increment in displacements,

[\frac{6}{h^{2}} M + \frac{3}{h} C + K] {δ x}_{B} = {δ f}_{B}^{u} - {δ F}_{B} + [3 M + \frac{h}{2} C] {\ddot{x}}_{B} + [\frac{6}{h} M + 3 C] {\dot{x}}_{B}

(26)

Where β = 1/6 and γ = 1/2 solving this linear system of equation for

{δ x}_{B}

the displacements are updated with,

x_{B + 1} = x_{B} + {δ x}_{B}

(27)

The velocities are updated with,

{\dot{x}}_{B + 1} = - 2 {\dot{x}}_{B} - \frac{h}{2} {\ddot{x}}_{B} + \frac{3}{h} {δ x}_{B}

(28)

Acceleration satisfies the equation of motion,

{\ddot{x}}_{B + 1} = - M^{- 1} [C {\dot{x}}_{B + 1} + K x_{B + 1} + F (x_{B + 1}, {\dot{x}}_{B + 1}) - f_{B + 1}^{u}]

(29)

Similar relations may be found for other values of β and γ.¹⁹

Experimental method for the vibration and noise investigation

Experimental setup

The Figure 3 shows the experimental arrangement for the response parameter data collection of car with use of dual channel FFT analyzer having 20 kHz bandwidth, absolute accuracy of ±0.05 dB, amplitude accuracy ±1% of range, absolute maximum rating 3 g, 5–500 Hz.

Figure 3.

Experimental setup for vibration measurement.

As per the sources of vibration the four points identified²⁰ on the car chassis for vibration analysis as is shown in Figure 4. Point 1 is near right front suspension and engine mount, Point 2 is near left front suspension and engine mount, Point 3 is near right rear suspension, Point 4 is near left rear suspension.

Figure 4.

Outline of research for vibration analysis.

Peak acceleration levels measured with FFT analyzer at identified points 1, 2, 3 and 4 under stationary and in running condition at the four driving speeds with respect to gear shifting signal on display panel of vehicle that is first gear at idle, shifting to second gear at speed of 16 kmph, shifting to third gear at speed 29 kmph and shifting to third gear at speed of 49 kmph.

Experiment investigation of noise

For internal noise measurement dual channel FFT analyzer used with microphone of normal sensitivity 50 mV/Pa at 250 Hz, frequency response ±2 dB, dynamic range 14 to 135 dB. Microphone was placed at centre position of front and back seat.

Pass by noise measured with digital sound level meter with A & C frequency weighting, range 30 to 130 dB as per the International standard, ISO 10,844. For the noise analysis of vehicle the outline is selected as shown in Figure 5.^7,21 Pass by noise data of vehicle were collected at two locations with gradient 1.15° and 11.31°^22,23 on state highway (MH SH-44) near Konchi Maharashtra state, India were no other objects in their environment. Weather conditions: no wind, no rain, and dry road surface. The air humidity was not measured. The noise sampling survey has been carried out between 12.30 p.m. and 3.30 p.m. during weekdays at all selected locations.

Figure 5.

Outline for pass by noise measurement.

Factorial design analysis

The one quarter fraction of 2^k experiment (2^k−2 FFD)

The one quarter fraction of the 2^k design is called a 2^k−2 FFD and this design contains 2^k−2 runs. The one quarter fraction of the 2^k has two generators say A₁ and A₂. The signs of A₁ and A₂ are either + or − which produces one of the one quarter fraction. There are in all four fractions associated with the choice of ±A₁ and ±A₂ which produces the same aliasing pattern. The aliases of any particular effect are obtained by multiplying that effect to all members of the defining relations.²⁴

Main effect analysis

The main effect of a factor on the response is the average of the response of a factor to all tests at a certain level. Changing the level of a single factor and using average of the effects of all possible combinations of each level and other factors on the results provides the main effect.²⁵

\begin{array}{l} \bar{z} (x) = \bar{z} (x_{1}, x_{2},,,, x_{n}) \\ \bar{z} (x) = β_{0} + \sum_{l = 1}^{n} β_{l} x_{l} + \sum_{l = 1}^{n} β_{ll} x_{l}^{2} + \sum_{l = 1}^{n - 1} \sum_{l < k = 2}^{n} β_{lk} x_{l} x_{k} \end{array}

(30)

Analysis of contribution

Contribution analysis mainly uses regression of DOE to calculate the contribution. Design variable in the high discrete or high non linear analysis are screened to reduce calculation cost and improve the efficiency of optimization. The design variables have different design spaces and the contribution values vary from the design spaces requiring normalization of the sample data inputs using following equation,

x_{l}^{#} = \frac{x_{l -} \bar{x}}{σ} = (x_{l} - \frac{1}{P} \sum_{l = 1}^{P} x_{l}) * {[\sqrt{\frac{1}{P}} \sum_{l = 1}^{P} {(x_{l} - \bar{x})}^{2}]}^{- 1}

(31)

Where

σ

is standard deviation, P is total number of sample data,

\bar{x}

is a mean of sample data.

x_{l}^{#}

denotes the normalized data.

Results and discussion

Discussion on the numerical investigation results

The road surface is considered as sine wave with wave length λ and velocity v then frequency of excitation may be expressed as,

Frequency (f) = v / λ

(32)

Input excitation y and velocity of excitation $\dot{y}$ for the road surface is expressed as,

y = Ysinwt, \dot{y} = Ywcoswt

(33)

To correlate numerical model with practical model, a car considered with following details, Gross weight = 1164 kg, pay load capacity = 410 kg, sprung mass m_B = 1000 kg, m_F = 100 kg, m_R = 100 kg, I₁ = 20, I₂ = 20, I₃ = 500 and I₄ = 1200 kg.m², k₁ = 2 × 10⁵, k₂ = 1250, k₃ = 850 N/m, equivalent spring stiffness = 4200 N/m, c₁ = 50, c₂ = 2100, c₃ = 1700 Ns/m, equivalent damping coefficient = 7600 Ns/m, l₁ = 1.0, l₂ = 1.5 and b = 0.75 m.^1,13,21

The idle condition

The model analyzed in idle condition a input excitation y₁ = y₂ = 0.1 m given to front axle that is at point 1 and point 2, the amplitude, acceleration and natural frequencies obtained as shown in Table 1.

Table 1.

Acceleration and natural frequency of the model.

Sr. No.	Motion	Amplitude (m)	Acceleration (m/s²)	Frequency Hz
1	x_F	0.1	164.299	0.4560
2	x_R	0.000972	1.6413	0.5095
3	x_B	−0.0455	8.4436	10.1073
4	ϕ_B	−0.03104	7.4476	10.1287
5	θ_F	0.1	360.937	0.4837
6	θ_R	−0.00458	12.1655	16.9504
7	θ_B	−0.04605	6.9716	16.9861

The amplitude of vibration of front axle x_F is maximum as front axle is directly exposed to input excitations. The pitching displacement ϕ_B and rolling displacement θ_B of sprung mass increased with increase in displacement of front and rear axle as shown in Figure 6. The acceleration of vibration of front axle is maximum as front axle is directly exposed to input excitations. The acceleration of chassis or sprung mass M_B is increased with increase in acceleration of front and rear axle. In this research a car chassis M_B were analyzed for response parameters that is peak amplitude and peak acceleration of vibration of a chassis in vertical motion that is x_B = −0.0455 m and ${\ddot{x}}_{B} = 8.4436 {m / s}^{2}$ with frequency of 10.1073 Hz for idle condition at point 1 and 2, excitation of y₃ = y₄ = 0.1 m at point 3 and 4 of rear axle x_B = −0.0316 m and ${\ddot{x}}_{B} = 8.7509 {m / s}^{2}$ with frequency of 10.1073 Hz. The natural frequencies as shown in Table 1 are the natural frequencies of the total system. The resonance may occur whenever the exciting frequency matches with any one of these natural frequencies. Similar process followed for obtaining values of response parameters in running condition at point 1, 2, 3 and 4 at speed 16, 29 and 49 kmph respectively and results are shown in Table 4.

Figure 6.

Amplitude of vibration of the car model in idle condition.

Discussion on the experimental investigation results

The idle condition

For the measurement of response parameters in idle condition the accelerometer of FFT analyzer were mounted on the car chassis at point 1, 2, 3 and 4 and results shown in Figure 7. Acceleration of the chassis at Point 1, 2, 3 and 4 at idle condition is observed 8.829 m/s². Same process followed for the running condition and results are shown in Table 2.

Figure 7.

Acceleration of the chassis ẍ_B at point 1, 2, 3 and 4 at idle condition.

Table 2.

Design variable low and high level values.

Sr. no.	Design variables	Low level	High level
1	Sprung mass (kg)	1000	2000
2	Equivalent spring stiffness (N/m)	4200	8300
3	Equivalent damping coefficient (Ns/m)	760	7600
4	Amplitude of input excitation (m)	0.05	0.1
5	Velocity of input excitation (m/s)	0.9308	1.7095
6	Wavelength of frequency of excitation (m)	1.5	5

Discussion on noise results

Pass by noise data were collected as per outline shown in Figure 5, it shows that with increase in road gradient from 1.15° to 11.31° pass by noise increased by 6.76%, 5.90% and 5.16% at a speed of 16, 29 and 49 kmph respectively and internal noise level increased by 6.33%, 4.66% and 2.62 % at a speed of 16, 29 and 49 kmph respectively.

Discussion on the DOE analysis results

Parameters selected for the DOE analysis

As per the numerical and experimental analysis of vibration of the chassis the parameters selected with values for DOE analysis are as shown in Table 2. Sprung mass or net mass (M_B), equivalent stiffness of spring of front and rear suspension (K = 2*stiffness of front suspension spring +2*stiffness of rear suspension spring), equivalent damping coefficient of damper of front and rear suspension (C = 2*damping coefficient of front suspension damper + 2*damping coefficient of rear suspension damper), amplitude of input excitation (y), velocity of input excitation ( $\dot{y}$ ) is considered for speed 16 kmph at which shifting gear from first to second and 49 kmph at which shifting the gear from third to top. Last parameter is wavelength (λ) of frequency of excitation. All remaining parameters are not changed during the analysis as these parameters are considered while investigating the values of six design variables. The DOE analysis of 6 factors 2 level for the design of one quarter fraction of 2⁶ experiments (2⁶⁻² FFD) were performed. The Minitab software used for the DOE analysis and response optimization with 16 trials.

Main effect analysis

Experimental designs calculate the main effect of a factor of a response by constructing a multiple quadratic regression model based on the results of input factor and output response samples with equation (30).

Figure 8 shows the main effect plot for acceleration of chassis ${\ddot{x}}_{B}$ . Acceleration of vibration decreases with increase in net mass, stiffness from low level to high level whereas with increase in the value of damping coefficient and amplitude of excitation from low level to high level the acceleration increases.

Figure 8.

Main effect plot for acceleration ẍ_B.

Analysis of variance (ANOVA)

Analysis of variance is a statistical method that separates observed variance data into different components to use for additional tests. If no true variance exists between the groups F-ratio should equal close to 1. The ANOVA test allows a comparison of more than two groups at the same time to determine relationship between them. The ANOVA coefficient F is the ration of mean sum of squares due to treatment to mean sum of squares due to error. Based on models with greater R² values the models are accepted. Equations (34) and (35) shows the regression equation for the prediction of x_B and ${\ddot{x}}_{B}$ respectively, having good relationship with the design variables.

x_{B} = 0.05332 - 0.000013 M_{B} + 0.000002 K - 0.000008 C - 0.9850 y - 0.00527 \dot{y} + 0.005068 λ + 0.000000 M_{B} * C + 0.000117 M_{B} * y + 0.000004 M_{B} * \dot{y} + 0.000024 K * y - 0.000001 K * λ

(34)

{\ddot{X}}_{B} = 12.4 - 0.00759 M_{B} - 0.00326 K + 0.001289 C + 213.8 y - 7.24 λ + 0.000002 M_{B} * K - 0.1124 M_{B} * y + 0.00453 M_{B} * λ + 0.001078 K * λ - 0.000001 M_{B} * K * λ

(35)

Full factorial design with two level and three factors (driving speed, acceleration of vibration and road gradient) were performed for analyzing pass by noise and internal noise and regression equation for the prediction of pass by noise and internal noise are shown in equations (36) and (37).

P a s s b y n o i s e = 31.01 + 0.4576 Driving Speed + 0.893 Acceleration {\ddot{x}}_{B} + 0.3986 Gradient

(36)

I n t e r n a l N o i s e = 56.75 + 0.0979 Driving Speed + 0.454 Acceleration {\ddot{x}}_{B} + 0.2687 Gradient

(37)

Response optimization

The response optimization with backward elimination of terms methodology used for optimizing the vibration level of the chassis and response obtained for the 50 number of solutions to minimize the response parameters. Response Parameters target values are assigned as 1.12,666 m/s² and −0.08572 for goal of minimizing the vibration level and results shown in Table 3.

Table 3.

Result of experimental design of contribution.

Solution	MB (kg)	K (N/m)	C (Ns/m)	y (m)	ẙ (m/s)	λ (m)	ẍ_B (m/s∧2)	x_B (m)
1	2000	4200	5527.27	0.1	0.9308	1.5	5.92403	−0.0637026
2	2000	4200	7050.74	0.1	0.9308	1.5	7.88845	−0.0750937
3	2000	4200	6312.38	0.1	1.32015	1.5	6.93637	−0.0681446
4	2000	4200	7085.19	0.1	1.70286	1.5	7.93286	−0.0725191
5	1974.32	4200	3226.37	0.1	0.9308	1.5	3.17841	−0.0466155
.	.	.	.	.	.	.	.	.
.	.	.	.	.	.	.	.	.
50	1064.54	8047.26	3149.75	0.1	0.93547	1.5549	7.98093	−0.0380092

The Figure 9 shows the multiple response prediction for optimizing the response with minimum acceleration ${\ddot{x}}_{B}$ 5.9240 m/s² and amplitude x_B −0.0637 m with composite desirability of 0.7415.

Figure 9.

Multiple response prediction.

Validation of model

Comparison of numerical and experimental results

The error of 4.609%, 6.00%, 1.76% and 0.6452% observed in the numerical and experimental investigation of chassis vibration at front axle and the error of 0.9029%, 5.4249%, 0.6035% and 2.497% observed at rear axle at speed 16, 29 and 49 kmph respectively. Comparison of numerical and experimental investigated values indicates minimum error of 0.6035% and average error of 2.81% on various points on chassis, which shows high accuracy of the method for vibration analysis and validates the result as shown in Table 4.

Table 4.

Comparison of vibration analysis with numerical and experimental method.

Sr. no.	Speed kmph	Numerical method				Experimental method				Error in %
		Acceleration (m/s²)				Acceleration (m/s²)				Error in %
		Point 1	Point 2	Point 3	Point 4	Point 1	Point 2	Point 3	Point 4	Point 1	Point 2	Point 3	Point4
1	Idle	8.44	8.44	8.75	8.75	8.829	8.829	8.829	8.829	4.609	4.609	0.902	0.902
2	16	16.48	16.48	16.59	16.59	15.49	15.49	15.69	15.69	6.007	6.007	5.424	5.424
3	29	16.47	16.47	16.57	16.57	16.18	16.18	16.67	16.67	1.760	1.760	0.603	0.603
4	49	17.05	17.05	17.22	17.22	17.16	17.16	17.65	17.65	0.645	0.645	2.497	2.497

Comparison of vibration and noise attenuation before and after the optimization

To verify the effectiveness of the multi response optimization, the corrected values of design variables as M_B = 1000 kg, K = 4200 N/m, C = 5527.87 Ns/m, y = 0.1 m, $\dot{y}$ = 0.9308 m/s and λ = 1.5 m substituted in the initial model and responses compare with the previous model. The comparison results shows that the response performance index of vibration that is peak acceleration level improved by 65.61%, peak amplitude level improved by 17.70% and frequency of vibration improved by 0.19% after optimization as shown in Figures 10 and 11. Also response performance index of pass by noise and internal noise improved by 19.41% and 7.13% respectively with reduction in peak acceleration ${\ddot{x}}_{B}$ .^26–29

Figure 10.

Peak amplitude x_B index response improvement of mass M_B.

Figure 11.

Peak acceleration index response improvement of mass M_B.

Conclusions

This paper presents a numerical and experimental investigation of vibration of a car chassis and its effect with road gradient on noise and response optimization.

The peak vibration level increased by 4.45% at point 1 and 2 of front side of chassis and 6.24 % at point 3 and 4 of rear side of chassis with respect to increase in the driving speed from 16 kmph to 29 kmph. It increased by 6.05% at point 1 and 2 of front side of chassis and 5.87% at point 3 and 4 of rear side of chassis with respect to increase in the driving speed from 29 kmph to 49 kmph. It increased by 10.78% at point 1 and 2 of front side of chassis and 11.10% at point 3 and 4 of rear side of chassis with respect to increase in the driving speed from 16 kmph to 49 kmph. Experimental results of noise shows that with increase in road gradient from 1.15° to 11.31° pass by noise increased by 6.76%, 5.90% and 5.16% at a speed of 16, 29 and 49 kmph respectively and internal noise level increased by 6.33%, 4.66% and 2.62% at a speed of 16, 29 and 49 kmph respectively. Main effect analysis of design variables shows that the peak amplitude of vibration decreases with increase in sprung mass and stiffness of suspension system and increases with increase in damping coefficient of suspension system, amplitude of input excitation and speed of input excitation. Peak acceleration of vibration decreases with increase in sprung mass and stiffness of suspension system and increases with increase in damping coefficient of suspension system, amplitude of input excitation and wavelength of input excitation. Comparison of numerical and experimental investigated values indicates minimum error of 0.6035% and average error of 2.81% on various points on chassis, which shows high accuracy of the method for vibration analysis and validates the result. The comparative analysis of the response parameter levels before and after the optimization shows the reduction of 65.61% in peak acceleration level and reduction of 17.70% in the peak amplitude level of a car chassis with respect to change in the driving speed. Performance index of pass by noise and internal noise improved by 19.41% and 7.13% respectively with reduction in peak acceleration. Vibration of the chassis has a significant effect on the internal and exterior noise level. The design and optimization method proposed in this study has a significant effect on the vibration reduction of the car chassis which provides a reference for the optimization of the vibration level to improve the performance of car suspension and passenger comfort.

Footnotes

Appendix

Acknowledgements

In this research work we kindly acknowledge the vehicle testing support from Bridgestone SGS Wheels Rahata, Maharashtra, India. Support from Heatcon Systems, MIDC, Ahmednagar, Maharashtra, India for investigation of factors affecting the noise and vibration of vehicle. We kindly acknowledge support from Mr Sanjay Belkar, professor at Pravara Rural Engineering College Loni, Ahmednagar, Maharashtra, India also Dr Dabber P.S. Acharya Institute of technology, Bangalore, Karnataka, India.

Author contributions

The authors confirm contribution to the paper as follows: study conception and design: 1. Author, 2. Author; data collection: 1. Author; analysis and interpretation of results: 1. Author, 2. Author; draft manuscript preparation: 1. Author, 2. Author. All authors reviewed the results and approved the final version of the manuscript.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Mosin Shaikh

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Numerical and experimental investigation of vibration of a car chassis and its effect with road gradient on noise and response optimization with factorial design method

Abstract

Keywords

Introduction

Methods

Numerical method for the vibration investigation

Mathematical modelling of car

Equation of motion of the car model

Newmark beta method

Experimental method for the vibration and noise investigation

Experimental setup

Experiment investigation of noise

Factorial design analysis

The one quarter fraction of 2k experiment (2k−2 FFD)

Main effect analysis

Analysis of contribution

Results and discussion

Discussion on the numerical investigation results

The idle condition

Discussion on the experimental investigation results

The idle condition

Discussion on noise results

Discussion on the DOE analysis results

Parameters selected for the DOE analysis

Main effect analysis

Analysis of variance (ANOVA)

Response optimization

Validation of model

Comparison of numerical and experimental results

Comparison of vibration and noise attenuation before and after the optimization

Conclusions

Footnotes

Appendix

Acknowledgements

Author contributions

Declaration of conflicting interests

Funding

ORCID iD

References

The one quarter fraction of 2^k experiment (2^k−2 FFD)